From b4bb7cdbf4947ba6a1360a593f681b5a1a4fee4a Mon Sep 17 00:00:00 2001 From: Douglas Ezra Morrison Date: Tue, 21 May 2024 13:06:49 -0700 Subject: [PATCH] Built site for gh-pages --- .nojekyll | 2 +- Linear-models-overview.html | 52 +- .../figure-html/unnamed-chunk-104-1.png | Bin 85821 -> 86321 bytes .../figure-html/unnamed-chunk-98-1.png | Bin 44330 -> 45078 bytes Regression-Models-for-Epidemiology.pdf | Bin 2938956 -> 2950796 bytes count-regression.html | 969 +++-- .../figure-html/fig-needles-1.png | Bin 110707 -> 132376 bytes .../tbl-pois-model-diagnostics-1.png | Bin 156959 -> 169566 bytes coxph-model-building.html | 4 +- images/Songofthecount.jpg | Bin 0 -> 22606 bytes index.html | 2 +- intro-MLEs.html | 12 +- intro-to-survival-analysis.html | 3235 +++++++++-------- .../figure-html/fig-cuhaz-exp-1.png | Bin 0 -> 106280 bytes logistic-regression.html | 74 +- .../figure-html/fig-beetles-glm-diag-1.png | Bin 214684 -> 216615 bytes .../figure-html/fig-diagnostics-glm1-1.png | Bin 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Published
-

Last modified: 2024-05-16: 13:44:16 (PM)

+

Last modified: 2024-05-18: 7:52:50 (AM)

@@ -906,7 +906,7 @@
Show R code
coef(bw_lm1)
 #> (Intercept)     sexmale         age 
-#> -1773.32184   163.03930   120.89433
+#> -1773.3 163.0 120.9
@@ -929,7 +929,7 @@ \begin{aligned} E[Y|A = 0, A = 36] &= \beta_0 + \beta_M \cdot 0+ \beta_A \cdot 36 \\ -&= 2578.873934 +&= 2578.8739 \end{aligned} \]

@@ -960,7 +960,7 @@ \begin{aligned} E[Y|M = 1, A = 36] &= \beta_0 + \beta_M \cdot 1+ \beta_A \cdot 36 \\ -&= 2741.91323693 +&= 2741.9132 \end{aligned} \]

@@ -980,9 +980,9 @@ \begin{aligned} & E[Y|M = 1, A = 36] - E[Y|M = 0, A = 36]\\ &= -2741.91323693 - 2578.873934\\ +2741.9132 - 2578.8739\\ &= -163.03930293 +163.0393 \end{aligned} \]

Shortcut:

@@ -992,7 +992,7 @@ &= (\beta_0 + \beta_M \cdot 1+ \beta_A \cdot 36) - (\beta_0 + \beta_M \cdot 0+ \beta_A \cdot 36) \\ &= \beta_M \\ -&= 163.03930293 +&= 163.0393 \end{aligned} \]

@@ -1131,7 +1131,7 @@

\[ E[Y|A = 0, X_2 = 36] = \beta_0 + \beta_M \cdot 0+ \beta_A \cdot 36 + \beta_{AM} \cdot (0 * 36) -= 2552.73333333 += 2552.7333 \]


@@ -1153,7 +1153,7 @@ \begin{aligned} E[Y|A = 0, X_2 = 36] &= \beta_0 + \beta_M \cdot 1+ \beta_A \cdot 36 + \beta_{AM} \cdot 1 \cdot 36\\ -&= 2762.70689655 +&= 2762.7069 \end{aligned} \]

@@ -1169,7 +1169,7 @@ &= (\beta_0 + \beta_M \cdot 1+ \beta_A \cdot 36 + \beta_{AM} \cdot 1 \cdot 36)\\ &\ \ \ \ \ -(\beta_0 + \beta_M \cdot 0+ \beta_A \cdot 36 + \beta_{AM} \cdot 0 \cdot 36) \\ &= \beta_{S} + \beta_{AM}\cdot 36\\ -&= 209.97356322 +&= 209.9736 \end{aligned} \]

@@ -1602,7 +1602,7 @@

Note

Research question: is there really an interaction between sex and age?

\(H_0: \beta_{AM} = 0\)

\(H_A: \beta_{AM} \neq 0\)

-

\(P(|\hat\beta_{AM}| > |-18.41724138| \mid H_0)\) = ?

+

\(P(|\hat\beta_{AM}| > |-18.4172| \mid H_0)\) = ?

2.4.2 Wald tests and CIs

R can give you Wald tests for single coefficients and corresponding CIs:

@@ -1726,17 +1726,17 @@

Note

\begin{aligned} &P\left( | \hat \beta_{AM} | > -| -18.41724138| \middle| H_0 +| -18.4172| \middle| H_0 \right) \\ &= \Pr \left( \left| \frac{\hat\beta_{AM}}{\hat{SE}(\hat\beta_{AM})} \right| > -\left| \frac{-18.41724138}{41.75581745} \right| \middle| H_0 +\left| \frac{-18.4172}{41.7558} \right| \middle| H_0 \right)\\ &= \Pr \left( -\left| T_{20} \right| > 0.44107007 | H_0 +\left| T_{20} \right| > 0.4411 | H_0 \right)\\ -&= 0.66389342 +&= 0.6639 \end{aligned} \]

@@ -2329,12 +2329,12 @@

Note

-

The likelihood ratio statistic for this test is \[\lambda = 2 * (\ell_{\text{full}} - \ell) = 10.35537421\] where:

+

The likelihood ratio statistic for this test is \[\lambda = 2 * (\ell_{\text{full}} - \ell) = 10.3554\] where:

This statistic is called the deviance or residual deviance for our two-slopes and two-intercepts model; it tells us how much the likelihood of that model deviates from the likelihood of the maximal model.

The corresponding p-value tells us whether there we have enough evidence to detect that our two-slopes, two-intercepts model is a worse fit for the data than the maximal model; in other words, it tells us if there’s evidence that we missed any important patterns. (Remember, a nonsignificant p-value could mean that we didn’t miss anything and a more complicated model is unnecessary, or it could mean we just don’t have enough data to tell the difference between these models.)

@@ -2413,12 +2413,12 @@

Note

-

The likelihood ratio statistic for the test comparing the null model to the maximal model is \[\lambda = 2 * (\ell_{\text{full}} - \ell_{0}) = 35.10673188\] where:

+

The likelihood ratio statistic for the test comparing the null model to the maximal model is \[\lambda = 2 * (\ell_{\text{full}} - \ell_{0}) = 35.1067\] where:

In R, this test is:

@@ -3926,7 +3926,7 @@

Note

full -6.946 +6.958 0.4805 0.3831 5.956 @@ -3934,7 +3934,7 @@

Note

reduced -6.627 +6.778 0.4454 0.3802 5.971 @@ -4071,10 +4071,10 @@

Note

Show R code
coef(cvfit, s = "lambda.1se")
 #> 4 x 1 sparse Matrix of class "dgCMatrix"
 #>                  s1
-#> (Intercept) 34.4241
+#> (Intercept) 34.1090
 #> age          .     
-#> weight      -0.0662
-#> protein      0.6607
+#> weight -0.1041 +#> protein 0.9441

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  • 4.8 model
  • +
  • 4.8 Covariate counts:
  • +
  • 4.9 model
  • @@ -700,7 +701,8 @@ factor() |> relevel(ref = "one drug used"), homeless = ifelse(homeless, "homeless", "not homeless") |> - factor() |> relevel(ref = "not homeless")) + factor() |> relevel(ref = "not homeless")) |> + mutate(sex = factor(sex) |> relevel(ref = "M")) needles
    @@ -709,7 +711,7 @@
    @@ -726,12 +728,12 @@ aes( x = age, y = shared_syr, - col = sex, - shape = ethn + shape = sex, + col = ethn ) ) + geom_point( - size = 4, + size = 3, alpha = .5) + facet_grid( cols = vars(polydrug), @@ -748,10 +750,28 @@
    -
  • 4.7 Example: needle-sharing
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    -4.8 model

    +

    +4.8 Covariate counts:

    -
    Show R code
    glm1 = glm(
    +
    Show R code
    needles |> 
    +  dplyr::select(sex, homeless, polydrug) |> 
    +  summary()
    +#>     sex             homeless                 polydrug  
    +#>  M    :97   not homeless:63   one drug used      :109  
    +#>  F    :30   homeless    :61   multiple drugs used: 19  
    +#>  Trans: 1   NA's        : 4
    +
    +
    +

    There’s only one individual with sex = Trans, which unfortunately isn’t enough data to analyze. We will remove that individual:

    +
    +
    Show R code
    
    +needles = needles |> filter(sex != "Trans")
    +
    +
    +

    +4.9 model

    +
    +
    Show R code
    glm1 = glm(
       data = needles,
       family = stats::poisson,
       shared_syr ~ age + sex + homeless*polydrug
    @@ -770,7 +790,7 @@
     
     
     
    -
    +
     
     
     
    @@ -785,10 +805,10 @@
     
     
     (Intercept)
    -8.95
    -2.26
    -(5.46, 14.68)
    -8.69
    +4.52
    +1.15
    +(2.74, 7.45)
    +5.90
     < .001
     
     
    @@ -800,22 +820,14 @@
     < .001
     
     
    -sex (M)
    -0.50
    -0.06
    -(0.40, 0.63)
    --5.88
    +sex (F)
    +1.98
    +0.23
    +(1.58, 2.49)
    +5.88
     < .001
     
     
    -sex (Trans)
    -7.99e-08
    -1.02e-04
    -(0.00, Inf)
    --0.01
    -0.990
    -
    -
     homeless (homeless)
     3.58
     0.45
    @@ -823,7 +835,7 @@
     10.06
     < .001
     
    -
    +
     polydrug (multiple drugs used)
     1.45e-07
     5.82e-05
    @@ -831,7 +843,7 @@
     -0.04
     0.969
     
    -
    +
     homeless (homeless) × polydrug (multiple drugs used)
     1.27e+06
     5.12e+08
    @@ -847,7 +859,7 @@
     
    -
    Show R code
    library(ggfortify)
    +
    Show R code
    @@ -868,7 +880,7 @@
    Table 4.4: Negative binomial model for needle-sharing data
    -
    Show R code
    library(MASS) #need this for glm.nb()
    +
    Show R code
    library(MASS) #need this for glm.nb()
     glm1.nb = glm.nb(
       data = needles,
       shared_syr ~ age + sex + homeless*polydrug
    @@ -880,31 +892,29 @@
     #>     data = needles, init.theta = 0.08436295825, link = log)
     #> 
     #> Coefficients:
    -#>                                                 Estimate  Std. Error z value
    -#> (Intercept)                                   2.0557e+00  1.7133e+00  1.1999
    -#> age                                          -2.7630e-02  3.8153e-02 -0.7242
    -#> sexM                                         -1.0648e+00  8.0684e-01 -1.3197
    -#> sexTrans                                     -2.5336e+01  1.1484e+05 -0.0002
    -#> homelesshomeless                              1.6546e+00  7.2207e-01  2.2916
    -#> polydrugmultiple drugs used                  -2.4562e+01  3.6055e+04 -0.0007
    -#> homelesshomeless:polydrugmultiple drugs used  2.3178e+01  3.6055e+04  0.0006
    +#>                                               Estimate Std. Error z value
    +#> (Intercept)                                   9.91e-01   1.71e+00    0.58
    +#> age                                          -2.76e-02   3.82e-02   -0.72
    +#> sexF                                          1.06e+00   8.07e-01    1.32
    +#> homelesshomeless                              1.65e+00   7.22e-01    2.29
    +#> polydrugmultiple drugs used                  -2.46e+01   3.61e+04    0.00
    +#> homelesshomeless:polydrugmultiple drugs used  2.32e+01   3.61e+04    0.00
     #>                                              Pr(>|z|)  
    -#> (Intercept)                                   0.23019  
    -#> age                                           0.46895  
    -#> sexM                                          0.18694  
    -#> sexTrans                                      0.99982  
    -#> homelesshomeless                              0.02193 *
    -#> polydrugmultiple drugs used                   0.99946  
    -#> homelesshomeless:polydrugmultiple drugs used  0.99949  
    +#> (Intercept)                                     0.563  
    +#> age                                             0.469  
    +#> sexF                                            0.187  
    +#> homelesshomeless                                0.022 *
    +#> polydrugmultiple drugs used                     0.999  
    +#> homelesshomeless:polydrugmultiple drugs used    0.999  
     #> ---
     #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
     #> 
     #> (Dispersion parameter for Negative Binomial(0.0844) family taken to be 1)
     #> 
    -#>     Null deviance: 69.8027  on 120  degrees of freedom
    -#> Residual deviance: 57.7820  on 114  degrees of freedom
    +#>     Null deviance: 69.193  on 119  degrees of freedom
    +#> Residual deviance: 57.782  on 114  degrees of freedom
     #>   (7 observations deleted due to missingness)
    -#> AIC: 317.506
    +#> AIC: 315.5
     #> 
     #> Number of Fisher Scoring iterations: 1
     #> 
    @@ -918,7 +928,7 @@
     
    -
    Show R code
    tibble(name = names(coef(glm1)), poisson = coef(glm1), nb = coef(glm1.nb))
    +
    Show R code
    tibble(name = names(coef(glm1)), poisson = coef(glm1), nb = coef(glm1.nb))
    Table 4.5: Poisson versus Negative Binomial Regression coefficient estimates @@ -926,7 +936,7 @@
    @@ -934,6 +944,325 @@
    +

    zero-inflation

    +
    +
    Show R code
    library(glmmTMB)
    +zinf_fit1 = glmmTMB(
    +  family = "poisson",
    +  data  = needles,
    +  formula = shared_syr ~ age + sex + homeless*polydrug,
    +  ziformula = ~ age + sex + homeless + polydrug # fit won't converge with interaction
    +)
    +
    +zinf_fit1 |> 
    +  parameters(exponentiate = TRUE) |> 
    +  print_md()
    +
    +
    +Table 4.6: Zero-inflated poisson model +
    +
    + + ++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    # Fixed Effects
    ParameterIRRSE95% CIzp
    (Intercept)3.160.82(1.90, 5.25)4.44< .001
    age1.015.88e-03(1.00, 1.02)1.740.081
    sex [F]3.430.44(2.67, 4.40)9.68< .001
    homeless [homeless]3.440.47(2.63, 4.50)9.03< .001
    polydrug [multiple drugs used]1.85e-091.21e-05(0.00, Inf)-3.08e-030.998
    homeless [homeless] × polydrug [multiple drugs used]1.38e+089.04e+11(0.00, Inf)2.87e-030.998
    + + ++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    # Zero-Inflation
    ParameterOdds RatioSE95% CIzp
    (Intercept)0.490.54(0.06, 4.25)-0.650.514
    age1.050.03(1.00, 1.10)1.950.051
    sex [F]1.440.84(0.46, 4.50)0.620.533
    homeless [homeless]0.680.34(0.26, 1.80)-0.780.436
    polydrug [multiple drugs used]1.150.91(0.24, 5.43)0.180.858
    +
    +
    +
    +
    +
    +

    zero-inflated negative binomial model

    +
    +
    Show R code
    library(glmmTMB)
    +zinf_fit1 = glmmTMB(
    +  family = nbinom2,
    +  data  = needles,
    +  formula = shared_syr ~ age + sex + homeless*polydrug,
    +  ziformula = ~ age + sex + homeless + polydrug # fit won't converge with interaction
    +)
    +
    +zinf_fit1 |> 
    +  parameters(exponentiate = TRUE) |> 
    +  print_md()
    +
    +
    +Table 4.7: Zero-inflated negative binomial model +
    +
    + + ++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    # Fixed Effects
    ParameterIRRSE95% CIzp
    (Intercept)1.061.48(0.07, 16.52)0.040.969
    age1.020.03(0.96, 1.08)0.530.599
    sex [F]6.866.36(1.12, 42.16)2.080.038
    homeless [homeless]6.444.59(1.60, 26.01)2.620.009
    polydrug [multiple drugs used]8.25e-107.07e-06(0.00, Inf)-2.44e-030.998
    homeless [homeless] × polydrug [multiple drugs used]2.36e+082.02e+12(0.00, Inf)2.25e-030.998
    + + ++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    # Zero-Inflation
    ParameterOdds RatioSE95% CIzp
    (Intercept)0.100.20(1.47e-03, 6.14)-1.110.269
    age1.070.04(0.99, 1.15)1.780.075
    sex [F]2.722.40(0.48, 15.33)1.130.258
    homeless [homeless]1.150.86(0.27, 4.96)0.190.853
    polydrug [multiple drugs used]0.750.86(0.08, 7.12)-0.250.799
    + + + + + + + + + + + + +
    # Dispersion
    ParameterCoefficient95% CI
    (Intercept)0.44(0.11, 1.71)
    +
    +
    +
    +
    +
    @@ -946,7 +1275,7 @@ Vittinghoff, Eric, David V Glidden, Stephen C Shiboski, and Charles E McCulloch. 2012. Regression Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Measures Models. 2nd ed. Springer. https://doi.org/10.1007/978-1-4614-1353-0.
    -
    diff --git a/intro-to-survival-analysis.html b/intro-to-survival-analysis.html index edd913406..caffe24f1 100644 --- a/intro-to-survival-analysis.html +++ b/intro-to-survival-analysis.html @@ -359,13 +359,14 @@
  • 5.3.4 The Hazard Function
  • 5.3.5 The Cumulative Hazard Function
  • 5.3.6 Some Key Mathematical Relationships among Survival Concepts
  • +
  • 5.3.7 Likelihood with censoring
  • 5.4 Parametric Models for Time-to-Event Outcomes
  • @@ -432,7 +433,7 @@
    Published
    -

    Last modified: 2024-05-07: 14:03:57 (PM)

    +

    Last modified: 2024-05-21: 13:03:25 (PM)

    @@ -503,7 +504,7 @@ }

    5.1 Overview

    -

    +

    5.1.1 Time-to-event outcomes

    Survival analysis is a framework for modeling time-to-event outcomes. It is used in:

      @@ -526,7 +527,7 @@

    5.2 Time-to-event outcome distributions

    -

    +

    5.2.1 Distributions of Time-to-Event Data

    • The distribution of event times is asymmetric and can be long-tailed, and starts at 0 (that is, \(P(T<0) = 0\)).
    • @@ -550,15 +551,17 @@
    • If an individual has a negative HIV test at time \(t_1\) and a positive HIV test at time \(t_2\), then the infection event is interval censored.

    5.3 Distribution functions for time-to-event variables

    -

    +

    5.3.1 The Probability Density Function (PDF)

    For a time-to-event variable \(T\) with a continuous distribution, the probability density function is defined as usual:

    \[f(t)\stackrel{\text{def}}{=}p(t) \stackrel{\text{def}}{=}p(T=t)\]

    Typically, this density is assumed to be 0 for all \(t<0\); that is, \(f(t) = 0, \forall t<0\). In other words, the range of \(T\) is typically \([0,\infty)\).

    +

    Example 5.1 (exponential distribution) Recall from Epi 202: the pdf of the exponential distribution family of models is:

    \[p(T=t) = \mathbb{1}_{t \ge 0} \cdot \lambda \text{e}^{-\lambda t}\]

    where \(\lambda > 0\).

    +

    Here are some examples of exponential pdfs:

    @@ -575,7 +578,7 @@

    \[ \begin{aligned} F(t) &\stackrel{\text{def}}{=}\Pr(T \le t)\\ -&=\int_{u=0}^t f(u) du +&=\int_{u=-\infty}^t f(u) du \end{aligned} \]

    @@ -583,6 +586,7 @@

    \[ P(T\le t) = \mathbb{1}_{t \ge 0} \cdot (1- \text{e}^{-\lambda t}) \] where \(\lambda > 0\).

    +

    Here are some examples of exponential cdfs:

    @@ -592,7 +596,6 @@
    -

    5.3.3 The Survival Function

    For survival data, a more important quantity is the survival function:

    @@ -603,13 +606,20 @@ &=1-F(t)\\ \end{aligned} \]

    -

    The survival function \(S(t)\) is the probability that the event time is later than \(t\). If the event in a clinical trial is death, then \(S(t)\) is the expected fraction of the original population at time 0 who have survived up to time \(t\) and are still alive at time \(t\); that is, if \(X_t\) represents survival status at time \(t\), with \(X_t = 1\) denoting alive at time \(t\) and \(X_t = 0\) denoting deceased at time \(t\), then:

    +
    +
    +

    The survival function \(S(t)\) is the probability that the event time is later than \(t\). If the event in a clinical trial is death, then \(S(t)\) is the expected fraction of the original population at time 0 who have survived up to time \(t\) and are still alive at time \(t\); that is:

    +
    +

    If \(X_t\) represents survival status at time \(t\), with \(X_t = 1\) denoting alive at time \(t\) and \(X_t = 0\) denoting deceased at time \(t\), then:

    \[S(t) = \mathbb{E}\left[X_t\right]\]

    +

    Example 5.3 (exponential distribution) Since \(S(t) = 1 - F(t)\), the survival function of the exponential distribution family of models is:

    \[ P(T> t) = \left\{ {{\text{e}^{-\lambda t}, t\ge0} \atop {1, t \le 0}}\right. \] where \(\lambda > 0\).

    +
    +

    Here are some examples of exponential pdfs:

    @@ -619,7 +629,6 @@
    -

    5.3.4 The Hazard Function

    Another important quantity is the hazard function:

    @@ -628,19 +637,14 @@

    \[h(t) \stackrel{\text{def}}{=}p(T=t|T\ge t)\]

    If \(T\) represents the time at which an event occurs, then \(h(t)\) is the probability that the event occurs at time \(t\), given that it has not occurred prior to time \(t\).

    +

    The hazard function has an important relationship to the density and survival functions, which we can use to derive the hazard function for a given probability distribution.

    Theorem 5.1 \[h(t)=\frac{f(t)}{S(t)}\]

    -
    -

    Proof.

    -
    -

    Lemma 5.1 (Joint probability of a variable with itself) \[p(T=t, T\ge t) = p(T=t)\]

    -
    -

    Proof. Recall from Epi 202: if \(A\) and \(B\) are statistical events and \(A\subseteq B\), then \(p(A, B) = p(A)\). In particular, \(\{T=t\} \subseteq \{T\geq t\}\), so \(p(T=t, T\ge t) = p(T=t)\).

    -
    -

    Hence:

    -

    \[ +


    +
    +

    Proof. \[ \begin{aligned} h(t) &=p(T=t|T\ge t)\\ &=\frac{p(T=t, T\ge t)}{p(T \ge t)}\\ @@ -649,7 +653,7 @@ \end{aligned} \]

    -
    +

    Example 5.4 (exponential distribution) The hazard function of the exponential distribution family of models is:

    \[ @@ -659,8 +663,9 @@ &= \frac{\mathbb{1}_{t \ge 0}\cdot \lambda \text{e}^{-\lambda t}}{\text{e}^{-\lambda t}}\\ &=\mathbb{1}_{t \ge 0}\cdot \lambda \end{aligned} -\]

    -

    Figure 5.1 shows some examples of exponential hazard functions:

    +\] Figure 5.1 shows some examples of exponential hazard functions.

    +
    +
    @@ -673,10 +678,12 @@
    -
    +

    We can also view the hazard function as the derivative of the negative of the logarithm of the survival function:

    Theorem 5.2 \[h(t) = \frac{\partial}{\partial t}\left\{-\text{log}\left\{S(t)\right\}\right\}\]

    +
    +

    Proof. \[ \begin{aligned} @@ -689,14 +696,16 @@ \end{aligned} \]

    -

    5.3.5 The Cumulative Hazard Function

    Since \(h(t) = \frac{\partial}{\partial t}\left\{-\text{log}\left\{S(t)\right\}\right\}\) (see Theorem 5.2), we also have:

    Corollary 5.1 \[S(t) = \text{exp}\left\{-\int_{u=0}^t h(u)du\right\} \tag{5.1}\]

    +
    +

    The integral in Equation 5.1 is important enough to have its own name: cumulative hazard.

    +

    Definition 5.2 (cumulative hazard) The cumulative hazard function \(H(t)\) is defined as:

    \[H(t) \stackrel{\text{def}}{=}\int_{u=0}^t h(u) du\]

    @@ -707,13 +716,18 @@

    \[ H(t) = \mathbb{1}_{t \ge 0}\cdot \lambda t \]

    -

    Here are some examples of exponential cumulative hazard functions:

    +

    Figure 5.2 shows some examples of exponential cumulative hazard functions.

    +
    +
    -
    -

    -
    +
    +
    +Figure 5.2: Examples of exponential cumulative hazard functions +
    +
    +
    @@ -730,6 +744,7 @@ \xleftarrow[-\text{log}\left\{S(t)\right\}]{} S(t) \xleftarrow[1-F(t)]{} F(t) \]

    +

    Identities:

    \[ \begin{aligned} @@ -743,6 +758,7 @@ f(t) &= h(t)\cdot S(t)\\ \end{aligned} \]

    +

    Some proofs (others left as exercises):

    \[ \begin{aligned} @@ -751,6 +767,7 @@ &= -f(t)\\ \end{aligned} \]

    +

    \[ \begin{aligned} \frac{\partial}{\partial t}\text{log}\left\{S(t)\right\} @@ -759,6 +776,7 @@ &= -h(t)\\ \end{aligned} \]

    +

    \[ \begin{aligned} H(t) @@ -772,10 +790,11 @@ &=-\text{log}\left\{S(t)\right\} \end{aligned} \]

    -

    Equivalently:

    +
    +

    Corollary:

    \[S(t) = \text{exp}\left\{-H(t)\right\}\]

    +

    Example: Time to death the US in 2004

    -
    Daily hazard rates for US Females in 2004

    The first day is the most dangerous:

    @@ -785,7 +804,7 @@
    -
    Daily hazard rates for US Males and Females in 2004
    +

    Exercise: hypothesize why these curves differ where they do?

    @@ -795,7 +814,7 @@
    -
    Survival curve for US females
    +

    Exercise: compare and contrast this curve with the corresponding hazard curve.

    @@ -805,20 +824,8 @@
    -

    Likelihood with censoring *

    -
    -
    -
    - -
    -
    -Note -
    -
    -
    -

    This subsection was not presented in class in 2023; it is not necessary to understand for the qualifying exam.

    -
    -
    +

    +5.3.7 Likelihood with censoring

    If an event time \(T\) is observed exactly as \(T=t\), then the likelihood of that observation is just its probability density function:

    \[ \begin{aligned} @@ -833,6 +840,7 @@ &= \text{log}\left\{h_T(t)\right\} - H_T(t)\\ \end{aligned} \]

    +

    If instead the event time \(T\) is censored and only known to be after time \(y\), then the likelihood of that censored observation is instead the survival function evaluated at the censoring time:

    \[ \begin{aligned} @@ -845,9 +853,12 @@ &=-H(y)\\ \end{aligned} \]

    +
    +

    What’s written above is incomplete. We also observed whether or not the observation was censored. Let \(C\) denote the time when censoring would occur (if the event did not occur first); let \(f_C(y)\) and \(S_C(y)\) be the corresponding density and survival functions for the censoring event.

    Let \(Y\) denote the time when observation ended (either by censoring or by the event of interest occurring), and let \(D\) be an indicator variable for the event occurring at \(Y\) (so \(D=0\) represents a censored observation and \(D=1\) represents an uncensored observation). In other words, let \(Y \stackrel{\text{def}}{=}\min(T,C)\) and \(D \stackrel{\text{def}}{=}\mathbb 1{\{T<=C\}}\).

    Then the complete likelihood of the observed data \((Y,D)\) is:

    +

    \[ \begin{aligned} \mathcal L(y,d) @@ -856,8 +867,11 @@ \left[p(T>y,C=y)\right]^{1-d}\\ \end{aligned} \]

    +
    +

    Typically, survival analyses assume that \(C\) and \(T\) are mutually independent; this assumption is called “non-informative” censoring.

    Then the joint likelihood \(p(Y,D)\) factors into the product \(p(Y), p(D)\), and the likelihood reduces to:

    +

    \[ \begin{aligned} \mathcal L(y,d) @@ -873,7 +887,10 @@ \left(f_C(y)^{1-d} \cdot S_C(y)^{d}\right) \end{aligned} \]

    +
    +

    The corresponding log-likelihood is:

    +

    \[ \begin{aligned} \ell(y,d) @@ -899,7 +916,10 @@

  • \(\theta = (\theta_T, \theta_C)\) be the combined vector of all parameters.
  • -

    Then corresponding score function is:

    +
    +
    +

    The corresponding score function is:

    +

    \[ \begin{aligned} \ell'(y,d) @@ -929,7 +949,10 @@ \right)\\ \end{aligned} \]

    +
    +

    As long as \(\theta_C\) and \(\theta_T\) don’t share any parameters, then if censoring is non-informative, the partial derivative with respect to \(\theta_T\) is:

    +

    \[ \begin{aligned} \ell'_{\theta_T}(y,d) @@ -961,7 +984,11 @@ f_T(y)^d \cdot S_T(y)^{1-d} \right\}\\ \end{aligned} -\] Thus, the MLE for \(\theta_T\) won’t depend on \(\theta_C\), and we can ignore the distribution of \(C\) when estimating the parameters of \(f_T(t)=p(T=t)\).

    +\]

    +
    +
    +

    Thus, the MLE for \(\theta_T\) won’t depend on \(\theta_C\), and we can ignore the distribution of \(C\) when estimating the parameters of \(f_T(t)=p(T=t)\).

    +

    Then:

    \[ \begin{aligned} @@ -973,8 +1000,13 @@ &= S_T(y) \cdot h_T(y)^d \\ \end{aligned} \]

    +

    That is, if the event occurred at time \(y\) (i.e., if \(d=1\)), then the likelihood of \((Y,D) = (y,d)\) is equal to the hazard function at \(y\) times the survival function at \(y\). Otherwise, the likelihood is equal to just the survival function at \(y\).

    +
    +
    +

    The corresponding log-likelihood is:

    +

    \[ \begin{aligned} \ell(y,d) @@ -985,21 +1017,10 @@ &= -H_T(y) + d\cdot \text{log}\left\{h_T(y)\right\}\\ \end{aligned} \]

    +

    In other words, the log-likelihood contribution from a single observation \((Y,D) = (y,d)\) is equal to the negative cumulative hazard at \(y\), plus the log of the hazard at \(y\) if the event occurred at time \(y\).

    -
    -
    -
    - -
    -
    -Note -
    -
    -
    -

    End of extra section.

    -
    -

    +

    5.4 Parametric Models for Time-to-Event Outcomes

    5.4.1 Exponential Distribution

    @@ -1010,6 +1031,7 @@
  • The mean survival time is \(\lambda^{-1}\)
  • +

    Mathematical details of exponential distribution

    \[ \begin{aligned} @@ -1022,7 +1044,7 @@ h(t) &= -\frac{f(t)}{S(t)} = -\frac{\lambda \text{e}^{-\lambda t}}{\text{e}^{-\lambda t}}=\lambda \end{aligned} \]

    -

    Estimation of \(\lambda\) +

    Estimating \(\lambda\)

    • Suppose we have \(m\) exponential survival times of \(t_1, t_2,\ldots,t_m\) and \(k\) right-censored values at \(u_1,u_2,\ldots,u_k\).

    • @@ -1034,8 +1056,12 @@ \text{log}\left\{\lambda\right\}-\lambda t_i \] and the likelihood of a censored value is the probability of that outcome (survival greater than \(u_j\)) so the log-likelihood is

      \[ -\text{log}\left\{\lambda \text{e}^{u_j}\right\} =-\lambda u_j. +\begin{aligned} +\ell_j(\lambda) &= \text{log}\left\{\lambda \text{e}^{u_j}\right\} +\\ &= -\lambda u_j +\end{aligned} \]

      +

      Let \(T=\sum t_i\) and \(U=\sum u_j\). Then:

      \[ \begin{aligned} @@ -1050,7 +1076,7 @@ &= \frac{T+U}{m} \end{aligned} \]

      -

    Fisher Information and Standard Error

    +

    Fisher Information and Standard Error

    \[ \begin{aligned} E[-\ell''] @@ -1063,26 +1089,17 @@ &\approx \lambda/\sqrt{m} \end{aligned} \]

    +

    \(\hat\lambda\) depends on the censoring times of the censored observations, but \(\text{Var}\left(\hat\lambda\right)\) only depends on the number of uncensored observations, \(m\), and not on the number of censored observations (\(k\)).

    -

    Other Parametric Survival Distributions

    + +

    +5.4.2 Other Parametric Survival Distributions

    • Any density on \([0,\infty)\) can be a survival distribution, but the most useful ones are all skew right.
    • -
    • The commonest generalization of the exponential is the Weibull.
    • +
    • The most frequently used generalization of the exponential is the Weibull.
    • Other common choices are the gamma, log-normal, log-logistic, Gompertz, inverse Gaussian, and Pareto.
    • Most of what we do going forward is non-parametric or semi-parametric, but sometimes these parametric distributions provide a useful approach.
    • -

    -5.4.2 Weibull Distribution

    -

    \[ -\begin{aligned} -p(t)&= \alpha\lambda x^{\alpha-1}\text{e}^{-\lambda x^\alpha}\\ -h(t)&=\alpha\lambda x^{\alpha-1}\\ -S(t)&=\text{e}^{-\lambda x^\alpha}\\ -E(T)&= \Gamma(1+1/\alpha)\cdot \lambda^{-1/\alpha} -\end{aligned} -\]

    -

    When \(\alpha=1\) this is the exponential. When \(\alpha>1\) the hazard is increasing and when \(\alpha < 1\) the hazard is decreasing. This provides more flexibility than the exponential.

    -

    We will see more of this distribution later.

    -

    +

    5.5 Nonparametric Survival Analysis

    5.5.1 Basic ideas

    @@ -2927,7 +2944,7 @@

    Examples

    ## Overview -### Time-to-event outcomes +### Time-to-event outcomes {.smaller} **Survival analysis** is a framework for modeling *time-to-event* outcomes. It is used in: @@ -2944,7 +2961,7 @@

    Examples

    ## Time-to-event outcome distributions -### Distributions of Time-to-Event Data +### Distributions of Time-to-Event Data {.smaller} - The distribution of event times is asymmetric and can be long-tailed, and starts at 0 (that is, $P(T<0) = 0$). @@ -2983,7 +3000,7 @@

    Examples

    ## Distribution functions for time-to-event variables -### The Probability Density Function (PDF) +### The Probability Density Function (PDF) {.smaller} For a time-to-event variable $T$ with a continuous distribution, the **probability density function** is defined as usual: @@ -2994,1583 +3011,1637 @@

    Examples

    $f(t) = 0, \forall t<0$. In other words, the range of $T$ is typically $[0,\infty)$. -:::{#exm-exp-pdf} -#### exponential distribution - -Recall from Epi 202: the pdf of the exponential distribution family of -models is: - -$$p(T=t) = \1{t \ge 0} \cdot \lambda \ef{-\lambda t}$$ +--- + +:::{#exm-exp-pdf} +#### exponential distribution + +Recall from Epi 202: the pdf of the exponential distribution family of +models is: -where $\lambda > 0$. +$$p(T=t) = \1{t \ge 0} \cdot \lambda \ef{-\lambda t}$$ -Here are some examples of exponential pdfs: +where $\lambda > 0$. -```{r, echo = FALSE} -library(ggplot2) -ggplot() + - geom_function( - aes(col = "0.5"), - fun = \(x) dexp(x, rate = 0.5)) + - geom_function( - aes(col = "p = 1"), - fun = \(x) dexp(x, rate = 1)) + - geom_function( - aes(col = "p = 1.5"), - fun = \(x) dexp(x, rate = 1.5)) + - geom_function( - aes(col = "p = 5"), - fun = \(x) dexp(x, rate = 5)) + - theme_bw() + - ylab("p(T=t)") + - guides(col = guide_legend(title = expr(lambda))) + - xlab("Time (t)") + - xlim(0, 2.5) + - theme( - axis.title.x = - element_text( - angle = 0, - vjust = 1, - hjust = 1), - axis.title.y = - element_text( - angle = 0, - vjust = 1, - hjust = 1)) -``` - -::: - -### The Cumulative Distribution Function (CDF) +--- + +Here are some examples of exponential pdfs: + +```{r, echo = FALSE} +library(ggplot2) +ggplot() + + geom_function( + aes(col = "0.5"), + fun = \(x) dexp(x, rate = 0.5)) + + geom_function( + aes(col = "p = 1"), + fun = \(x) dexp(x, rate = 1)) + + geom_function( + aes(col = "p = 1.5"), + fun = \(x) dexp(x, rate = 1.5)) + + geom_function( + aes(col = "p = 5"), + fun = \(x) dexp(x, rate = 5)) + + theme_bw() + + ylab("p(T=t)") + + guides(col = guide_legend(title = expr(lambda))) + + xlab("Time (t)") + + xlim(0, 2.5) + + theme( + axis.title.x = + element_text( + angle = 0, + vjust = 1, + hjust = 1), + axis.title.y = + element_text( + angle = 0, + vjust = 1, + hjust = 1)) +``` -The **cumulative distribution function** is defined as: +::: -$$ -\begin{aligned} -F(t) &\eqdef \Pr(T \le t)\\ -&=\int_{u=0}^t f(u) du -\end{aligned} -$$ - -:::{#exm-exp-cdf} -##### exponential distribution - -Recall from Epi 202: the cdf of the exponential distribution family of -models is: - -$$ -P(T\le t) = \mathbb{1}_{t \ge 0} \cdot (1- \text{e}^{-\lambda t}) -$$ where $\lambda > 0$. +### The Cumulative Distribution Function (CDF) + +The **cumulative distribution function** is defined as: + +$$ +\begin{aligned} +F(t) &\eqdef \Pr(T \le t)\\ +&=\int_{u=-\infty}^t f(u) du +\end{aligned} +$$ + +:::{#exm-exp-cdf} +##### exponential distribution + +Recall from Epi 202: the cdf of the exponential distribution family of +models is: -Here are some examples of exponential cdfs: - -```{r, echo = FALSE} -library(ggplot2) -ggplot() + - geom_function( - aes(col = "0.5"), - fun = \(x) pexp(x, rate = 0.5)) + - geom_function( - aes(col = "p = 1"), - fun = \(x) pexp(x, rate = 1)) + +$$ +P(T\le t) = \mathbb{1}_{t \ge 0} \cdot (1- \text{e}^{-\lambda t}) +$$ where $\lambda > 0$. + +::: + +Here are some examples of exponential cdfs: + +```{r, echo = FALSE} +library(ggplot2) +ggplot() + geom_function( - aes(col = "p = 1.5"), - fun = \(x) pexp(x, rate = 1.5)) + + aes(col = "0.5"), + fun = \(x) pexp(x, rate = 0.5)) + geom_function( - aes(col = "p = 5"), - fun = \(x) pexp(x, rate = 5)) + - theme_bw() + - ylab("p(T<=t)") + - guides(col = guide_legend(title = expr(lambda))) + - xlab("Time (t)") + - xlim(0, 2.5) + - theme( - axis.title.x = - element_text( - angle = 0, - vjust = 1, - hjust = 1), - axis.title.y = - element_text( - angle = 0, - vjust = 1, - hjust = 1)) -``` - -::: - -### The Survival Function - -For survival data, a more important quantity is the **survival -function**: - -$$ -\begin{aligned} -S(t) &\eqdef \Pr(T > t)\\ -&=\int_{u=t}^\infty p(u) du\\ -&=1-F(t)\\ -\end{aligned} -$$ - -The survival function $S(t)$ is the probability that the event time is -later than $t$. If the event in a clinical trial is death, then $S(t)$ is -the expected fraction of the original population at time 0 who have survived up to time $t$ and are still alive at time $t$; that is, if $X_t$ represents survival status at time $t$, with $X_t = 1$ denoting alive at time $t$ and $X_t = 0$ denoting deceased at time $t$, then: + aes(col = "p = 1"), + fun = \(x) pexp(x, rate = 1)) + + geom_function( + aes(col = "p = 1.5"), + fun = \(x) pexp(x, rate = 1.5)) + + geom_function( + aes(col = "p = 5"), + fun = \(x) pexp(x, rate = 5)) + + theme_bw() + + ylab("p(T<=t)") + + guides(col = guide_legend(title = expr(lambda))) + + xlab("Time (t)") + + xlim(0, 2.5) + + theme( + axis.title.x = + element_text( + angle = 0, + vjust = 1, + hjust = 1), + axis.title.y = + element_text( + angle = 0, + vjust = 1, + hjust = 1)) +``` + +### The Survival Function + +For survival data, a more important quantity is the **survival +function**: + +$$ +\begin{aligned} +S(t) &\eqdef \Pr(T > t)\\ +&=\int_{u=t}^\infty p(u) du\\ +&=1-F(t)\\ +\end{aligned} +$$ -$$S(t) = \E{X_t}$$ +--- -:::{#exm-exp-survfn} -##### exponential distribution - -Since $S(t) = 1 - F(t)$, the survival function of the exponential -distribution family of models is: +::: notes +The survival function $S(t)$ is the probability that the event time is +later than $t$. If the event in a clinical trial is death, then $S(t)$ is +the expected fraction of the original population at time 0 who have survived up to time $t$ and are still alive at time $t$; that is: +::: -$$ -P(T> t) = \left\{ {{\text{e}^{-\lambda t}, t\ge0} \atop {1, t \le 0}}\right. -$$ where $\lambda > 0$. +If $X_t$ represents survival status at time $t$, with $X_t = 1$ denoting alive at time $t$ and $X_t = 0$ denoting deceased at time $t$, then: + +$$S(t) = \E{X_t}$$ -Here are some examples of exponential pdfs: +--- -```{r, echo = FALSE} -library(ggplot2) -ggplot() + - geom_function( - aes(col = "0.5"), - fun = \(x) pexp(x, lower = FALSE, rate = 0.5)) + - geom_function( - aes(col = "p = 1"), - fun = \(x) pexp(x, lower = FALSE, rate = 1)) + - geom_function( - aes(col = "p = 1.5"), - fun = \(x) pexp(x, lower = FALSE, rate = 1.5)) + - geom_function( - aes(col = "p = 5"), - fun = \(x) pexp(x, lower = FALSE, rate = 5)) + - theme_bw() + - ylab("p(T<=t)") + - guides(col = guide_legend(title = expr(lambda))) + - xlab("Time (t)") + - xlim(0, 2.5) + - theme( - axis.title.x = - element_text( - angle = 0, - vjust = 1, - hjust = 1), - axis.title.y = - element_text( - angle = 0, - vjust = 1, - hjust = 1)) -``` - -::: - -### The Hazard Function - -Another important quantity is the **hazard function**: - -:::{#def-hazard} - -{{< include _def-hazard.qmd >}} - -::: - -The hazard function has an important relationship to the density and survival functions, which we can use to derive the hazard function for a given probability distribution. - -:::{#thm-hazard1} +:::{#exm-exp-survfn} +##### exponential distribution + +Since $S(t) = 1 - F(t)$, the survival function of the exponential +distribution family of models is: + +$$ +P(T> t) = \left\{ {{\text{e}^{-\lambda t}, t\ge0} \atop {1, t \le 0}}\right. +$$ where $\lambda > 0$. + +::: + +--- + +Here are some examples of exponential pdfs: + +```{r, echo = FALSE} +library(ggplot2) +ggplot() + + geom_function( + aes(col = "0.5"), + fun = \(x) pexp(x, lower = FALSE, rate = 0.5)) + + geom_function( + aes(col = "p = 1"), + fun = \(x) pexp(x, lower = FALSE, rate = 1)) + + geom_function( + aes(col = "p = 1.5"), + fun = \(x) pexp(x, lower = FALSE, rate = 1.5)) + + geom_function( + aes(col = "p = 5"), + fun = \(x) pexp(x, lower = FALSE, rate = 5)) + + theme_bw() + + ylab("p(T<=t)") + + guides(col = guide_legend(title = expr(lambda))) + + xlab("Time (t)") + + xlim(0, 2.5) + + theme( + axis.title.x = + element_text( + angle = 0, + vjust = 1, + hjust = 1), + axis.title.y = + element_text( + angle = 0, + vjust = 1, + hjust = 1)) +``` -$$h(t)=\frac{f(t)}{S(t)}$$ +### The Hazard Function -::::{.proof} +Another important quantity is the **hazard function**: - -:::::{#lem-joint-prob-same-var} - -#### Joint probability of a variable with itself - -$$p(T=t, T\ge t) = p(T=t)$$ - -::::::{.proof} -Recall from Epi 202: if $A$ and $B$ are statistical events and $A\subseteq B$, then $p(A, B) = p(A)$. In particular, $\{T=t\} \subseteq \{T\geq t\}$, so $p(T=t, T\ge t) = p(T=t)$. -:::::: -::::: -Hence: - -$$ -\begin{aligned} -h(t) &=p(T=t|T\ge t)\\ -&=\frac{p(T=t, T\ge t)}{p(T \ge t)}\\ -&=\frac{p(T=t)}{p(T \ge t)}\\ -&=\frac{f(t)}{S(t)} -\end{aligned} -$$ - -:::: - -::: - -:::{#exm-exp-haz} -##### exponential distribution - -The hazard function of the exponential distribution family of models is: - -$$ -\begin{aligned} -P(T=t|T \ge t) -&= \frac{f(t)}{S(t)}\\ -&= \frac{\mathbb{1}_{t \ge 0}\cdot \lambda \text{e}^{-\lambda t}}{\text{e}^{-\lambda t}}\\ -&=\mathbb{1}_{t \ge 0}\cdot \lambda -\end{aligned} -$$ - -@fig-exp-hazard shows some examples of exponential hazard functions: - -```{r, echo = FALSE} -#| fig-cap: "Examples of hazard functions for exponential distributions" -#| label: fig-exp-hazard -library(ggplot2) -ggplot() + - geom_hline( - aes(col = "0.5",yintercept = 0.5)) + - geom_hline( - aes(col = "p = 1", yintercept = 1)) + - geom_hline( - aes(col = "p = 1.5", yintercept = 1.5)) + - geom_hline( - aes(col = "p = 5", yintercept = 5)) + - theme_bw() + - ylab("p(T=t | T<=t)") + - ylim(0,5) + - guides(col = guide_legend(title = expr(lambda))) + - xlab("Time (t)") + - xlim(0, 2.5) + - theme( - axis.title.x = - element_text( - angle = 0, - vjust = 1, - hjust = 1), - axis.title.y = - element_text( - angle = 0, - vjust = 1, - hjust = 1)) -``` - -::: - -We can also view the hazard function as the derivative of the negative of the logarithm of the survival function: - -:::{#thm-h-logS} - -$$h(t) = \deriv{t}\cb{-\log{S(t)}}$$ - -::::{.proof} -$$ -\begin{aligned} -h(t) -&= \frac{f(t)}{S(t)}\\ -&= \frac{-S'(t)}{S(t)}\\ -&= -\frac{S'(t)}{S(t)}\\ -&=-\deriv{t}\log{S(t)}\\ -&=\deriv{t}\cb{-\log{S(t)}} -\end{aligned} -$$ -:::: -::: - -### The Cumulative Hazard Function - -Since $h(t) = \deriv{t}\cb{-\log{S(t)}}$ (see @thm-h-logS), we also have: - -:::{#cor-surv-int-haz} -$$S(t) = \exp{-\int_{u=0}^t h(u)du}$${#eq-surv-int-haz} -::: - -The integral in @eq-surv-int-haz is important enough to have its own name: **cumulative hazard**. - -:::{#def-cumhaz} - -##### cumulative hazard - -The **cumulative hazard function** $H(t)$ is defined as: - -$$H(t) \eqdef \int_{u=0}^t h(u) du$$ - -::: - -As we will see below, $H(t)$ is tractable to estimate, and we can then -derive an estimate of the hazard function using an approximate derivative -of the estimated cumulative hazard. - -:::{#exm-exp-cumhaz} - -The cumulative hazard function of the exponential distribution family of -models is: +:::{#def-hazard} + +{{< include _def-hazard.qmd >}} + +::: + +--- + +The hazard function has an important relationship to the density and survival functions, which we can use to derive the hazard function for a given probability distribution. + +:::{#thm-hazard1} + +$$h(t)=\frac{f(t)}{S(t)}$$ +::: + +--- + +::::{.proof} + +$$ +\begin{aligned} +h(t) &=p(T=t|T\ge t)\\ +&=\frac{p(T=t, T\ge t)}{p(T \ge t)}\\ +&=\frac{p(T=t)}{p(T \ge t)}\\ +&=\frac{f(t)}{S(t)} +\end{aligned} +$$ + +:::: + +--- + +:::{#exm-exp-haz} +##### exponential distribution + +The hazard function of the exponential distribution family of models is: + +$$ +\begin{aligned} +P(T=t|T \ge t) +&= \frac{f(t)}{S(t)}\\ +&= \frac{\mathbb{1}_{t \ge 0}\cdot \lambda \text{e}^{-\lambda t}}{\text{e}^{-\lambda t}}\\ +&=\mathbb{1}_{t \ge 0}\cdot \lambda +\end{aligned} +$$ +@fig-exp-hazard shows some examples of exponential hazard functions. + +::: + +--- + +```{r, echo = FALSE} +#| fig-cap: "Examples of hazard functions for exponential distributions" +#| label: fig-exp-hazard +library(ggplot2) +ggplot() + + geom_hline( + aes(col = "0.5",yintercept = 0.5)) + + geom_hline( + aes(col = "p = 1", yintercept = 1)) + + geom_hline( + aes(col = "p = 1.5", yintercept = 1.5)) + + geom_hline( + aes(col = "p = 5", yintercept = 5)) + + theme_bw() + + ylab("p(T=t | T<=t)") + + ylim(0,5) + + guides(col = guide_legend(title = expr(lambda))) + + xlab("Time (t)") + + xlim(0, 2.5) + + theme( + axis.title.x = + element_text( + angle = 0, + vjust = 1, + hjust = 1), + axis.title.y = + element_text( + angle = 0, + vjust = 1, + hjust = 1)) +``` + +--- + +We can also view the hazard function as the derivative of the negative of the logarithm of the survival function: + +:::{#thm-h-logS} + +$$h(t) = \deriv{t}\cb{-\log{S(t)}}$$ +::: + +--- + +::::{.proof} +$$ +\begin{aligned} +h(t) +&= \frac{f(t)}{S(t)}\\ +&= \frac{-S'(t)}{S(t)}\\ +&= -\frac{S'(t)}{S(t)}\\ +&=-\deriv{t}\log{S(t)}\\ +&=\deriv{t}\cb{-\log{S(t)}} +\end{aligned} +$$ +:::: + +### The Cumulative Hazard Function + +Since $h(t) = \deriv{t}\cb{-\log{S(t)}}$ (see @thm-h-logS), we also have: + +:::{#cor-surv-int-haz} +$$S(t) = \exp{-\int_{u=0}^t h(u)du}$${#eq-surv-int-haz} +::: + +--- + +::: notes +The integral in @eq-surv-int-haz is important enough to have its own name: **cumulative hazard**. +::: + +:::{#def-cumhaz} + +##### cumulative hazard -$$ -H(t) = \mathbb{1}_{t \ge 0}\cdot \lambda t -$$ +The **cumulative hazard function** $H(t)$ is defined as: + +$$H(t) \eqdef \int_{u=0}^t h(u) du$$ -Here are some examples of exponential cumulative hazard functions: +::: -```{r, echo = FALSE} -library(ggplot2) -ggplot() + - geom_abline( - aes(col = "0.5",intercept = 0, slope = 0.5)) + - geom_abline( - aes(col = "p = 1", intercept = 0, slope = 1)) + - geom_abline( - aes(col = "p = 1.5", intercept = 0, slope = 1.5)) + - geom_abline( - aes(col = "p = 5", intercept = 0, slope = 5)) + - theme_bw() + - ylab("H(t)") + - ylim(0,5) + - guides(col = guide_legend(title = expr(lambda))) + - xlab("Time (t)") + - xlim(0, 2.5) + - theme( - axis.title.x = - element_text( - angle = 0, - vjust = 1, - hjust = 1), - axis.title.y = - element_text( - angle = 0, - vjust = 1, - hjust = 1)) -``` - -::: - -### Some Key Mathematical Relationships among Survival Concepts - -Diagram: - -$$ -h(t) \xrightarrow[]{\int_{u=0}^t h(u)du} H(t) -\xrightarrow[]{\exp{-H(t)}} S(t) -\xrightarrow[]{1-S(t)} F(t) -$$ - -$$ -h(t) \xleftarrow[\deriv{t}H(t)]{} H(t) -\xleftarrow[-\log{S(t)}]{} S(t) -\xleftarrow[1-F(t)]{} F(t) -$$ - - -Identities: +As we will see below, $H(t)$ is tractable to estimate, and we can then +derive an estimate of the hazard function using an approximate derivative +of the estimated cumulative hazard. + +:::{#exm-exp-cumhaz} + +The cumulative hazard function of the exponential distribution family of +models is: + +$$ +H(t) = \mathbb{1}_{t \ge 0}\cdot \lambda t +$$ + +@fig-cuhaz-exp shows some examples of exponential cumulative hazard functions. + +::: + +--- + +```{r, echo = FALSE} +#| fig-cap: "Examples of exponential cumulative hazard functions" +#| label: fig-cuhaz-exp +library(ggplot2) +ggplot() + + geom_abline( + aes(col = "0.5",intercept = 0, slope = 0.5)) + + geom_abline( + aes(col = "p = 1", intercept = 0, slope = 1)) + + geom_abline( + aes(col = "p = 1.5", intercept = 0, slope = 1.5)) + + geom_abline( + aes(col = "p = 5", intercept = 0, slope = 5)) + + theme_bw() + + ylab("H(t)") + + ylim(0,5) + + guides(col = guide_legend(title = expr(lambda))) + + xlab("Time (t)") + + xlim(0, 2.5) + + theme( + axis.title.x = + element_text( + angle = 0, + vjust = 1, + hjust = 1), + axis.title.y = + element_text( + angle = 0, + vjust = 1, + hjust = 1)) +``` -$$ -\begin{aligned} -S(t) &= 1 - F(t)\\ -&= \text{exp}\left\{-H(t)\right\}\\ -S'(t) &= -f(t)\\ -H(t) &= -\text{log}\left\{S(t)\right\}\\ -H'(t) &= h(t)\\ -h(t) &= \frac{f(t)}{S(t)}\\ - &= -\deriv{t}\log{S(t)} \\ -f(t) &= h(t)\cdot S(t)\\ -\end{aligned} -$$ - -::: {.content-visible unless-format="revealjs"} -Some proofs (others left as exercises): +### Some Key Mathematical Relationships among Survival Concepts + +Diagram: + +$$ +h(t) \xrightarrow[]{\int_{u=0}^t h(u)du} H(t) +\xrightarrow[]{\exp{-H(t)}} S(t) +\xrightarrow[]{1-S(t)} F(t) +$$ + +$$ +h(t) \xleftarrow[\deriv{t}H(t)]{} H(t) +\xleftarrow[-\log{S(t)}]{} S(t) +\xleftarrow[1-F(t)]{} F(t) +$$ -$$ -\begin{aligned} -S'(t) &= \deriv{t}(1-F(t))\\ -&= -F'(t)\\ -&= -f(t)\\ -\end{aligned} -$$ - -$$ -\begin{aligned} -\deriv{t}\log{S(t)} -&= \frac{S'(t)}{S(t)}\\ -&= -\frac{f(t)}{S(t)}\\ -&= -h(t)\\ -\end{aligned} -$$ - -$$ -\begin{aligned} -H(t) -&\eqdef \int_{u=0}^t h(u) du\\ -&= \int_0^t -\deriv{u}\text{log}\left\{S(u)\right\} du\\ -&= \left[-\text{log}\left\{S(u)\right\}\right]_{u=0}^{u=t}\\ -&= \left[\text{log}\left\{S(u)\right\}\right]_{u=t}^{u=0}\\ -&= \text{log}\left\{S(0)\right\} - \text{log}\left\{S(t)\right\}\\ -&= \text{log}\left\{1\right\} - \text{log}\left\{S(t)\right\}\\ -&= 0 - \text{log}\left\{S(t)\right\}\\ -&=-\text{log}\left\{S(t)\right\} -\end{aligned} -$$ - -Equivalently: - -$$S(t) = \text{exp}\left\{-H(t)\right\}$$ -::: - -#### Example: Time to death the US in 2004 - -##### Daily hazard rates for US Females in 2004 - -The first day is the most dangerous: - -```{r, echo = FALSE} -#| fig-cap: "Daily Hazard Rates in 2004 for US Females" -#| fig-pos: "H" -#| fig-height: 6 - -# download `survexp.rda` from: -# paste0( -# "https://github.com/therneau/survival/raw/", -# "f3ac93704949ff26e07720b56f2b18ffa8066470/", -# "data/survexp.rda") - -#(newer versions of `survival` don't have the first-year breakdown; see: -# https://cran.r-project.org/web/packages/survival/news.html) - -fs::path( - here::here(), - "data", - "survexp.rda") |> - load() -s1 <- survexp.us[,"female","2004"] -age1 <- c( - 0.5/365.25, - 4/365.25, - 17.5/365.25, - 196.6/365.25, - 1:109+0.5) -s2 <- 365.25*s1[5:113] -s2 <- c(s1[1], 6*s1[2], 22*s1[3], 337.25*s1[4], s2) -cols <- rep(1,113) -cols[1] <- 2 -cols[2] <- 3 -cols[3] <- 4 - -plot(age1,s1,type="b",lwd=2,xlab="Age",ylab="Daily Hazard Rate",col=cols) - -text(10,.003,"First Day",col=2) -text(18,.00030,"Rest of First Week",col=3) -text(18,.00015,"Rest of First month",col=4) -``` - -##### Daily hazard rates for US Males and Females in 2004 - -Exercise: hypothesize why these curves differ where they do? - -```{r,echo = FALSE} -#| fig-cap: "Daily Hazard Rates in 2004 for US Males and Females 1-40" -#| fig-pos: "H" -yrs=1:40 -s1 <- survexp.us[5:113,"male","2004"] -s2 <- survexp.us[5:113,"female","2004"] - -age1 <- 1:109 - -plot(age1[yrs],s1[yrs],type="l",lwd=2,xlab="Age",ylab="Daily Hazard Rate") -lines(age1[yrs],s2[yrs],col=2,lwd=2) -legend(5,5e-6,c("Males","Females"),col=1:2,lwd=2) - -``` - -##### Survival curve for US females - -Exercise: compare and contrast this curve with the corresponding hazard -curve. - -```{r, echo = FALSE} -#| fig-cap: "Survival Curve in 2004 for US Females" -#| fig-pos: "H" - -s1 <- survexp.us[,"female","2004"] - -s2 <- 365.25*s1[5:113] -s2 <- c(s1[1], 6*s1[2], 21*s1[3], 337.25*s1[4], s2) -cs2 <- cumsum(s2) -age2 <- c(1/365.25, 7/365.25, 28/365.25, 1:110) -plot(age2,exp(-cs2),type="l",lwd=2,xlab="Age",ylab="Survival") - -``` + +--- + +Identities: + +$$ +\begin{aligned} +S(t) &= 1 - F(t)\\ +&= \text{exp}\left\{-H(t)\right\}\\ +S'(t) &= -f(t)\\ +H(t) &= -\text{log}\left\{S(t)\right\}\\ +H'(t) &= h(t)\\ +h(t) &= \frac{f(t)}{S(t)}\\ + &= -\deriv{t}\log{S(t)} \\ +f(t) &= h(t)\cdot S(t)\\ +\end{aligned} +$$ + +--- + +Some proofs (others left as exercises): + +$$ +\begin{aligned} +S'(t) &= \deriv{t}(1-F(t))\\ +&= -F'(t)\\ +&= -f(t)\\ +\end{aligned} +$$ + +--- + +$$ +\begin{aligned} +\deriv{t}\log{S(t)} +&= \frac{S'(t)}{S(t)}\\ +&= -\frac{f(t)}{S(t)}\\ +&= -h(t)\\ +\end{aligned} +$$ + +--- + +$$ +\begin{aligned} +H(t) +&\eqdef \int_{u=0}^t h(u) du\\ +&= \int_0^t -\deriv{u}\text{log}\left\{S(u)\right\} du\\ +&= \left[-\text{log}\left\{S(u)\right\}\right]_{u=0}^{u=t}\\ +&= \left[\text{log}\left\{S(u)\right\}\right]_{u=t}^{u=0}\\ +&= \text{log}\left\{S(0)\right\} - \text{log}\left\{S(t)\right\}\\ +&= \text{log}\left\{1\right\} - \text{log}\left\{S(t)\right\}\\ +&= 0 - \text{log}\left\{S(t)\right\}\\ +&=-\text{log}\left\{S(t)\right\} +\end{aligned} +$$ + +--- + +Corollary: + +$$S(t) = \text{exp}\left\{-H(t)\right\}$$ + +--- + +#### Example: Time to death the US in 2004 + +The first day is the most dangerous: + +```{r, echo = FALSE} +#| fig-cap: "Daily Hazard Rates in 2004 for US Females" +#| fig-pos: "H" +#| fig-height: 6 + +# download `survexp.rda` from: +# paste0( +# "https://github.com/therneau/survival/raw/", +# "f3ac93704949ff26e07720b56f2b18ffa8066470/", +# "data/survexp.rda") + +#(newer versions of `survival` don't have the first-year breakdown; see: +# https://cran.r-project.org/web/packages/survival/news.html) + +fs::path( + here::here(), + "data", + "survexp.rda") |> + load() +s1 <- survexp.us[,"female","2004"] +age1 <- c( + 0.5/365.25, + 4/365.25, + 17.5/365.25, + 196.6/365.25, + 1:109+0.5) +s2 <- 365.25*s1[5:113] +s2 <- c(s1[1], 6*s1[2], 22*s1[3], 337.25*s1[4], s2) +cols <- rep(1,113) +cols[1] <- 2 +cols[2] <- 3 +cols[3] <- 4 + +plot(age1,s1,type="b",lwd=2,xlab="Age",ylab="Daily Hazard Rate",col=cols) + +text(10,.003,"First Day",col=2) +text(18,.00030,"Rest of First Week",col=3) +text(18,.00015,"Rest of First month",col=4) +``` + +--- + +Exercise: hypothesize why these curves differ where they do? + +```{r,echo = FALSE} +#| fig-cap: "Daily Hazard Rates in 2004 for US Males and Females 1-40" +#| fig-pos: "H" +yrs=1:40 +s1 <- survexp.us[5:113,"male","2004"] +s2 <- survexp.us[5:113,"female","2004"] -#### Likelihood with censoring \* +age1 <- 1:109 -::: callout-note -This subsection was not presented in class in 2023; it is not necessary -to understand for the qualifying exam. -::: - -If an event time $T$ is observed exactly as $T=t$, then the likelihood -of that observation is just its probability density function: +plot(age1[yrs],s1[yrs],type="l",lwd=2,xlab="Age",ylab="Daily Hazard Rate") +lines(age1[yrs],s2[yrs],col=2,lwd=2) +legend(5,5e-6,c("Males","Females"),col=1:2,lwd=2) + +``` + +--- -$$ -\begin{aligned} -\mathcal L(t) -&= p(T=t)\\ -&\eqdef f_T(t)\\ -&= h_T(t)S_T(t)\\ -\ell(t) -&\eqdef \text{log}\left\{\mathcal L(t)\right\}\\ -&= \text{log}\left\{h_T(t)S_T(t)\right\}\\ -&= \text{log}\left\{h_T(t)\right\} + \text{log}\left\{S_T(t)\right\}\\ -&= \text{log}\left\{h_T(t)\right\} - H_T(t)\\ -\end{aligned} -$$ - -If instead the event time $T$ is censored and only known to be after -time $y$, then the likelihood of that censored observation is instead -the survival function evaluated at the censoring time: - -$$ -\begin{aligned} -\mathcal L(y) -&=p_T(T>y)\\ -&\eqdef S_T(y)\\ -\ell(y) -&\eqdef \text{log}\left\{\mathcal L(y)\right\}\\ -&=\text{log}\left\{S(y)\right\}\\ -&=-H(y)\\ -\end{aligned} -$$ - -What's written above is incomplete. We also observed whether or not the -observation was censored. Let $C$ denote the time when censoring would -occur (if the event did not occur first); let $f_C(y)$ and $S_C(y)$ be -the corresponding density and survival functions for the censoring -event. - -Let $Y$ denote the time when observation ended (either by censoring or -by the event of interest occurring), and let $D$ be an indicator -variable for the event occurring at $Y$ (so $D=0$ represents a censored -observation and $D=1$ represents an uncensored observation). In other -words, let $Y \eqdef \min(T,C)$ and -$D \eqdef \mathbb 1{\{T<=C\}}$. - -Then the complete likelihood of the observed data $(Y,D)$ is: - -$$ -\begin{aligned} -\mathcal L(y,d) -&= p(Y=y, D=d)\\ -&= \left[p(T=y,C> y)\right]^d \cdot -\left[p(T>y,C=y)\right]^{1-d}\\ -\end{aligned} -$$ - -Typically, survival analyses assume that $C$ and $T$ are mutually -independent; this assumption is called "non-informative" censoring. - -Then the joint likelihood $p(Y,D)$ factors into the product -$p(Y), p(D)$, and the likelihood reduces to: - -$$ -\begin{aligned} -\mathcal L(y,d) -&= \left[p(T=y,C> y)\right]^d\cdot -\left[p(T>y,C=y)\right]^{1-d}\\ -&= \left[p(T=y)p(C> y)\right]^d\cdot -\left[p(T>y)p(C=y)\right]^{1-d}\\ -&= \left[f_T(y)S_C(y)\right]^d\cdot -\left[S(y)f_C(y)\right]^{1-d}\\ -&= \left[f_T(y)^d S_C(y)^d\right]\cdot -\left[S_T(y)^{1-d}f_C(y)^{1-d}\right]\\ -&= \left(f_T(y)^d \cdot S_T(y)^{1-d}\right)\cdot -\left(f_C(y)^{1-d} \cdot S_C(y)^{d}\right) -\end{aligned} -$$ - -The corresponding log-likelihood is: - -$$ -\begin{aligned} -\ell(y,d) -&= \text{log}\left\{\mathcal L(y,d) \right\}\\ -&= \text{log}\left\{ -\left(f_T(y)^d \cdot S_T(y)^{1-d}\right)\cdot -\left(f_C(y)^{1-d} \cdot S_C(y)^{d}\right) -\right\}\\ -&= \text{log}\left\{ -f_T(y)^d \cdot S_T(y)^{1-d} -\right\} -+ -\text{log}\left\{ -f_C(y)^{1-d} \cdot S_C(y)^{d} -\right\}\\ -\end{aligned} -$$ Let - -- $\theta_T$ represent the parameters of $p_T(t)$, -- $\theta_C$ represent the parameters of $p_C(c)$, -- $\theta = (\theta_T, \theta_C)$ be the combined vector of all - parameters. - -Then corresponding score function is: - -$$ -\begin{aligned} -\ell'(y,d) -&= \deriv{\theta} -\left[ -\text{log}\left\{ -f_T(y)^d \cdot S_T(y)^{1-d} -\right\} -+ -\text{log}\left\{ -f_C(y)^{1-d} \cdot S_C(y)^{d} -\right\} -\right]\\ -&= -\left( -\deriv{\theta} -\text{log}\left\{ -f_T(y)^d \cdot S_T(y)^{1-d} -\right\} -\right) -+ -\left( -\deriv{\theta} -\text{log}\left\{ -f_C(y)^{1-d} \cdot S_C(y)^{d} -\right\} -\right)\\ -\end{aligned} -$$ - -As long as $\theta_C$ and $\theta_T$ don't share any parameters, then if -censoring is non-informative, the partial derivative with respect to -$\theta_T$ is: - -$$ -\begin{aligned} -\ell'_{\theta_T}(y,d) -&\eqdef \deriv{\theta_T}\ell(y,d)\\ -&= -\left( -\deriv{\theta_T} -\text{log}\left\{ -f_T(y)^d \cdot S_T(y)^{1-d} -\right\} -\right) -+ -\left( -\deriv{\theta_T} -\text{log}\left\{ -f_C(y)^{1-d} \cdot S_C(y)^{d} -\right\} -\right)\\ -&= -\left( -\deriv{\theta_T} -\text{log}\left\{ -f_T(y)^d \cdot S_T(y)^{1-d} -\right\} -\right) + 0\\ -&= -\deriv{\theta_T} -\text{log}\left\{ -f_T(y)^d \cdot S_T(y)^{1-d} -\right\}\\ -\end{aligned} -$$ Thus, the MLE for $\theta_T$ won't depend on $\theta_C$, and we can -ignore the distribution of $C$ when estimating the parameters of -$f_T(t)=p(T=t)$. - -Then: - -$$ -\begin{aligned} -\mathcal L(y,d) -&= f_T(y)^d \cdot S_T(y)^{1-d}\\ -&= \left(h_T(y)^d S_T(y)^d\right) \cdot S_T(y)^{1-d}\\ -&= h_T(y)^d \cdot S_T(y)^d \cdot S_T(y)^{1-d}\\ -&= h_T(y)^d \cdot S_T(y)\\ -&= S_T(y) \cdot h_T(y)^d \\ -\end{aligned} +Exercise: compare and contrast this curve with the corresponding hazard +curve. + +```{r, echo = FALSE} +#| fig-cap: "Survival Curve in 2004 for US Females" +#| fig-pos: "H" + +s1 <- survexp.us[,"female","2004"] + +s2 <- 365.25*s1[5:113] +s2 <- c(s1[1], 6*s1[2], 21*s1[3], 337.25*s1[4], s2) +cs2 <- cumsum(s2) +age2 <- c(1/365.25, 7/365.25, 28/365.25, 1:110) +plot(age2,exp(-cs2),type="l",lwd=2,xlab="Age",ylab="Survival") + +``` + +--- + +### Likelihood with censoring + +If an event time $T$ is observed exactly as $T=t$, then the likelihood +of that observation is just its probability density function: + +$$ +\begin{aligned} +\mathcal L(t) +&= p(T=t)\\ +&\eqdef f_T(t)\\ +&= h_T(t)S_T(t)\\ +\ell(t) +&\eqdef \text{log}\left\{\mathcal L(t)\right\}\\ +&= \text{log}\left\{h_T(t)S_T(t)\right\}\\ +&= \text{log}\left\{h_T(t)\right\} + \text{log}\left\{S_T(t)\right\}\\ +&= \text{log}\left\{h_T(t)\right\} - H_T(t)\\ +\end{aligned} +$$ + +--- + +If instead the event time $T$ is censored and only known to be after +time $y$, then the likelihood of that censored observation is instead +the survival function evaluated at the censoring time: + +$$ +\begin{aligned} +\mathcal L(y) +&=p_T(T>y)\\ +&\eqdef S_T(y)\\ +\ell(y) +&\eqdef \text{log}\left\{\mathcal L(y)\right\}\\ +&=\text{log}\left\{S(y)\right\}\\ +&=-H(y)\\ +\end{aligned} +$$ + +--- + +::: notes +What's written above is incomplete. We also observed whether or not the +observation was censored. Let $C$ denote the time when censoring would +occur (if the event did not occur first); let $f_C(y)$ and $S_C(y)$ be +the corresponding density and survival functions for the censoring +event. + +Let $Y$ denote the time when observation ended (either by censoring or +by the event of interest occurring), and let $D$ be an indicator +variable for the event occurring at $Y$ (so $D=0$ represents a censored +observation and $D=1$ represents an uncensored observation). In other +words, let $Y \eqdef \min(T,C)$ and +$D \eqdef \mathbb 1{\{T<=C\}}$. + +Then the complete likelihood of the observed data $(Y,D)$ is: +::: + +$$ +\begin{aligned} +\mathcal L(y,d) +&= p(Y=y, D=d)\\ +&= \left[p(T=y,C> y)\right]^d \cdot +\left[p(T>y,C=y)\right]^{1-d}\\ +\end{aligned} +$$ + +--- + +::: notes +Typically, survival analyses assume that $C$ and $T$ are mutually +independent; this assumption is called "non-informative" censoring. + +Then the joint likelihood $p(Y,D)$ factors into the product +$p(Y), p(D)$, and the likelihood reduces to: +::: + +$$ +\begin{aligned} +\mathcal L(y,d) +&= \left[p(T=y,C> y)\right]^d\cdot +\left[p(T>y,C=y)\right]^{1-d}\\ +&= \left[p(T=y)p(C> y)\right]^d\cdot +\left[p(T>y)p(C=y)\right]^{1-d}\\ +&= \left[f_T(y)S_C(y)\right]^d\cdot +\left[S(y)f_C(y)\right]^{1-d}\\ +&= \left[f_T(y)^d S_C(y)^d\right]\cdot +\left[S_T(y)^{1-d}f_C(y)^{1-d}\right]\\ +&= \left(f_T(y)^d \cdot S_T(y)^{1-d}\right)\cdot +\left(f_C(y)^{1-d} \cdot S_C(y)^{d}\right) +\end{aligned} +$$ + +--- + +::: notes +The corresponding log-likelihood is: +::: + +$$ +\begin{aligned} +\ell(y,d) +&= \text{log}\left\{\mathcal L(y,d) \right\}\\ +&= \text{log}\left\{ +\left(f_T(y)^d \cdot S_T(y)^{1-d}\right)\cdot +\left(f_C(y)^{1-d} \cdot S_C(y)^{d}\right) +\right\}\\ +&= \text{log}\left\{ +f_T(y)^d \cdot S_T(y)^{1-d} +\right\} ++ +\text{log}\left\{ +f_C(y)^{1-d} \cdot S_C(y)^{d} +\right\}\\ +\end{aligned} +$$ Let + +- $\theta_T$ represent the parameters of $p_T(t)$, +- $\theta_C$ represent the parameters of $p_C(c)$, +- $\theta = (\theta_T, \theta_C)$ be the combined vector of all + parameters. + +--- + +::: notes +The corresponding score function is: +::: + +$$ +\begin{aligned} +\ell'(y,d) +&= \deriv{\theta} +\left[ +\text{log}\left\{ +f_T(y)^d \cdot S_T(y)^{1-d} +\right\} ++ +\text{log}\left\{ +f_C(y)^{1-d} \cdot S_C(y)^{d} +\right\} +\right]\\ +&= +\left( +\deriv{\theta} +\text{log}\left\{ +f_T(y)^d \cdot S_T(y)^{1-d} +\right\} +\right) ++ +\left( +\deriv{\theta} +\text{log}\left\{ +f_C(y)^{1-d} \cdot S_C(y)^{d} +\right\} +\right)\\ +\end{aligned} +$$ + +--- + +::: notes +As long as $\theta_C$ and $\theta_T$ don't share any parameters, then if +censoring is non-informative, the partial derivative with respect to +$\theta_T$ is: +::: + $$ - -That is, if the event occurred at time $y$ (i.e., if $d=1$), then the -likelihood of $(Y,D) = (y,d)$ is equal to the hazard function at $y$ -times the survival function at $y$. Otherwise, the likelihood is equal -to just the survival function at $y$. - -The corresponding log-likelihood is: - -$$ -\begin{aligned} -\ell(y,d) -&=\text{log}\left\{\mathcal L(y,d)\right\}\\ -&= \text{log}\left\{S_T(y) \cdot h_T(y)^d\right\}\\ -&= \text{log}\left\{S_T(y)\right\} + \text{log}\left\{h_T(y)^d\right\}\\ -&= \text{log}\left\{S_T(y)\right\} + d\cdot \text{log}\left\{h_T(y)\right\}\\ -&= -H_T(y) + d\cdot \text{log}\left\{h_T(y)\right\}\\ -\end{aligned} -$$ - -In other words, the log-likelihood contribution from a single -observation $(Y,D) = (y,d)$ is equal to the negative cumulative hazard -at $y$, plus the log of the hazard at $y$ if the event occurred at time -$y$. - -::: callout-note -End of extra section. -::: - -## Parametric Models for Time-to-Event Outcomes - -### Exponential Distribution +\begin{aligned} +\ell'_{\theta_T}(y,d) +&\eqdef \deriv{\theta_T}\ell(y,d)\\ +&= +\left( +\deriv{\theta_T} +\text{log}\left\{ +f_T(y)^d \cdot S_T(y)^{1-d} +\right\} +\right) ++ +\left( +\deriv{\theta_T} +\text{log}\left\{ +f_C(y)^{1-d} \cdot S_C(y)^{d} +\right\} +\right)\\ +&= +\left( +\deriv{\theta_T} +\text{log}\left\{ +f_T(y)^d \cdot S_T(y)^{1-d} +\right\} +\right) + 0\\ +&= +\deriv{\theta_T} +\text{log}\left\{ +f_T(y)^d \cdot S_T(y)^{1-d} +\right\}\\ +\end{aligned} +$$ -- The exponential distribution is the base distribution for survival - analysis. -- The distribution has a constant hazard $\lambda$ -- The mean survival time is $\lambda^{-1}$ - -#### Mathematical details of exponential distribution - -$$ -\begin{aligned} -f(t) &= \lambda \text{e}^{-\lambda t}\\ -E(t) &= \lambda^{-1}\\ -Var(t) &= \lambda^{-2}\\ -F(t) &= 1-\text{e}^{-\lambda x}\\ -S(t)&= \text{e}^{-\lambda x}\\ -\ln(S(t))&=-\lambda x\\ -h(t) &= -\frac{f(t)}{S(t)} = -\frac{\lambda \text{e}^{-\lambda t}}{\text{e}^{-\lambda t}}=\lambda -\end{aligned} -$$ - -#### Estimation of $\lambda$ +--- + +::: notes +Thus, the MLE for $\theta_T$ won't depend on $\theta_C$, and we can +ignore the distribution of $C$ when estimating the parameters of +$f_T(t)=p(T=t)$. +::: + +Then: + +$$ +\begin{aligned} +\mathcal L(y,d) +&= f_T(y)^d \cdot S_T(y)^{1-d}\\ +&= \left(h_T(y)^d S_T(y)^d\right) \cdot S_T(y)^{1-d}\\ +&= h_T(y)^d \cdot S_T(y)^d \cdot S_T(y)^{1-d}\\ +&= h_T(y)^d \cdot S_T(y)\\ +&= S_T(y) \cdot h_T(y)^d \\ +\end{aligned} +$$ -- Suppose we have $m$ exponential survival times of - $t_1, t_2,\ldots,t_m$ and $k$ right-censored values at - $u_1,u_2,\ldots,u_k$. - -- A survival time of $t_i=10$ means that subject $i$ died at time 10. - A right-censored time $u_i=10$ means that at time 10, subject $i$ - was still alive and that we have no further follow-up. - -- For the moment we will assume that the survival distribution is - exponential and that all the subjects have the same parameter - $\lambda$. - -We have $m$ exponential survival times of $t_1, t_2,\ldots,t_m$ and $k$ -right-censored values at $u_1,u_2,\ldots,u_k$. The log-likelihood of an -observed survival time $t_i$ is $$ -\text{log}\left\{\lambda \text{e}^{-\lambda t_i}\right\} = -\text{log}\left\{\lambda\right\}-\lambda t_i -$$ and the likelihood of a censored value is the probability of that -outcome (survival greater than $u_j$) so the log-likelihood is - -$$ -\text{log}\left\{\lambda \text{e}^{u_j}\right\} =-\lambda u_j. +::: notes +That is, if the event occurred at time $y$ (i.e., if $d=1$), then the +likelihood of $(Y,D) = (y,d)$ is equal to the hazard function at $y$ +times the survival function at $y$. Otherwise, the likelihood is equal +to just the survival function at $y$. +::: + +--- + +::: notes +The corresponding log-likelihood is: +::: + +$$ +\begin{aligned} +\ell(y,d) +&=\text{log}\left\{\mathcal L(y,d)\right\}\\ +&= \text{log}\left\{S_T(y) \cdot h_T(y)^d\right\}\\ +&= \text{log}\left\{S_T(y)\right\} + \text{log}\left\{h_T(y)^d\right\}\\ +&= \text{log}\left\{S_T(y)\right\} + d\cdot \text{log}\left\{h_T(y)\right\}\\ +&= -H_T(y) + d\cdot \text{log}\left\{h_T(y)\right\}\\ +\end{aligned} $$ -Let $T=\sum t_i$ and $U=\sum u_j$. Then: - -$$ -\begin{aligned} -\ell(\lambda) &= \sum_{i=1}^m( \ln \lambda-\lambda t_i) + \sum_{j=1}^k (-\lambda u_j)\\ -&= m \ln \lambda -(T+U)\lambda\\ -\ell'(\lambda) -&=m\lambda^{-1} -(T+U)\\ -\hat{\lambda} &= \frac{m}{T+U}\\ -\ell''&=-m/\lambda^2\\ -&< 0\\ -\hat E[T] &= \hat\lambda^{-1}\\ -&= \frac{T+U}{m} -\end{aligned} -$$ +::: notes +In other words, the log-likelihood contribution from a single +observation $(Y,D) = (y,d)$ is equal to the negative cumulative hazard +at $y$, plus the log of the hazard at $y$ if the event occurred at time +$y$. +::: + +## Parametric Models for Time-to-Event Outcomes + +### Exponential Distribution + +- The exponential distribution is the base distribution for survival + analysis. +- The distribution has a constant hazard $\lambda$ +- The mean survival time is $\lambda^{-1}$ -#### Fisher Information and Standard Error +--- -$$ -\begin{aligned} -E[-\ell''] -& = m/\lambda^2\\ -\text{Var}\left(\hat\lambda\right) -&\approx \left(E[-\ell'']\right)^{-1}\\ -&=\lambda^2/m\\ -\text{SE}\left(\hat\lambda\right) -&= \sqrt{\text{Var}\left(\hat\lambda\right)}\\ -&\approx \lambda/\sqrt{m} -\end{aligned} -$$ - -$\hat\lambda$ depends on the censoring times of the censored -observations, but $\text{Var}\left(\hat\lambda\right)$ only depends on -the number of uncensored observations, $m$, and not on the number of -censored observations ($k$). +#### Mathematical details of exponential distribution + +$$ +\begin{aligned} +f(t) &= \lambda \text{e}^{-\lambda t}\\ +E(t) &= \lambda^{-1}\\ +Var(t) &= \lambda^{-2}\\ +F(t) &= 1-\text{e}^{-\lambda x}\\ +S(t)&= \text{e}^{-\lambda x}\\ +\ln(S(t))&=-\lambda x\\ +h(t) &= -\frac{f(t)}{S(t)} = -\frac{\lambda \text{e}^{-\lambda t}}{\text{e}^{-\lambda t}}=\lambda +\end{aligned} +$$ + +--- + +#### Estimating $\lambda$ {.smaller} -#### Other Parametric Survival Distributions - -- Any density on $[0,\infty)$ can be a survival distribution, but the - most useful ones are all skew right. -- The commonest generalization of the exponential is the Weibull. -- Other common choices are the gamma, log-normal, log-logistic, - Gompertz, inverse Gaussian, and Pareto. -- Most of what we do going forward is non-parametric or - semi-parametric, but sometimes these parametric distributions - provide a useful approach. - -### Weibull Distribution - -$$ -\begin{aligned} -p(t)&= \alpha\lambda x^{\alpha-1}\text{e}^{-\lambda x^\alpha}\\ -h(t)&=\alpha\lambda x^{\alpha-1}\\ -S(t)&=\text{e}^{-\lambda x^\alpha}\\ -E(T)&= \Gamma(1+1/\alpha)\cdot \lambda^{-1/\alpha} -\end{aligned} +- Suppose we have $m$ exponential survival times of + $t_1, t_2,\ldots,t_m$ and $k$ right-censored values at + $u_1,u_2,\ldots,u_k$. + +- A survival time of $t_i=10$ means that subject $i$ died at time 10. + A right-censored time $u_i=10$ means that at time 10, subject $i$ + was still alive and that we have no further follow-up. + +- For the moment we will assume that the survival distribution is + exponential and that all the subjects have the same parameter + $\lambda$. + +We have $m$ exponential survival times of $t_1, t_2,\ldots,t_m$ and $k$ +right-censored values at $u_1,u_2,\ldots,u_k$. The log-likelihood of an +observed survival time $t_i$ is $$ +\text{log}\left\{\lambda \text{e}^{-\lambda t_i}\right\} = +\text{log}\left\{\lambda\right\}-\lambda t_i +$$ and the likelihood of a censored value is the probability of that +outcome (survival greater than $u_j$) so the log-likelihood is + $$ - -When $\alpha=1$ this is the exponential. When $\alpha>1$ the hazard is -increasing and when $\alpha < 1$ the hazard is decreasing. This provides -more flexibility than the exponential. - -We will see more of this distribution later. - -## Nonparametric Survival Analysis - -### Basic ideas - -- Mostly, we work without a parametric model. - -- The first task is to estimate a survival function from data listing - survival times, and censoring times for censored data. - -- For example one patient may have relapsed at 10 months. Another - might have been followed for 32 months without a relapse having - occurred (censored). - -- The minimum information we need for each patient is a time and a - censoring variable which is 1 if the event occurred at the indicated - time and 0 if this is a censoring time. +\ba +\ell_j(\lambda) &= \text{log}\left\{\lambda \text{e}^{u_j}\right\} +\\ &= -\lambda u_j +\ea +$$ + +--- + +Let $T=\sum t_i$ and $U=\sum u_j$. Then: + +$$ +\begin{aligned} +\ell(\lambda) &= \sum_{i=1}^m( \ln \lambda-\lambda t_i) + \sum_{j=1}^k (-\lambda u_j)\\ +&= m \ln \lambda -(T+U)\lambda\\ +\ell'(\lambda) +&=m\lambda^{-1} -(T+U)\\ +\hat{\lambda} &= \frac{m}{T+U}\\ +\ell''&=-m/\lambda^2\\ +&< 0\\ +\hat E[T] &= \hat\lambda^{-1}\\ +&= \frac{T+U}{m} +\end{aligned} +$$ -## Example: clinical trial for pediatric acute leukemia +--- -### Overview of study +#### Fisher Information and Standard Error -This is from a clinical trial in 1963 for 6-MP treatment vs. placebo for -Acute Leukemia in 42 children. - -- Pairs of children: - - - matched by remission status at the time of treatment (`remstat`: - `1` = partial, `2` = complete) - - randomized to 6-MP (exit times in `t2`) or placebo (exit times - in `t1`) - -- Followed until relapse or end of study. - -- All of the placebo group relapsed, but some of the 6-MP group were - censored (which means they were still in remission); indicated by - `relapse` variable (`0` = censored, `1` = relapse). - -- 6-MP = 6-Mercaptopurine (Purinethol) is an anti-cancer - ("antineoplastic" or "cytotoxic") chemotherapy drug used currently - for Acute lymphoblastic leukemia (ALL). It is classified as an - antimetabolite. - -### Study design {.smaller} - -Clinical trial in 1963 for 6-MP treatment vs. placebo for Acute Leukemia -in 42 children. Pairs of children matched by remission status at the -time of treatment (1 = partial or 2 = complete) and randomized to 6-MP -or placebo. Followed until relapse or end of study. All of the placebo -group relapsed, but some of the 6-MP group were censored. - -```{r} -library(KMsurv) -data(drug6mp) -drug6mp |> tibble() |> print() -``` +$$ +\begin{aligned} +E[-\ell''] +& = m/\lambda^2\\ +\text{Var}\left(\hat\lambda\right) +&\approx \left(E[-\ell'']\right)^{-1}\\ +&=\lambda^2/m\\ +\text{SE}\left(\hat\lambda\right) +&= \sqrt{\text{Var}\left(\hat\lambda\right)}\\ +&\approx \lambda/\sqrt{m} +\end{aligned} +$$ + +::: notes +$\hat\lambda$ depends on the censoring times of the censored +observations, but $\text{Var}\left(\hat\lambda\right)$ only depends on +the number of uncensored observations, $m$, and not on the number of +censored observations ($k$). +::: + +--- + +### Other Parametric Survival Distributions + +- Any density on $[0,\infty)$ can be a survival distribution, but the + most useful ones are all skew right. +- The most frequently used generalization of the exponential is the [Weibull](probability.qmd#sec-weibull). +- Other common choices are the gamma, log-normal, log-logistic, + Gompertz, inverse Gaussian, and Pareto. +- Most of what we do going forward is non-parametric or + semi-parametric, but sometimes these parametric distributions + provide a useful approach. + +## Nonparametric Survival Analysis + +### Basic ideas -### Data documentation for `drug6mp` +- Mostly, we work without a parametric model. -```{r, printr.help.sections = c("description", "format")} -library(printr) # inserts help-file output into markdown output -library(KMsurv) -?drug6mp -``` - -### Descriptive Statistics - -- The average time in each group is not useful. Some of the 6-MP - patients have not relapsed at the time recorded, while all of the - placebo patients have relapsed. -- The median time is not really useful either because so many of the - 6-MP patients have not relapsed (12/21). -- Both are biased down in the 6-MP group. Remember that lower times - are worse since they indicate sooner recurrence. -- We can compute the average hazard rate, which is the estimate of the - exponential parameter: number of relapses divided by the sum of the - times. -- For the placebo, that is just the reciprocal of the mean time = - 1/8.667 = 0.115. -- For the 6-MP group this is 9/359 = 0.025 -- The estimated average hazard in the placebo group is 4.6 times as - large (if the hazard is constant over time). - -## The Kaplan-Meier Product Limit Estimator - -- The estimated survival function for the placebo patients is easy to - compute. For any time $t$ in months, $S(t)$ is the fraction of - patients with times greater than $t$. - -- For the 6-MP patients, we cannot ignore the censored data because we - know that the time to relapse is greater than the censoring time. - -- For any time $t$ in months, we know that 6-MP patients with times - greater than $t$ have not relapsed, and those with relapse time less - than $t$ have relapsed, but we don't know if patients with censored - time less than $t$ have relapsed or not. +- The first task is to estimate a survival function from data listing + survival times, and censoring times for censored data. + +- For example one patient may have relapsed at 10 months. Another + might have been followed for 32 months without a relapse having + occurred (censored). + +- The minimum information we need for each patient is a time and a + censoring variable which is 1 if the event occurred at the indicated + time and 0 if this is a censoring time. + +## Example: clinical trial for pediatric acute leukemia + +### Overview of study + +This is from a clinical trial in 1963 for 6-MP treatment vs. placebo for +Acute Leukemia in 42 children. + +- Pairs of children: + + - matched by remission status at the time of treatment (`remstat`: + `1` = partial, `2` = complete) + - randomized to 6-MP (exit times in `t2`) or placebo (exit times + in `t1`) + +- Followed until relapse or end of study. + +- All of the placebo group relapsed, but some of the 6-MP group were + censored (which means they were still in remission); indicated by + `relapse` variable (`0` = censored, `1` = relapse). + +- 6-MP = 6-Mercaptopurine (Purinethol) is an anti-cancer + ("antineoplastic" or "cytotoxic") chemotherapy drug used currently + for Acute lymphoblastic leukemia (ALL). It is classified as an + antimetabolite. + +### Study design {.smaller} -- The procedure we usually use is the Kaplan-Meier product-limit - estimator of the survival function. - -- The Kaplan-Meier estimator is a step function (like the empirical - cdf), which changes value only at the event times, not at the - censoring times. - -- At each event time $t$, we compute the at-risk group size $Y$, which - is all those observations whose event time or censoring time is at - least $t$. - -- If $d$ of the observations have an event time (not a censoring time) - of $t$, then the group of survivors immediately following time $t$ - is reduced by the fraction $$\frac{Y-d}{Y}=1-\frac{d}{Y}$$ - -If the event times are $t_i$ with events per time of $d_i$ -($1\le i \le k$), then $$\hat S(t) = \prod_{t_i < t}[1-d_i/Y_i]$$ where -$Y_i$ is the set of observations whose time (event or censored) is -$\ge t_i$, the group at risk at time $t_i$. - -If there are no censored data, and there are $n$ data points, then just -after (say) the third event time - -$$ -\begin{aligned} -\hat S(t) -&= \prod_{t_i < t}[1-d_i/Y_i]\\ -&=[\frac{n-d_1}{n}][\frac{n-d_1-d_2}{n-d_1}][\frac{n-d_1-d_2-d_3}{n-d_1-d_2}]\\ -&=\frac{n-d_1-d_2-d_3}{n}\\ -&=1-\frac{d_1+d_2+d_3}{n}\\ -&=1-\hat F(t)\\ -\end{aligned} -$$ - -where $\hat F(t)$ is the usual empirical CDF estimate. - -### Kaplan-Meier curve for `drug6mp` data - -Here is the Kaplan-Meier estimated survival curve for the patients who -received 6-MP in the `drug6mp` dataset (we will see code to produce -figures like this one shortly): +Clinical trial in 1963 for 6-MP treatment vs. placebo for Acute Leukemia +in 42 children. Pairs of children matched by remission status at the +time of treatment (1 = partial or 2 = complete) and randomized to 6-MP +or placebo. Followed until relapse or end of study. All of the placebo +group relapsed, but some of the 6-MP group were censored. + +```{r} +library(KMsurv) +data(drug6mp) +drug6mp |> tibble() |> print() +``` + +### Data documentation for `drug6mp` + +```{r, printr.help.sections = c("description", "format")} +library(printr) # inserts help-file output into markdown output +library(KMsurv) +?drug6mp +``` + +### Descriptive Statistics + +- The average time in each group is not useful. Some of the 6-MP + patients have not relapsed at the time recorded, while all of the + placebo patients have relapsed. +- The median time is not really useful either because so many of the + 6-MP patients have not relapsed (12/21). +- Both are biased down in the 6-MP group. Remember that lower times + are worse since they indicate sooner recurrence. +- We can compute the average hazard rate, which is the estimate of the + exponential parameter: number of relapses divided by the sum of the + times. +- For the placebo, that is just the reciprocal of the mean time = + 1/8.667 = 0.115. +- For the 6-MP group this is 9/359 = 0.025 +- The estimated average hazard in the placebo group is 4.6 times as + large (if the hazard is constant over time). + +## The Kaplan-Meier Product Limit Estimator -```{r, echo = FALSE} -#| fig-cap: "Kaplan-Meier Survival Curve for 6-MP Patients" - -require(KMsurv) -data(drug6mp) - -library(survival) -drug6mp_km_model1 = - drug6mp |> - mutate(surv = Surv(t2, relapse)) |> - survfit(formula = surv ~ 1, data = _) +- The estimated survival function for the placebo patients is easy to + compute. For any time $t$ in months, $S(t)$ is the fraction of + patients with times greater than $t$. + +- For the 6-MP patients, we cannot ignore the censored data because we + know that the time to relapse is greater than the censoring time. + +- For any time $t$ in months, we know that 6-MP patients with times + greater than $t$ have not relapsed, and those with relapse time less + than $t$ have relapsed, but we don't know if patients with censored + time less than $t$ have relapsed or not. -library(ggfortify) -drug6mp_km_model1 |> - autoplot( - mark.time = TRUE, - conf.int = FALSE) + - expand_limits(y = 0) + - xlab('Time since diagnosis (months)') + - ylab("KM Survival Curve") - -``` +- The procedure we usually use is the Kaplan-Meier product-limit + estimator of the survival function. + +- The Kaplan-Meier estimator is a step function (like the empirical + cdf), which changes value only at the event times, not at the + censoring times. + +- At each event time $t$, we compute the at-risk group size $Y$, which + is all those observations whose event time or censoring time is at + least $t$. -### Kaplan-Meier calculations {.smaller} - -Let's compute these estimates and build the chart by hand: +- If $d$ of the observations have an event time (not a censoring time) + of $t$, then the group of survivors immediately following time $t$ + is reduced by the fraction $$\frac{Y-d}{Y}=1-\frac{d}{Y}$$ -```{r} -library(KMsurv) -library(dplyr) -data(drug6mp) - -drug6mp.v2 = - drug6mp |> - as_tibble() |> - mutate( - remstat = remstat |> - case_match( - 1 ~ "partial", - 2 ~ "complete" - ), - # renaming to "outcome" while relabeling is just a style choice: - outcome = relapse |> - case_match( - 0 ~ "censored", - 1 ~ "relapsed" - ) - ) - -km.6mp = - drug6mp.v2 |> - summarize( - .by = t2, - Relapses = sum(outcome == "relapsed"), - Censored = sum(outcome == "censored")) |> - # here we add a start time row, so the graph starts at time 0: - bind_rows( - tibble( - t2 = 0, - Relapses = 0, - Censored = 0) - ) |> - # sort in time order: - arrange(t2) |> - mutate( - Exiting = Relapses + Censored, - `Study Size` = sum(Exiting), - Exited = cumsum(Exiting) |> dplyr::lag(default = 0), - `At Risk` = `Study Size` - Exited, - Hazard = Relapses / `At Risk`, - `KM Factor` = 1 - Hazard, - `Cumulative Hazard` = cumsum(`Hazard`), - `KM Survival Curve` = cumprod(`KM Factor`) - ) +If the event times are $t_i$ with events per time of $d_i$ +($1\le i \le k$), then $$\hat S(t) = \prod_{t_i < t}[1-d_i/Y_i]$$ where +$Y_i$ is the set of observations whose time (event or censored) is +$\ge t_i$, the group at risk at time $t_i$. + +If there are no censored data, and there are $n$ data points, then just +after (say) the third event time + +$$ +\begin{aligned} +\hat S(t) +&= \prod_{t_i < t}[1-d_i/Y_i]\\ +&=[\frac{n-d_1}{n}][\frac{n-d_1-d_2}{n-d_1}][\frac{n-d_1-d_2-d_3}{n-d_1-d_2}]\\ +&=\frac{n-d_1-d_2-d_3}{n}\\ +&=1-\frac{d_1+d_2+d_3}{n}\\ +&=1-\hat F(t)\\ +\end{aligned} +$$ + +where $\hat F(t)$ is the usual empirical CDF estimate. + +### Kaplan-Meier curve for `drug6mp` data + +Here is the Kaplan-Meier estimated survival curve for the patients who +received 6-MP in the `drug6mp` dataset (we will see code to produce +figures like this one shortly): + +```{r, echo = FALSE} +#| fig-cap: "Kaplan-Meier Survival Curve for 6-MP Patients" + +require(KMsurv) +data(drug6mp) + +library(survival) +drug6mp_km_model1 = + drug6mp |> + mutate(surv = Surv(t2, relapse)) |> + survfit(formula = surv ~ 1, data = _) + +library(ggfortify) +drug6mp_km_model1 |> + autoplot( + mark.time = TRUE, + conf.int = FALSE) + + expand_limits(y = 0) + + xlab('Time since diagnosis (months)') + + ylab("KM Survival Curve") + +``` -library(pander) -pander(km.6mp) -``` +### Kaplan-Meier calculations {.smaller} + +Let's compute these estimates and build the chart by hand: -#### Summary - -For the 6-MP patients at time 6 months, there are 21 patients at risk. -At $t=6$ there are 3 relapses and 1 censored observations. - -The Kaplan-Meier factor is $(21-3)/21 = 0.857$. The number at risk for -the next time ($t=7$) is $21-3-1=17$. - -At time 7 months, there are 17 patients at risk. At $t=7$ there is 1 -relapse and 0 censored observations. The Kaplan-Meier factor is -$(17-1)/17 = 0.941$. The Kaplan Meier estimate is -$0.857\times0.941=0.807$. The number at risk for the next time ($t=9$) -is $17-1=16$. - -Now, let's graph this estimated survival curve using `ggplot()`: - -```{r "estimated survival curve"} -library(ggplot2) -conflicts_prefer(dplyr::filter) -km.6mp |> - ggplot(aes(x = t2, y = `KM Survival Curve`)) + - geom_step() + - geom_point(data = km.6mp |> filter(Censored > 0), shape = 3) + - expand_limits(y = c(0,1), x = 0) + - xlab('Time since diagnosis (months)') + - ylab("KM Survival Curve") + - scale_y_continuous(labels = scales::percent) -``` - -## Using the `survival` package in R - -We don't have to do these calculations by hand every time; the -`survival` package and several others have functions available to -automate many of these tasks (full list: -<https://cran.r-project.org/web/views/Survival.html>). - -### The `Surv` function - -To use the `survival` package, the first step is telling R how to -combine the exit time and exit reason (censoring versus event) columns. -The `Surv()` function accomplishes this task. - -#### Example: `Surv()` with `drug6mp` data - -```{r} -#| code-fold: show -#| code-line-numbers: "5-7" -library(survival) -drug6mp.v3 = - drug6mp.v2 |> - mutate( - surv2 = Surv( - time = t2, - event = (outcome == "relapsed"))) - -print(drug6mp.v3) +```{r} +library(KMsurv) +library(dplyr) +data(drug6mp) + +drug6mp.v2 = + drug6mp |> + as_tibble() |> + mutate( + remstat = remstat |> + case_match( + 1 ~ "partial", + 2 ~ "complete" + ), + # renaming to "outcome" while relabeling is just a style choice: + outcome = relapse |> + case_match( + 0 ~ "censored", + 1 ~ "relapsed" + ) + ) + +km.6mp = + drug6mp.v2 |> + summarize( + .by = t2, + Relapses = sum(outcome == "relapsed"), + Censored = sum(outcome == "censored")) |> + # here we add a start time row, so the graph starts at time 0: + bind_rows( + tibble( + t2 = 0, + Relapses = 0, + Censored = 0) + ) |> + # sort in time order: + arrange(t2) |> + mutate( + Exiting = Relapses + Censored, + `Study Size` = sum(Exiting), + Exited = cumsum(Exiting) |> dplyr::lag(default = 0), + `At Risk` = `Study Size` - Exited, + Hazard = Relapses / `At Risk`, + `KM Factor` = 1 - Hazard, + `Cumulative Hazard` = cumsum(`Hazard`), + `KM Survival Curve` = cumprod(`KM Factor`) + ) + +library(pander) +pander(km.6mp) +``` + +#### Summary + +For the 6-MP patients at time 6 months, there are 21 patients at risk. +At $t=6$ there are 3 relapses and 1 censored observations. -``` - -The output of `Surv()` is a vector of objects with class `Surv`. When we -print this vector: - -- observations where the event was observed are printed as the event - time (for example, `surv2 = 10` on line 1) - -- observations where the event was right-censored are printed as the - censoring time with a plus sign (`+`; for example, `surv2 = 32+` on - line 3). - -### The `survfit` function - -Once we have constructed our `Surv` variable, we can calculate the -Kaplan-Meier estimate of the survival curve using the `survfit()` -function. - -::: callout-note -The documentation for `?survfit` isn't too helpful; the -`survfit.formula` documentation is better. - -```{r, printr.help.sections = c("description", "usage")} -#| include: false -?survfit.formula -``` -::: - -#### Example: `survfit()` with `drug6mp` data - -Here we use `survfit()` to create a `survfit` object, which contains the -Kaplan-Meier estimate: +The Kaplan-Meier factor is $(21-3)/21 = 0.857$. The number at risk for +the next time ($t=7$) is $21-3-1=17$. + +At time 7 months, there are 17 patients at risk. At $t=7$ there is 1 +relapse and 0 censored observations. The Kaplan-Meier factor is +$(17-1)/17 = 0.941$. The Kaplan Meier estimate is +$0.857\times0.941=0.807$. The number at risk for the next time ($t=9$) +is $17-1=16$. + +Now, let's graph this estimated survival curve using `ggplot()`: + +```{r "estimated survival curve"} +library(ggplot2) +conflicts_prefer(dplyr::filter) +km.6mp |> + ggplot(aes(x = t2, y = `KM Survival Curve`)) + + geom_step() + + geom_point(data = km.6mp |> filter(Censored > 0), shape = 3) + + expand_limits(y = c(0,1), x = 0) + + xlab('Time since diagnosis (months)') + + ylab("KM Survival Curve") + + scale_y_continuous(labels = scales::percent) +``` + +## Using the `survival` package in R + +We don't have to do these calculations by hand every time; the +`survival` package and several others have functions available to +automate many of these tasks (full list: +<https://cran.r-project.org/web/views/Survival.html>). + +### The `Surv` function -```{r} -#| code-fold: show -drug6mp.km_model = survfit( - formula = surv2 ~ 1, - data = drug6mp.v3) -``` - -`print.survfit()` just gives some summary statistics: - -```{r} -#| code-fold: show -print(drug6mp.km_model) -``` - -`summary.survfit()` shows us the underlying Kaplan-Meier table: - -```{r} -#| code-fold: show -summary(drug6mp.km_model) -``` - -`summary.survfit()` shows us the underlying Kaplan-Meier table: +To use the `survival` package, the first step is telling R how to +combine the exit time and exit reason (censoring versus event) columns. +The `Surv()` function accomplishes this task. + +#### Example: `Surv()` with `drug6mp` data + +```{r} +#| code-fold: show +#| code-line-numbers: "5-7" +library(survival) +drug6mp.v3 = + drug6mp.v2 |> + mutate( + surv2 = Surv( + time = t2, + event = (outcome == "relapsed"))) + +print(drug6mp.v3) + +``` + +The output of `Surv()` is a vector of objects with class `Surv`. When we +print this vector: + +- observations where the event was observed are printed as the event + time (for example, `surv2 = 10` on line 1) -```{r} -#| code-fold: show -summary(drug6mp.km_model) -``` - -We can specify which time points we want using the `times` argument: - -```{r} -#| code-fold: show -summary( - drug6mp.km_model, - times = c(0, drug6mp.v3$t2)) - +- observations where the event was right-censored are printed as the + censoring time with a plus sign (`+`; for example, `surv2 = 32+` on + line 3). + +### The `survfit` function + +Once we have constructed our `Surv` variable, we can calculate the +Kaplan-Meier estimate of the survival curve using the `survfit()` +function. + +::: callout-note +The documentation for `?survfit` isn't too helpful; the +`survfit.formula` documentation is better. + +```{r, printr.help.sections = c("description", "usage")} +#| include: false +?survfit.formula ``` - -```{r, printr.help.sections = c("description", "usage", "arguments")} -#| code-fold: show -?summary.survfit -``` - -### Plotting estimated survival functions - -We can plot `survfit` objects with `plot()`, `autoplot()`, or -`ggsurvplot()`: - -```{r} -#| code-fold: show -#| fig-cap: "Kaplan-Meier Survival Curve for 6-MP Patients" - -library(ggfortify) -autoplot(drug6mp.km_model) - -# not shown: -# plot(drug6mp.km_model) - -# library(survminer) -# ggsurvplot(drug6mp.km_model) - -``` - -#### quantiles of survival curve - -We can extract quantiles with `quantile()`: - -```{r} -#| code-line-numbers: "2" -drug6mp.km_model |> - quantile(p = c(.25, .5)) |> - as_tibble() |> - mutate(p = c(.25, .5)) |> - relocate(p, .before = everything()) +::: + +#### Example: `survfit()` with `drug6mp` data + +Here we use `survfit()` to create a `survfit` object, which contains the +Kaplan-Meier estimate: + +```{r} +#| code-fold: show +drug6mp.km_model = survfit( + formula = surv2 ~ 1, + data = drug6mp.v3) +``` + +`print.survfit()` just gives some summary statistics: + +```{r} +#| code-fold: show +print(drug6mp.km_model) +``` + +`summary.survfit()` shows us the underlying Kaplan-Meier table: + +```{r} +#| code-fold: show +summary(drug6mp.km_model) +``` + +`summary.survfit()` shows us the underlying Kaplan-Meier table: + +```{r} +#| code-fold: show +summary(drug6mp.km_model) ``` -### Two-sample tests +We can specify which time points we want using the `times` argument: -#### The `survdiff` function - -```{r, printr.help.sections = c("description", "usage")} -?survdiff -``` - -#### Example: `survdiff()` with `drug6mp` data +```{r} +#| code-fold: show +summary( + drug6mp.km_model, + times = c(0, drug6mp.v3$t2)) -Now we are going to compare the placebo and 6-MP data. We need to -reshape the data to make it usable with the standard `survival` -workflow: - -```{r} -library(survival) +``` + +```{r, printr.help.sections = c("description", "usage", "arguments")} +#| code-fold: show +?summary.survfit +``` + +### Plotting estimated survival functions -drug6mp.v4 = - drug6mp.v3 |> - select(pair, remstat, t1, t2, outcome) |> - # here we are going to change the data from a wide format to long: - pivot_longer( - cols = c(t1, t2), - names_to = "treatment", - values_to = "exit_time") |> - mutate( - treatment = treatment |> - case_match( - "t1" ~ "placebo", - "t2" ~ "6-MP" - ), - outcome = if_else( - treatment == "placebo", - "relapsed", - outcome - ), - surv = Surv( - time = exit_time, - event = (outcome == "relapsed")) - ) -``` - -Using this long data format, we can fit a Kaplan-Meier curve for each -treatment group simultaneously: - -```{r} -drug6mp.km_model2 = - survfit( - formula = surv ~ treatment, - data = drug6mp.v4) -``` +We can plot `survfit` objects with `plot()`, `autoplot()`, or +`ggsurvplot()`: + +```{r} +#| code-fold: show +#| fig-cap: "Kaplan-Meier Survival Curve for 6-MP Patients" + +library(ggfortify) +autoplot(drug6mp.km_model) + +# not shown: +# plot(drug6mp.km_model) + +# library(survminer) +# ggsurvplot(drug6mp.km_model) + +``` + +#### quantiles of survival curve + +We can extract quantiles with `quantile()`: + +```{r} +#| code-line-numbers: "2" +drug6mp.km_model |> + quantile(p = c(.25, .5)) |> + as_tibble() |> + mutate(p = c(.25, .5)) |> + relocate(p, .before = everything()) +``` + +### Two-sample tests -We can plot the curves in the same graph: +#### The `survdiff` function -```{r} -drug6mp.km_model2 |> autoplot() -``` +```{r, printr.help.sections = c("description", "usage")} +?survdiff +``` + +#### Example: `survdiff()` with `drug6mp` data -We can also perform something like a t-test, where the null hypothesis -is that the curves are the same: - -```{r} -survdiff( - formula = surv ~ treatment, - data = drug6mp.v4) -``` - -By default, `survdiff()` ignores any pairing, but we can use `strata()` -to perform something similar to a paired t-test: - -```{r} -survdiff( - formula = surv ~ treatment + strata(pair), - data = drug6mp.v4) - -``` - -Interestingly, accounting for pairing reduces the significant of the -difference. - -## Example: Bone Marrow Transplant Data - -(Copelan et al., 1991) - -##### Treatment {.unnumbered} - -- **allogeneic** (from a donor) **bone marrow transplant therapy** +Now we are going to compare the placebo and 6-MP data. We need to +reshape the data to make it usable with the standard `survival` +workflow: + +```{r} +library(survival) + +drug6mp.v4 = + drug6mp.v3 |> + select(pair, remstat, t1, t2, outcome) |> + # here we are going to change the data from a wide format to long: + pivot_longer( + cols = c(t1, t2), + names_to = "treatment", + values_to = "exit_time") |> + mutate( + treatment = treatment |> + case_match( + "t1" ~ "placebo", + "t2" ~ "6-MP" + ), + outcome = if_else( + treatment == "placebo", + "relapsed", + outcome + ), + surv = Surv( + time = exit_time, + event = (outcome == "relapsed")) + ) +``` -##### Inclusion criteria {.unnumbered} - -- **acute myeloid leukemia (AML)** -- **acute lymphoblastic leukemia (ALL).** - -##### Possible intermediate events {.unnumbered} - -- **graft vs. host disease (GVHD)**: an immunological rejection - response to the transplant -- **platelet recovery**: a return of platelet count to normal levels. - -One or the other, both in either order, or neither may occur. - -##### End point events - -- relapse of the disease -- death - -Any or all of these events may be censored. +Using this long data format, we can fit a Kaplan-Meier curve for each +treatment group simultaneously: + +```{r} +drug6mp.km_model2 = + survfit( + formula = surv ~ treatment, + data = drug6mp.v4) +``` + +We can plot the curves in the same graph: + +```{r} +drug6mp.km_model2 |> autoplot() +``` -### `KMsurv::bmt` data in R - -```{r} -library(KMsurv) -?bmt -``` - -### Analysis plan +We can also perform something like a t-test, where the null hypothesis +is that the curves are the same: + +```{r} +survdiff( + formula = surv ~ treatment, + data = drug6mp.v4) +``` -- We concentrate for now on disease-free survival (`t2` and `d3`) for - the three risk groups, ALL, AML Low Risk, and AML High Risk. -- We will construct the Kaplan-Meier survival curves, compare them, - and test for differences. -- We will construct the cumulative hazard curves and compare them. -- We will estimate the hazard functions, interpret, and compare them. - -### Survival Function Estimate and Variance - -$$\hat S(t) = \prod_{t_i < t}\left[1-\frac{d_i}{Y_i}\right]$$ where -$Y_i$ is the group at risk at time $t_i$. +By default, `survdiff()` ignores any pairing, but we can use `strata()` +to perform something similar to a paired t-test: + +```{r} +survdiff( + formula = surv ~ treatment + strata(pair), + data = drug6mp.v4) + +``` -The estimated variance of $\hat S(t)$ is (Greenwood's formula) -$$\hat{\text{Var}}\left(\hat S (t)\right) = \hat S(t)^2\sum_{t_i <t}\frac{d_i}{Y_i(Y_i-d_i)}$$ -which we can use for confidence intervals for a survival function or a -difference of survival functions. +Interestingly, accounting for pairing reduces the significant of the +difference. + +## Example: Bone Marrow Transplant Data -##### Kaplan-Meier survival curves +(Copelan et al., 1991) -```{r} -#| code-summary: "code to preprocess and model bmt data" -library(KMsurv) -library(survival) -data(bmt) - -bmt = - bmt |> - as_tibble() |> - mutate( - group = - group |> - factor( - labels = c("ALL","Low Risk AML","High Risk AML")), - surv = Surv(t2,d3)) - -km_model1 = survfit( - formula = surv ~ group, - data = bmt) -``` - -```{r "KM survival curves for bmt data"} -#| fig-cap: "Disease-Free Survival by Disease Group" +##### Treatment {.unnumbered} + +- **allogeneic** (from a donor) **bone marrow transplant therapy** + +##### Inclusion criteria {.unnumbered} + +- **acute myeloid leukemia (AML)** +- **acute lymphoblastic leukemia (ALL).** + +##### Possible intermediate events {.unnumbered} + +- **graft vs. host disease (GVHD)**: an immunological rejection + response to the transplant +- **platelet recovery**: a return of platelet count to normal levels. + +One or the other, both in either order, or neither may occur. + +##### End point events + +- relapse of the disease +- death + +Any or all of these events may be censored. + +### `KMsurv::bmt` data in R -library(ggfortify) -autoplot( - km_model1, - conf.int = TRUE, - ylab = "Pr(disease-free survival)", - xlab = "Time since transplant (days)") + - theme_bw() + - theme(legend.position="bottom") +```{r} +library(KMsurv) +?bmt +``` + +### Analysis plan -``` - -#### Understanding Greenwood's formula (optional) - -To see where Greenwood's formula comes from, let $x_i=Y_i-d_i$. We -approximate the solution treating each time as independent, with $Y_i$ -fixed and ignore randomness in times of failure and we treat $x_i$ as -independent binomials $\text{Bin}(Y_i,p_i)$. Letting $S(t)$ be the -"true" survival function - -$$ -\begin{aligned} -\hat S(t) &=\prod_{t_i<t}x_i/Y_i\\ -S(t)&=\prod_{t_i<t}p_i -\end{aligned} -$$ +- We concentrate for now on disease-free survival (`t2` and `d3`) for + the three risk groups, ALL, AML Low Risk, and AML High Risk. +- We will construct the Kaplan-Meier survival curves, compare them, + and test for differences. +- We will construct the cumulative hazard curves and compare them. +- We will estimate the hazard functions, interpret, and compare them. + +### Survival Function Estimate and Variance + +$$\hat S(t) = \prod_{t_i < t}\left[1-\frac{d_i}{Y_i}\right]$$ where +$Y_i$ is the group at risk at time $t_i$. + +The estimated variance of $\hat S(t)$ is (Greenwood's formula) +$$\hat{\text{Var}}\left(\hat S (t)\right) = \hat S(t)^2\sum_{t_i <t}\frac{d_i}{Y_i(Y_i-d_i)}$$ +which we can use for confidence intervals for a survival function or a +difference of survival functions. -$$ -\begin{aligned} -\frac{\hat S(t)}{S(t)} -&= \prod_{t_i<t} \frac{x_i}{p_iY_i}=\prod_{t_i<t} \frac{\hat p_i}{p_i}\\ -&=\prod_{t_i<t} \left( 1+\frac{\hat p_i-p_i}{p_i} \right)\\ -&\approx 1+\sum_{t_i<t} \frac{\hat p_i-p_i}{p_i} \\ -\end{aligned} -$$ - -$$ -\begin{aligned} -\text{Var}\left(\frac{\hat S(t)}{S(t)}\right) -&\approx \text{Var}\left(1+\sum_{t_i<t} \frac{\hat p_i-p_i}{p_i}\right) \\ -&=\sum_{t_i<t} \frac{1}{p_i^2}\frac{p_i(1-p_i)}{Y_i} \\ -&= \sum_{t_i<t} \frac{(1-p_i)}{p_iY_i} -\approx\sum_{t_i<t} \frac{(1-x_i/Y_i)}{x_i}\\ -&=\sum_{t_i<t} \frac{Y_i-x_i}{x_iY_i}=\sum_{t_i<t} \frac{d_i}{Y_i(Y_i-d_i)}\\ -\text{Var}\left(\hat S(t)\right) -&\approx \hat S(t)^2\sum_{t_i<t} \frac{d_i}{Y_i(Y_i-d_i)} -\end{aligned} -$$ - -### Test for differences among the disease groups - -Here we compute a chi-square test for assocation between disease group -(`group`) and disease-free survival: - -```{r} -survdiff(surv ~ group, data = bmt) -``` - -### Cumulative Hazard - -$$ -\begin{aligned} -h(t) -&\eqdef P(T=t|T\ge t)\\ -&= \frac{p(T=t)}{P(T\ge t)}\\ -&= -\deriv{t}\text{log}\left\{S(t)\right\} -\end{aligned} -$$ - -The **cumulative hazard** (or **integrated hazard**) function is - -$$H(t)\eqdef \int_0^t h(t) dt$$ Since -$h(t) = -\deriv{t}\text{log}\left\{S(t)\right\}$ as shown above, we -have: - -$$ -H(t)=-\text{log}\left\{S\right\}(t) +##### Kaplan-Meier survival curves + +```{r} +#| code-summary: "code to preprocess and model bmt data" +library(KMsurv) +library(survival) +data(bmt) + +bmt = + bmt |> + as_tibble() |> + mutate( + group = + group |> + factor( + labels = c("ALL","Low Risk AML","High Risk AML")), + surv = Surv(t2,d3)) + +km_model1 = survfit( + formula = surv ~ group, + data = bmt) +``` + +```{r "KM survival curves for bmt data"} +#| fig-cap: "Disease-Free Survival by Disease Group" + +library(ggfortify) +autoplot( + km_model1, + conf.int = TRUE, + ylab = "Pr(disease-free survival)", + xlab = "Time since transplant (days)") + + theme_bw() + + theme(legend.position="bottom") + +``` + +#### Understanding Greenwood's formula (optional) + +To see where Greenwood's formula comes from, let $x_i=Y_i-d_i$. We +approximate the solution treating each time as independent, with $Y_i$ +fixed and ignore randomness in times of failure and we treat $x_i$ as +independent binomials $\text{Bin}(Y_i,p_i)$. Letting $S(t)$ be the +"true" survival function + +$$ +\begin{aligned} +\hat S(t) &=\prod_{t_i<t}x_i/Y_i\\ +S(t)&=\prod_{t_i<t}p_i +\end{aligned} $$ -So we can estimate $H(t)$ as: - -$$ -\begin{aligned} -\hat H(t) -&= -\text{log}\left\{\hat S(t)\right\}\\ -&= -\text{log}\left\{\prod_{t_i < t}\left[1-\frac{d_i}{Y_i}\right]\right\}\\ -&= -\sum_{t_i < t}\text{log}\left\{1-\frac{d_i}{Y_i}\right\}\\ -\end{aligned} +$$ +\begin{aligned} +\frac{\hat S(t)}{S(t)} +&= \prod_{t_i<t} \frac{x_i}{p_iY_i}=\prod_{t_i<t} \frac{\hat p_i}{p_i}\\ +&=\prod_{t_i<t} \left( 1+\frac{\hat p_i-p_i}{p_i} \right)\\ +&\approx 1+\sum_{t_i<t} \frac{\hat p_i-p_i}{p_i} \\ +\end{aligned} +$$ + $$ - -This is the **Kaplan-Meier (product-limit) estimate of cumulative -hazard**. - -#### Example: Cumulative Hazard Curves for Bone-Marrow Transplant (`bmt`) data - -```{r} -#| fig-cap: "Disease-Free Cumulative Hazard by Disease Group" - -autoplot( - fun = "cumhaz", - km_model1, - conf.int = FALSE, - ylab = "Cumulative hazard (disease-free survival)", - xlab = "Time since transplant (days)") + - theme_bw() + - theme(legend.position="bottom") -``` - -## Nelson-Aalen Estimates of Cumulative Hazard and Survival +\begin{aligned} +\text{Var}\left(\frac{\hat S(t)}{S(t)}\right) +&\approx \text{Var}\left(1+\sum_{t_i<t} \frac{\hat p_i-p_i}{p_i}\right) \\ +&=\sum_{t_i<t} \frac{1}{p_i^2}\frac{p_i(1-p_i)}{Y_i} \\ +&= \sum_{t_i<t} \frac{(1-p_i)}{p_iY_i} +\approx\sum_{t_i<t} \frac{(1-x_i/Y_i)}{x_i}\\ +&=\sum_{t_i<t} \frac{Y_i-x_i}{x_iY_i}=\sum_{t_i<t} \frac{d_i}{Y_i(Y_i-d_i)}\\ +\text{Var}\left(\hat S(t)\right) +&\approx \hat S(t)^2\sum_{t_i<t} \frac{d_i}{Y_i(Y_i-d_i)} +\end{aligned} +$$ + +### Test for differences among the disease groups + +Here we compute a chi-square test for assocation between disease group +(`group`) and disease-free survival: + +```{r} +survdiff(surv ~ group, data = bmt) +``` -The point hazard at time $t_i$ can be estimated by $d_i/Y_i$, which -leads to the **Nelson-Aalen estimator of the cumulative hazard**: - -$$\hat H_{NA}(t) \eqdef \sum_{t_i < t}\frac{d_i}{Y_i}$$ - -The variance of this estimator is approximately: - -$$ -\begin{aligned} -\hat{\text{Var}}\left(\hat H_{NA} (t)\right) -&= \sum_{t_i <t}\frac{(d_i/Y_i)(1-d_i/Y_i)}{Y_i}\\ -&\approx \sum_{t_i <t}\frac{d_i}{Y_i^2} -\end{aligned} -$$ - -Since $S(t)=\text{exp}\left\{-H(t)\right\}$, the Nelson-Aalen cumulative -hazard estimate can be converted into an alternate estimate of the -survival function: - +### Cumulative Hazard + +$$ +\begin{aligned} +h(t) +&\eqdef P(T=t|T\ge t)\\ +&= \frac{p(T=t)}{P(T\ge t)}\\ +&= -\deriv{t}\text{log}\left\{S(t)\right\} +\end{aligned} +$$ + +The **cumulative hazard** (or **integrated hazard**) function is + +$$H(t)\eqdef \int_0^t h(t) dt$$ Since +$h(t) = -\deriv{t}\text{log}\left\{S(t)\right\}$ as shown above, we +have: + +$$ +H(t)=-\text{log}\left\{S\right\}(t) $$ -\begin{aligned} -\hat S_{NA}(t) -&= \text{exp}\left\{-\hat H_{NA}(t)\right\}\\ -&= \text{exp}\left\{-\sum_{t_i < t}\frac{d_i}{Y_i}\right\}\\ -&= \prod_{t_i < t}\text{exp}\left\{-\frac{d_i}{Y_i}\right\}\\ -\end{aligned} -$$ - -Compare these with the corresponding Kaplan-Meier estimates: - + +So we can estimate $H(t)$ as: + +$$ +\begin{aligned} +\hat H(t) +&= -\text{log}\left\{\hat S(t)\right\}\\ +&= -\text{log}\left\{\prod_{t_i < t}\left[1-\frac{d_i}{Y_i}\right]\right\}\\ +&= -\sum_{t_i < t}\text{log}\left\{1-\frac{d_i}{Y_i}\right\}\\ +\end{aligned} $$ -\begin{aligned} -\hat H_{KM}(t) &= -\sum_{t_i < t}\text{log}\left\{1-\frac{d_i}{Y_i}\right\}\\ -\hat S_{KM}(t) &= \prod_{t_i < t}\left[1-\frac{d_i}{Y_i}\right] -\end{aligned} -$$ + +This is the **Kaplan-Meier (product-limit) estimate of cumulative +hazard**. + +#### Example: Cumulative Hazard Curves for Bone-Marrow Transplant (`bmt`) data -The product limit estimate and the Nelson-Aalen estimate often do not -differ by much. The latter is considered more accurate in small samples -and also directly estimates the cumulative hazard. The -`"fleming-harrington"` method for `survfit()` reduces to Nelson-Aalen -when the data are unweighted. We can also estimate the cumulative hazard -as the negative log of the KM survival function estimate. - -### Application to `bmt` dataset - -```{r} -na_fit = survfit( - formula = surv ~ group, - type = "fleming-harrington", - data = bmt) +```{r} +#| fig-cap: "Disease-Free Cumulative Hazard by Disease Group" + +autoplot( + fun = "cumhaz", + km_model1, + conf.int = FALSE, + ylab = "Cumulative hazard (disease-free survival)", + xlab = "Time since transplant (days)") + + theme_bw() + + theme(legend.position="bottom") +``` + +## Nelson-Aalen Estimates of Cumulative Hazard and Survival -km_fit = survfit( - formula = surv ~ group, - type = "kaplan-meier", - data = bmt) +The point hazard at time $t_i$ can be estimated by $d_i/Y_i$, which +leads to the **Nelson-Aalen estimator of the cumulative hazard**: + +$$\hat H_{NA}(t) \eqdef \sum_{t_i < t}\frac{d_i}{Y_i}$$ -km_and_na = - bind_rows( - .id = "model", - "Kaplan-Meier" = km_fit |> fortify(surv.connect = TRUE), - "Nelson-Aalen" = na_fit |> fortify(surv.connect = TRUE) - ) |> - as_tibble() - -``` +The variance of this estimator is approximately: + +$$ +\begin{aligned} +\hat{\text{Var}}\left(\hat H_{NA} (t)\right) +&= \sum_{t_i <t}\frac{(d_i/Y_i)(1-d_i/Y_i)}{Y_i}\\ +&\approx \sum_{t_i <t}\frac{d_i}{Y_i^2} +\end{aligned} +$$ -```{r} -#| fig-cap: "Kaplan-Meier and Nelson-Aalen Survival Function Estimates, stratified by disease group" - -km_and_na |> - ggplot(aes(x = time, y = surv, col = model)) + - geom_step() + - facet_grid(. ~ strata) + - theme_bw() + - ylab("S(t) = P(T>=t)") + - xlab("Survival time (t, days)") + - theme(legend.position = "bottom") - -``` - -The Kaplan-Meier and Nelson-Aalen survival estimates are very similar -for this dataset. +Since $S(t)=\text{exp}\left\{-H(t)\right\}$, the Nelson-Aalen cumulative +hazard estimate can be converted into an alternate estimate of the +survival function: + +$$ +\begin{aligned} +\hat S_{NA}(t) +&= \text{exp}\left\{-\hat H_{NA}(t)\right\}\\ +&= \text{exp}\left\{-\sum_{t_i < t}\frac{d_i}{Y_i}\right\}\\ +&= \prod_{t_i < t}\text{exp}\left\{-\frac{d_i}{Y_i}\right\}\\ +\end{aligned} +$$ + +Compare these with the corresponding Kaplan-Meier estimates: + +$$ +\begin{aligned} +\hat H_{KM}(t) &= -\sum_{t_i < t}\text{log}\left\{1-\frac{d_i}{Y_i}\right\}\\ +\hat S_{KM}(t) &= \prod_{t_i < t}\left[1-\frac{d_i}{Y_i}\right] +\end{aligned} +$$ + +The product limit estimate and the Nelson-Aalen estimate often do not +differ by much. The latter is considered more accurate in small samples +and also directly estimates the cumulative hazard. The +`"fleming-harrington"` method for `survfit()` reduces to Nelson-Aalen +when the data are unweighted. We can also estimate the cumulative hazard +as the negative log of the KM survival function estimate. + +### Application to `bmt` dataset + +```{r} +na_fit = survfit( + formula = surv ~ group, + type = "fleming-harrington", + data = bmt) + +km_fit = survfit( + formula = surv ~ group, + type = "kaplan-meier", + data = bmt) + +km_and_na = + bind_rows( + .id = "model", + "Kaplan-Meier" = km_fit |> fortify(surv.connect = TRUE), + "Nelson-Aalen" = na_fit |> fortify(surv.connect = TRUE) + ) |> + as_tibble() + +``` + +```{r} +#| fig-cap: "Kaplan-Meier and Nelson-Aalen Survival Function Estimates, stratified by disease group" + +km_and_na |> + ggplot(aes(x = time, y = surv, col = model)) + + geom_step() + + facet_grid(. ~ strata) + + theme_bw() + + ylab("S(t) = P(T>=t)") + + xlab("Survival time (t, days)") + + theme(legend.position = "bottom") + +``` + +The Kaplan-Meier and Nelson-Aalen survival estimates are very similar +for this dataset.
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b/logistic-regression.html index 4159f9eb2..6a7475626 100644 --- a/logistic-regression.html +++ b/logistic-regression.html @@ -779,11 +779,11 @@

    -

    We might summarize this result by saying that “the estimated probability of MI among OC users was 3.71428571 as high as the estimated probability among OC non-users.

    +

    We might summarize this result by saying that “the estimated probability of MI among OC users was 3.7143 as high as the estimated probability among OC non-users.


    @@ -831,7 +831,7 @@

    \[ \begin{aligned} \rho(OC, \neg OC) -&= 3.71428571 +&= 3.7143 \end{aligned} \]

    In comparison, the risk ratio for non-OC versus OC is:

    @@ -840,7 +840,7 @@

    Exercise 3.6 (Computing odds from probabilities) Estimate the odds of MI, for non-OC users.

    Solution. \[ -\omega(\neg OC) = 7.00490343\times 10^{-4} +\omega(\neg OC) = 7.0049\times 10^{-4} \]

    @@ -969,7 +969,7 @@

    Example 3.6 (calculating odds using the shortcut) In Example 3.5, we calculated \[ \begin{aligned} \omega(OC) -&=0.00260678 +&=0.0026 \end{aligned} \]

    Let’s recalculate this result using our shortcut.

    @@ -980,7 +980,7 @@

    Same answer as in Example 3.5!

    @@ -1023,7 +1023,7 @@

    Odds of rare events

    For rare events (small \(\pi\)), odds and probabilities are nearly equal, because \(1-\pi \approx 1\) (see Figure 3.1).

    -

    For example, in Example 3.5, the probability and odds differ by \(6.77762182\times 10^{-6}\).

    +

    For example, in Example 3.5, the probability and odds differ by \(6.7776\times 10^{-6}\).


    Exercise 3.8 What odds value corresponds to the probability \(\pi = 0.2\), and what is the numerical difference between these two values?

    @@ -1221,7 +1221,7 @@

    @@ -1392,7 +1392,7 @@

    Case 1: rare events
    pi2/pi1 #> [1] 2 odds(pi2)/odds(pi1) -#> [1] 2.0204082
    +#> [1] 2.02
    @@ -1442,9 +1442,9 @@

    Odds R

    Solution.

      -
    • \(\omega(OC|MI) = P(OC|MI)/(1 – P(OC|MI) = \frac{13}{7} = 1.85714286\)

    • -
    • \(\omega(OC|\neg MI) = P(OC|\neg MI)/(1 – P(OC|\neg MI) = \frac{4987}{9993} = 0.49904933\)

    • -
    • \(\theta(OC,MI) = \frac{\omega(OC|MI)}{\omega(OC|\neg MI)} = \frac{13/7}{4987/9993} = 3.72136125\)

    • +
    • \(\omega(OC|MI) = P(OC|MI)/(1 – P(OC|MI) = \frac{13}{7} = 1.8571\)

    • +
    • \(\omega(OC|\neg MI) = P(OC|\neg MI)/(1 – P(OC|\neg MI) = \frac{4987}{9993} = 0.499\)

    • +
    • \(\theta(OC,MI) = \frac{\omega(OC|MI)}{\omega(OC|\neg MI)} = \frac{13/7}{4987/9993} = 3.7214\)

    This is the same estimate we calculated in Example 3.7.

    @@ -1943,22 +1943,16 @@

    Show R code
    fitted.values(beetles_glm_grouped)
    -#>           1           2           3           4           5           6 
    -#> 0.058601026 0.164027869 0.362119006 0.605314906 0.795171772 0.903235819 
    -#>           7           8 
    -#> 0.955196107 0.979049344
    -predict(beetles_glm_grouped, type = "response")
    -#>           1           2           3           4           5           6 
    -#> 0.058601026 0.164027869 0.362119006 0.605314906 0.795171772 0.903235819 
    -#>           7           8 
    -#> 0.955196107 0.979049344
    -predict(beetles_glm_grouped, type = "link")
    -#>           1           2           3           4           5           6 
    -#> -2.77661484 -1.62855893 -0.56617883  0.42766061  1.35638644  2.23370678 
    -#>           7           8 
    -#>  3.05962163  3.84441209
    -
    -fit_y = beetles$n * fitted.values(beetles_glm_grouped)
    +#> 1 2 3 4 5 6 7 8 +#> 0.0586 0.1640 0.3621 0.6053 0.7952 0.9032 0.9552 0.9790 +predict(beetles_glm_grouped, type = "response") +#> 1 2 3 4 5 6 7 8 +#> 0.0586 0.1640 0.3621 0.6053 0.7952 0.9032 0.9552 0.9790 +predict(beetles_glm_grouped, type = "link") +#> 1 2 3 4 5 6 7 8 +#> -2.7766 -1.6286 -0.5662 0.4277 1.3564 2.2337 3.0596 3.8444 + +fit_y = beetles$n * fitted.values(beetles_glm_grouped)


    @@ -2100,9 +2094,9 @@

    
     logLik(beetles_glm_grouped)
    -#> 'log Lik.' -18.715135 (df=2)
    +#> 'log Lik.' -18.72 (df=2)
     logLik(beetles_glm_ungrouped)
    -#> 'log Lik.' -186.2354 (df=2)
    +#> 'log Lik.' -186.2 (df=2)
    @@ -2787,7 +2781,7 @@

    @@ -3516,7 +3510,7 @@

    < -+ @@ -3543,7 +3537,7 @@

    <

    - + @@ -3709,8 +3703,8 @@

    <
    ggplotly(HL_plot)
    -
    - +
    +
    @@ -3861,8 +3855,8 @@

    wcgs_response_resid_plot |> ggplotly()
    -
    - +
    +

    We can see a slight fan-shape here: observations on the right have larger variance (as expected since \(var(\bar y) = \pi(1-\pi)/n\) is maximized when \(\pi = 0.5\)).

    @@ -3998,8 +3992,8 @@

    Re
    wcgs_resid_plot1 |> ggplotly()
    -
    - +
    +
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  • B.2.1 The Bernoulli distribution
  • B.2.2 The Poisson distribution
  • -
  • B.2.3 Accounting for exposure
  • -
  • B.2.4 The Negative-Binomial distribution
  • +
  • B.2.3 The Negative-Binomial distribution
  • +
  • B.2.4 Weibull Distribution
  • @@ -373,7 +373,7 @@
    Published
    -

    Last modified: 2024-05-16: 19:20:14 (PM)

    +

    Last modified: 2024-05-21: 12:59:24 (PM)

    @@ -469,8 +469,9 @@ B.1.2 Count variables
  • Definition B.2 (Count variable) A count variable is a random variable whose possible values are some subset of the non-negative integers; that is, a random variable \(X\) such that:

    -

    \[\mathcal{R}{X} \in \mathbb{N}\]

    +

    \[\mathcal{R}(X) \in \mathbb{N}\]

    +

    Exercise B.2 What are some examples of count variables?

    @@ -489,11 +490,15 @@

    Exposure magnitude

    -

    Definition B.3 (Exposure magnitude)  

    -
    -

    For many count outcomes, there is some sense of exposure magnitude, population size, or duration of observation (Table B.1).

    +

    Definition B.3 (Exposure magnitude) For many count outcomes, there is some sense of exposure magnitude, population size, or duration of observation.

    +
    +
    +

    Exercise B.3 What are some examples of exposure magnitudes?

    +
    +
    +

    Solution.

    Table B.1: Examples of exposure units @@ -509,8 +514,8 @@

    - - + + @@ -529,6 +534,7 @@ +

    Exposure units are similar to the number of trials in a binomial distribution, but in non-binomial count outcomes, there can be more than one event per unit of exposure.

    We can use \(t\) to represent continuous-valued exposures/observation durations, and \(n\) to represent discrete-valued exposures.

    @@ -537,14 +543,28 @@

    Definition B.4 (Event rate)  

    -

    When the concept of an exposure magnitude is meaningful, the mean of the outcome \(Y\) is typically modeled as an event rate (denoted \(\lambda\)) times the exposure magnitude (\(t\)). That is:

    +

    For a count outcome \(Y\) with exposure magnitude \(t\), the event rate (denoted \(\lambda\)) is defined as the mean of \(Y\) divided by the the exposure magnitude. That is:

    -

    \[\mathbb{E}[Y|T=t] \stackrel{\text{def}}{=}\mu = \lambda \cdot t\]

    +

    \[\mu \stackrel{\text{def}}{=}\mathbb{E}[Y|T=t]\]

    \[\lambda \stackrel{\text{def}}{=}\frac{\mu}{t} \tag{B.1}\]

    +
    +

    Event rate is somewhat analogous to odds in binary outcome models; it typically serves as an intermediate transformation between the mean of the outcome and the linear component of the model. However, in contrast with the odds function, the transformation \(\lambda = \mu/t\) is not considered part of the Poisson model’s link function, and it treats the exposure magnitude covariate differently from the other covariates.

    +
    +
    +
    +

    Theorem B.1 (Transformation function from event rate to mean) For a count variable with mean \(\mu\), event rate \(\lambda\), and exposure magnitude \(t\):

    +

    \[\therefore\mu = \lambda \cdot t \tag{B.2}\]

    +
    +
    +
    +

    Solution. Start from definition of event rate and use algebra to solve for \(\mu\).

    +
    +
    +

    Equation B.2 is analogous to the inverse-odds function for binary variables.


    -

    Theorem B.1 When the exposure magnitude is 0, there is no opportunity for events to occur:

    +

    Theorem B.2 When the exposure magnitude is 0, there is no opportunity for events to occur:

    \[\mathbb{E}[Y|T=0] = 0\]


    @@ -595,11 +615,11 @@

    Definition B.6 (Poisson distribution) \[\mathcal{R}(Y) = \left\{0, 1, 2, ...\right\} = \mathbb{N}\]

    -

    \[\text{P}(Y = y) = \frac{\mu^{y} e^{-\mu}}{y!}, y \in \mathbb{N} \tag{B.2}\]

    +

    \[\text{P}(Y = y) = \frac{\mu^{y} e^{-\mu}}{y!}, y \in \mathbb{N} \tag{B.3}\]

    (see Figure B.2)

    -

    \[\text{P}(Y \le y) = e^{-\mu} \sum_{j=0}^{\left \lfloor{y}\right \rfloor}\frac{\mu^j}{j!} \tag{B.3}\]

    +

    \[\text{P}(Y \le y) = e^{-\mu} \sum_{j=0}^{\left \lfloor{y}\right \rfloor}\frac{\mu^j}{j!} \tag{B.4}\]

    (see Figure B.3)

    @@ -676,7 +696,7 @@

    -

    Exercise B.3 (Poisson distribution functions) Let \(X \sim \text{Pois}(\mu = 3.75)\).

    +

    Exercise B.4 (Poisson distribution functions) Let \(X \sim \text{Pois}(\mu = 3.75)\).

    Compute:

    • \(\text{P}(X = 4 | \mu = 3.75)\)
    • @@ -688,24 +708,24 @@

      Solution.

        -
      • \(\text{P}(X=4) = 0.19378025\)
      • -
      • \(\text{P}(X\le 7) = 0.96237866\)
      • -
      • \(\text{P}(X > 5) = 0.17711717\)
      • +
      • \(\text{P}(X=4) = 0.1938\)
      • +
      • \(\text{P}(X\le 7) = 0.9624\)
      • +
      • \(\text{P}(X > 5) = 0.1771\)

      -

      Theorem B.2 (Properties of the Poisson distribution) If \(X \sim \text{Pois}(\mu)\), then:

      +

      Theorem B.3 (Properties of the Poisson distribution) If \(X \sim \text{Pois}(\mu)\), then:

      • \(\mathbb{E}[Y] = \mu\)
      • \(\text{Var}(Y) = \mu\)
      -

      Exercise B.4 Prove Theorem B.2.

      +

      Exercise B.5 Prove Theorem B.3.


      -
      +

      Solution. \[ \begin{aligned} \text{E}[X] @@ -724,8 +744,8 @@ \end{aligned} \]

      -

      -B.2.3 Accounting for exposure

      +
      +

      Accounting for exposure

      If the exposures/observation durations, denoted \(T=t\) or \(N=n\), vary between observations, we model:

      \[\mu = \lambda\cdot t\]

      \(\lambda\) is interpreted as the “expected event rate per unit of exposure”; that is,

      @@ -744,28 +764,60 @@
    -

    Theorem B.3 If \(\mu = \lambda\cdot t\), then:

    +

    Theorem B.4 If \(\mu = \lambda\cdot t\), then:

    \[\text{log}\left\{\mu \right\}= \text{log}\left\{\lambda\right\} + \text{log}\left\{t\right\}\]

    Definition B.7 (Offset) When the linear component of a model involves a term without an unknown coefficient, that term is called an offset.

    -

    -B.2.4 The Negative-Binomial distribution

    -{{}} +

    +B.2.3 The Negative-Binomial distribution

    +
    +

    Definition B.8 (Negative binomial distribution) \[ +\text{P}(Y=y) += \frac{\mu^y}{y!} +\cdot \frac{\Gamma(\rho + y)}{\Gamma(\rho) \cdot (\rho + \mu)^y} +\cdot \left(1+\frac{\mu}{\rho}\right)^{-\rho} +\]

    +

    where \(\rho\) is an overdispersion parameter and \(\Gamma(x) = (x-1)!\) for integers \(x\).

    +
    +
    +

    You don’t need to memorize or understand this expression.

    +

    As \(\rho \rightarrow \infty\), the second term converges to 1 and the third term converges to \(\text{exp}\left\{-\mu\right\}\), which brings us back to the Poisson distribution.

    +
    +
    +
    +

    Theorem B.5 If \(Y \sim \text{NegBin}(\mu, \rho)\), then:

    +
      +
    • \(\mathbb{E}[Y] = \mu\)
    • +
    • \(\text{Var}\left(Y\right) = \mu + \frac{\mu^2}{\rho} > \mu\)
    • +
    +
    +

    +B.2.4 Weibull Distribution

    +

    \[ +\begin{aligned} +p(t)&= \alpha\lambda x^{\alpha-1}\text{e}^{-\lambda x^\alpha}\\ +h(t)&=\alpha\lambda x^{\alpha-1}\\ +S(t)&=\text{e}^{-\lambda x^\alpha}\\ +E(T)&= \Gamma(1+1/\alpha)\cdot \lambda^{-1/\alpha} +\end{aligned} +\]

    +

    When \(\alpha=1\) this is the exponential. When \(\alpha>1\) the hazard is increasing and when \(\alpha < 1\) the hazard is decreasing. This provides more flexibility than the exponential.

    +

    We will see more of this distribution later.

    B.3 Characteristics of probability distributions

    -

    Definition B.8 (Density function) The density function \(f(t)\) or \(\text{p}(T=t)\) for a random variable \(T\) at value \(t\) can be defined as the derivative of the cumulative probability function \(P(T\le t)\); that is:

    +

    Definition B.9 (Density function) The density function \(f(t)\) or \(\text{p}(T=t)\) for a random variable \(T\) at value \(t\) can be defined as the derivative of the cumulative probability function \(P(T\le t)\); that is:

    \[f(t) \stackrel{\text{def}}{=}\frac{\partial}{\partial t} \Pr(T\le t)\]

    -

    Definition B.9 (Hazard function) The hazard function for a random variable \(T\) at value \(t\) is the conditional density of \(T\) at \(t\), given \(T\ge t\); that is:

    +

    Definition B.10 (Hazard function) The hazard function for a random variable \(T\) at value \(t\) is the conditional density of \(T\) at \(t\), given \(T\ge t\); that is:

    \[h(t) \stackrel{\text{def}}{=}p(T=t|T\ge t)\]

    If \(T\) represents the time at which an event occurs, then \(h(t)\) is the probability that the event occurs at time \(t\), given that it has not occurred prior to time \(t\).

    -

    Definition B.10 (Expectation, expected value, population mean ) The expectation, expected value, or population mean of a continuous random variable \(X\), denoted \(\mathbb{E}\left[X\right]\), \(\mu(X)\), or \(\mu_X\), is the weighted mean of \(X\)’s possible values, weighted by the probability density function of those values:

    +

    Definition B.11 (Expectation, expected value, population mean ) The expectation, expected value, or population mean of a continuous random variable \(X\), denoted \(\mathbb{E}\left[X\right]\), \(\mu(X)\), or \(\mu_X\), is the weighted mean of \(X\)’s possible values, weighted by the probability density function of those values:

    \[\mathbb{E}\left[X\right] = \int_{x\in \mathcal{R}(X)} x \cdot \text{p}(X=x)dx\]

    The expectation, expected value, or population mean of a discrete random variable \(X\), denoted \(\mathbb{E}\left[X\right]\), \(\mu(X)\), or \(\mu_X\), is the mean of \(X\)’s possible values, weighted by the probability mass function of those values:

    \[\mathbb{E}\left[X\right] = \sum_{x \in \mathcal{R}(X)} x \cdot \text{P}(X=x)\]

    @@ -773,7 +825,7 @@

    -

    Theorem B.4 (Expectation of the Bernoulli distribution) The expectation of a Bernoulli random variable with parameter \(\pi\) is:

    +

    Theorem B.6 (Expectation of the Bernoulli distribution) The expectation of a Bernoulli random variable with parameter \(\pi\) is:

    \[\mathbb{E}\left[X\right] = \pi\]


    @@ -794,14 +846,14 @@
    -

    The symbol for independence, \(⫫\), is essentially just \(\prod\) upside-down. So the symbol can remind you of its definition (Definition B.17).

    +

    The symbol for independence, \(⫫\), is essentially just \(\prod\) upside-down. So the symbol can remind you of its definition (Definition B.18).


    -

    Definition B.18 (Conditional independence) A set of random variables \(Y_1, \ldots, Y_n\) are conditionally statistically independent given a set of covariates \(X_1, \ldots, X_n\) if the joint probability of the \(Y_i\)s given the \(X_i\)s is equal to the product of their marginal probabilities:

    +

    Definition B.19 (Conditional independence) A set of random variables \(Y_1, \ldots, Y_n\) are conditionally statistically independent given a set of covariates \(X_1, \ldots, X_n\) if the joint probability of the \(Y_i\)s given the \(X_i\)s is equal to the product of their marginal probabilities:

    \[\Pr(Y_1=y_1, \ldots, Y_n = y_n|X_1=x_1, \ldots, X_n = x_n) = \prod_{i=1}^n{\Pr(Y_i=y_i|X_i=x_i)}\]


    -

    Definition B.19 (Identically distributed) A set of random variables \(X_1, \ldots, X_n\) are identically distributed if they have the same range \(\mathcal{R}(X)\) and if their marginal distributions \(\text{P}(X_1=x_1), ..., \text{P}(X_n=x_n)\) are all equal to some shared distribution \(\text{P}(X=x)\):

    +

    Definition B.20 (Identically distributed) A set of random variables \(X_1, \ldots, X_n\) are identically distributed if they have the same range \(\mathcal{R}(X)\) and if their marginal distributions \(\text{P}(X_1=x_1), ..., \text{P}(X_n=x_n)\) are all equal to some shared distribution \(\text{P}(X=x)\):

    \[ \forall i\in \left\{1:n\right\}, \forall x \in \mathcal{R}(X): \text{P}(X_i=x) = \text{P}(X=x) \]


    -

    Definition B.20 (Conditionally identically distributed) A set of random variables \(Y_1, \ldots, Y_n\) are conditionally identically distributed given a set of covariates \(X_1, \ldots, X_n\) if \(Y_1, \ldots, Y_n\) have the same range \(\mathcal{R}(X)\) and if the distributions \(\text{P}(Y_i=y_i|X_i =x_i)\) are all equal to the same distribution \(\text{P}(Y=y|X=x)\):

    +

    Definition B.21 (Conditionally identically distributed) A set of random variables \(Y_1, \ldots, Y_n\) are conditionally identically distributed given a set of covariates \(X_1, \ldots, X_n\) if \(Y_1, \ldots, Y_n\) have the same range \(\mathcal{R}(X)\) and if the distributions \(\text{P}(Y_i=y_i|X_i =x_i)\) are all equal to the same distribution \(\text{P}(Y=y|X=x)\):

    \[ \text{P}(Y_i=y|X_i=x) = \text{P}(Y=y|X=x) \]


    -

    Definition B.21 (Independent and identically distributed) A set of random variables \(X_1, \ldots, X_n\) are independent and identically distributed (shorthand: “\(X_i\ \text{iid}\)”) if they are statistically independent and identically distributed.

    +

    Definition B.22 (Independent and identically distributed) A set of random variables \(X_1, \ldots, X_n\) are independent and identically distributed (shorthand: “\(X_i\ \text{iid}\)”) if they are statistically independent and identically distributed.


    -

    Definition B.22 (Conditionally independent and identically distributed) A set of random variables \(Y_1, \ldots, Y_n\) are conditionally independent and identically distributed (shorthand: “\(Y_i | X_i\ \text{ciid}\)” or just “\(Y_i |X_i\ \text{iid}\)”) given a set of covariates \(X_1, \ldots, X_n\) if \(Y_1, \ldots, Y_n\) are conditionally independent given \(X_1, \ldots, X_n\) and \(Y_1, \ldots, Y_n\) are identically distributed given \(X_1, \ldots, X_n\).

    +

    Definition B.23 (Conditionally independent and identically distributed) A set of random variables \(Y_1, \ldots, Y_n\) are conditionally independent and identically distributed (shorthand: “\(Y_i | X_i\ \text{ciid}\)” or just “\(Y_i |X_i\ \text{iid}\)”) given a set of covariates \(X_1, \ldots, X_n\) if \(Y_1, \ldots, Y_n\) are conditionally independent given \(X_1, \ldots, X_n\) and \(Y_1, \ldots, Y_n\) are identically distributed given \(X_1, \ldots, X_n\).

    @@ -1614,366 +1666,383 @@ ### The Negative-Binomial distribution {#sec-nb-dist} -{{<< include negbinom.qmd >}} +{{< include negbinom.qmd >}} -## Characteristics of probability distributions +### Weibull Distribution {#sec-weibull} -:::{#def-density} -### Density function - -The density function $f(t)$ or $\p(T=t)$ for a random variable $T$ at value $t$ can be defined as the derivative of the cumulative probability function $P(T\le t)$; that is: - -$$f(t) \eqdef \deriv{t} \Pr(T\le t)$$ - -::: +$$ +\begin{aligned} +p(t)&= \alpha\lambda x^{\alpha-1}\text{e}^{-\lambda x^\alpha}\\ +h(t)&=\alpha\lambda x^{\alpha-1}\\ +S(t)&=\text{e}^{-\lambda x^\alpha}\\ +E(T)&= \Gamma(1+1/\alpha)\cdot \lambda^{-1/\alpha} +\end{aligned} +$$ -:::{#def-hazard} - -{{< include _def-hazard.qmd >}} +When $\alpha=1$ this is the exponential. When $\alpha>1$ the hazard is +increasing and when $\alpha < 1$ the hazard is decreasing. This provides +more flexibility than the exponential. -::: +We will see more of this distribution later. -:::{#def-expectation} -### Expectation, expected value, population mean \index{expectation} \index{expected value} - -The **expectation**, **expected value**, or **population mean** of a *continuous* random variable $X$, denoted $\E{X}$, $\mu(X)$, or $\mu_X$, is the weighted mean of $X$'s possible values, weighted by the probability density function of those values: +## Characteristics of probability distributions + +:::{#def-density} +### Density function -$$\E{X} = \int_{x\in \rangef{X}} x \cdot \p(X=x)dx$$ +The density function $f(t)$ or $\p(T=t)$ for a random variable $T$ at value $t$ can be defined as the derivative of the cumulative probability function $P(T\le t)$; that is: -The **expectation**, **expected value**, or **population mean** of a *discrete* random variable $X$, denoted $\E{X}$, $\mu(X)$, or $\mu_X$, is the mean of $X$'s possible values, weighted by the probability mass function of those values: +$$f(t) \eqdef \deriv{t} \Pr(T\le t)$$ -$$\E{X} = \sum_{x \in \rangef{X}} x \cdot \P(X=x)$$ +::: -(c.f. <https://en.wikipedia.org/wiki/Expected_value>) +:::{#def-hazard} -::: +{{< include _def-hazard.qmd >}} ---- +::: -:::{#thm-bernoulli-mean} -#### Expectation of the Bernoulli distribution +:::{#def-expectation} +### Expectation, expected value, population mean \index{expectation} \index{expected value} -The expectation of a Bernoulli random variable with parameter $\pi$ is: +The **expectation**, **expected value**, or **population mean** of a *continuous* random variable $X$, denoted $\E{X}$, $\mu(X)$, or $\mu_X$, is the weighted mean of $X$'s possible values, weighted by the probability density function of those values: -$$\E{X} = \pi$$ -::: - ---- - -:::{.proof} - -$$ -\ba -\E{X} -&= \sum_{x\in \rangef{X}} x \cd \P(X=x) -\\&= \sum_{x\in \set{0,1}} x \cd \P(X=x) -\\&= \paren{0 \cd \P(X=0)} + \paren{1 \cd \P(X=1)} -\\&= \paren{0 \cd (1-\pi)} + \paren{1 \cd \pi} -\\&= 0 + \pi -\\&= \pi -\ea -$$ - -::: - ---- - -### Variance and related characteristics - -:::{#def-variance} -#### Variance - -The variance of a random variable $X$ is the expectation of the squared difference between $X$ and $\E{X}$; that is: - -$$ -\Var{X} \eqdef \E{(X-\E{X})^2} -$$ - -::: +$$\E{X} = \int_{x\in \rangef{X}} x \cdot \p(X=x)dx$$ + +The **expectation**, **expected value**, or **population mean** of a *discrete* random variable $X$, denoted $\E{X}$, $\mu(X)$, or $\mu_X$, is the mean of $X$'s possible values, weighted by the probability mass function of those values: + +$$\E{X} = \sum_{x \in \rangef{X}} x \cdot \P(X=x)$$ + +(c.f. <https://en.wikipedia.org/wiki/Expected_value>) + +::: + +--- + +:::{#thm-bernoulli-mean} +#### Expectation of the Bernoulli distribution + +The expectation of a Bernoulli random variable with parameter $\pi$ is: + +$$\E{X} = \pi$$ +::: + +--- + +:::{.proof} + +$$ +\ba +\E{X} +&= \sum_{x\in \rangef{X}} x \cd \P(X=x) +\\&= \sum_{x\in \set{0,1}} x \cd \P(X=x) +\\&= \paren{0 \cd \P(X=0)} + \paren{1 \cd \P(X=1)} +\\&= \paren{0 \cd (1-\pi)} + \paren{1 \cd \pi} +\\&= 0 + \pi +\\&= \pi +\ea +$$ ---- +::: -:::{#thm-variance} -#### Simplified expression for variance - -$$\Var{X}=\E{X^2} - \sqf{\E{X}}$$ - ---- +--- + +### Variance and related characteristics + +:::{#def-variance} +#### Variance -::::{.proof} -By linearity of expectation, we have: - -$$ -\begin{aligned} -\Var{X} -&\eqdef \E{(X-\E{X})^2}\\ -&=\E{X^2 - 2X\E{X} + \sqf{\E{X}}}\\ -&=\E{X^2} - \E{2X\E{X}} + \E{\sqf{\E{X}}}\\ -&=\E{X^2} - 2\E{X}\E{X} + \sqf{\E{X}}\\ -&=\E{X^2} - \sqf{\E{X}}\\ -\end{aligned} -$$ -:::: +The variance of a random variable $X$ is the expectation of the squared difference between $X$ and $\E{X}$; that is: + +$$ +\Var{X} \eqdef \E{(X-\E{X})^2} +$$ + +::: + +--- + +:::{#thm-variance} +#### Simplified expression for variance + +$$\Var{X}=\E{X^2} - \sqf{\E{X}}$$ -::: +--- ---- - -::: {#def-precision} -#### Precision - -The **precision** of a random variable $X$, often denoted $\tau(X)$, $\tau_X$, or shorthanded as $\tau$, is -the inverse of that random variable's variance; that is: - -$$\tau(X) \eqdef \inv{\Var{X}}$$ -::: - -::: {#def-sd} - -#### Standard deviation +::::{.proof} +By linearity of expectation, we have: + +$$ +\begin{aligned} +\Var{X} +&\eqdef \E{(X-\E{X})^2}\\ +&=\E{X^2 - 2X\E{X} + \sqf{\E{X}}}\\ +&=\E{X^2} - \E{2X\E{X}} + \E{\sqf{\E{X}}}\\ +&=\E{X^2} - 2\E{X}\E{X} + \sqf{\E{X}}\\ +&=\E{X^2} - \sqf{\E{X}}\\ +\end{aligned} +$$ +:::: -The standard deviation of a random variable $X$ is the square-root of the variance of $X$: +::: -$$\SD{X} \eqdef \sqrt{\Var{X}}$$ +--- -::: - ---- - -:::{#def-cov} -#### Covariance - -For any two one-dimensional random variables, $X,Y$: +::: {#def-precision} +#### Precision + +The **precision** of a random variable $X$, often denoted $\tau(X)$, $\tau_X$, or shorthanded as $\tau$, is +the inverse of that random variable's variance; that is: + +$$\tau(X) \eqdef \inv{\Var{X}}$$ +::: -$$\Cov{X,Y} \eqdef \Expf{(X - \E X)(Y - \E Y)}$$ +::: {#def-sd} -::: +#### Standard deviation ---- +The standard deviation of a random variable $X$ is the square-root of the variance of $X$: -:::{#thm-alt-cov} -$$\Cov{X,Y}= \E{XY} - \E{X} \E{Y}$$ +$$\SD{X} \eqdef \sqrt{\Var{X}}$$ + ::: --- -:::{.proof} -Left to the reader. -::: - ---- - -:::{#lem-cov-xx} - -##### The covariance of a variable with itself is its variance - -For any random variable $X$: - -$$\Cov{X,X} = \Var{X}$$ - -::: - -:::{.proof} -$$ -\ba -\Cov{X,X} &= E[XX] - E[X]E[X] -\\ &= E[X^2]-(E[X])^2 -\\ &= \Var{X} -\ea -$$ -::: - ---- - -:::{#def-cov-vec-x} - -#### Variance/covariance of a $p \times 1$ random vector - -For a $p \times 1$ dimensional random vector $X$, - +:::{#def-cov} +#### Covariance + +For any two one-dimensional random variables, $X,Y$: + +$$\Cov{X,Y} \eqdef \Expf{(X - \E X)(Y - \E Y)}$$ + +::: + +--- + +:::{#thm-alt-cov} +$$\Cov{X,Y}= \E{XY} - \E{X} \E{Y}$$ +::: + +--- + +:::{.proof} +Left to the reader. +::: + +--- + +:::{#lem-cov-xx} + +##### The covariance of a variable with itself is its variance + +For any random variable $X$: + +$$\Cov{X,X} = \Var{X}$$ + +::: + +:::{.proof} $$ -\begin{aligned} -\text{Var}(X) -&\eqdef \text{Cov}(X)\\ -&\eqdef E[ \left( X - E\lbrack X\rbrack \right)^{\top}\left( X - E\lbrack X\rbrack \right) ]\\ +\ba +\Cov{X,X} &= E[XX] - E[X]E[X] +\\ &= E[X^2]-(E[X])^2 +\\ &= \Var{X} \ea $$ - -::: - ---- - -:::{#thm-vcov-vec} - -#### Alternate expression for variance of a random vector - -$$ -\ba -\Var{X} -&= E[ X^{\top}X ] - {E\lbrack X\rbrack}^{\top}E\lbrack X\rbrack -\end{aligned} -$$ -::: - ---- - -:::{.proof} -$$ -\ba -\Var{X} -&= E[ \left( X^{\top} - E\lbrack X\rbrack^{\top} \right)\left( X - E\lbrack X\rbrack \right) ]\\ -&= E[ X^{\top}X - E\lbrack X\rbrack^{\top}X - X^{\top}E\lbrack X\rbrack + E\lbrack X\rbrack^{\top}E\lbrack X\rbrack ]\\ -&= E[ X^{\top}X ] - E\lbrack X\rbrack^{\top}E\lbrack X\rbrack - {E\lbrack X\rbrack}^{\top}E\lbrack X\rbrack + E\lbrack X\rbrack^{\top}E\lbrack X\rbrack\\ -&= E[ X^{\top}X ] - 2{E\lbrack X\rbrack}^{\top}E\lbrack X\rbrack + E\lbrack X\rbrack^{\top}E\lbrack X\rbrack\\ -&= E[ X^{\top}X ] - {E\lbrack X\rbrack}^{\top}E\lbrack X\rbrack -\end{aligned} -$$ -::: - ---- +::: + +--- + +:::{#def-cov-vec-x} + +#### Variance/covariance of a $p \times 1$ random vector + +For a $p \times 1$ dimensional random vector $X$, + +$$ +\begin{aligned} +\text{Var}(X) +&\eqdef \text{Cov}(X)\\ +&\eqdef E[ \left( X - E\lbrack X\rbrack \right)^{\top}\left( X - E\lbrack X\rbrack \right) ]\\ +\ea +$$ + +::: + +--- + +:::{#thm-vcov-vec} + +#### Alternate expression for variance of a random vector + +$$ +\ba +\Var{X} +&= E[ X^{\top}X ] - {E\lbrack X\rbrack}^{\top}E\lbrack X\rbrack +\end{aligned} +$$ +::: -:::{#thm-var-lincom} +--- -#### Variance of a linear combination - -For any set of random variables $\Xin$ and corresponding constants $a_1, ... ,a_n$: - -$$\Var{\sumin a_i X_i} = \sumin \sumn{j} a_i a_j \Cov{X_i,X_j}$$ -::: - ---- - -:::{.proof} - -Left to the reader... +:::{.proof} +$$ +\ba +\Var{X} +&= E[ \left( X^{\top} - E\lbrack X\rbrack^{\top} \right)\left( X - E\lbrack X\rbrack \right) ]\\ +&= E[ X^{\top}X - E\lbrack X\rbrack^{\top}X - X^{\top}E\lbrack X\rbrack + E\lbrack X\rbrack^{\top}E\lbrack X\rbrack ]\\ +&= E[ X^{\top}X ] - E\lbrack X\rbrack^{\top}E\lbrack X\rbrack - {E\lbrack X\rbrack}^{\top}E\lbrack X\rbrack + E\lbrack X\rbrack^{\top}E\lbrack X\rbrack\\ +&= E[ X^{\top}X ] - 2{E\lbrack X\rbrack}^{\top}E\lbrack X\rbrack + E\lbrack X\rbrack^{\top}E\lbrack X\rbrack\\ +&= E[ X^{\top}X ] - {E\lbrack X\rbrack}^{\top}E\lbrack X\rbrack +\end{aligned} +$$ +::: -::: +--- ---- +:::{#thm-var-lincom} -:::{#lem-var-lincom2} +#### Variance of a linear combination -For any two random variables $X$ and $Y$ and scalars $a$ and $b$: +For any set of random variables $\Xin$ and corresponding constants $a_1, ... ,a_n$: -$$\Var{aX + bY} = a^2 \Var{X} + b^2 \Var{Y} + 2(a \cd b) \Cov{X,Y}$$ - -::: - ---- - -:::{.proof} - -Apply @thm-var-lincom with $n=2$, $X_1 = X$, and $X_2 = Y$. - -Or, see <https://statproofbook.github.io/P/var-lincomb.html> - -::: - ---- - -:::{#def-homosked} -### homoskedastic, heteroskedastic +$$\Var{\sumin a_i X_i} = \sumin \sumn{j} a_i a_j \Cov{X_i,X_j}$$ +::: + +--- + +:::{.proof} + +Left to the reader... + +::: + +--- + +:::{#lem-var-lincom2} + +For any two random variables $X$ and $Y$ and scalars $a$ and $b$: + +$$\Var{aX + bY} = a^2 \Var{X} + b^2 \Var{Y} + 2(a \cd b) \Cov{X,Y}$$ -A random variable $Y$ is **homoskedastic** (with respect to covariates $X$) if the variance of $Y$ does not vary with $X$: +::: -$$\Varr(Y|X=x) = \ss, \forall x$$ +--- -Otherwise it is **heteroskedastic**. +:::{.proof} -::: +Apply @thm-var-lincom with $n=2$, $X_1 = X$, and $X_2 = Y$. ---- +Or, see <https://statproofbook.github.io/P/var-lincomb.html> -:::{#def-indpt} +::: -### Statistical independence +--- -A set of random variables $\X1n$ are **statistically independent** -if their joint probability is equal to the product of their marginal probabilities: +:::{#def-homosked} +### homoskedastic, heteroskedastic -$$\Pr(\Xx1n) = \prodi1n{\Pr(X_i=x_i)}$$ +A random variable $Y$ is **homoskedastic** (with respect to covariates $X$) if the variance of $Y$ does not vary with $X$: -::: +$$\Varr(Y|X=x) = \ss, \forall x$$ -::: notes +Otherwise it is **heteroskedastic**. -::::{.callout-tip} -The symbol for independence, $\ind$, is essentially just $\prod$ upside-down. -So the symbol can remind you of its definition (@def-indpt). -:::: - -::: - ---- - -:::{#def-cind} +::: + +--- + +:::{#def-indpt} + +### Statistical independence + +A set of random variables $\X1n$ are **statistically independent** +if their joint probability is equal to the product of their marginal probabilities: -### Conditional independence +$$\Pr(\Xx1n) = \prodi1n{\Pr(X_i=x_i)}$$ -A set of random variables $\dsn{Y}$ are **conditionally statistically independent** -given a set of covariates $\X1n$ -if the joint probability of the $Y_i$s given the $X_i$s is equal to -the product of their marginal probabilities: - -$$\Pr(\dsvn{Y}{y}|\dsvn{X}{x}) = \prodi1n{\Pr(Y_i=y_i|X_i=x_i)}$$ - -::: +::: + +::: notes + +::::{.callout-tip} +The symbol for independence, $\ind$, is essentially just $\prod$ upside-down. +So the symbol can remind you of its definition (@def-indpt). +:::: ---- +::: -:::{#def-ident} +--- -#### Identically distributed +:::{#def-cind} -A set of random variables $\X1n$ are **identically distributed** -if they have the same range $\rangef{X}$ and if -their marginal distributions $\P(X_1=x_1), ..., \P(X_n=x_n)$ are all -equal to some shared distribution $\P(X=x)$: - -$$ -\forall i\in \set{1:n}, \forall x \in \rangef{X}: \P(X_i=x) = \P(X=x) -$$ +### Conditional independence + +A set of random variables $\dsn{Y}$ are **conditionally statistically independent** +given a set of covariates $\X1n$ +if the joint probability of the $Y_i$s given the $X_i$s is equal to +the product of their marginal probabilities: + +$$\Pr(\dsvn{Y}{y}|\dsvn{X}{x}) = \prodi1n{\Pr(Y_i=y_i|X_i=x_i)}$$ ::: --- -:::{#def-cident} +:::{#def-ident} -#### Conditionally identically distributed +#### Identically distributed -A set of random variables $\dsn{Y}$ are **conditionally identically distributed** -given a set of covariates $\X1n$ -if $\dsn{Y}$ have the same range $\rangef{X}$ and if -the distributions $\P(Y_i=y_i|X_i =x_i)$ are all -equal to the same distribution $\P(Y=y|X=x)$: - -$$ -\P(Y_i=y|X_i=x) = \P(Y=y|X=x) -$$ - -::: - ---- - -:::{#def-iid} -#### Independent and identically distributed +A set of random variables $\X1n$ are **identically distributed** +if they have the same range $\rangef{X}$ and if +their marginal distributions $\P(X_1=x_1), ..., \P(X_n=x_n)$ are all +equal to some shared distribution $\P(X=x)$: + +$$ +\forall i\in \set{1:n}, \forall x \in \rangef{X}: \P(X_i=x) = \P(X=x) +$$ + +::: + +--- + +:::{#def-cident} + +#### Conditionally identically distributed -A set of random variables $\dsn{X}$ are **independent and identically distributed** -(shorthand: "$X_i\ \iid$") if they are statistically independent and identically distributed. - -::: - ---- - -:::{#def-iid} -#### Conditionally independent and identically distributed - -A set of random variables $\dsn{Y}$ are **conditionally independent and identically distributed** (shorthand: "$Y_i | X_i\ \ciid$" or just "$Y_i |X_i\ \iid$") given a set of covariates $\dsn{X}$ -if $\dsn{Y}$ are conditionally independent given $\dsn{X}$ and $\dsn{Y}$ are identically distributed given -$\dsn{X}$. +A set of random variables $\dsn{Y}$ are **conditionally identically distributed** +given a set of covariates $\X1n$ +if $\dsn{Y}$ have the same range $\rangef{X}$ and if +the distributions $\P(Y_i=y_i|X_i =x_i)$ are all +equal to the same distribution $\P(Y=y|X=x)$: + +$$ +\P(Y_i=y|X_i=x) = \P(Y=y|X=x) +$$ + +::: + +--- -::: +:::{#def-iid} +#### Independent and identically distributed + +A set of random variables $\dsn{X}$ are **independent and identically distributed** +(shorthand: "$X_i\ \iid$") if they are statistically independent and identically distributed. + +::: + +--- + +:::{#def-iid} +#### Conditionally independent and identically distributed + +A set of random variables $\dsn{Y}$ are **conditionally independent and identically distributed** (shorthand: "$Y_i | X_i\ \ciid$" or just "$Y_i |X_i\ \iid$") given a set of covariates $\dsn{X}$ +if $\dsn{Y}$ are conditionally independent given $\dsn{X}$ and $\dsn{Y}$ are identically distributed given +$\dsn{X}$. + +:::
    [0,0.1]23052,305 129 135.2
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    ").addClass(errClass).css("position", "absolute") - .css("top", el.offsetTop) - .css("left", el.offsetLeft) - // setting width can push out the page size, forcing otherwise - // unnecessary scrollbars to appear and making it impossible for - // the element to shrink; so use max-width instead - .css("maxWidth", el.offsetWidth) - .css("height", el.offsetHeight); - errorDiv.text(err.message); - $el.after(errorDiv); - - // Really dumb way to keep the size/position of the error in sync with - // the parent element as the window is resized or whatever. - var intId = setInterval(function() { - if (!errorDiv[0].parentElement) { - clearInterval(intId); - return; - } - errorDiv - .css("top", el.offsetTop) - .css("left", el.offsetLeft) - .css("maxWidth", el.offsetWidth) - .css("height", el.offsetHeight); - }, 500); - } - } - }, - clearError: function(el) { - var $el = $(el); - var display = $el.data("restore-display-mode"); - $el.data("restore-display-mode", null); - - if (display === "inline" || display === "inline-block") { - if (display) - $el.css("display", display); - $(el.nextSibling).filter(".htmlwidgets-error").remove(); - } else if (display === "block"){ - $el.css("visibility", "inherit"); - $(el.nextSibling).filter(".htmlwidgets-error").remove(); - } - }, - sizing: {} - }; - - // Called by widget bindings to register a new type of widget. The definition - // object can contain the following properties: - // - name (required) - A string indicating the binding name, which will be - // used by default as the CSS classname to look for. - // - initialize (optional) - A function(el) that will be called once per - // widget element; if a value is returned, it will be passed as the third - // value to renderValue. - // - renderValue (required) - A function(el, data, initValue) that will be - // called with data. Static contexts will cause this to be called once per - // element; Shiny apps will cause this to be called multiple times per - // element, as the data changes. - window.HTMLWidgets.widget = function(definition) { - if (!definition.name) { - throw new Error("Widget must have a name"); - } - if (!definition.type) { - throw new Error("Widget must have a type"); - } - // Currently we only support output widgets - if (definition.type !== "output") { - throw new Error("Unrecognized widget type '" + definition.type + "'"); - } - // TODO: Verify that .name is a valid CSS classname - - // Support new-style instance-bound definitions. Old-style class-bound - // definitions have one widget "object" per widget per type/class of - // widget; the renderValue and resize methods on such widget objects - // take el and instance arguments, because the widget object can't - // store them. New-style instance-bound definitions have one widget - // object per widget instance; the definition that's passed in doesn't - // provide renderValue or resize methods at all, just the single method - // factory(el, width, height) - // which returns an object that has renderValue(x) and resize(w, h). - // This enables a far more natural programming style for the widget - // author, who can store per-instance state using either OO-style - // instance fields or functional-style closure variables (I guess this - // is in contrast to what can only be called C-style pseudo-OO which is - // what we required before). - if (definition.factory) { - definition = createLegacyDefinitionAdapter(definition); - } - - if (!definition.renderValue) { - throw new Error("Widget must have a renderValue function"); - } - - // For static rendering (non-Shiny), use a simple widget registration - // scheme. We also use this scheme for Shiny apps/documents that also - // contain static widgets. - window.HTMLWidgets.widgets = window.HTMLWidgets.widgets || []; - // Merge defaults into the definition; don't mutate the original definition. - var staticBinding = extend({}, defaults, definition); - overrideMethod(staticBinding, "find", function(superfunc) { - return function(scope) { - var results = superfunc(scope); - // Filter out Shiny outputs, we only want the static kind - return filterByClass(results, "html-widget-output", false); - }; - }); - window.HTMLWidgets.widgets.push(staticBinding); - - if (shinyMode) { - // Shiny is running. Register the definition with an output binding. - // The definition itself will not be the output binding, instead - // we will make an output binding object that delegates to the - // definition. This is because we foolishly used the same method - // name (renderValue) for htmlwidgets definition and Shiny bindings - // but they actually have quite different semantics (the Shiny - // bindings receive data that includes lots of metadata that it - // strips off before calling htmlwidgets renderValue). We can't - // just ignore the difference because in some widgets it's helpful - // to call this.renderValue() from inside of resize(), and if - // we're not delegating, then that call will go to the Shiny - // version instead of the htmlwidgets version. - - // Merge defaults with definition, without mutating either. - var bindingDef = extend({}, defaults, definition); - - // This object will be our actual Shiny binding. - var shinyBinding = new Shiny.OutputBinding(); - - // With a few exceptions, we'll want to simply use the bindingDef's - // version of methods if they are available, otherwise fall back to - // Shiny's defaults. NOTE: If Shiny's output bindings gain additional - // methods in the future, and we want them to be overrideable by - // HTMLWidget binding definitions, then we'll need to add them to this - // list. - delegateMethod(shinyBinding, bindingDef, "getId"); - delegateMethod(shinyBinding, bindingDef, "onValueChange"); - delegateMethod(shinyBinding, bindingDef, "onValueError"); - delegateMethod(shinyBinding, bindingDef, "renderError"); - delegateMethod(shinyBinding, bindingDef, "clearError"); - delegateMethod(shinyBinding, bindingDef, "showProgress"); - - // The find, renderValue, and resize are handled differently, because we - // want to actually decorate the behavior of the bindingDef methods. - - shinyBinding.find = function(scope) { - var results = bindingDef.find(scope); - - // Only return elements that are Shiny outputs, not static ones - var dynamicResults = results.filter(".html-widget-output"); - - // It's possible that whatever caused Shiny to think there might be - // new dynamic outputs, also caused there to be new static outputs. - // Since there might be lots of different htmlwidgets bindings, we - // schedule execution for later--no need to staticRender multiple - // times. - if (results.length !== dynamicResults.length) - scheduleStaticRender(); - - return dynamicResults; - }; - - // Wrap renderValue to handle initialization, which unfortunately isn't - // supported natively by Shiny at the time of this writing. - - shinyBinding.renderValue = function(el, data) { - Shiny.renderDependencies(data.deps); - // Resolve strings marked as javascript literals to objects - if (!(data.evals instanceof Array)) data.evals = [data.evals]; - for (var i = 0; data.evals && i < data.evals.length; i++) { - window.HTMLWidgets.evaluateStringMember(data.x, data.evals[i]); - } - if (!bindingDef.renderOnNullValue) { - if (data.x === null) { - el.style.visibility = "hidden"; - return; - } else { - el.style.visibility = "inherit"; - } - } - if (!elementData(el, "initialized")) { - initSizing(el); - - elementData(el, "initialized", true); - if (bindingDef.initialize) { - var rect = el.getBoundingClientRect(); - var result = bindingDef.initialize(el, rect.width, rect.height); - elementData(el, "init_result", result); - } - } - bindingDef.renderValue(el, data.x, elementData(el, "init_result")); - evalAndRun(data.jsHooks.render, elementData(el, "init_result"), [el, data.x]); - }; - - // Only override resize if bindingDef implements it - if (bindingDef.resize) { - shinyBinding.resize = function(el, width, height) { - // Shiny can call resize before initialize/renderValue have been - // called, which doesn't make sense for widgets. - if (elementData(el, "initialized")) { - bindingDef.resize(el, width, height, elementData(el, "init_result")); - } - }; - } - - Shiny.outputBindings.register(shinyBinding, bindingDef.name); - } - }; - - var scheduleStaticRenderTimerId = null; - function scheduleStaticRender() { - if (!scheduleStaticRenderTimerId) { - scheduleStaticRenderTimerId = setTimeout(function() { - scheduleStaticRenderTimerId = null; - window.HTMLWidgets.staticRender(); - }, 1); - } - } - - // Render static widgets after the document finishes loading - // Statically render all elements that are of this widget's class - window.HTMLWidgets.staticRender = function() { - var bindings = window.HTMLWidgets.widgets || []; - forEach(bindings, function(binding) { - var matches = binding.find(document.documentElement); - forEach(matches, function(el) { - var sizeObj = initSizing(el, binding); - - var getSize = function(el) { - if (sizeObj) { - return {w: sizeObj.getWidth(), h: sizeObj.getHeight()} - } else { - var rect = el.getBoundingClientRect(); - return {w: rect.width, h: rect.height} - } - }; - - if (hasClass(el, "html-widget-static-bound")) - return; - el.className = el.className + " html-widget-static-bound"; - - var initResult; - if (binding.initialize) { - var size = getSize(el); - initResult = binding.initialize(el, size.w, size.h); - elementData(el, "init_result", initResult); - } - - if (binding.resize) { - var lastSize = getSize(el); - var resizeHandler = function(e) { - var size = getSize(el); - if (size.w === 0 && size.h === 0) - return; - if (size.w === lastSize.w && size.h === lastSize.h) - return; - lastSize = size; - binding.resize(el, size.w, size.h, initResult); - }; - - on(window, "resize", resizeHandler); - - // This is needed for cases where we're running in a Shiny - // app, but the widget itself is not a Shiny output, but - // rather a simple static widget. One example of this is - // an rmarkdown document that has runtime:shiny and widget - // that isn't in a render function. Shiny only knows to - // call resize handlers for Shiny outputs, not for static - // widgets, so we do it ourselves. - if (window.jQuery) { - window.jQuery(document).on( - "shown.htmlwidgets shown.bs.tab.htmlwidgets shown.bs.collapse.htmlwidgets", - resizeHandler - ); - window.jQuery(document).on( - "hidden.htmlwidgets hidden.bs.tab.htmlwidgets hidden.bs.collapse.htmlwidgets", - resizeHandler - ); - } - - // This is needed for the specific case of ioslides, which - // flips slides between display:none and display:block. - // Ideally we would not have to have ioslide-specific code - // here, but rather have ioslides raise a generic event, - // but the rmarkdown package just went to CRAN so the - // window to getting that fixed may be long. - if (window.addEventListener) { - // It's OK to limit this to window.addEventListener - // browsers because ioslides itself only supports - // such browsers. - on(document, "slideenter", resizeHandler); - on(document, "slideleave", resizeHandler); - } - } - - var scriptData = document.querySelector("script[data-for='" + el.id + "'][type='application/json']"); - if (scriptData) { - var data = JSON.parse(scriptData.textContent || scriptData.text); - // Resolve strings marked as javascript literals to objects - if (!(data.evals instanceof Array)) data.evals = [data.evals]; - for (var k = 0; data.evals && k < data.evals.length; k++) { - window.HTMLWidgets.evaluateStringMember(data.x, data.evals[k]); - } - binding.renderValue(el, data.x, initResult); - evalAndRun(data.jsHooks.render, initResult, [el, data.x]); - } - }); - }); - - invokePostRenderHandlers(); - } - - - function has_jQuery3() { - if (!window.jQuery) { - return false; - } - var $version = window.jQuery.fn.jquery; - var $major_version = parseInt($version.split(".")[0]); - return $major_version >= 3; - } - - /* - / Shiny 1.4 bumped jQuery from 1.x to 3.x which means jQuery's - / on-ready handler (i.e., $(fn)) is now asyncronous (i.e., it now - / really means $(setTimeout(fn)). - / https://jquery.com/upgrade-guide/3.0/#breaking-change-document-ready-handlers-are-now-asynchronous - / - / Since Shiny uses $() to schedule initShiny, shiny>=1.4 calls initShiny - / one tick later than it did before, which means staticRender() is - / called renderValue() earlier than (advanced) widget authors might be expecting. - / https://github.com/rstudio/shiny/issues/2630 - / - / For a concrete example, leaflet has some methods (e.g., updateBounds) - / which reference Shiny methods registered in initShiny (e.g., setInputValue). - / Since leaflet is privy to this life-cycle, it knows to use setTimeout() to - / delay execution of those methods (until Shiny methods are ready) - / https://github.com/rstudio/leaflet/blob/18ec981/javascript/src/index.js#L266-L268 - / - / Ideally widget authors wouldn't need to use this setTimeout() hack that - / leaflet uses to call Shiny methods on a staticRender(). In the long run, - / the logic initShiny should be broken up so that method registration happens - / right away, but binding happens later. - */ - function maybeStaticRenderLater() { - if (shinyMode && has_jQuery3()) { - window.jQuery(window.HTMLWidgets.staticRender); - } else { - window.HTMLWidgets.staticRender(); - } - } - - if (document.addEventListener) { - document.addEventListener("DOMContentLoaded", function() { - document.removeEventListener("DOMContentLoaded", arguments.callee, false); - maybeStaticRenderLater(); - }, false); - } else if (document.attachEvent) { - document.attachEvent("onreadystatechange", function() { - if (document.readyState === "complete") { - document.detachEvent("onreadystatechange", arguments.callee); - maybeStaticRenderLater(); - } - }); - } - - - window.HTMLWidgets.getAttachmentUrl = function(depname, key) { - // If no key, default to the first item - if (typeof(key) === "undefined") - key = 1; - - var link = document.getElementById(depname + "-" + key + "-attachment"); - if (!link) { - throw new Error("Attachment " + depname + "/" + key + " not found in document"); - } - return link.getAttribute("href"); - }; - - window.HTMLWidgets.dataframeToD3 = function(df) { - var names = []; - var length; - for (var name in df) { - if (df.hasOwnProperty(name)) - names.push(name); - if (typeof(df[name]) !== "object" || typeof(df[name].length) === "undefined") { - throw new Error("All fields must be arrays"); - } else if (typeof(length) !== "undefined" && length !== df[name].length) { - throw new Error("All fields must be arrays of the same length"); - } - length = df[name].length; - } - var results = []; - var item; - for (var row = 0; row < length; row++) { - item = {}; - for (var col = 0; col < names.length; col++) { - item[names[col]] = df[names[col]][row]; - } - results.push(item); - } - return results; - }; - - window.HTMLWidgets.transposeArray2D = function(array) { - if (array.length === 0) return array; - var newArray = array[0].map(function(col, i) { - return array.map(function(row) { - return row[i] - }) - }); - return newArray; - }; - // Split value at splitChar, but allow splitChar to be escaped - // using escapeChar. Any other characters escaped by escapeChar - // will be included as usual (including escapeChar itself). - function splitWithEscape(value, splitChar, escapeChar) { - var results = []; - var escapeMode = false; - var currentResult = ""; - for (var pos = 0; pos < value.length; pos++) { - if (!escapeMode) { - if (value[pos] === splitChar) { - results.push(currentResult); - currentResult = ""; - } else if (value[pos] === escapeChar) { - escapeMode = true; - } else { - currentResult += value[pos]; - } - } else { - currentResult += value[pos]; - escapeMode = false; - } - } - if (currentResult !== "") { - results.push(currentResult); - } - return results; - } - // Function authored by Yihui/JJ Allaire - window.HTMLWidgets.evaluateStringMember = function(o, member) { - var parts = splitWithEscape(member, '.', '\\'); - for (var i = 0, l = parts.length; i < l; i++) { - var part = parts[i]; - // part may be a character or 'numeric' member name - if (o !== null && typeof o === "object" && part in o) { - if (i == (l - 1)) { // if we are at the end of the line then evalulate - if (typeof o[part] === "string") - o[part] = tryEval(o[part]); - } else { // otherwise continue to next embedded object - o = o[part]; - } - } - } - }; - - // Retrieve the HTMLWidget instance (i.e. the return value of an - // HTMLWidget binding's initialize() or factory() function) - // associated with an element, or null if none. - window.HTMLWidgets.getInstance = function(el) { - return elementData(el, "init_result"); - }; - - // Finds the first element in the scope that matches the selector, - // and returns the HTMLWidget instance (i.e. the return value of - // an HTMLWidget binding's initialize() or factory() function) - // associated with that element, if any. If no element matches the - // selector, or the first matching element has no HTMLWidget - // instance associated with it, then null is returned. - // - // The scope argument is optional, and defaults to window.document. - window.HTMLWidgets.find = function(scope, selector) { - if (arguments.length == 1) { - selector = scope; - scope = document; - } - - var el = scope.querySelector(selector); - if (el === null) { - return null; - } else { - return window.HTMLWidgets.getInstance(el); - } - }; - - // Finds all elements in the scope that match the selector, and - // returns the HTMLWidget instances (i.e. the return values of - // an HTMLWidget binding's initialize() or factory() function) - // associated with the elements, in an array. If elements that - // match the selector don't have an associated HTMLWidget - // instance, the returned array will contain nulls. - // - // The scope argument is optional, and defaults to window.document. - window.HTMLWidgets.findAll = function(scope, selector) { - if (arguments.length == 1) { - selector = scope; - scope = document; - } - - var nodes = scope.querySelectorAll(selector); - var results = []; - for (var i = 0; i < nodes.length; i++) { - results.push(window.HTMLWidgets.getInstance(nodes[i])); - } - return results; - }; - - var postRenderHandlers = []; - function invokePostRenderHandlers() { - while (postRenderHandlers.length) { - var handler = postRenderHandlers.shift(); - if (handler) { - handler(); - } - } - } - - // Register the given callback function to be invoked after the - // next time static widgets are rendered. - window.HTMLWidgets.addPostRenderHandler = function(callback) { - postRenderHandlers.push(callback); - }; - - // Takes a new-style instance-bound definition, and returns an - // old-style class-bound definition. This saves us from having - // to rewrite all the logic in this file to accomodate both - // types of definitions. - function createLegacyDefinitionAdapter(defn) { - var result = { - name: defn.name, - type: defn.type, - initialize: function(el, width, height) { - return defn.factory(el, width, height); - }, - renderValue: function(el, x, instance) { - return instance.renderValue(x); - }, - resize: function(el, width, height, instance) { - return instance.resize(width, height); - } - }; - - if (defn.find) - result.find = defn.find; - if (defn.renderError) - result.renderError = defn.renderError; - if (defn.clearError) - result.clearError = defn.clearError; - - return result; - } -})(); diff --git a/site_libs/plotly-binding-4.10.1/plotly.js b/site_libs/plotly-binding-4.10.1/plotly.js deleted file mode 100644 index 1155269d3..000000000 --- a/site_libs/plotly-binding-4.10.1/plotly.js +++ /dev/null @@ -1,939 +0,0 @@ - -HTMLWidgets.widget({ - name: "plotly", - type: "output", - - initialize: function(el, width, height) { - return {}; - }, - - resize: function(el, width, height, instance) { - if (instance.autosize) { - var width = instance.width || width; - var height = instance.height || height; - Plotly.relayout(el.id, {width: width, height: height}); - } - }, - - renderValue: function(el, x, instance) { - - // Plotly.relayout() mutates the plot input object, so make sure to - // keep a reference to the user-supplied width/height *before* - // we call Plotly.plot(); - var lay = x.layout || {}; - instance.width = lay.width; - instance.height = lay.height; - instance.autosize = lay.autosize || true; - - /* - / 'inform the world' about highlighting options this is so other - / crosstalk libraries have a chance to respond to special settings - / such as persistent selection. - / AFAIK, leaflet is the only library with such intergration - / https://github.com/rstudio/leaflet/pull/346/files#diff-ad0c2d51ce5fdf8c90c7395b102f4265R154 - */ - var ctConfig = crosstalk.var('plotlyCrosstalkOpts').set(x.highlight); - - if (typeof(window) !== "undefined") { - // make sure plots don't get created outside the network (for on-prem) - window.PLOTLYENV = window.PLOTLYENV || {}; - window.PLOTLYENV.BASE_URL = x.base_url; - - // Enable persistent selection when shift key is down - // https://stackoverflow.com/questions/1828613/check-if-a-key-is-down - var persistOnShift = function(e) { - if (!e) window.event; - if (e.shiftKey) { - x.highlight.persistent = true; - x.highlight.persistentShift = true; - } else { - x.highlight.persistent = false; - x.highlight.persistentShift = false; - } - }; - - // Only relevant if we haven't forced persistent mode at command line - if (!x.highlight.persistent) { - window.onmousemove = persistOnShift; - } - } - - var graphDiv = document.getElementById(el.id); - - // TODO: move the control panel injection strategy inside here... - HTMLWidgets.addPostRenderHandler(function() { - - // lower the z-index of the modebar to prevent it from highjacking hover - // (TODO: do this via CSS?) - // https://github.com/ropensci/plotly/issues/956 - // https://www.w3schools.com/jsref/prop_style_zindex.asp - var modebars = document.querySelectorAll(".js-plotly-plot .plotly .modebar"); - for (var i = 0; i < modebars.length; i++) { - modebars[i].style.zIndex = 1; - } - }); - - // inject a "control panel" holding selectize/dynamic color widget(s) - if ((x.selectize || x.highlight.dynamic) && !instance.plotly) { - var flex = document.createElement("div"); - flex.class = "plotly-crosstalk-control-panel"; - flex.style = "display: flex; flex-wrap: wrap"; - - // inject the colourpicker HTML container into the flexbox - if (x.highlight.dynamic) { - var pickerDiv = document.createElement("div"); - - var pickerInput = document.createElement("input"); - pickerInput.id = el.id + "-colourpicker"; - pickerInput.placeholder = "asdasd"; - - var pickerLabel = document.createElement("label"); - pickerLabel.for = pickerInput.id; - pickerLabel.innerHTML = "Brush color  "; - - pickerDiv.appendChild(pickerLabel); - pickerDiv.appendChild(pickerInput); - flex.appendChild(pickerDiv); - } - - // inject selectize HTML containers (one for every crosstalk group) - if (x.selectize) { - var ids = Object.keys(x.selectize); - - for (var i = 0; i < ids.length; i++) { - var container = document.createElement("div"); - container.id = ids[i]; - container.style = "width: 80%; height: 10%"; - container.class = "form-group crosstalk-input-plotly-highlight"; - - var label = document.createElement("label"); - label.for = ids[i]; - label.innerHTML = x.selectize[ids[i]].group; - label.class = "control-label"; - - var selectDiv = document.createElement("div"); - var select = document.createElement("select"); - select.multiple = true; - - selectDiv.appendChild(select); - container.appendChild(label); - container.appendChild(selectDiv); - flex.appendChild(container); - } - } - - // finally, insert the flexbox inside the htmlwidget container, - // but before the plotly graph div - graphDiv.parentElement.insertBefore(flex, graphDiv); - - if (x.highlight.dynamic) { - var picker = $("#" + pickerInput.id); - var colors = x.highlight.color || []; - // TODO: let users specify options? - var opts = { - value: colors[0], - showColour: "both", - palette: "limited", - allowedCols: colors.join(" "), - width: "20%", - height: "10%" - }; - picker.colourpicker({changeDelay: 0}); - picker.colourpicker("settings", opts); - picker.colourpicker("value", opts.value); - // inform crosstalk about a change in the current selection colour - var grps = x.highlight.ctGroups || []; - for (var i = 0; i < grps.length; i++) { - crosstalk.group(grps[i]).var('plotlySelectionColour') - .set(picker.colourpicker('value')); - } - picker.on("change", function() { - for (var i = 0; i < grps.length; i++) { - crosstalk.group(grps[i]).var('plotlySelectionColour') - .set(picker.colourpicker('value')); - } - }); - } - } - - // if no plot exists yet, create one with a particular configuration - if (!instance.plotly) { - - var plot = Plotly.newPlot(graphDiv, x); - instance.plotly = true; - - } else if (x.layout.transition) { - - var plot = Plotly.react(graphDiv, x); - - } else { - - // this is essentially equivalent to Plotly.newPlot(), but avoids creating - // a new webgl context - // https://github.com/plotly/plotly.js/blob/2b24f9def901831e61282076cf3f835598d56f0e/src/plot_api/plot_api.js#L531-L532 - - // TODO: restore crosstalk selections? - Plotly.purge(graphDiv); - // TODO: why is this necessary to get crosstalk working? - graphDiv.data = undefined; - graphDiv.layout = undefined; - var plot = Plotly.newPlot(graphDiv, x); - } - - // Trigger plotly.js calls defined via `plotlyProxy()` - plot.then(function() { - if (HTMLWidgets.shinyMode) { - Shiny.addCustomMessageHandler("plotly-calls", function(msg) { - var gd = document.getElementById(msg.id); - if (!gd) { - throw new Error("Couldn't find plotly graph with id: " + msg.id); - } - // This isn't an official plotly.js method, but it's the only current way to - // change just the configuration of a plot - // https://community.plot.ly/t/update-config-function/9057 - if (msg.method == "reconfig") { - Plotly.react(gd, gd.data, gd.layout, msg.args); - return; - } - if (!Plotly[msg.method]) { - throw new Error("Unknown method " + msg.method); - } - var args = [gd].concat(msg.args); - Plotly[msg.method].apply(null, args); - }); - } - - // plotly's mapbox API doesn't currently support setting bounding boxes - // https://www.mapbox.com/mapbox-gl-js/example/fitbounds/ - // so we do this manually... - // TODO: make sure this triggers on a redraw and relayout as well as on initial draw - var mapboxIDs = graphDiv._fullLayout._subplots.mapbox || []; - for (var i = 0; i < mapboxIDs.length; i++) { - var id = mapboxIDs[i]; - var mapOpts = x.layout[id] || {}; - var args = mapOpts._fitBounds || {}; - if (!args) { - continue; - } - var mapObj = graphDiv._fullLayout[id]._subplot.map; - mapObj.fitBounds(args.bounds, args.options); - } - - }); - - // Attach attributes (e.g., "key", "z") to plotly event data - function eventDataWithKey(eventData) { - if (eventData === undefined || !eventData.hasOwnProperty("points")) { - return null; - } - return eventData.points.map(function(pt) { - var obj = { - curveNumber: pt.curveNumber, - pointNumber: pt.pointNumber, - x: pt.x, - y: pt.y - }; - - // If 'z' is reported with the event data, then use it! - if (pt.hasOwnProperty("z")) { - obj.z = pt.z; - } - - if (pt.hasOwnProperty("customdata")) { - obj.customdata = pt.customdata; - } - - /* - TL;DR: (I think) we have to select the graph div (again) to attach keys... - - Why? Remember that crosstalk will dynamically add/delete traces - (see traceManager.prototype.updateSelection() below) - For this reason, we can't simply grab keys from x.data (like we did previously) - Moreover, we can't use _fullData, since that doesn't include - unofficial attributes. It's true that click/hover events fire with - pt.data, but drag events don't... - */ - var gd = document.getElementById(el.id); - var trace = gd.data[pt.curveNumber]; - - if (!trace._isSimpleKey) { - var attrsToAttach = ["key"]; - } else { - // simple keys fire the whole key - obj.key = trace.key; - var attrsToAttach = []; - } - - for (var i = 0; i < attrsToAttach.length; i++) { - var attr = trace[attrsToAttach[i]]; - if (Array.isArray(attr)) { - if (typeof pt.pointNumber === "number") { - obj[attrsToAttach[i]] = attr[pt.pointNumber]; - } else if (Array.isArray(pt.pointNumber)) { - obj[attrsToAttach[i]] = attr[pt.pointNumber[0]][pt.pointNumber[1]]; - } else if (Array.isArray(pt.pointNumbers)) { - obj[attrsToAttach[i]] = pt.pointNumbers.map(function(idx) { return attr[idx]; }); - } - } - } - return obj; - }); - } - - - var legendEventData = function(d) { - // if legendgroup is not relevant just return the trace - var trace = d.data[d.curveNumber]; - if (!trace.legendgroup) return trace; - - // if legendgroup was specified, return all traces that match the group - var legendgrps = d.data.map(function(trace){ return trace.legendgroup; }); - var traces = []; - for (i = 0; i < legendgrps.length; i++) { - if (legendgrps[i] == trace.legendgroup) { - traces.push(d.data[i]); - } - } - - return traces; - }; - - - // send user input event data to shiny - if (HTMLWidgets.shinyMode && Shiny.setInputValue) { - - // Some events clear other input values - // TODO: always register these? - var eventClearMap = { - plotly_deselect: ["plotly_selected", "plotly_selecting", "plotly_brushed", "plotly_brushing", "plotly_click"], - plotly_unhover: ["plotly_hover"], - plotly_doubleclick: ["plotly_click"] - }; - - Object.keys(eventClearMap).map(function(evt) { - graphDiv.on(evt, function() { - var inputsToClear = eventClearMap[evt]; - inputsToClear.map(function(input) { - Shiny.setInputValue(input + "-" + x.source, null, {priority: "event"}); - }); - }); - }); - - var eventDataFunctionMap = { - plotly_click: eventDataWithKey, - plotly_sunburstclick: eventDataWithKey, - plotly_hover: eventDataWithKey, - plotly_unhover: eventDataWithKey, - // If 'plotly_selected' has already been fired, and you click - // on the plot afterwards, this event fires `undefined`?!? - // That might be considered a plotly.js bug, but it doesn't make - // sense for this input change to occur if `d` is falsy because, - // even in the empty selection case, `d` is truthy (an object), - // and the 'plotly_deselect' event will reset this input - plotly_selected: function(d) { if (d) { return eventDataWithKey(d); } }, - plotly_selecting: function(d) { if (d) { return eventDataWithKey(d); } }, - plotly_brushed: function(d) { - if (d) { return d.range ? d.range : d.lassoPoints; } - }, - plotly_brushing: function(d) { - if (d) { return d.range ? d.range : d.lassoPoints; } - }, - plotly_legendclick: legendEventData, - plotly_legenddoubleclick: legendEventData, - plotly_clickannotation: function(d) { return d.fullAnnotation } - }; - - var registerShinyValue = function(event) { - var eventDataPreProcessor = eventDataFunctionMap[event] || function(d) { return d ? d : el.id }; - // some events are unique to the R package - var plotlyJSevent = (event == "plotly_brushed") ? "plotly_selected" : (event == "plotly_brushing") ? "plotly_selecting" : event; - // register the event - graphDiv.on(plotlyJSevent, function(d) { - Shiny.setInputValue( - event + "-" + x.source, - JSON.stringify(eventDataPreProcessor(d)), - {priority: "event"} - ); - }); - } - - var shinyEvents = x.shinyEvents || []; - shinyEvents.map(registerShinyValue); - } - - // Given an array of {curveNumber: x, pointNumber: y} objects, - // return a hash of { - // set1: {value: [key1, key2, ...], _isSimpleKey: false}, - // set2: {value: [key3, key4, ...], _isSimpleKey: false} - // } - function pointsToKeys(points) { - var keysBySet = {}; - for (var i = 0; i < points.length; i++) { - - var trace = graphDiv.data[points[i].curveNumber]; - if (!trace.key || !trace.set) { - continue; - } - - // set defaults for this keySet - // note that we don't track the nested property (yet) since we always - // emit the union -- http://cpsievert.github.io/talks/20161212b/#21 - keysBySet[trace.set] = keysBySet[trace.set] || { - value: [], - _isSimpleKey: trace._isSimpleKey - }; - - // Use pointNumber by default, but aggregated traces should emit pointNumbers - var ptNum = points[i].pointNumber; - var hasPtNum = typeof ptNum === "number"; - var ptNum = hasPtNum ? ptNum : points[i].pointNumbers; - - // selecting a point of a "simple" trace means: select the - // entire key attached to this trace, which is useful for, - // say clicking on a fitted line to select corresponding observations - var key = trace._isSimpleKey ? trace.key : Array.isArray(ptNum) ? ptNum.map(function(idx) { return trace.key[idx]; }) : trace.key[ptNum]; - // http://stackoverflow.com/questions/10865025/merge-flatten-an-array-of-arrays-in-javascript - var keyFlat = trace._isNestedKey ? [].concat.apply([], key) : key; - - // TODO: better to only add new values? - keysBySet[trace.set].value = keysBySet[trace.set].value.concat(keyFlat); - } - - return keysBySet; - } - - - x.highlight.color = x.highlight.color || []; - // make sure highlight color is an array - if (!Array.isArray(x.highlight.color)) { - x.highlight.color = [x.highlight.color]; - } - - var traceManager = new TraceManager(graphDiv, x.highlight); - - // Gather all *unique* sets. - var allSets = []; - for (var curveIdx = 0; curveIdx < x.data.length; curveIdx++) { - var newSet = x.data[curveIdx].set; - if (newSet) { - if (allSets.indexOf(newSet) === -1) { - allSets.push(newSet); - } - } - } - - // register event listeners for all sets - for (var i = 0; i < allSets.length; i++) { - - var set = allSets[i]; - var selection = new crosstalk.SelectionHandle(set); - var filter = new crosstalk.FilterHandle(set); - - var filterChange = function(e) { - removeBrush(el); - traceManager.updateFilter(set, e.value); - }; - filter.on("change", filterChange); - - - var selectionChange = function(e) { - - // Workaround for 'plotly_selected' now firing previously selected - // points (in addition to new ones) when holding shift key. In our case, - // we just want the new keys - if (x.highlight.on === "plotly_selected" && x.highlight.persistentShift) { - // https://stackoverflow.com/questions/1187518/how-to-get-the-difference-between-two-arrays-in-javascript - Array.prototype.diff = function(a) { - return this.filter(function(i) {return a.indexOf(i) < 0;}); - }; - e.value = e.value.diff(e.oldValue); - } - - // array of "event objects" tracking the selection history - // this is used to avoid adding redundant selections - var selectionHistory = crosstalk.var("plotlySelectionHistory").get() || []; - - // Construct an event object "defining" the current event. - var event = { - receiverID: traceManager.gd.id, - plotlySelectionColour: crosstalk.group(set).var("plotlySelectionColour").get() - }; - event[set] = e.value; - // TODO: is there a smarter way to check object equality? - if (selectionHistory.length > 0) { - var ev = JSON.stringify(event); - for (var i = 0; i < selectionHistory.length; i++) { - var sel = JSON.stringify(selectionHistory[i]); - if (sel == ev) { - return; - } - } - } - - // accumulate history for persistent selection - if (!x.highlight.persistent) { - selectionHistory = [event]; - } else { - selectionHistory.push(event); - } - crosstalk.var("plotlySelectionHistory").set(selectionHistory); - - // do the actual updating of traces, frames, and the selectize widget - traceManager.updateSelection(set, e.value); - // https://github.com/selectize/selectize.js/blob/master/docs/api.md#methods_items - if (x.selectize) { - if (!x.highlight.persistent || e.value === null) { - selectize.clear(true); - } - selectize.addItems(e.value, true); - selectize.close(); - } - } - selection.on("change", selectionChange); - - // Set a crosstalk variable selection value, triggering an update - var turnOn = function(e) { - if (e) { - var selectedKeys = pointsToKeys(e.points); - // Keys are group names, values are array of selected keys from group. - for (var set in selectedKeys) { - if (selectedKeys.hasOwnProperty(set)) { - selection.set(selectedKeys[set].value, {sender: el}); - } - } - } - }; - if (x.highlight.debounce > 0) { - turnOn = debounce(turnOn, x.highlight.debounce); - } - graphDiv.on(x.highlight.on, turnOn); - - graphDiv.on(x.highlight.off, function turnOff(e) { - // remove any visual clues - removeBrush(el); - // remove any selection history - crosstalk.var("plotlySelectionHistory").set(null); - // trigger the actual removal of selection traces - selection.set(null, {sender: el}); - }); - - // register a callback for selectize so that there is bi-directional - // communication between the widget and direct manipulation events - if (x.selectize) { - var selectizeID = Object.keys(x.selectize)[i]; - var items = x.selectize[selectizeID].items; - var first = [{value: "", label: "(All)"}]; - var opts = { - options: first.concat(items), - searchField: "label", - valueField: "value", - labelField: "label", - maxItems: 50 - }; - var select = $("#" + selectizeID).find("select")[0]; - var selectize = $(select).selectize(opts)[0].selectize; - // NOTE: this callback is triggered when *directly* altering - // dropdown items - selectize.on("change", function() { - var currentItems = traceManager.groupSelections[set] || []; - if (!x.highlight.persistent) { - removeBrush(el); - for (var i = 0; i < currentItems.length; i++) { - selectize.removeItem(currentItems[i], true); - } - } - var newItems = selectize.items.filter(function(idx) { - return currentItems.indexOf(idx) < 0; - }); - if (newItems.length > 0) { - traceManager.updateSelection(set, newItems); - } else { - // Item has been removed... - // TODO: this logic won't work for dynamically changing palette - traceManager.updateSelection(set, null); - traceManager.updateSelection(set, selectize.items); - } - }); - } - } // end of selectionChange - - } // end of renderValue -}); // end of widget definition - -/** - * @param graphDiv The Plotly graph div - * @param highlight An object with options for updating selection(s) - */ -function TraceManager(graphDiv, highlight) { - // The Plotly graph div - this.gd = graphDiv; - - // Preserve the original data. - // TODO: try using Lib.extendFlat() as done in - // https://github.com/plotly/plotly.js/pull/1136 - this.origData = JSON.parse(JSON.stringify(graphDiv.data)); - - // avoid doing this over and over - this.origOpacity = []; - for (var i = 0; i < this.origData.length; i++) { - this.origOpacity[i] = this.origData[i].opacity === 0 ? 0 : (this.origData[i].opacity || 1); - } - - // key: group name, value: null or array of keys representing the - // most recently received selection for that group. - this.groupSelections = {}; - - // selection parameters (e.g., transient versus persistent selection) - this.highlight = highlight; -} - -TraceManager.prototype.close = function() { - // TODO: Unhook all event handlers -}; - -TraceManager.prototype.updateFilter = function(group, keys) { - - if (typeof(keys) === "undefined" || keys === null) { - - this.gd.data = JSON.parse(JSON.stringify(this.origData)); - - } else { - - var traces = []; - for (var i = 0; i < this.origData.length; i++) { - var trace = this.origData[i]; - if (!trace.key || trace.set !== group) { - continue; - } - var matchFunc = getMatchFunc(trace); - var matches = matchFunc(trace.key, keys); - - if (matches.length > 0) { - if (!trace._isSimpleKey) { - // subsetArrayAttrs doesn't mutate trace (it makes a modified clone) - trace = subsetArrayAttrs(trace, matches); - } - traces.push(trace); - } - } - this.gd.data = traces; - } - - Plotly.redraw(this.gd); - - // NOTE: we purposely do _not_ restore selection(s), since on filter, - // axis likely will update, changing the pixel -> data mapping, leading - // to a likely mismatch in the brush outline and highlighted marks - -}; - -TraceManager.prototype.updateSelection = function(group, keys) { - - if (keys !== null && !Array.isArray(keys)) { - throw new Error("Invalid keys argument; null or array expected"); - } - - // if selection has been cleared, or if this is transient - // selection, delete the "selection traces" - var nNewTraces = this.gd.data.length - this.origData.length; - if (keys === null || !this.highlight.persistent && nNewTraces > 0) { - var tracesToRemove = []; - for (var i = 0; i < this.gd.data.length; i++) { - if (this.gd.data[i]._isCrosstalkTrace) tracesToRemove.push(i); - } - Plotly.deleteTraces(this.gd, tracesToRemove); - this.groupSelections[group] = keys; - } else { - // add to the groupSelection, rather than overwriting it - // TODO: can this be removed? - this.groupSelections[group] = this.groupSelections[group] || []; - for (var i = 0; i < keys.length; i++) { - var k = keys[i]; - if (this.groupSelections[group].indexOf(k) < 0) { - this.groupSelections[group].push(k); - } - } - } - - if (keys === null) { - - Plotly.restyle(this.gd, {"opacity": this.origOpacity}); - - } else if (keys.length >= 1) { - - // placeholder for new "selection traces" - var traces = []; - // this variable is set in R/highlight.R - var selectionColour = crosstalk.group(group).var("plotlySelectionColour").get() || - this.highlight.color[0]; - - for (var i = 0; i < this.origData.length; i++) { - // TODO: try using Lib.extendFlat() as done in - // https://github.com/plotly/plotly.js/pull/1136 - var trace = JSON.parse(JSON.stringify(this.gd.data[i])); - if (!trace.key || trace.set !== group) { - continue; - } - // Get sorted array of matching indices in trace.key - var matchFunc = getMatchFunc(trace); - var matches = matchFunc(trace.key, keys); - - if (matches.length > 0) { - // If this is a "simple" key, that means select the entire trace - if (!trace._isSimpleKey) { - trace = subsetArrayAttrs(trace, matches); - } - // reach into the full trace object so we can properly reflect the - // selection attributes in every view - var d = this.gd._fullData[i]; - - /* - / Recursively inherit selection attributes from various sources, - / in order of preference: - / (1) official plotly.js selected attribute - / (2) highlight(selected = attrs_selected(...)) - */ - // TODO: it would be neat to have a dropdown to dynamically specify these! - $.extend(true, trace, this.highlight.selected); - - // if it is defined, override color with the "dynamic brush color"" - if (d.marker) { - trace.marker = trace.marker || {}; - trace.marker.color = selectionColour || trace.marker.color || d.marker.color; - } - if (d.line) { - trace.line = trace.line || {}; - trace.line.color = selectionColour || trace.line.color || d.line.color; - } - if (d.textfont) { - trace.textfont = trace.textfont || {}; - trace.textfont.color = selectionColour || trace.textfont.color || d.textfont.color; - } - if (d.fillcolor) { - // TODO: should selectionColour inherit alpha from the existing fillcolor? - trace.fillcolor = selectionColour || trace.fillcolor || d.fillcolor; - } - // attach a sensible name/legendgroup - trace.name = trace.name || keys.join("
    "); - trace.legendgroup = trace.legendgroup || keys.join("
    "); - - // keep track of mapping between this new trace and the trace it targets - // (necessary for updating frames to reflect the selection traces) - trace._originalIndex = i; - trace._newIndex = this.gd._fullData.length + traces.length; - trace._isCrosstalkTrace = true; - traces.push(trace); - } - } - - if (traces.length > 0) { - - Plotly.addTraces(this.gd, traces).then(function(gd) { - // incrementally add selection traces to frames - // (this is heavily inspired by Plotly.Plots.modifyFrames() - // in src/plots/plots.js) - var _hash = gd._transitionData._frameHash; - var _frames = gd._transitionData._frames || []; - - for (var i = 0; i < _frames.length; i++) { - - // add to _frames[i].traces *if* this frame references selected trace(s) - var newIndices = []; - for (var j = 0; j < traces.length; j++) { - var tr = traces[j]; - if (_frames[i].traces.indexOf(tr._originalIndex) > -1) { - newIndices.push(tr._newIndex); - _frames[i].traces.push(tr._newIndex); - } - } - - // nothing to do... - if (newIndices.length === 0) { - continue; - } - - var ctr = 0; - var nFrameTraces = _frames[i].data.length; - - for (var j = 0; j < nFrameTraces; j++) { - var frameTrace = _frames[i].data[j]; - if (!frameTrace.key || frameTrace.set !== group) { - continue; - } - - var matchFunc = getMatchFunc(frameTrace); - var matches = matchFunc(frameTrace.key, keys); - - if (matches.length > 0) { - if (!trace._isSimpleKey) { - frameTrace = subsetArrayAttrs(frameTrace, matches); - } - var d = gd._fullData[newIndices[ctr]]; - if (d.marker) { - frameTrace.marker = d.marker; - } - if (d.line) { - frameTrace.line = d.line; - } - if (d.textfont) { - frameTrace.textfont = d.textfont; - } - ctr = ctr + 1; - _frames[i].data.push(frameTrace); - } - } - - // update gd._transitionData._frameHash - _hash[_frames[i].name] = _frames[i]; - } - - }); - - // dim traces that have a set matching the set of selection sets - var tracesToDim = [], - opacities = [], - sets = Object.keys(this.groupSelections), - n = this.origData.length; - - for (var i = 0; i < n; i++) { - var opacity = this.origOpacity[i] || 1; - // have we already dimmed this trace? Or is this even worth doing? - if (opacity !== this.gd._fullData[i].opacity || this.highlight.opacityDim === 1) { - continue; - } - // is this set an element of the set of selection sets? - var matches = findMatches(sets, [this.gd.data[i].set]); - if (matches.length) { - tracesToDim.push(i); - opacities.push(opacity * this.highlight.opacityDim); - } - } - - if (tracesToDim.length > 0) { - Plotly.restyle(this.gd, {"opacity": opacities}, tracesToDim); - // turn off the selected/unselected API - Plotly.restyle(this.gd, {"selectedpoints": null}); - } - - } - - } -}; - -/* -Note: in all of these match functions, we assume needleSet (i.e. the selected keys) -is a 1D (or flat) array. The real difference is the meaning of haystack. -findMatches() does the usual thing you'd expect for -linked brushing on a scatterplot matrix. findSimpleMatches() returns a match iff -haystack is a subset of the needleSet. findNestedMatches() returns -*/ - -function getMatchFunc(trace) { - return (trace._isNestedKey) ? findNestedMatches : - (trace._isSimpleKey) ? findSimpleMatches : findMatches; -} - -// find matches for "flat" keys -function findMatches(haystack, needleSet) { - var matches = []; - haystack.forEach(function(obj, i) { - if (obj === null || needleSet.indexOf(obj) >= 0) { - matches.push(i); - } - }); - return matches; -} - -// find matches for "simple" keys -function findSimpleMatches(haystack, needleSet) { - var match = haystack.every(function(val) { - return val === null || needleSet.indexOf(val) >= 0; - }); - // yes, this doesn't make much sense other than conforming - // to the output type of the other match functions - return (match) ? [0] : [] -} - -// find matches for a "nested" haystack (2D arrays) -function findNestedMatches(haystack, needleSet) { - var matches = []; - for (var i = 0; i < haystack.length; i++) { - var hay = haystack[i]; - var match = hay.every(function(val) { - return val === null || needleSet.indexOf(val) >= 0; - }); - if (match) { - matches.push(i); - } - } - return matches; -} - -function isPlainObject(obj) { - return ( - Object.prototype.toString.call(obj) === '[object Object]' && - Object.getPrototypeOf(obj) === Object.prototype - ); -} - -function subsetArrayAttrs(obj, indices) { - var newObj = {}; - Object.keys(obj).forEach(function(k) { - var val = obj[k]; - - if (k.charAt(0) === "_") { - newObj[k] = val; - } else if (k === "transforms" && Array.isArray(val)) { - newObj[k] = val.map(function(transform) { - return subsetArrayAttrs(transform, indices); - }); - } else if (k === "colorscale" && Array.isArray(val)) { - newObj[k] = val; - } else if (isPlainObject(val)) { - newObj[k] = subsetArrayAttrs(val, indices); - } else if (Array.isArray(val)) { - newObj[k] = subsetArray(val, indices); - } else { - newObj[k] = val; - } - }); - return newObj; -} - -function subsetArray(arr, indices) { - var result = []; - for (var i = 0; i < indices.length; i++) { - result.push(arr[indices[i]]); - } - return result; -} - -// Convenience function for removing plotly's brush -function removeBrush(el) { - var outlines = el.querySelectorAll(".select-outline"); - for (var i = 0; i < outlines.length; i++) { - outlines[i].remove(); - } -} - - -// https://davidwalsh.name/javascript-debounce-function - -// Returns a function, that, as long as it continues to be invoked, will not -// be triggered. The function will be called after it stops being called for -// N milliseconds. If `immediate` is passed, trigger the function on the -// leading edge, instead of the trailing. -function debounce(func, wait, immediate) { - var timeout; - return function() { - var context = this, args = arguments; - var later = function() { - timeout = null; - if (!immediate) func.apply(context, args); - }; - var callNow = immediate && !timeout; - clearTimeout(timeout); - timeout = setTimeout(later, wait); - if (callNow) func.apply(context, args); - }; -}; diff --git a/site_libs/plotly-binding-4.10.2/plotly.js b/site_libs/plotly-binding-4.10.2/plotly.js deleted file mode 100644 index 7a2a143b6..000000000 --- a/site_libs/plotly-binding-4.10.2/plotly.js +++ /dev/null @@ -1,941 +0,0 @@ - -HTMLWidgets.widget({ - name: "plotly", - type: "output", - - initialize: function(el, width, height) { - return {}; - }, - - resize: function(el, width, height, instance) { - if (instance.autosize) { - var width = instance.width || width; - var height = instance.height || height; - Plotly.relayout(el.id, {width: width, height: height}); - } - }, - - renderValue: function(el, x, instance) { - - // Plotly.relayout() mutates the plot input object, so make sure to - // keep a reference to the user-supplied width/height *before* - // we call Plotly.plot(); - var lay = x.layout || {}; - instance.width = lay.width; - instance.height = lay.height; - instance.autosize = lay.autosize || true; - - /* - / 'inform the world' about highlighting options this is so other - / crosstalk libraries have a chance to respond to special settings - / such as persistent selection. - / AFAIK, leaflet is the only library with such intergration - / https://github.com/rstudio/leaflet/pull/346/files#diff-ad0c2d51ce5fdf8c90c7395b102f4265R154 - */ - var ctConfig = crosstalk.var('plotlyCrosstalkOpts').set(x.highlight); - - if (typeof(window) !== "undefined") { - // make sure plots don't get created outside the network (for on-prem) - window.PLOTLYENV = window.PLOTLYENV || {}; - window.PLOTLYENV.BASE_URL = x.base_url; - - // Enable persistent selection when shift key is down - // https://stackoverflow.com/questions/1828613/check-if-a-key-is-down - var persistOnShift = function(e) { - if (!e) window.event; - if (e.shiftKey) { - x.highlight.persistent = true; - x.highlight.persistentShift = true; - } else { - x.highlight.persistent = false; - x.highlight.persistentShift = false; - } - }; - - // Only relevant if we haven't forced persistent mode at command line - if (!x.highlight.persistent) { - window.onmousemove = persistOnShift; - } - } - - var graphDiv = document.getElementById(el.id); - - // TODO: move the control panel injection strategy inside here... - HTMLWidgets.addPostRenderHandler(function() { - - // lower the z-index of the modebar to prevent it from highjacking hover - // (TODO: do this via CSS?) - // https://github.com/ropensci/plotly/issues/956 - // https://www.w3schools.com/jsref/prop_style_zindex.asp - var modebars = document.querySelectorAll(".js-plotly-plot .plotly .modebar"); - for (var i = 0; i < modebars.length; i++) { - modebars[i].style.zIndex = 1; - } - }); - - // inject a "control panel" holding selectize/dynamic color widget(s) - if ((x.selectize || x.highlight.dynamic) && !instance.plotly) { - var flex = document.createElement("div"); - flex.class = "plotly-crosstalk-control-panel"; - flex.style = "display: flex; flex-wrap: wrap"; - - // inject the colourpicker HTML container into the flexbox - if (x.highlight.dynamic) { - var pickerDiv = document.createElement("div"); - - var pickerInput = document.createElement("input"); - pickerInput.id = el.id + "-colourpicker"; - pickerInput.placeholder = "asdasd"; - - var pickerLabel = document.createElement("label"); - pickerLabel.for = pickerInput.id; - pickerLabel.innerHTML = "Brush color  "; - - pickerDiv.appendChild(pickerLabel); - pickerDiv.appendChild(pickerInput); - flex.appendChild(pickerDiv); - } - - // inject selectize HTML containers (one for every crosstalk group) - if (x.selectize) { - var ids = Object.keys(x.selectize); - - for (var i = 0; i < ids.length; i++) { - var container = document.createElement("div"); - container.id = ids[i]; - container.style = "width: 80%; height: 10%"; - container.class = "form-group crosstalk-input-plotly-highlight"; - - var label = document.createElement("label"); - label.for = ids[i]; - label.innerHTML = x.selectize[ids[i]].group; - label.class = "control-label"; - - var selectDiv = document.createElement("div"); - var select = document.createElement("select"); - select.multiple = true; - - selectDiv.appendChild(select); - container.appendChild(label); - container.appendChild(selectDiv); - flex.appendChild(container); - } - } - - // finally, insert the flexbox inside the htmlwidget container, - // but before the plotly graph div - graphDiv.parentElement.insertBefore(flex, graphDiv); - - if (x.highlight.dynamic) { - var picker = $("#" + pickerInput.id); - var colors = x.highlight.color || []; - // TODO: let users specify options? - var opts = { - value: colors[0], - showColour: "both", - palette: "limited", - allowedCols: colors.join(" "), - width: "20%", - height: "10%" - }; - picker.colourpicker({changeDelay: 0}); - picker.colourpicker("settings", opts); - picker.colourpicker("value", opts.value); - // inform crosstalk about a change in the current selection colour - var grps = x.highlight.ctGroups || []; - for (var i = 0; i < grps.length; i++) { - crosstalk.group(grps[i]).var('plotlySelectionColour') - .set(picker.colourpicker('value')); - } - picker.on("change", function() { - for (var i = 0; i < grps.length; i++) { - crosstalk.group(grps[i]).var('plotlySelectionColour') - .set(picker.colourpicker('value')); - } - }); - } - } - - // if no plot exists yet, create one with a particular configuration - if (!instance.plotly) { - - var plot = Plotly.newPlot(graphDiv, x); - instance.plotly = true; - - } else if (x.layout.transition) { - - var plot = Plotly.react(graphDiv, x); - - } else { - - // this is essentially equivalent to Plotly.newPlot(), but avoids creating - // a new webgl context - // https://github.com/plotly/plotly.js/blob/2b24f9def901831e61282076cf3f835598d56f0e/src/plot_api/plot_api.js#L531-L532 - - // TODO: restore crosstalk selections? - Plotly.purge(graphDiv); - // TODO: why is this necessary to get crosstalk working? - graphDiv.data = undefined; - graphDiv.layout = undefined; - var plot = Plotly.newPlot(graphDiv, x); - } - - // Trigger plotly.js calls defined via `plotlyProxy()` - plot.then(function() { - if (HTMLWidgets.shinyMode) { - Shiny.addCustomMessageHandler("plotly-calls", function(msg) { - var gd = document.getElementById(msg.id); - if (!gd) { - throw new Error("Couldn't find plotly graph with id: " + msg.id); - } - // This isn't an official plotly.js method, but it's the only current way to - // change just the configuration of a plot - // https://community.plot.ly/t/update-config-function/9057 - if (msg.method == "reconfig") { - Plotly.react(gd, gd.data, gd.layout, msg.args); - return; - } - if (!Plotly[msg.method]) { - throw new Error("Unknown method " + msg.method); - } - var args = [gd].concat(msg.args); - Plotly[msg.method].apply(null, args); - }); - } - - // plotly's mapbox API doesn't currently support setting bounding boxes - // https://www.mapbox.com/mapbox-gl-js/example/fitbounds/ - // so we do this manually... - // TODO: make sure this triggers on a redraw and relayout as well as on initial draw - var mapboxIDs = graphDiv._fullLayout._subplots.mapbox || []; - for (var i = 0; i < mapboxIDs.length; i++) { - var id = mapboxIDs[i]; - var mapOpts = x.layout[id] || {}; - var args = mapOpts._fitBounds || {}; - if (!args) { - continue; - } - var mapObj = graphDiv._fullLayout[id]._subplot.map; - mapObj.fitBounds(args.bounds, args.options); - } - - }); - - // Attach attributes (e.g., "key", "z") to plotly event data - function eventDataWithKey(eventData) { - if (eventData === undefined || !eventData.hasOwnProperty("points")) { - return null; - } - return eventData.points.map(function(pt) { - var obj = { - curveNumber: pt.curveNumber, - pointNumber: pt.pointNumber, - x: pt.x, - y: pt.y - }; - - // If 'z' is reported with the event data, then use it! - if (pt.hasOwnProperty("z")) { - obj.z = pt.z; - } - - if (pt.hasOwnProperty("customdata")) { - obj.customdata = pt.customdata; - } - - /* - TL;DR: (I think) we have to select the graph div (again) to attach keys... - - Why? Remember that crosstalk will dynamically add/delete traces - (see traceManager.prototype.updateSelection() below) - For this reason, we can't simply grab keys from x.data (like we did previously) - Moreover, we can't use _fullData, since that doesn't include - unofficial attributes. It's true that click/hover events fire with - pt.data, but drag events don't... - */ - var gd = document.getElementById(el.id); - var trace = gd.data[pt.curveNumber]; - - if (!trace._isSimpleKey) { - var attrsToAttach = ["key"]; - } else { - // simple keys fire the whole key - obj.key = trace.key; - var attrsToAttach = []; - } - - for (var i = 0; i < attrsToAttach.length; i++) { - var attr = trace[attrsToAttach[i]]; - if (Array.isArray(attr)) { - if (typeof pt.pointNumber === "number") { - obj[attrsToAttach[i]] = attr[pt.pointNumber]; - } else if (Array.isArray(pt.pointNumber)) { - obj[attrsToAttach[i]] = attr[pt.pointNumber[0]][pt.pointNumber[1]]; - } else if (Array.isArray(pt.pointNumbers)) { - obj[attrsToAttach[i]] = pt.pointNumbers.map(function(idx) { return attr[idx]; }); - } - } - } - return obj; - }); - } - - - var legendEventData = function(d) { - // if legendgroup is not relevant just return the trace - var trace = d.data[d.curveNumber]; - if (!trace.legendgroup) return trace; - - // if legendgroup was specified, return all traces that match the group - var legendgrps = d.data.map(function(trace){ return trace.legendgroup; }); - var traces = []; - for (i = 0; i < legendgrps.length; i++) { - if (legendgrps[i] == trace.legendgroup) { - traces.push(d.data[i]); - } - } - - return traces; - }; - - - // send user input event data to shiny - if (HTMLWidgets.shinyMode && Shiny.setInputValue) { - - // Some events clear other input values - // TODO: always register these? - var eventClearMap = { - plotly_deselect: ["plotly_selected", "plotly_selecting", "plotly_brushed", "plotly_brushing", "plotly_click"], - plotly_unhover: ["plotly_hover"], - plotly_doubleclick: ["plotly_click"] - }; - - Object.keys(eventClearMap).map(function(evt) { - graphDiv.on(evt, function() { - var inputsToClear = eventClearMap[evt]; - inputsToClear.map(function(input) { - Shiny.setInputValue(input + "-" + x.source, null, {priority: "event"}); - }); - }); - }); - - var eventDataFunctionMap = { - plotly_click: eventDataWithKey, - plotly_sunburstclick: eventDataWithKey, - plotly_hover: eventDataWithKey, - plotly_unhover: eventDataWithKey, - // If 'plotly_selected' has already been fired, and you click - // on the plot afterwards, this event fires `undefined`?!? - // That might be considered a plotly.js bug, but it doesn't make - // sense for this input change to occur if `d` is falsy because, - // even in the empty selection case, `d` is truthy (an object), - // and the 'plotly_deselect' event will reset this input - plotly_selected: function(d) { if (d) { return eventDataWithKey(d); } }, - plotly_selecting: function(d) { if (d) { return eventDataWithKey(d); } }, - plotly_brushed: function(d) { - if (d) { return d.range ? d.range : d.lassoPoints; } - }, - plotly_brushing: function(d) { - if (d) { return d.range ? d.range : d.lassoPoints; } - }, - plotly_legendclick: legendEventData, - plotly_legenddoubleclick: legendEventData, - plotly_clickannotation: function(d) { return d.fullAnnotation } - }; - - var registerShinyValue = function(event) { - var eventDataPreProcessor = eventDataFunctionMap[event] || function(d) { return d ? d : el.id }; - // some events are unique to the R package - var plotlyJSevent = (event == "plotly_brushed") ? "plotly_selected" : (event == "plotly_brushing") ? "plotly_selecting" : event; - // register the event - graphDiv.on(plotlyJSevent, function(d) { - Shiny.setInputValue( - event + "-" + x.source, - JSON.stringify(eventDataPreProcessor(d)), - {priority: "event"} - ); - }); - } - - var shinyEvents = x.shinyEvents || []; - shinyEvents.map(registerShinyValue); - } - - // Given an array of {curveNumber: x, pointNumber: y} objects, - // return a hash of { - // set1: {value: [key1, key2, ...], _isSimpleKey: false}, - // set2: {value: [key3, key4, ...], _isSimpleKey: false} - // } - function pointsToKeys(points) { - var keysBySet = {}; - for (var i = 0; i < points.length; i++) { - - var trace = graphDiv.data[points[i].curveNumber]; - if (!trace.key || !trace.set) { - continue; - } - - // set defaults for this keySet - // note that we don't track the nested property (yet) since we always - // emit the union -- http://cpsievert.github.io/talks/20161212b/#21 - keysBySet[trace.set] = keysBySet[trace.set] || { - value: [], - _isSimpleKey: trace._isSimpleKey - }; - - // Use pointNumber by default, but aggregated traces should emit pointNumbers - var ptNum = points[i].pointNumber; - var hasPtNum = typeof ptNum === "number"; - var ptNum = hasPtNum ? ptNum : points[i].pointNumbers; - - // selecting a point of a "simple" trace means: select the - // entire key attached to this trace, which is useful for, - // say clicking on a fitted line to select corresponding observations - var key = trace._isSimpleKey ? trace.key : Array.isArray(ptNum) ? ptNum.map(function(idx) { return trace.key[idx]; }) : trace.key[ptNum]; - // http://stackoverflow.com/questions/10865025/merge-flatten-an-array-of-arrays-in-javascript - var keyFlat = trace._isNestedKey ? [].concat.apply([], key) : key; - - // TODO: better to only add new values? - keysBySet[trace.set].value = keysBySet[trace.set].value.concat(keyFlat); - } - - return keysBySet; - } - - - x.highlight.color = x.highlight.color || []; - // make sure highlight color is an array - if (!Array.isArray(x.highlight.color)) { - x.highlight.color = [x.highlight.color]; - } - - var traceManager = new TraceManager(graphDiv, x.highlight); - - // Gather all *unique* sets. - var allSets = []; - for (var curveIdx = 0; curveIdx < x.data.length; curveIdx++) { - var newSet = x.data[curveIdx].set; - if (newSet) { - if (allSets.indexOf(newSet) === -1) { - allSets.push(newSet); - } - } - } - - // register event listeners for all sets - for (var i = 0; i < allSets.length; i++) { - - var set = allSets[i]; - var selection = new crosstalk.SelectionHandle(set); - var filter = new crosstalk.FilterHandle(set); - - var filterChange = function(e) { - removeBrush(el); - traceManager.updateFilter(set, e.value); - }; - filter.on("change", filterChange); - - - var selectionChange = function(e) { - - // Workaround for 'plotly_selected' now firing previously selected - // points (in addition to new ones) when holding shift key. In our case, - // we just want the new keys - if (x.highlight.on === "plotly_selected" && x.highlight.persistentShift) { - // https://stackoverflow.com/questions/1187518/how-to-get-the-difference-between-two-arrays-in-javascript - Array.prototype.diff = function(a) { - return this.filter(function(i) {return a.indexOf(i) < 0;}); - }; - e.value = e.value.diff(e.oldValue); - } - - // array of "event objects" tracking the selection history - // this is used to avoid adding redundant selections - var selectionHistory = crosstalk.var("plotlySelectionHistory").get() || []; - - // Construct an event object "defining" the current event. - var event = { - receiverID: traceManager.gd.id, - plotlySelectionColour: crosstalk.group(set).var("plotlySelectionColour").get() - }; - event[set] = e.value; - // TODO: is there a smarter way to check object equality? - if (selectionHistory.length > 0) { - var ev = JSON.stringify(event); - for (var i = 0; i < selectionHistory.length; i++) { - var sel = JSON.stringify(selectionHistory[i]); - if (sel == ev) { - return; - } - } - } - - // accumulate history for persistent selection - if (!x.highlight.persistent) { - selectionHistory = [event]; - } else { - selectionHistory.push(event); - } - crosstalk.var("plotlySelectionHistory").set(selectionHistory); - - // do the actual updating of traces, frames, and the selectize widget - traceManager.updateSelection(set, e.value); - // https://github.com/selectize/selectize.js/blob/master/docs/api.md#methods_items - if (x.selectize) { - if (!x.highlight.persistent || e.value === null) { - selectize.clear(true); - } - selectize.addItems(e.value, true); - selectize.close(); - } - } - selection.on("change", selectionChange); - - // Set a crosstalk variable selection value, triggering an update - var turnOn = function(e) { - if (e) { - var selectedKeys = pointsToKeys(e.points); - // Keys are group names, values are array of selected keys from group. - for (var set in selectedKeys) { - if (selectedKeys.hasOwnProperty(set)) { - selection.set(selectedKeys[set].value, {sender: el}); - } - } - } - }; - if (x.highlight.debounce > 0) { - turnOn = debounce(turnOn, x.highlight.debounce); - } - graphDiv.on(x.highlight.on, turnOn); - - graphDiv.on(x.highlight.off, function turnOff(e) { - // remove any visual clues - removeBrush(el); - // remove any selection history - crosstalk.var("plotlySelectionHistory").set(null); - // trigger the actual removal of selection traces - selection.set(null, {sender: el}); - }); - - // register a callback for selectize so that there is bi-directional - // communication between the widget and direct manipulation events - if (x.selectize) { - var selectizeID = Object.keys(x.selectize)[i]; - var options = x.selectize[selectizeID]; - var first = [{value: "", label: "(All)"}]; - var opts = $.extend({ - options: first.concat(options.items), - searchField: "label", - valueField: "value", - labelField: "label", - maxItems: 50 - }, - options - ); - var select = $("#" + selectizeID).find("select")[0]; - var selectize = $(select).selectize(opts)[0].selectize; - // NOTE: this callback is triggered when *directly* altering - // dropdown items - selectize.on("change", function() { - var currentItems = traceManager.groupSelections[set] || []; - if (!x.highlight.persistent) { - removeBrush(el); - for (var i = 0; i < currentItems.length; i++) { - selectize.removeItem(currentItems[i], true); - } - } - var newItems = selectize.items.filter(function(idx) { - return currentItems.indexOf(idx) < 0; - }); - if (newItems.length > 0) { - traceManager.updateSelection(set, newItems); - } else { - // Item has been removed... - // TODO: this logic won't work for dynamically changing palette - traceManager.updateSelection(set, null); - traceManager.updateSelection(set, selectize.items); - } - }); - } - } // end of selectionChange - - } // end of renderValue -}); // end of widget definition - -/** - * @param graphDiv The Plotly graph div - * @param highlight An object with options for updating selection(s) - */ -function TraceManager(graphDiv, highlight) { - // The Plotly graph div - this.gd = graphDiv; - - // Preserve the original data. - // TODO: try using Lib.extendFlat() as done in - // https://github.com/plotly/plotly.js/pull/1136 - this.origData = JSON.parse(JSON.stringify(graphDiv.data)); - - // avoid doing this over and over - this.origOpacity = []; - for (var i = 0; i < this.origData.length; i++) { - this.origOpacity[i] = this.origData[i].opacity === 0 ? 0 : (this.origData[i].opacity || 1); - } - - // key: group name, value: null or array of keys representing the - // most recently received selection for that group. - this.groupSelections = {}; - - // selection parameters (e.g., transient versus persistent selection) - this.highlight = highlight; -} - -TraceManager.prototype.close = function() { - // TODO: Unhook all event handlers -}; - -TraceManager.prototype.updateFilter = function(group, keys) { - - if (typeof(keys) === "undefined" || keys === null) { - - this.gd.data = JSON.parse(JSON.stringify(this.origData)); - - } else { - - var traces = []; - for (var i = 0; i < this.origData.length; i++) { - var trace = this.origData[i]; - if (!trace.key || trace.set !== group) { - continue; - } - var matchFunc = getMatchFunc(trace); - var matches = matchFunc(trace.key, keys); - - if (matches.length > 0) { - if (!trace._isSimpleKey) { - // subsetArrayAttrs doesn't mutate trace (it makes a modified clone) - trace = subsetArrayAttrs(trace, matches); - } - traces.push(trace); - } - } - this.gd.data = traces; - } - - Plotly.redraw(this.gd); - - // NOTE: we purposely do _not_ restore selection(s), since on filter, - // axis likely will update, changing the pixel -> data mapping, leading - // to a likely mismatch in the brush outline and highlighted marks - -}; - -TraceManager.prototype.updateSelection = function(group, keys) { - - if (keys !== null && !Array.isArray(keys)) { - throw new Error("Invalid keys argument; null or array expected"); - } - - // if selection has been cleared, or if this is transient - // selection, delete the "selection traces" - var nNewTraces = this.gd.data.length - this.origData.length; - if (keys === null || !this.highlight.persistent && nNewTraces > 0) { - var tracesToRemove = []; - for (var i = 0; i < this.gd.data.length; i++) { - if (this.gd.data[i]._isCrosstalkTrace) tracesToRemove.push(i); - } - Plotly.deleteTraces(this.gd, tracesToRemove); - this.groupSelections[group] = keys; - } else { - // add to the groupSelection, rather than overwriting it - // TODO: can this be removed? - this.groupSelections[group] = this.groupSelections[group] || []; - for (var i = 0; i < keys.length; i++) { - var k = keys[i]; - if (this.groupSelections[group].indexOf(k) < 0) { - this.groupSelections[group].push(k); - } - } - } - - if (keys === null) { - - Plotly.restyle(this.gd, {"opacity": this.origOpacity}); - - } else if (keys.length >= 1) { - - // placeholder for new "selection traces" - var traces = []; - // this variable is set in R/highlight.R - var selectionColour = crosstalk.group(group).var("plotlySelectionColour").get() || - this.highlight.color[0]; - - for (var i = 0; i < this.origData.length; i++) { - // TODO: try using Lib.extendFlat() as done in - // https://github.com/plotly/plotly.js/pull/1136 - var trace = JSON.parse(JSON.stringify(this.gd.data[i])); - if (!trace.key || trace.set !== group) { - continue; - } - // Get sorted array of matching indices in trace.key - var matchFunc = getMatchFunc(trace); - var matches = matchFunc(trace.key, keys); - - if (matches.length > 0) { - // If this is a "simple" key, that means select the entire trace - if (!trace._isSimpleKey) { - trace = subsetArrayAttrs(trace, matches); - } - // reach into the full trace object so we can properly reflect the - // selection attributes in every view - var d = this.gd._fullData[i]; - - /* - / Recursively inherit selection attributes from various sources, - / in order of preference: - / (1) official plotly.js selected attribute - / (2) highlight(selected = attrs_selected(...)) - */ - // TODO: it would be neat to have a dropdown to dynamically specify these! - $.extend(true, trace, this.highlight.selected); - - // if it is defined, override color with the "dynamic brush color"" - if (d.marker) { - trace.marker = trace.marker || {}; - trace.marker.color = selectionColour || trace.marker.color || d.marker.color; - } - if (d.line) { - trace.line = trace.line || {}; - trace.line.color = selectionColour || trace.line.color || d.line.color; - } - if (d.textfont) { - trace.textfont = trace.textfont || {}; - trace.textfont.color = selectionColour || trace.textfont.color || d.textfont.color; - } - if (d.fillcolor) { - // TODO: should selectionColour inherit alpha from the existing fillcolor? - trace.fillcolor = selectionColour || trace.fillcolor || d.fillcolor; - } - // attach a sensible name/legendgroup - trace.name = trace.name || keys.join("
    "); - trace.legendgroup = trace.legendgroup || keys.join("
    "); - - // keep track of mapping between this new trace and the trace it targets - // (necessary for updating frames to reflect the selection traces) - trace._originalIndex = i; - trace._newIndex = this.gd._fullData.length + traces.length; - trace._isCrosstalkTrace = true; - traces.push(trace); - } - } - - if (traces.length > 0) { - - Plotly.addTraces(this.gd, traces).then(function(gd) { - // incrementally add selection traces to frames - // (this is heavily inspired by Plotly.Plots.modifyFrames() - // in src/plots/plots.js) - var _hash = gd._transitionData._frameHash; - var _frames = gd._transitionData._frames || []; - - for (var i = 0; i < _frames.length; i++) { - - // add to _frames[i].traces *if* this frame references selected trace(s) - var newIndices = []; - for (var j = 0; j < traces.length; j++) { - var tr = traces[j]; - if (_frames[i].traces.indexOf(tr._originalIndex) > -1) { - newIndices.push(tr._newIndex); - _frames[i].traces.push(tr._newIndex); - } - } - - // nothing to do... - if (newIndices.length === 0) { - continue; - } - - var ctr = 0; - var nFrameTraces = _frames[i].data.length; - - for (var j = 0; j < nFrameTraces; j++) { - var frameTrace = _frames[i].data[j]; - if (!frameTrace.key || frameTrace.set !== group) { - continue; - } - - var matchFunc = getMatchFunc(frameTrace); - var matches = matchFunc(frameTrace.key, keys); - - if (matches.length > 0) { - if (!trace._isSimpleKey) { - frameTrace = subsetArrayAttrs(frameTrace, matches); - } - var d = gd._fullData[newIndices[ctr]]; - if (d.marker) { - frameTrace.marker = d.marker; - } - if (d.line) { - frameTrace.line = d.line; - } - if (d.textfont) { - frameTrace.textfont = d.textfont; - } - ctr = ctr + 1; - _frames[i].data.push(frameTrace); - } - } - - // update gd._transitionData._frameHash - _hash[_frames[i].name] = _frames[i]; - } - - }); - - // dim traces that have a set matching the set of selection sets - var tracesToDim = [], - opacities = [], - sets = Object.keys(this.groupSelections), - n = this.origData.length; - - for (var i = 0; i < n; i++) { - var opacity = this.origOpacity[i] || 1; - // have we already dimmed this trace? Or is this even worth doing? - if (opacity !== this.gd._fullData[i].opacity || this.highlight.opacityDim === 1) { - continue; - } - // is this set an element of the set of selection sets? - var matches = findMatches(sets, [this.gd.data[i].set]); - if (matches.length) { - tracesToDim.push(i); - opacities.push(opacity * this.highlight.opacityDim); - } - } - - if (tracesToDim.length > 0) { - Plotly.restyle(this.gd, {"opacity": opacities}, tracesToDim); - // turn off the selected/unselected API - Plotly.restyle(this.gd, {"selectedpoints": null}); - } - - } - - } -}; - -/* -Note: in all of these match functions, we assume needleSet (i.e. the selected keys) -is a 1D (or flat) array. The real difference is the meaning of haystack. -findMatches() does the usual thing you'd expect for -linked brushing on a scatterplot matrix. findSimpleMatches() returns a match iff -haystack is a subset of the needleSet. findNestedMatches() returns -*/ - -function getMatchFunc(trace) { - return (trace._isNestedKey) ? findNestedMatches : - (trace._isSimpleKey) ? findSimpleMatches : findMatches; -} - -// find matches for "flat" keys -function findMatches(haystack, needleSet) { - var matches = []; - haystack.forEach(function(obj, i) { - if (obj === null || needleSet.indexOf(obj) >= 0) { - matches.push(i); - } - }); - return matches; -} - -// find matches for "simple" keys -function findSimpleMatches(haystack, needleSet) { - var match = haystack.every(function(val) { - return val === null || needleSet.indexOf(val) >= 0; - }); - // yes, this doesn't make much sense other than conforming - // to the output type of the other match functions - return (match) ? [0] : [] -} - -// find matches for a "nested" haystack (2D arrays) -function findNestedMatches(haystack, needleSet) { - var matches = []; - for (var i = 0; i < haystack.length; i++) { - var hay = haystack[i]; - var match = hay.every(function(val) { - return val === null || needleSet.indexOf(val) >= 0; - }); - if (match) { - matches.push(i); - } - } - return matches; -} - -function isPlainObject(obj) { - return ( - Object.prototype.toString.call(obj) === '[object Object]' && - Object.getPrototypeOf(obj) === Object.prototype - ); -} - -function subsetArrayAttrs(obj, indices) { - var newObj = {}; - Object.keys(obj).forEach(function(k) { - var val = obj[k]; - - if (k.charAt(0) === "_") { - newObj[k] = val; - } else if (k === "transforms" && Array.isArray(val)) { - newObj[k] = val.map(function(transform) { - return subsetArrayAttrs(transform, indices); - }); - } else if (k === "colorscale" && Array.isArray(val)) { - newObj[k] = val; - } else if (isPlainObject(val)) { - newObj[k] = subsetArrayAttrs(val, indices); - } else if (Array.isArray(val)) { - newObj[k] = subsetArray(val, indices); - } else { - newObj[k] = val; - } - }); - return newObj; -} - -function subsetArray(arr, indices) { - var result = []; - for (var i = 0; i < indices.length; i++) { - result.push(arr[indices[i]]); - } - return result; -} - -// Convenience function for removing plotly's brush -function removeBrush(el) { - var outlines = el.querySelectorAll(".select-outline"); - for (var i = 0; i < outlines.length; i++) { - outlines[i].remove(); - } -} - - -// https://davidwalsh.name/javascript-debounce-function - -// Returns a function, that, as long as it continues to be invoked, will not -// be triggered. The function will be called after it stops being called for -// N milliseconds. 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-{"version":3,"file":"reveal.esm.js","sources":["../js/utils/util.js","../js/utils/device.js","../node_modules/fitty/dist/fitty.module.js","../js/controllers/slidecontent.js","../js/controllers/slidenumber.js","../js/utils/color.js","../js/controllers/backgrounds.js","../js/utils/constants.js","../js/controllers/autoanimate.js","../js/controllers/fragments.js","../js/controllers/overview.js","../js/controllers/keyboard.js","../js/controllers/location.js","../js/controllers/controls.js","../js/controllers/progress.js","../js/controllers/pointer.js","../js/utils/loader.js","../js/controllers/plugins.js","../js/controllers/print.js","../js/controllers/touch.js","../js/controllers/focus.js","../js/controllers/notes.js","../js/components/playback.js","../js/config.js","../js/reveal.js","../js/index.js"],"sourcesContent":["/**\n * Extend object a with the properties of object b.\n * If there's a conflict, object b takes precedence.\n *\n * @param {object} a\n * @param {object} b\n */\nexport const extend = ( a, b ) => {\n\n\tfor( let i in b ) {\n\t\ta[ i ] = b[ i ];\n\t}\n\n\treturn a;\n\n}\n\n/**\n * querySelectorAll but returns an Array.\n */\nexport const queryAll = ( el, selector ) => {\n\n\treturn Array.from( el.querySelectorAll( selector ) );\n\n}\n\n/**\n * classList.toggle() with cross browser support\n */\nexport const toggleClass = ( el, className, value ) => {\n\tif( value ) {\n\t\tel.classList.add( className );\n\t}\n\telse {\n\t\tel.classList.remove( className );\n\t}\n}\n\n/**\n * Utility for deserializing a value.\n *\n * @param {*} value\n * @return {*}\n */\nexport const deserialize = ( value ) => {\n\n\tif( typeof value === 'string' ) {\n\t\tif( value === 'null' ) return null;\n\t\telse if( value === 'true' ) return true;\n\t\telse if( value === 'false' ) return false;\n\t\telse if( value.match( /^-?[\\d\\.]+$/ ) ) return parseFloat( value );\n\t}\n\n\treturn value;\n\n}\n\n/**\n * Measures the distance in pixels between point a\n * and point b.\n *\n * @param {object} a point with x/y properties\n * @param {object} b point with x/y properties\n *\n * @return {number}\n */\nexport const distanceBetween = ( a, b ) => {\n\n\tlet dx = a.x - b.x,\n\t\tdy = a.y - b.y;\n\n\treturn Math.sqrt( dx*dx + dy*dy );\n\n}\n\n/**\n * Applies a CSS transform to the target element.\n *\n * @param {HTMLElement} element\n * @param {string} transform\n */\nexport const transformElement = ( element, transform ) => {\n\n\telement.style.transform = transform;\n\n}\n\n/**\n * Element.matches with IE support.\n *\n * @param {HTMLElement} target The element to match\n * @param {String} selector The CSS selector to match\n * the element against\n *\n * @return {Boolean}\n */\nexport const matches = ( target, selector ) => {\n\n\tlet matchesMethod = target.matches || target.matchesSelector || target.msMatchesSelector;\n\n\treturn !!( matchesMethod && matchesMethod.call( target, selector ) );\n\n}\n\n/**\n * Find the closest parent that matches the given\n * selector.\n *\n * @param {HTMLElement} target The child element\n * @param {String} selector The CSS selector to match\n * the parents against\n *\n * @return {HTMLElement} The matched parent or null\n * if no matching parent was found\n */\nexport const closest = ( target, selector ) => {\n\n\t// Native Element.closest\n\tif( typeof target.closest === 'function' ) {\n\t\treturn target.closest( selector );\n\t}\n\n\t// Polyfill\n\twhile( target ) {\n\t\tif( matches( target, selector ) ) {\n\t\t\treturn target;\n\t\t}\n\n\t\t// Keep searching\n\t\ttarget = target.parentNode;\n\t}\n\n\treturn null;\n\n}\n\n/**\n * Handling the fullscreen functionality via the fullscreen API\n *\n * @see http://fullscreen.spec.whatwg.org/\n * @see https://developer.mozilla.org/en-US/docs/DOM/Using_fullscreen_mode\n */\nexport const enterFullscreen = element => {\n\n\telement = element || document.documentElement;\n\n\t// Check which implementation is available\n\tlet requestMethod = element.requestFullscreen ||\n\t\t\t\t\t\telement.webkitRequestFullscreen ||\n\t\t\t\t\t\telement.webkitRequestFullScreen ||\n\t\t\t\t\t\telement.mozRequestFullScreen ||\n\t\t\t\t\t\telement.msRequestFullscreen;\n\n\tif( requestMethod ) {\n\t\trequestMethod.apply( element );\n\t}\n\n}\n\n/**\n * Creates an HTML element and returns a reference to it.\n * If the element already exists the existing instance will\n * be returned.\n *\n * @param {HTMLElement} container\n * @param {string} tagname\n * @param {string} classname\n * @param {string} innerHTML\n *\n * @return {HTMLElement}\n */\nexport const createSingletonNode = ( container, tagname, classname, innerHTML='' ) => {\n\n\t// Find all nodes matching the description\n\tlet nodes = container.querySelectorAll( '.' + classname );\n\n\t// Check all matches to find one which is a direct child of\n\t// the specified container\n\tfor( let i = 0; i < nodes.length; i++ ) {\n\t\tlet testNode = nodes[i];\n\t\tif( testNode.parentNode === container ) {\n\t\t\treturn testNode;\n\t\t}\n\t}\n\n\t// If no node was found, create it now\n\tlet node = document.createElement( tagname );\n\tnode.className = classname;\n\tnode.innerHTML = innerHTML;\n\tcontainer.appendChild( node );\n\n\treturn node;\n\n}\n\n/**\n * Injects the given CSS styles into the DOM.\n *\n * @param {string} value\n */\nexport const createStyleSheet = ( value ) => {\n\n\tlet tag = document.createElement( 'style' );\n\ttag.type = 'text/css';\n\n\tif( value && value.length > 0 ) {\n\t\tif( tag.styleSheet ) {\n\t\t\ttag.styleSheet.cssText = value;\n\t\t}\n\t\telse {\n\t\t\ttag.appendChild( document.createTextNode( value ) );\n\t\t}\n\t}\n\n\tdocument.head.appendChild( tag );\n\n\treturn tag;\n\n}\n\n/**\n * Returns a key:value hash of all query params.\n */\nexport const getQueryHash = () => {\n\n\tlet query = {};\n\n\tlocation.search.replace( /[A-Z0-9]+?=([\\w\\.%-]*)/gi, a => {\n\t\tquery[ a.split( '=' ).shift() ] = a.split( '=' ).pop();\n\t} );\n\n\t// Basic deserialization\n\tfor( let i in query ) {\n\t\tlet value = query[ i ];\n\n\t\tquery[ i ] = deserialize( unescape( value ) );\n\t}\n\n\t// Do not accept new dependencies via query config to avoid\n\t// the potential of malicious script injection\n\tif( typeof query['dependencies'] !== 'undefined' ) delete query['dependencies'];\n\n\treturn query;\n\n}\n\n/**\n * Returns the remaining height within the parent of the\n * target element.\n *\n * remaining height = [ configured parent height ] - [ current parent height ]\n *\n * @param {HTMLElement} element\n * @param {number} [height]\n */\nexport const getRemainingHeight = ( element, height = 0 ) => {\n\n\tif( element ) {\n\t\tlet newHeight, oldHeight = element.style.height;\n\n\t\t// Change the .stretch element height to 0 in order find the height of all\n\t\t// the other elements\n\t\telement.style.height = '0px';\n\n\t\t// In Overview mode, the parent (.slide) height is set of 700px.\n\t\t// Restore it temporarily to its natural height.\n\t\telement.parentNode.style.height = 'auto';\n\n\t\tnewHeight = height - element.parentNode.offsetHeight;\n\n\t\t// Restore the old height, just in case\n\t\telement.style.height = oldHeight + 'px';\n\n\t\t// Clear the parent (.slide) height. .removeProperty works in IE9+\n\t\telement.parentNode.style.removeProperty('height');\n\n\t\treturn newHeight;\n\t}\n\n\treturn height;\n\n}\n\nconst fileExtensionToMimeMap = {\n\t'mp4': 'video/mp4',\n\t'm4a': 'video/mp4',\n\t'ogv': 'video/ogg',\n\t'mpeg': 'video/mpeg',\n\t'webm': 'video/webm'\n}\n\n/**\n * Guess the MIME type for common file formats.\n */\nexport const getMimeTypeFromFile = ( filename='' ) => {\n\treturn fileExtensionToMimeMap[filename.split('.').pop()]\n}","const UA = navigator.userAgent;\n\nexport const isMobile = /(iphone|ipod|ipad|android)/gi.test( UA ) ||\n\t\t\t\t\t\t( navigator.platform === 'MacIntel' && navigator.maxTouchPoints > 1 ); // iPadOS\n\nexport const isChrome = /chrome/i.test( UA ) && !/edge/i.test( UA );\n\nexport const isAndroid = /android/gi.test( UA );","/*\n * fitty v2.3.3 - Snugly resizes text to fit its parent container\n * Copyright (c) 2020 Rik Schennink (https://pqina.nl/)\n */\n'use strict';\n\nObject.defineProperty(exports, \"__esModule\", {\n value: true\n});\n\nvar _extends = Object.assign || function (target) { for (var i = 1; i < arguments.length; i++) { var source = arguments[i]; for (var key in source) { if (Object.prototype.hasOwnProperty.call(source, key)) { target[key] = source[key]; } } } return target; };\n\nexports.default = function (w) {\n\n // no window, early exit\n if (!w) return;\n\n // node list to array helper method\n var toArray = function toArray(nl) {\n return [].slice.call(nl);\n };\n\n // states\n var DrawState = {\n IDLE: 0,\n DIRTY_CONTENT: 1,\n DIRTY_LAYOUT: 2,\n DIRTY: 3\n };\n\n // all active fitty elements\n var fitties = [];\n\n // group all redraw calls till next frame, we cancel each frame request when a new one comes in. If no support for request animation frame, this is an empty function and supports for fitty stops.\n var redrawFrame = null;\n var requestRedraw = 'requestAnimationFrame' in w ? function () {\n w.cancelAnimationFrame(redrawFrame);\n redrawFrame = w.requestAnimationFrame(function () {\n return redraw(fitties.filter(function (f) {\n return f.dirty && f.active;\n }));\n });\n } : function () {};\n\n // sets all fitties to dirty so they are redrawn on the next redraw loop, then calls redraw\n var redrawAll = function redrawAll(type) {\n return function () {\n fitties.forEach(function (f) {\n return f.dirty = type;\n });\n requestRedraw();\n };\n };\n\n // redraws fitties so they nicely fit their parent container\n var redraw = function redraw(fitties) {\n\n // getting info from the DOM at this point should not trigger a reflow, let's gather as much intel as possible before triggering a reflow\n\n // check if styles of all fitties have been computed\n fitties.filter(function (f) {\n return !f.styleComputed;\n }).forEach(function (f) {\n f.styleComputed = computeStyle(f);\n });\n\n // restyle elements that require pre-styling, this triggers a reflow, please try to prevent by adding CSS rules (see docs)\n fitties.filter(shouldPreStyle).forEach(applyStyle);\n\n // we now determine which fitties should be redrawn\n var fittiesToRedraw = fitties.filter(shouldRedraw);\n\n // we calculate final styles for these fitties\n fittiesToRedraw.forEach(calculateStyles);\n\n // now we apply the calculated styles from our previous loop\n fittiesToRedraw.forEach(function (f) {\n applyStyle(f);\n markAsClean(f);\n });\n\n // now we dispatch events for all restyled fitties\n fittiesToRedraw.forEach(dispatchFitEvent);\n };\n\n var markAsClean = function markAsClean(f) {\n return f.dirty = DrawState.IDLE;\n };\n\n var calculateStyles = function calculateStyles(f) {\n\n // get available width from parent node\n f.availableWidth = f.element.parentNode.clientWidth;\n\n // the space our target element uses\n f.currentWidth = f.element.scrollWidth;\n\n // remember current font size\n f.previousFontSize = f.currentFontSize;\n\n // let's calculate the new font size\n f.currentFontSize = Math.min(Math.max(f.minSize, f.availableWidth / f.currentWidth * f.previousFontSize), f.maxSize);\n\n // if allows wrapping, only wrap when at minimum font size (otherwise would break container)\n f.whiteSpace = f.multiLine && f.currentFontSize === f.minSize ? 'normal' : 'nowrap';\n };\n\n // should always redraw if is not dirty layout, if is dirty layout, only redraw if size has changed\n var shouldRedraw = function shouldRedraw(f) {\n return f.dirty !== DrawState.DIRTY_LAYOUT || f.dirty === DrawState.DIRTY_LAYOUT && f.element.parentNode.clientWidth !== f.availableWidth;\n };\n\n // every fitty element is tested for invalid styles\n var computeStyle = function computeStyle(f) {\n\n // get style properties\n var style = w.getComputedStyle(f.element, null);\n\n // get current font size in pixels (if we already calculated it, use the calculated version)\n f.currentFontSize = parseFloat(style.getPropertyValue('font-size'));\n\n // get display type and wrap mode\n f.display = style.getPropertyValue('display');\n f.whiteSpace = style.getPropertyValue('white-space');\n };\n\n // determines if this fitty requires initial styling, can be prevented by applying correct styles through CSS\n var shouldPreStyle = function shouldPreStyle(f) {\n\n var preStyle = false;\n\n // if we already tested for prestyling we don't have to do it again\n if (f.preStyleTestCompleted) return false;\n\n // should have an inline style, if not, apply\n if (!/inline-/.test(f.display)) {\n preStyle = true;\n f.display = 'inline-block';\n }\n\n // to correctly calculate dimensions the element should have whiteSpace set to nowrap\n if (f.whiteSpace !== 'nowrap') {\n preStyle = true;\n f.whiteSpace = 'nowrap';\n }\n\n // we don't have to do this twice\n f.preStyleTestCompleted = true;\n\n return preStyle;\n };\n\n // apply styles to single fitty\n var applyStyle = function applyStyle(f) {\n f.element.style.whiteSpace = f.whiteSpace;\n f.element.style.display = f.display;\n f.element.style.fontSize = f.currentFontSize + 'px';\n };\n\n // dispatch a fit event on a fitty\n var dispatchFitEvent = function dispatchFitEvent(f) {\n f.element.dispatchEvent(new CustomEvent('fit', {\n detail: {\n oldValue: f.previousFontSize,\n newValue: f.currentFontSize,\n scaleFactor: f.currentFontSize / f.previousFontSize\n }\n }));\n };\n\n // fit method, marks the fitty as dirty and requests a redraw (this will also redraw any other fitty marked as dirty)\n var fit = function fit(f, type) {\n return function () {\n f.dirty = type;\n if (!f.active) return;\n requestRedraw();\n };\n };\n\n var init = function init(f) {\n\n // save some of the original CSS properties before we change them\n f.originalStyle = {\n whiteSpace: f.element.style.whiteSpace,\n display: f.element.style.display,\n fontSize: f.element.style.fontSize\n };\n\n // should we observe DOM mutations\n observeMutations(f);\n\n // this is a new fitty so we need to validate if it's styles are in order\n f.newbie = true;\n\n // because it's a new fitty it should also be dirty, we want it to redraw on the first loop\n f.dirty = true;\n\n // we want to be able to update this fitty\n fitties.push(f);\n };\n\n var destroy = function destroy(f) {\n return function () {\n\n // remove from fitties array\n fitties = fitties.filter(function (_) {\n return _.element !== f.element;\n });\n\n // stop observing DOM\n if (f.observeMutations) f.observer.disconnect();\n\n // reset the CSS properties we changes\n f.element.style.whiteSpace = f.originalStyle.whiteSpace;\n f.element.style.display = f.originalStyle.display;\n f.element.style.fontSize = f.originalStyle.fontSize;\n };\n };\n\n // add a new fitty, does not redraw said fitty\n var subscribe = function subscribe(f) {\n return function () {\n if (f.active) return;\n f.active = true;\n requestRedraw();\n };\n };\n\n // remove an existing fitty\n var unsubscribe = function unsubscribe(f) {\n return function () {\n return f.active = false;\n };\n };\n\n var observeMutations = function observeMutations(f) {\n\n // no observing?\n if (!f.observeMutations) return;\n\n // start observing mutations\n f.observer = new MutationObserver(fit(f, DrawState.DIRTY_CONTENT));\n\n // start observing\n f.observer.observe(f.element, f.observeMutations);\n };\n\n // default mutation observer settings\n var mutationObserverDefaultSetting = {\n subtree: true,\n childList: true,\n characterData: true\n };\n\n // default fitty options\n var defaultOptions = {\n minSize: 16,\n maxSize: 512,\n multiLine: true,\n observeMutations: 'MutationObserver' in w ? mutationObserverDefaultSetting : false\n };\n\n // array of elements in, fitty instances out\n function fittyCreate(elements, options) {\n\n // set options object\n var fittyOptions = _extends({}, defaultOptions, options);\n\n // create fitties\n var publicFitties = elements.map(function (element) {\n\n // create fitty instance\n var f = _extends({}, fittyOptions, {\n\n // internal options for this fitty\n element: element,\n active: true\n });\n\n // initialise this fitty\n init(f);\n\n // expose API\n return {\n element: element,\n fit: fit(f, DrawState.DIRTY),\n unfreeze: subscribe(f),\n freeze: unsubscribe(f),\n unsubscribe: destroy(f)\n };\n });\n\n // call redraw on newly initiated fitties\n requestRedraw();\n\n // expose fitties\n return publicFitties;\n }\n\n // fitty creation function\n function fitty(target) {\n var options = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : {};\n\n\n // if target is a string\n return typeof target === 'string' ?\n\n // treat it as a querySelector\n fittyCreate(toArray(document.querySelectorAll(target)), options) :\n\n // create single fitty\n fittyCreate([target], options)[0];\n }\n\n // handles viewport changes, redraws all fitties, but only does so after a timeout\n var resizeDebounce = null;\n var onWindowResized = function onWindowResized() {\n w.clearTimeout(resizeDebounce);\n resizeDebounce = w.setTimeout(redrawAll(DrawState.DIRTY_LAYOUT), fitty.observeWindowDelay);\n };\n\n // define observe window property, so when we set it to true or false events are automatically added and removed\n var events = ['resize', 'orientationchange'];\n Object.defineProperty(fitty, 'observeWindow', {\n set: function set(enabled) {\n var method = (enabled ? 'add' : 'remove') + 'EventListener';\n events.forEach(function (e) {\n w[method](e, onWindowResized);\n });\n }\n });\n\n // fitty global properties (by setting observeWindow to true the events above get added)\n fitty.observeWindow = true;\n fitty.observeWindowDelay = 100;\n\n // public fit all method, will force redraw no matter what\n fitty.fitAll = redrawAll(DrawState.DIRTY);\n\n // export our fitty function, we don't want to keep it to our selves\n return fitty;\n}(typeof window === 'undefined' ? null : window);","import { extend, queryAll, closest, getMimeTypeFromFile } from '../utils/util.js'\nimport { isMobile } from '../utils/device.js'\n\nimport fitty from 'fitty';\n\n/**\n * Handles loading, unloading and playback of slide\n * content such as images, videos and iframes.\n */\nexport default class SlideContent {\n\n\tconstructor( Reveal ) {\n\n\t\tthis.Reveal = Reveal;\n\n\t\tthis.startEmbeddedIframe = this.startEmbeddedIframe.bind( this );\n\n\t}\n\n\t/**\n\t * Should the given element be preloaded?\n\t * Decides based on local element attributes and global config.\n\t *\n\t * @param {HTMLElement} element\n\t */\n\tshouldPreload( element ) {\n\n\t\t// Prefer an explicit global preload setting\n\t\tlet preload = this.Reveal.getConfig().preloadIframes;\n\n\t\t// If no global setting is available, fall back on the element's\n\t\t// own preload setting\n\t\tif( typeof preload !== 'boolean' ) {\n\t\t\tpreload = element.hasAttribute( 'data-preload' );\n\t\t}\n\n\t\treturn preload;\n\t}\n\n\t/**\n\t * Called when the given slide is within the configured view\n\t * distance. Shows the slide element and loads any content\n\t * that is set to load lazily (data-src).\n\t *\n\t * @param {HTMLElement} slide Slide to show\n\t */\n\tload( slide, options = {} ) {\n\n\t\t// Show the slide element\n\t\tslide.style.display = this.Reveal.getConfig().display;\n\n\t\t// Media elements with data-src attributes\n\t\tqueryAll( slide, 'img[data-src], video[data-src], audio[data-src], iframe[data-src]' ).forEach( element => {\n\t\t\tif( element.tagName !== 'IFRAME' || this.shouldPreload( element ) ) {\n\t\t\t\telement.setAttribute( 'src', element.getAttribute( 'data-src' ) );\n\t\t\t\telement.setAttribute( 'data-lazy-loaded', '' );\n\t\t\t\telement.removeAttribute( 'data-src' );\n\t\t\t}\n\t\t} );\n\n\t\t// Media elements with children\n\t\tqueryAll( slide, 'video, audio' ).forEach( media => {\n\t\t\tlet sources = 0;\n\n\t\t\tqueryAll( media, 'source[data-src]' ).forEach( source => {\n\t\t\t\tsource.setAttribute( 'src', source.getAttribute( 'data-src' ) );\n\t\t\t\tsource.removeAttribute( 'data-src' );\n\t\t\t\tsource.setAttribute( 'data-lazy-loaded', '' );\n\t\t\t\tsources += 1;\n\t\t\t} );\n\n\t\t\t// Enable inline video playback in mobile Safari\n\t\t\tif( isMobile && media.tagName === 'VIDEO' ) {\n\t\t\t\tmedia.setAttribute( 'playsinline', '' );\n\t\t\t}\n\n\t\t\t// If we rewrote sources for this video/audio element, we need\n\t\t\t// to manually tell it to load from its new origin\n\t\t\tif( sources > 0 ) {\n\t\t\t\tmedia.load();\n\t\t\t}\n\t\t} );\n\n\n\t\t// Show the corresponding background element\n\t\tlet background = slide.slideBackgroundElement;\n\t\tif( background ) {\n\t\t\tbackground.style.display = 'block';\n\n\t\t\tlet backgroundContent = slide.slideBackgroundContentElement;\n\t\t\tlet backgroundIframe = slide.getAttribute( 'data-background-iframe' );\n\n\t\t\t// If the background contains media, load it\n\t\t\tif( background.hasAttribute( 'data-loaded' ) === false ) {\n\t\t\t\tbackground.setAttribute( 'data-loaded', 'true' );\n\n\t\t\t\tlet backgroundImage = slide.getAttribute( 'data-background-image' ),\n\t\t\t\t\tbackgroundVideo = slide.getAttribute( 'data-background-video' ),\n\t\t\t\t\tbackgroundVideoLoop = slide.hasAttribute( 'data-background-video-loop' ),\n\t\t\t\t\tbackgroundVideoMuted = slide.hasAttribute( 'data-background-video-muted' );\n\n\t\t\t\t// Images\n\t\t\t\tif( backgroundImage ) {\n\t\t\t\t\t// base64\n\t\t\t\t\tif( /^data:/.test( backgroundImage.trim() ) ) {\n\t\t\t\t\t\tbackgroundContent.style.backgroundImage = `url(${backgroundImage.trim()})`;\n\t\t\t\t\t}\n\t\t\t\t\t// URL(s)\n\t\t\t\t\telse {\n\t\t\t\t\t\tbackgroundContent.style.backgroundImage = backgroundImage.split( ',' ).map( background => {\n\t\t\t\t\t\t\treturn `url(${encodeURI(background.trim())})`;\n\t\t\t\t\t\t}).join( ',' );\n\t\t\t\t\t}\n\t\t\t\t}\n\t\t\t\t// Videos\n\t\t\t\telse if ( backgroundVideo && !this.Reveal.isSpeakerNotes() ) {\n\t\t\t\t\tlet video = document.createElement( 'video' );\n\n\t\t\t\t\tif( backgroundVideoLoop ) {\n\t\t\t\t\t\tvideo.setAttribute( 'loop', '' );\n\t\t\t\t\t}\n\n\t\t\t\t\tif( backgroundVideoMuted ) {\n\t\t\t\t\t\tvideo.muted = true;\n\t\t\t\t\t}\n\n\t\t\t\t\t// Enable inline playback in mobile Safari\n\t\t\t\t\t//\n\t\t\t\t\t// Mute is required for video to play when using\n\t\t\t\t\t// swipe gestures to navigate since they don't\n\t\t\t\t\t// count as direct user actions :'(\n\t\t\t\t\tif( isMobile ) {\n\t\t\t\t\t\tvideo.muted = true;\n\t\t\t\t\t\tvideo.setAttribute( 'playsinline', '' );\n\t\t\t\t\t}\n\n\t\t\t\t\t// Support comma separated lists of video sources\n\t\t\t\t\tbackgroundVideo.split( ',' ).forEach( source => {\n\t\t\t\t\t\tlet type = getMimeTypeFromFile( source );\n\t\t\t\t\t\tif( type ) {\n\t\t\t\t\t\t\tvideo.innerHTML += ``;\n\t\t\t\t\t\t}\n\t\t\t\t\t\telse {\n\t\t\t\t\t\t\tvideo.innerHTML += ``;\n\t\t\t\t\t\t}\n\t\t\t\t\t} );\n\n\t\t\t\t\tbackgroundContent.appendChild( video );\n\t\t\t\t}\n\t\t\t\t// Iframes\n\t\t\t\telse if( backgroundIframe && options.excludeIframes !== true ) {\n\t\t\t\t\tlet iframe = document.createElement( 'iframe' );\n\t\t\t\t\tiframe.setAttribute( 'allowfullscreen', '' );\n\t\t\t\t\tiframe.setAttribute( 'mozallowfullscreen', '' );\n\t\t\t\t\tiframe.setAttribute( 'webkitallowfullscreen', '' );\n\t\t\t\t\tiframe.setAttribute( 'allow', 'autoplay' );\n\n\t\t\t\t\tiframe.setAttribute( 'data-src', backgroundIframe );\n\n\t\t\t\t\tiframe.style.width = '100%';\n\t\t\t\t\tiframe.style.height = '100%';\n\t\t\t\t\tiframe.style.maxHeight = '100%';\n\t\t\t\t\tiframe.style.maxWidth = '100%';\n\n\t\t\t\t\tbackgroundContent.appendChild( iframe );\n\t\t\t\t}\n\t\t\t}\n\n\t\t\t// Start loading preloadable iframes\n\t\t\tlet backgroundIframeElement = backgroundContent.querySelector( 'iframe[data-src]' );\n\t\t\tif( backgroundIframeElement ) {\n\n\t\t\t\t// Check if this iframe is eligible to be preloaded\n\t\t\t\tif( this.shouldPreload( background ) && !/autoplay=(1|true|yes)/gi.test( backgroundIframe ) ) {\n\t\t\t\t\tif( backgroundIframeElement.getAttribute( 'src' ) !== backgroundIframe ) {\n\t\t\t\t\t\tbackgroundIframeElement.setAttribute( 'src', backgroundIframe );\n\t\t\t\t\t}\n\t\t\t\t}\n\n\t\t\t}\n\n\t\t}\n\n\t\tthis.layout( slide );\n\n\t}\n\n\t/**\n\t * Applies JS-dependent layout helpers for the given slide,\n\t * if there are any.\n\t */\n\tlayout( slide ) {\n\n\t\t// Autosize text with the r-fit-text class based on the\n\t\t// size of its container. This needs to happen after the\n\t\t// slide is visible in order to measure the text.\n\t\tArray.from( slide.querySelectorAll( '.r-fit-text' ) ).forEach( element => {\n\t\t\tfitty( element, {\n\t\t\t\tminSize: 24,\n\t\t\t\tmaxSize: this.Reveal.getConfig().height * 0.8,\n\t\t\t\tobserveMutations: false,\n\t\t\t\tobserveWindow: false\n\t\t\t} );\n\t\t} );\n\n\t}\n\n\t/**\n\t * Unloads and hides the given slide. This is called when the\n\t * slide is moved outside of the configured view distance.\n\t *\n\t * @param {HTMLElement} slide\n\t */\n\tunload( slide ) {\n\n\t\t// Hide the slide element\n\t\tslide.style.display = 'none';\n\n\t\t// Hide the corresponding background element\n\t\tlet background = this.Reveal.getSlideBackground( slide );\n\t\tif( background ) {\n\t\t\tbackground.style.display = 'none';\n\n\t\t\t// Unload any background iframes\n\t\t\tqueryAll( background, 'iframe[src]' ).forEach( element => {\n\t\t\t\telement.removeAttribute( 'src' );\n\t\t\t} );\n\t\t}\n\n\t\t// Reset lazy-loaded media elements with src attributes\n\t\tqueryAll( slide, 'video[data-lazy-loaded][src], audio[data-lazy-loaded][src], iframe[data-lazy-loaded][src]' ).forEach( element => {\n\t\t\telement.setAttribute( 'data-src', element.getAttribute( 'src' ) );\n\t\t\telement.removeAttribute( 'src' );\n\t\t} );\n\n\t\t// Reset lazy-loaded media elements with children\n\t\tqueryAll( slide, 'video[data-lazy-loaded] source[src], audio source[src]' ).forEach( source => {\n\t\t\tsource.setAttribute( 'data-src', source.getAttribute( 'src' ) );\n\t\t\tsource.removeAttribute( 'src' );\n\t\t} );\n\n\t}\n\n\t/**\n\t * Enforces origin-specific format rules for embedded media.\n\t */\n\tformatEmbeddedContent() {\n\n\t\tlet _appendParamToIframeSource = ( sourceAttribute, sourceURL, param ) => {\n\t\t\tqueryAll( this.Reveal.getSlidesElement(), 'iframe['+ sourceAttribute +'*=\"'+ sourceURL +'\"]' ).forEach( el => {\n\t\t\t\tlet src = el.getAttribute( sourceAttribute );\n\t\t\t\tif( src && src.indexOf( param ) === -1 ) {\n\t\t\t\t\tel.setAttribute( sourceAttribute, src + ( !/\\?/.test( src ) ? '?' : '&' ) + param );\n\t\t\t\t}\n\t\t\t});\n\t\t};\n\n\t\t// YouTube frames must include \"?enablejsapi=1\"\n\t\t_appendParamToIframeSource( 'src', 'youtube.com/embed/', 'enablejsapi=1' );\n\t\t_appendParamToIframeSource( 'data-src', 'youtube.com/embed/', 'enablejsapi=1' );\n\n\t\t// Vimeo frames must include \"?api=1\"\n\t\t_appendParamToIframeSource( 'src', 'player.vimeo.com/', 'api=1' );\n\t\t_appendParamToIframeSource( 'data-src', 'player.vimeo.com/', 'api=1' );\n\n\t}\n\n\t/**\n\t * Start playback of any embedded content inside of\n\t * the given element.\n\t *\n\t * @param {HTMLElement} element\n\t */\n\tstartEmbeddedContent( element ) {\n\n\t\tif( element && !this.Reveal.isSpeakerNotes() ) {\n\n\t\t\t// Restart GIFs\n\t\t\tqueryAll( element, 'img[src$=\".gif\"]' ).forEach( el => {\n\t\t\t\t// Setting the same unchanged source like this was confirmed\n\t\t\t\t// to work in Chrome, FF & Safari\n\t\t\t\tel.setAttribute( 'src', el.getAttribute( 'src' ) );\n\t\t\t} );\n\n\t\t\t// HTML5 media elements\n\t\t\tqueryAll( element, 'video, audio' ).forEach( el => {\n\t\t\t\tif( closest( el, '.fragment' ) && !closest( el, '.fragment.visible' ) ) {\n\t\t\t\t\treturn;\n\t\t\t\t}\n\n\t\t\t\t// Prefer an explicit global autoplay setting\n\t\t\t\tlet autoplay = this.Reveal.getConfig().autoPlayMedia;\n\n\t\t\t\t// If no global setting is available, fall back on the element's\n\t\t\t\t// own autoplay setting\n\t\t\t\tif( typeof autoplay !== 'boolean' ) {\n\t\t\t\t\tautoplay = el.hasAttribute( 'data-autoplay' ) || !!closest( el, '.slide-background' );\n\t\t\t\t}\n\n\t\t\t\tif( autoplay && typeof el.play === 'function' ) {\n\n\t\t\t\t\t// If the media is ready, start playback\n\t\t\t\t\tif( el.readyState > 1 ) {\n\t\t\t\t\t\tthis.startEmbeddedMedia( { target: el } );\n\t\t\t\t\t}\n\t\t\t\t\t// Mobile devices never fire a loaded event so instead\n\t\t\t\t\t// of waiting, we initiate playback\n\t\t\t\t\telse if( isMobile ) {\n\t\t\t\t\t\tlet promise = el.play();\n\n\t\t\t\t\t\t// If autoplay does not work, ensure that the controls are visible so\n\t\t\t\t\t\t// that the viewer can start the media on their own\n\t\t\t\t\t\tif( promise && typeof promise.catch === 'function' && el.controls === false ) {\n\t\t\t\t\t\t\tpromise.catch( () => {\n\t\t\t\t\t\t\t\tel.controls = true;\n\n\t\t\t\t\t\t\t\t// Once the video does start playing, hide the controls again\n\t\t\t\t\t\t\t\tel.addEventListener( 'play', () => {\n\t\t\t\t\t\t\t\t\tel.controls = false;\n\t\t\t\t\t\t\t\t} );\n\t\t\t\t\t\t\t} );\n\t\t\t\t\t\t}\n\t\t\t\t\t}\n\t\t\t\t\t// If the media isn't loaded, wait before playing\n\t\t\t\t\telse {\n\t\t\t\t\t\tel.removeEventListener( 'loadeddata', this.startEmbeddedMedia ); // remove first to avoid dupes\n\t\t\t\t\t\tel.addEventListener( 'loadeddata', this.startEmbeddedMedia );\n\t\t\t\t\t}\n\n\t\t\t\t}\n\t\t\t} );\n\n\t\t\t// Normal iframes\n\t\t\tqueryAll( element, 'iframe[src]' ).forEach( el => {\n\t\t\t\tif( closest( el, '.fragment' ) && !closest( el, '.fragment.visible' ) ) {\n\t\t\t\t\treturn;\n\t\t\t\t}\n\n\t\t\t\tthis.startEmbeddedIframe( { target: el } );\n\t\t\t} );\n\n\t\t\t// Lazy loading iframes\n\t\t\tqueryAll( element, 'iframe[data-src]' ).forEach( el => {\n\t\t\t\tif( closest( el, '.fragment' ) && !closest( el, '.fragment.visible' ) ) {\n\t\t\t\t\treturn;\n\t\t\t\t}\n\n\t\t\t\tif( el.getAttribute( 'src' ) !== el.getAttribute( 'data-src' ) ) {\n\t\t\t\t\tel.removeEventListener( 'load', this.startEmbeddedIframe ); // remove first to avoid dupes\n\t\t\t\t\tel.addEventListener( 'load', this.startEmbeddedIframe );\n\t\t\t\t\tel.setAttribute( 'src', el.getAttribute( 'data-src' ) );\n\t\t\t\t}\n\t\t\t} );\n\n\t\t}\n\n\t}\n\n\t/**\n\t * Starts playing an embedded video/audio element after\n\t * it has finished loading.\n\t *\n\t * @param {object} event\n\t */\n\tstartEmbeddedMedia( event ) {\n\n\t\tlet isAttachedToDOM = !!closest( event.target, 'html' ),\n\t\t\tisVisible \t\t= !!closest( event.target, '.present' );\n\n\t\tif( isAttachedToDOM && isVisible ) {\n\t\t\tevent.target.currentTime = 0;\n\t\t\tevent.target.play();\n\t\t}\n\n\t\tevent.target.removeEventListener( 'loadeddata', this.startEmbeddedMedia );\n\n\t}\n\n\t/**\n\t * \"Starts\" the content of an embedded iframe using the\n\t * postMessage API.\n\t *\n\t * @param {object} event\n\t */\n\tstartEmbeddedIframe( event ) {\n\n\t\tlet iframe = event.target;\n\n\t\tif( iframe && iframe.contentWindow ) {\n\n\t\t\tlet isAttachedToDOM = !!closest( event.target, 'html' ),\n\t\t\t\tisVisible \t\t= !!closest( event.target, '.present' );\n\n\t\t\tif( isAttachedToDOM && isVisible ) {\n\n\t\t\t\t// Prefer an explicit global autoplay setting\n\t\t\t\tlet autoplay = this.Reveal.getConfig().autoPlayMedia;\n\n\t\t\t\t// If no global setting is available, fall back on the element's\n\t\t\t\t// own autoplay setting\n\t\t\t\tif( typeof autoplay !== 'boolean' ) {\n\t\t\t\t\tautoplay = iframe.hasAttribute( 'data-autoplay' ) || !!closest( iframe, '.slide-background' );\n\t\t\t\t}\n\n\t\t\t\t// YouTube postMessage API\n\t\t\t\tif( /youtube\\.com\\/embed\\//.test( iframe.getAttribute( 'src' ) ) && autoplay ) {\n\t\t\t\t\tiframe.contentWindow.postMessage( '{\"event\":\"command\",\"func\":\"playVideo\",\"args\":\"\"}', '*' );\n\t\t\t\t}\n\t\t\t\t// Vimeo postMessage API\n\t\t\t\telse if( /player\\.vimeo\\.com\\//.test( iframe.getAttribute( 'src' ) ) && autoplay ) {\n\t\t\t\t\tiframe.contentWindow.postMessage( '{\"method\":\"play\"}', '*' );\n\t\t\t\t}\n\t\t\t\t// Generic postMessage API\n\t\t\t\telse {\n\t\t\t\t\tiframe.contentWindow.postMessage( 'slide:start', '*' );\n\t\t\t\t}\n\n\t\t\t}\n\n\t\t}\n\n\t}\n\n\t/**\n\t * Stop playback of any embedded content inside of\n\t * the targeted slide.\n\t *\n\t * @param {HTMLElement} element\n\t */\n\tstopEmbeddedContent( element, options = {} ) {\n\n\t\toptions = extend( {\n\t\t\t// Defaults\n\t\t\tunloadIframes: true\n\t\t}, options );\n\n\t\tif( element && element.parentNode ) {\n\t\t\t// HTML5 media elements\n\t\t\tqueryAll( element, 'video, audio' ).forEach( el => {\n\t\t\t\tif( !el.hasAttribute( 'data-ignore' ) && typeof el.pause === 'function' ) {\n\t\t\t\t\tel.setAttribute('data-paused-by-reveal', '');\n\t\t\t\t\tel.pause();\n\t\t\t\t}\n\t\t\t} );\n\n\t\t\t// Generic postMessage API for non-lazy loaded iframes\n\t\t\tqueryAll( element, 'iframe' ).forEach( el => {\n\t\t\t\tif( el.contentWindow ) el.contentWindow.postMessage( 'slide:stop', '*' );\n\t\t\t\tel.removeEventListener( 'load', this.startEmbeddedIframe );\n\t\t\t});\n\n\t\t\t// YouTube postMessage API\n\t\t\tqueryAll( element, 'iframe[src*=\"youtube.com/embed/\"]' ).forEach( el => {\n\t\t\t\tif( !el.hasAttribute( 'data-ignore' ) && el.contentWindow && typeof el.contentWindow.postMessage === 'function' ) {\n\t\t\t\t\tel.contentWindow.postMessage( '{\"event\":\"command\",\"func\":\"pauseVideo\",\"args\":\"\"}', '*' );\n\t\t\t\t}\n\t\t\t});\n\n\t\t\t// Vimeo postMessage API\n\t\t\tqueryAll( element, 'iframe[src*=\"player.vimeo.com/\"]' ).forEach( el => {\n\t\t\t\tif( !el.hasAttribute( 'data-ignore' ) && el.contentWindow && typeof el.contentWindow.postMessage === 'function' ) {\n\t\t\t\t\tel.contentWindow.postMessage( '{\"method\":\"pause\"}', '*' );\n\t\t\t\t}\n\t\t\t});\n\n\t\t\tif( options.unloadIframes === true ) {\n\t\t\t\t// Unload lazy-loaded iframes\n\t\t\t\tqueryAll( element, 'iframe[data-src]' ).forEach( el => {\n\t\t\t\t\t// Only removing the src doesn't actually unload the frame\n\t\t\t\t\t// in all browsers (Firefox) so we set it to blank first\n\t\t\t\t\tel.setAttribute( 'src', 'about:blank' );\n\t\t\t\t\tel.removeAttribute( 'src' );\n\t\t\t\t} );\n\t\t\t}\n\t\t}\n\n\t}\n\n}\n","/**\n * Handles the display of reveal.js' optional slide number.\n */\nexport default class SlideNumber {\n\n\tconstructor( Reveal ) {\n\n\t\tthis.Reveal = Reveal;\n\n\t}\n\n\trender() {\n\n\t\tthis.element = document.createElement( 'div' );\n\t\tthis.element.className = 'slide-number';\n\t\tthis.Reveal.getRevealElement().appendChild( this.element );\n\n\t}\n\n\t/**\n\t * Called when the reveal.js config is updated.\n\t */\n\tconfigure( config, oldConfig ) {\n\n\t\tlet slideNumberDisplay = 'none';\n\t\tif( config.slideNumber && !this.Reveal.isPrintingPDF() ) {\n\t\t\tif( config.showSlideNumber === 'all' ) {\n\t\t\t\tslideNumberDisplay = 'block';\n\t\t\t}\n\t\t\telse if( config.showSlideNumber === 'speaker' && this.Reveal.isSpeakerNotes() ) {\n\t\t\t\tslideNumberDisplay = 'block';\n\t\t\t}\n\t\t}\n\n\t\tthis.element.style.display = slideNumberDisplay;\n\n\t}\n\n\t/**\n\t * Updates the slide number to match the current slide.\n\t */\n\tupdate() {\n\n\t\t// Update slide number if enabled\n\t\tif( this.Reveal.getConfig().slideNumber && this.element ) {\n\t\t\tthis.element.innerHTML = this.getSlideNumber();\n\t\t}\n\n\t}\n\n\t/**\n\t * Returns the HTML string corresponding to the current slide\n\t * number, including formatting.\n\t */\n\tgetSlideNumber( slide = this.Reveal.getCurrentSlide() ) {\n\n\t\tlet config = this.Reveal.getConfig();\n\t\tlet value;\n\t\tlet format = 'h.v';\n\n\t\tif ( typeof config.slideNumber === 'function' ) {\n\t\t\tvalue = config.slideNumber( slide );\n\t\t} else {\n\t\t\t// Check if a custom number format is available\n\t\t\tif( typeof config.slideNumber === 'string' ) {\n\t\t\t\tformat = config.slideNumber;\n\t\t\t}\n\n\t\t\t// If there are ONLY vertical slides in this deck, always use\n\t\t\t// a flattened slide number\n\t\t\tif( !/c/.test( format ) && this.Reveal.getHorizontalSlides().length === 1 ) {\n\t\t\t\tformat = 'c';\n\t\t\t}\n\n\t\t\t// Offset the current slide number by 1 to make it 1-indexed\n\t\t\tlet horizontalOffset = slide && slide.dataset.visibility === 'uncounted' ? 0 : 1;\n\n\t\t\tvalue = [];\n\t\t\tswitch( format ) {\n\t\t\t\tcase 'c':\n\t\t\t\t\tvalue.push( this.Reveal.getSlidePastCount( slide ) + horizontalOffset );\n\t\t\t\t\tbreak;\n\t\t\t\tcase 'c/t':\n\t\t\t\t\tvalue.push( this.Reveal.getSlidePastCount( slide ) + horizontalOffset, '/', this.Reveal.getTotalSlides() );\n\t\t\t\t\tbreak;\n\t\t\t\tdefault:\n\t\t\t\t\tlet indices = this.Reveal.getIndices( slide );\n\t\t\t\t\tvalue.push( indices.h + horizontalOffset );\n\t\t\t\t\tlet sep = format === 'h/v' ? '/' : '.';\n\t\t\t\t\tif( this.Reveal.isVerticalSlide( slide ) ) value.push( sep, indices.v + 1 );\n\t\t\t}\n\t\t}\n\n\t\tlet url = '#' + this.Reveal.location.getHash( slide );\n\t\treturn this.formatNumber( value[0], value[1], value[2], url );\n\n\t}\n\n\t/**\n\t * Applies HTML formatting to a slide number before it's\n\t * written to the DOM.\n\t *\n\t * @param {number} a Current slide\n\t * @param {string} delimiter Character to separate slide numbers\n\t * @param {(number|*)} b Total slides\n\t * @param {HTMLElement} [url='#'+locationHash()] The url to link to\n\t * @return {string} HTML string fragment\n\t */\n\tformatNumber( a, delimiter, b, url = '#' + this.Reveal.location.getHash() ) {\n\n\t\tif( typeof b === 'number' && !isNaN( b ) ) {\n\t\t\treturn `\n\t\t\t\t\t${a}\n\t\t\t\t\t${delimiter}\n\t\t\t\t\t${b}\n\t\t\t\t\t`;\n\t\t}\n\t\telse {\n\t\t\treturn `\n\t\t\t\t\t${a}\n\t\t\t\t\t`;\n\t\t}\n\n\t}\n\n\tdestroy() {\n\n\t\tthis.element.remove();\n\n\t}\n\n}","/**\n * Converts various color input formats to an {r:0,g:0,b:0} object.\n *\n * @param {string} color The string representation of a color\n * @example\n * colorToRgb('#000');\n * @example\n * colorToRgb('#000000');\n * @example\n * colorToRgb('rgb(0,0,0)');\n * @example\n * colorToRgb('rgba(0,0,0)');\n *\n * @return {{r: number, g: number, b: number, [a]: number}|null}\n */\nexport const colorToRgb = ( color ) => {\n\n\tlet hex3 = color.match( /^#([0-9a-f]{3})$/i );\n\tif( hex3 && hex3[1] ) {\n\t\thex3 = hex3[1];\n\t\treturn {\n\t\t\tr: parseInt( hex3.charAt( 0 ), 16 ) * 0x11,\n\t\t\tg: parseInt( hex3.charAt( 1 ), 16 ) * 0x11,\n\t\t\tb: parseInt( hex3.charAt( 2 ), 16 ) * 0x11\n\t\t};\n\t}\n\n\tlet hex6 = color.match( /^#([0-9a-f]{6})$/i );\n\tif( hex6 && hex6[1] ) {\n\t\thex6 = hex6[1];\n\t\treturn {\n\t\t\tr: parseInt( hex6.slice( 0, 2 ), 16 ),\n\t\t\tg: parseInt( hex6.slice( 2, 4 ), 16 ),\n\t\t\tb: parseInt( hex6.slice( 4, 6 ), 16 )\n\t\t};\n\t}\n\n\tlet rgb = color.match( /^rgb\\s*\\(\\s*(\\d+)\\s*,\\s*(\\d+)\\s*,\\s*(\\d+)\\s*\\)$/i );\n\tif( rgb ) {\n\t\treturn {\n\t\t\tr: parseInt( rgb[1], 10 ),\n\t\t\tg: parseInt( rgb[2], 10 ),\n\t\t\tb: parseInt( rgb[3], 10 )\n\t\t};\n\t}\n\n\tlet rgba = color.match( /^rgba\\s*\\(\\s*(\\d+)\\s*,\\s*(\\d+)\\s*,\\s*(\\d+)\\s*\\,\\s*([\\d]+|[\\d]*.[\\d]+)\\s*\\)$/i );\n\tif( rgba ) {\n\t\treturn {\n\t\t\tr: parseInt( rgba[1], 10 ),\n\t\t\tg: parseInt( rgba[2], 10 ),\n\t\t\tb: parseInt( rgba[3], 10 ),\n\t\t\ta: parseFloat( rgba[4] )\n\t\t};\n\t}\n\n\treturn null;\n\n}\n\n/**\n * Calculates brightness on a scale of 0-255.\n *\n * @param {string} color See colorToRgb for supported formats.\n * @see {@link colorToRgb}\n */\nexport const colorBrightness = ( color ) => {\n\n\tif( typeof color === 'string' ) color = colorToRgb( color );\n\n\tif( color ) {\n\t\treturn ( color.r * 299 + color.g * 587 + color.b * 114 ) / 1000;\n\t}\n\n\treturn null;\n\n}","import { queryAll } from '../utils/util.js'\nimport { colorToRgb, colorBrightness } from '../utils/color.js'\n\n/**\n * Creates and updates slide backgrounds.\n */\nexport default class Backgrounds {\n\n\tconstructor( Reveal ) {\n\n\t\tthis.Reveal = Reveal;\n\n\t}\n\n\trender() {\n\n\t\tthis.element = document.createElement( 'div' );\n\t\tthis.element.className = 'backgrounds';\n\t\tthis.Reveal.getRevealElement().appendChild( this.element );\n\n\t}\n\n\t/**\n\t * Creates the slide background elements and appends them\n\t * to the background container. One element is created per\n\t * slide no matter if the given slide has visible background.\n\t */\n\tcreate() {\n\n\t\t// Clear prior backgrounds\n\t\tthis.element.innerHTML = '';\n\t\tthis.element.classList.add( 'no-transition' );\n\n\t\t// Iterate over all horizontal slides\n\t\tthis.Reveal.getHorizontalSlides().forEach( slideh => {\n\n\t\t\tlet backgroundStack = this.createBackground( slideh, this.element );\n\n\t\t\t// Iterate over all vertical slides\n\t\t\tqueryAll( slideh, 'section' ).forEach( slidev => {\n\n\t\t\t\tthis.createBackground( slidev, backgroundStack );\n\n\t\t\t\tbackgroundStack.classList.add( 'stack' );\n\n\t\t\t} );\n\n\t\t} );\n\n\t\t// Add parallax background if specified\n\t\tif( this.Reveal.getConfig().parallaxBackgroundImage ) {\n\n\t\t\tthis.element.style.backgroundImage = 'url(\"' + this.Reveal.getConfig().parallaxBackgroundImage + '\")';\n\t\t\tthis.element.style.backgroundSize = this.Reveal.getConfig().parallaxBackgroundSize;\n\t\t\tthis.element.style.backgroundRepeat = this.Reveal.getConfig().parallaxBackgroundRepeat;\n\t\t\tthis.element.style.backgroundPosition = this.Reveal.getConfig().parallaxBackgroundPosition;\n\n\t\t\t// Make sure the below properties are set on the element - these properties are\n\t\t\t// needed for proper transitions to be set on the element via CSS. To remove\n\t\t\t// annoying background slide-in effect when the presentation starts, apply\n\t\t\t// these properties after short time delay\n\t\t\tsetTimeout( () => {\n\t\t\t\tthis.Reveal.getRevealElement().classList.add( 'has-parallax-background' );\n\t\t\t}, 1 );\n\n\t\t}\n\t\telse {\n\n\t\t\tthis.element.style.backgroundImage = '';\n\t\t\tthis.Reveal.getRevealElement().classList.remove( 'has-parallax-background' );\n\n\t\t}\n\n\t}\n\n\t/**\n\t * Creates a background for the given slide.\n\t *\n\t * @param {HTMLElement} slide\n\t * @param {HTMLElement} container The element that the background\n\t * should be appended to\n\t * @return {HTMLElement} New background div\n\t */\n\tcreateBackground( slide, container ) {\n\n\t\t// Main slide background element\n\t\tlet element = document.createElement( 'div' );\n\t\telement.className = 'slide-background ' + slide.className.replace( /present|past|future/, '' );\n\n\t\t// Inner background element that wraps images/videos/iframes\n\t\tlet contentElement = document.createElement( 'div' );\n\t\tcontentElement.className = 'slide-background-content';\n\n\t\telement.appendChild( contentElement );\n\t\tcontainer.appendChild( element );\n\n\t\tslide.slideBackgroundElement = element;\n\t\tslide.slideBackgroundContentElement = contentElement;\n\n\t\t// Syncs the background to reflect all current background settings\n\t\tthis.sync( slide );\n\n\t\treturn element;\n\n\t}\n\n\t/**\n\t * Renders all of the visual properties of a slide background\n\t * based on the various background attributes.\n\t *\n\t * @param {HTMLElement} slide\n\t */\n\tsync( slide ) {\n\n\t\tconst element = slide.slideBackgroundElement,\n\t\t\tcontentElement = slide.slideBackgroundContentElement;\n\n\t\tconst data = {\n\t\t\tbackground: slide.getAttribute( 'data-background' ),\n\t\t\tbackgroundSize: slide.getAttribute( 'data-background-size' ),\n\t\t\tbackgroundImage: slide.getAttribute( 'data-background-image' ),\n\t\t\tbackgroundVideo: slide.getAttribute( 'data-background-video' ),\n\t\t\tbackgroundIframe: slide.getAttribute( 'data-background-iframe' ),\n\t\t\tbackgroundColor: slide.getAttribute( 'data-background-color' ),\n\t\t\tbackgroundRepeat: slide.getAttribute( 'data-background-repeat' ),\n\t\t\tbackgroundPosition: slide.getAttribute( 'data-background-position' ),\n\t\t\tbackgroundTransition: slide.getAttribute( 'data-background-transition' ),\n\t\t\tbackgroundOpacity: slide.getAttribute( 'data-background-opacity' ),\n\t\t};\n\n\t\tconst dataPreload = slide.hasAttribute( 'data-preload' );\n\n\t\t// Reset the prior background state in case this is not the\n\t\t// initial sync\n\t\tslide.classList.remove( 'has-dark-background' );\n\t\tslide.classList.remove( 'has-light-background' );\n\n\t\telement.removeAttribute( 'data-loaded' );\n\t\telement.removeAttribute( 'data-background-hash' );\n\t\telement.removeAttribute( 'data-background-size' );\n\t\telement.removeAttribute( 'data-background-transition' );\n\t\telement.style.backgroundColor = '';\n\n\t\tcontentElement.style.backgroundSize = '';\n\t\tcontentElement.style.backgroundRepeat = '';\n\t\tcontentElement.style.backgroundPosition = '';\n\t\tcontentElement.style.backgroundImage = '';\n\t\tcontentElement.style.opacity = '';\n\t\tcontentElement.innerHTML = '';\n\n\t\tif( data.background ) {\n\t\t\t// Auto-wrap image urls in url(...)\n\t\t\tif( /^(http|file|\\/\\/)/gi.test( data.background ) || /\\.(svg|png|jpg|jpeg|gif|bmp)([?#\\s]|$)/gi.test( data.background ) ) {\n\t\t\t\tslide.setAttribute( 'data-background-image', data.background );\n\t\t\t}\n\t\t\telse {\n\t\t\t\telement.style.background = data.background;\n\t\t\t}\n\t\t}\n\n\t\t// Create a hash for this combination of background settings.\n\t\t// This is used to determine when two slide backgrounds are\n\t\t// the same.\n\t\tif( data.background || data.backgroundColor || data.backgroundImage || data.backgroundVideo || data.backgroundIframe ) {\n\t\t\telement.setAttribute( 'data-background-hash', data.background +\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tdata.backgroundSize +\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tdata.backgroundImage +\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tdata.backgroundVideo +\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tdata.backgroundIframe +\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tdata.backgroundColor +\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tdata.backgroundRepeat +\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tdata.backgroundPosition +\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tdata.backgroundTransition +\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tdata.backgroundOpacity );\n\t\t}\n\n\t\t// Additional and optional background properties\n\t\tif( data.backgroundSize ) element.setAttribute( 'data-background-size', data.backgroundSize );\n\t\tif( data.backgroundColor ) element.style.backgroundColor = data.backgroundColor;\n\t\tif( data.backgroundTransition ) element.setAttribute( 'data-background-transition', data.backgroundTransition );\n\n\t\tif( dataPreload ) element.setAttribute( 'data-preload', '' );\n\n\t\t// Background image options are set on the content wrapper\n\t\tif( data.backgroundSize ) contentElement.style.backgroundSize = data.backgroundSize;\n\t\tif( data.backgroundRepeat ) contentElement.style.backgroundRepeat = data.backgroundRepeat;\n\t\tif( data.backgroundPosition ) contentElement.style.backgroundPosition = data.backgroundPosition;\n\t\tif( data.backgroundOpacity ) contentElement.style.opacity = data.backgroundOpacity;\n\n\t\t// If this slide has a background color, we add a class that\n\t\t// signals if it is light or dark. If the slide has no background\n\t\t// color, no class will be added\n\t\tlet contrastColor = data.backgroundColor;\n\n\t\t// If no bg color was found, or it cannot be converted by colorToRgb, check the computed background\n\t\tif( !contrastColor || !colorToRgb( contrastColor ) ) {\n\t\t\tlet computedBackgroundStyle = window.getComputedStyle( element );\n\t\t\tif( computedBackgroundStyle && computedBackgroundStyle.backgroundColor ) {\n\t\t\t\tcontrastColor = computedBackgroundStyle.backgroundColor;\n\t\t\t}\n\t\t}\n\n\t\tif( contrastColor ) {\n\t\t\tconst rgb = colorToRgb( contrastColor );\n\n\t\t\t// Ignore fully transparent backgrounds. Some browsers return\n\t\t\t// rgba(0,0,0,0) when reading the computed background color of\n\t\t\t// an element with no background\n\t\t\tif( rgb && rgb.a !== 0 ) {\n\t\t\t\tif( colorBrightness( contrastColor ) < 128 ) {\n\t\t\t\t\tslide.classList.add( 'has-dark-background' );\n\t\t\t\t}\n\t\t\t\telse {\n\t\t\t\t\tslide.classList.add( 'has-light-background' );\n\t\t\t\t}\n\t\t\t}\n\t\t}\n\n\t}\n\n\t/**\n\t * Updates the background elements to reflect the current\n\t * slide.\n\t *\n\t * @param {boolean} includeAll If true, the backgrounds of\n\t * all vertical slides (not just the present) will be updated.\n\t */\n\tupdate( includeAll = false ) {\n\n\t\tlet currentSlide = this.Reveal.getCurrentSlide();\n\t\tlet indices = this.Reveal.getIndices();\n\n\t\tlet currentBackground = null;\n\n\t\t// Reverse past/future classes when in RTL mode\n\t\tlet horizontalPast = this.Reveal.getConfig().rtl ? 'future' : 'past',\n\t\t\thorizontalFuture = this.Reveal.getConfig().rtl ? 'past' : 'future';\n\n\t\t// Update the classes of all backgrounds to match the\n\t\t// states of their slides (past/present/future)\n\t\tArray.from( this.element.childNodes ).forEach( ( backgroundh, h ) => {\n\n\t\t\tbackgroundh.classList.remove( 'past', 'present', 'future' );\n\n\t\t\tif( h < indices.h ) {\n\t\t\t\tbackgroundh.classList.add( horizontalPast );\n\t\t\t}\n\t\t\telse if ( h > indices.h ) {\n\t\t\t\tbackgroundh.classList.add( horizontalFuture );\n\t\t\t}\n\t\t\telse {\n\t\t\t\tbackgroundh.classList.add( 'present' );\n\n\t\t\t\t// Store a reference to the current background element\n\t\t\t\tcurrentBackground = backgroundh;\n\t\t\t}\n\n\t\t\tif( includeAll || h === indices.h ) {\n\t\t\t\tqueryAll( backgroundh, '.slide-background' ).forEach( ( backgroundv, v ) => {\n\n\t\t\t\t\tbackgroundv.classList.remove( 'past', 'present', 'future' );\n\n\t\t\t\t\tif( v < indices.v ) {\n\t\t\t\t\t\tbackgroundv.classList.add( 'past' );\n\t\t\t\t\t}\n\t\t\t\t\telse if ( v > indices.v ) {\n\t\t\t\t\t\tbackgroundv.classList.add( 'future' );\n\t\t\t\t\t}\n\t\t\t\t\telse {\n\t\t\t\t\t\tbackgroundv.classList.add( 'present' );\n\n\t\t\t\t\t\t// Only if this is the present horizontal and vertical slide\n\t\t\t\t\t\tif( h === indices.h ) currentBackground = backgroundv;\n\t\t\t\t\t}\n\n\t\t\t\t} );\n\t\t\t}\n\n\t\t} );\n\n\t\t// Stop content inside of previous backgrounds\n\t\tif( this.previousBackground ) {\n\n\t\t\tthis.Reveal.slideContent.stopEmbeddedContent( this.previousBackground, { unloadIframes: !this.Reveal.slideContent.shouldPreload( this.previousBackground ) } );\n\n\t\t}\n\n\t\t// Start content in the current background\n\t\tif( currentBackground ) {\n\n\t\t\tthis.Reveal.slideContent.startEmbeddedContent( currentBackground );\n\n\t\t\tlet currentBackgroundContent = currentBackground.querySelector( '.slide-background-content' );\n\t\t\tif( currentBackgroundContent ) {\n\n\t\t\t\tlet backgroundImageURL = currentBackgroundContent.style.backgroundImage || '';\n\n\t\t\t\t// Restart GIFs (doesn't work in Firefox)\n\t\t\t\tif( /\\.gif/i.test( backgroundImageURL ) ) {\n\t\t\t\t\tcurrentBackgroundContent.style.backgroundImage = '';\n\t\t\t\t\twindow.getComputedStyle( currentBackgroundContent ).opacity;\n\t\t\t\t\tcurrentBackgroundContent.style.backgroundImage = backgroundImageURL;\n\t\t\t\t}\n\n\t\t\t}\n\n\t\t\t// Don't transition between identical backgrounds. This\n\t\t\t// prevents unwanted flicker.\n\t\t\tlet previousBackgroundHash = this.previousBackground ? this.previousBackground.getAttribute( 'data-background-hash' ) : null;\n\t\t\tlet currentBackgroundHash = currentBackground.getAttribute( 'data-background-hash' );\n\t\t\tif( currentBackgroundHash && currentBackgroundHash === previousBackgroundHash && currentBackground !== this.previousBackground ) {\n\t\t\t\tthis.element.classList.add( 'no-transition' );\n\t\t\t}\n\n\t\t\tthis.previousBackground = currentBackground;\n\n\t\t}\n\n\t\t// If there's a background brightness flag for this slide,\n\t\t// bubble it to the .reveal container\n\t\tif( currentSlide ) {\n\t\t\t[ 'has-light-background', 'has-dark-background' ].forEach( classToBubble => {\n\t\t\t\tif( currentSlide.classList.contains( classToBubble ) ) {\n\t\t\t\t\tthis.Reveal.getRevealElement().classList.add( classToBubble );\n\t\t\t\t}\n\t\t\t\telse {\n\t\t\t\t\tthis.Reveal.getRevealElement().classList.remove( classToBubble );\n\t\t\t\t}\n\t\t\t}, this );\n\t\t}\n\n\t\t// Allow the first background to apply without transition\n\t\tsetTimeout( () => {\n\t\t\tthis.element.classList.remove( 'no-transition' );\n\t\t}, 1 );\n\n\t}\n\n\t/**\n\t * Updates the position of the parallax background based\n\t * on the current slide index.\n\t */\n\tupdateParallax() {\n\n\t\tlet indices = this.Reveal.getIndices();\n\n\t\tif( this.Reveal.getConfig().parallaxBackgroundImage ) {\n\n\t\t\tlet horizontalSlides = this.Reveal.getHorizontalSlides(),\n\t\t\t\tverticalSlides = this.Reveal.getVerticalSlides();\n\n\t\t\tlet backgroundSize = this.element.style.backgroundSize.split( ' ' ),\n\t\t\t\tbackgroundWidth, backgroundHeight;\n\n\t\t\tif( backgroundSize.length === 1 ) {\n\t\t\t\tbackgroundWidth = backgroundHeight = parseInt( backgroundSize[0], 10 );\n\t\t\t}\n\t\t\telse {\n\t\t\t\tbackgroundWidth = parseInt( backgroundSize[0], 10 );\n\t\t\t\tbackgroundHeight = parseInt( backgroundSize[1], 10 );\n\t\t\t}\n\n\t\t\tlet slideWidth = this.element.offsetWidth,\n\t\t\t\thorizontalSlideCount = horizontalSlides.length,\n\t\t\t\thorizontalOffsetMultiplier,\n\t\t\t\thorizontalOffset;\n\n\t\t\tif( typeof this.Reveal.getConfig().parallaxBackgroundHorizontal === 'number' ) {\n\t\t\t\thorizontalOffsetMultiplier = this.Reveal.getConfig().parallaxBackgroundHorizontal;\n\t\t\t}\n\t\t\telse {\n\t\t\t\thorizontalOffsetMultiplier = horizontalSlideCount > 1 ? ( backgroundWidth - slideWidth ) / ( horizontalSlideCount-1 ) : 0;\n\t\t\t}\n\n\t\t\thorizontalOffset = horizontalOffsetMultiplier * indices.h * -1;\n\n\t\t\tlet slideHeight = this.element.offsetHeight,\n\t\t\t\tverticalSlideCount = verticalSlides.length,\n\t\t\t\tverticalOffsetMultiplier,\n\t\t\t\tverticalOffset;\n\n\t\t\tif( typeof this.Reveal.getConfig().parallaxBackgroundVertical === 'number' ) {\n\t\t\t\tverticalOffsetMultiplier = this.Reveal.getConfig().parallaxBackgroundVertical;\n\t\t\t}\n\t\t\telse {\n\t\t\t\tverticalOffsetMultiplier = ( backgroundHeight - slideHeight ) / ( verticalSlideCount-1 );\n\t\t\t}\n\n\t\t\tverticalOffset = verticalSlideCount > 0 ? verticalOffsetMultiplier * indices.v : 0;\n\n\t\t\tthis.element.style.backgroundPosition = horizontalOffset + 'px ' + -verticalOffset + 'px';\n\n\t\t}\n\n\t}\n\n\tdestroy() {\n\n\t\tthis.element.remove();\n\n\t}\n\n}\n","\nexport const SLIDES_SELECTOR = '.slides section';\nexport const HORIZONTAL_SLIDES_SELECTOR = '.slides>section';\nexport const VERTICAL_SLIDES_SELECTOR = '.slides>section.present>section';\n\n// Methods that may not be invoked via the postMessage API\nexport const POST_MESSAGE_METHOD_BLACKLIST = /registerPlugin|registerKeyboardShortcut|addKeyBinding|addEventListener/;\n\n// Regex for retrieving the fragment style from a class attribute\nexport const FRAGMENT_STYLE_REGEX = /fade-(down|up|right|left|out|in-then-out|in-then-semi-out)|semi-fade-out|current-visible|shrink|grow/;","import { queryAll, extend, createStyleSheet, matches, closest } from '../utils/util.js'\nimport { FRAGMENT_STYLE_REGEX } from '../utils/constants.js'\n\n// Counter used to generate unique IDs for auto-animated elements\nlet autoAnimateCounter = 0;\n\n/**\n * Automatically animates matching elements across\n * slides with the [data-auto-animate] attribute.\n */\nexport default class AutoAnimate {\n\n\tconstructor( Reveal ) {\n\n\t\tthis.Reveal = Reveal;\n\n\t}\n\n\t/**\n\t * Runs an auto-animation between the given slides.\n\t *\n\t * @param {HTMLElement} fromSlide\n\t * @param {HTMLElement} toSlide\n\t */\n\trun( fromSlide, toSlide ) {\n\n\t\t// Clean up after prior animations\n\t\tthis.reset();\n\n\t\tlet allSlides = this.Reveal.getSlides();\n\t\tlet toSlideIndex = allSlides.indexOf( toSlide );\n\t\tlet fromSlideIndex = allSlides.indexOf( fromSlide );\n\n\t\t// Ensure that both slides are auto-animate targets with the same data-auto-animate-id value\n\t\t// (including null if absent on both) and that data-auto-animate-restart isn't set on the\n\t\t// physically latter slide (independent of slide direction)\n\t\tif( fromSlide.hasAttribute( 'data-auto-animate' ) && toSlide.hasAttribute( 'data-auto-animate' )\n\t\t\t\t&& fromSlide.getAttribute( 'data-auto-animate-id' ) === toSlide.getAttribute( 'data-auto-animate-id' ) \n\t\t\t\t&& !( toSlideIndex > fromSlideIndex ? toSlide : fromSlide ).hasAttribute( 'data-auto-animate-restart' ) ) {\n\n\t\t\t// Create a new auto-animate sheet\n\t\t\tthis.autoAnimateStyleSheet = this.autoAnimateStyleSheet || createStyleSheet();\n\n\t\t\tlet animationOptions = this.getAutoAnimateOptions( toSlide );\n\n\t\t\t// Set our starting state\n\t\t\tfromSlide.dataset.autoAnimate = 'pending';\n\t\t\ttoSlide.dataset.autoAnimate = 'pending';\n\n\t\t\t// Flag the navigation direction, needed for fragment buildup\n\t\t\tanimationOptions.slideDirection = toSlideIndex > fromSlideIndex ? 'forward' : 'backward';\n\n\t\t\t// Inject our auto-animate styles for this transition\n\t\t\tlet css = this.getAutoAnimatableElements( fromSlide, toSlide ).map( elements => {\n\t\t\t\treturn this.autoAnimateElements( elements.from, elements.to, elements.options || {}, animationOptions, autoAnimateCounter++ );\n\t\t\t} );\n\n\t\t\t// Animate unmatched elements, if enabled\n\t\t\tif( toSlide.dataset.autoAnimateUnmatched !== 'false' && this.Reveal.getConfig().autoAnimateUnmatched === true ) {\n\n\t\t\t\t// Our default timings for unmatched elements\n\t\t\t\tlet defaultUnmatchedDuration = animationOptions.duration * 0.8,\n\t\t\t\t\tdefaultUnmatchedDelay = animationOptions.duration * 0.2;\n\n\t\t\t\tthis.getUnmatchedAutoAnimateElements( toSlide ).forEach( unmatchedElement => {\n\n\t\t\t\t\tlet unmatchedOptions = this.getAutoAnimateOptions( unmatchedElement, animationOptions );\n\t\t\t\t\tlet id = 'unmatched';\n\n\t\t\t\t\t// If there is a duration or delay set specifically for this\n\t\t\t\t\t// element our unmatched elements should adhere to those\n\t\t\t\t\tif( unmatchedOptions.duration !== animationOptions.duration || unmatchedOptions.delay !== animationOptions.delay ) {\n\t\t\t\t\t\tid = 'unmatched-' + autoAnimateCounter++;\n\t\t\t\t\t\tcss.push( `[data-auto-animate=\"running\"] [data-auto-animate-target=\"${id}\"] { transition: opacity ${unmatchedOptions.duration}s ease ${unmatchedOptions.delay}s; }` );\n\t\t\t\t\t}\n\n\t\t\t\t\tunmatchedElement.dataset.autoAnimateTarget = id;\n\n\t\t\t\t}, this );\n\n\t\t\t\t// Our default transition for unmatched elements\n\t\t\t\tcss.push( `[data-auto-animate=\"running\"] [data-auto-animate-target=\"unmatched\"] { transition: opacity ${defaultUnmatchedDuration}s ease ${defaultUnmatchedDelay}s; }` );\n\n\t\t\t}\n\n\t\t\t// Setting the whole chunk of CSS at once is the most\n\t\t\t// efficient way to do this. Using sheet.insertRule\n\t\t\t// is multiple factors slower.\n\t\t\tthis.autoAnimateStyleSheet.innerHTML = css.join( '' );\n\n\t\t\t// Start the animation next cycle\n\t\t\trequestAnimationFrame( () => {\n\t\t\t\tif( this.autoAnimateStyleSheet ) {\n\t\t\t\t\t// This forces our newly injected styles to be applied in Firefox\n\t\t\t\t\tgetComputedStyle( this.autoAnimateStyleSheet ).fontWeight;\n\n\t\t\t\t\ttoSlide.dataset.autoAnimate = 'running';\n\t\t\t\t}\n\t\t\t} );\n\n\t\t\tthis.Reveal.dispatchEvent({\n\t\t\t\ttype: 'autoanimate',\n\t\t\t\tdata: {\n\t\t\t\t\tfromSlide,\n\t\t\t\t\ttoSlide,\n\t\t\t\t\tsheet: this.autoAnimateStyleSheet\n\t\t\t\t}\n\t\t\t});\n\n\t\t}\n\n\t}\n\n\t/**\n\t * Rolls back all changes that we've made to the DOM so\n\t * that as part of animating.\n\t */\n\treset() {\n\n\t\t// Reset slides\n\t\tqueryAll( this.Reveal.getRevealElement(), '[data-auto-animate]:not([data-auto-animate=\"\"])' ).forEach( element => {\n\t\t\telement.dataset.autoAnimate = '';\n\t\t} );\n\n\t\t// Reset elements\n\t\tqueryAll( this.Reveal.getRevealElement(), '[data-auto-animate-target]' ).forEach( element => {\n\t\t\tdelete element.dataset.autoAnimateTarget;\n\t\t} );\n\n\t\t// Remove the animation sheet\n\t\tif( this.autoAnimateStyleSheet && this.autoAnimateStyleSheet.parentNode ) {\n\t\t\tthis.autoAnimateStyleSheet.parentNode.removeChild( this.autoAnimateStyleSheet );\n\t\t\tthis.autoAnimateStyleSheet = null;\n\t\t}\n\n\t}\n\n\t/**\n\t * Creates a FLIP animation where the `to` element starts out\n\t * in the `from` element position and animates to its original\n\t * state.\n\t *\n\t * @param {HTMLElement} from\n\t * @param {HTMLElement} to\n\t * @param {Object} elementOptions Options for this element pair\n\t * @param {Object} animationOptions Options set at the slide level\n\t * @param {String} id Unique ID that we can use to identify this\n\t * auto-animate element in the DOM\n\t */\n\tautoAnimateElements( from, to, elementOptions, animationOptions, id ) {\n\n\t\t// 'from' elements are given a data-auto-animate-target with no value,\n\t\t// 'to' elements are are given a data-auto-animate-target with an ID\n\t\tfrom.dataset.autoAnimateTarget = '';\n\t\tto.dataset.autoAnimateTarget = id;\n\n\t\t// Each element may override any of the auto-animate options\n\t\t// like transition easing, duration and delay via data-attributes\n\t\tlet options = this.getAutoAnimateOptions( to, animationOptions );\n\n\t\t// If we're using a custom element matcher the element options\n\t\t// may contain additional transition overrides\n\t\tif( typeof elementOptions.delay !== 'undefined' ) options.delay = elementOptions.delay;\n\t\tif( typeof elementOptions.duration !== 'undefined' ) options.duration = elementOptions.duration;\n\t\tif( typeof elementOptions.easing !== 'undefined' ) options.easing = elementOptions.easing;\n\n\t\tlet fromProps = this.getAutoAnimatableProperties( 'from', from, elementOptions ),\n\t\t\ttoProps = this.getAutoAnimatableProperties( 'to', to, elementOptions );\n\n\t\t// Maintain fragment visibility for matching elements when\n\t\t// we're navigating forwards, this way the viewer won't need\n\t\t// to step through the same fragments twice\n\t\tif( to.classList.contains( 'fragment' ) ) {\n\n\t\t\t// Don't auto-animate the opacity of fragments to avoid\n\t\t\t// conflicts with fragment animations\n\t\t\tdelete toProps.styles['opacity'];\n\n\t\t\tif( from.classList.contains( 'fragment' ) ) {\n\n\t\t\t\tlet fromFragmentStyle = ( from.className.match( FRAGMENT_STYLE_REGEX ) || [''] )[0];\n\t\t\t\tlet toFragmentStyle = ( to.className.match( FRAGMENT_STYLE_REGEX ) || [''] )[0];\n\n\t\t\t\t// Only skip the fragment if the fragment animation style\n\t\t\t\t// remains unchanged\n\t\t\t\tif( fromFragmentStyle === toFragmentStyle && animationOptions.slideDirection === 'forward' ) {\n\t\t\t\t\tto.classList.add( 'visible', 'disabled' );\n\t\t\t\t}\n\n\t\t\t}\n\n\t\t}\n\n\t\t// If translation and/or scaling are enabled, css transform\n\t\t// the 'to' element so that it matches the position and size\n\t\t// of the 'from' element\n\t\tif( elementOptions.translate !== false || elementOptions.scale !== false ) {\n\n\t\t\tlet presentationScale = this.Reveal.getScale();\n\n\t\t\tlet delta = {\n\t\t\t\tx: ( fromProps.x - toProps.x ) / presentationScale,\n\t\t\t\ty: ( fromProps.y - toProps.y ) / presentationScale,\n\t\t\t\tscaleX: fromProps.width / toProps.width,\n\t\t\t\tscaleY: fromProps.height / toProps.height\n\t\t\t};\n\n\t\t\t// Limit decimal points to avoid 0.0001px blur and stutter\n\t\t\tdelta.x = Math.round( delta.x * 1000 ) / 1000;\n\t\t\tdelta.y = Math.round( delta.y * 1000 ) / 1000;\n\t\t\tdelta.scaleX = Math.round( delta.scaleX * 1000 ) / 1000;\n\t\t\tdelta.scaleX = Math.round( delta.scaleX * 1000 ) / 1000;\n\n\t\t\tlet translate = elementOptions.translate !== false && ( delta.x !== 0 || delta.y !== 0 ),\n\t\t\t\tscale = elementOptions.scale !== false && ( delta.scaleX !== 0 || delta.scaleY !== 0 );\n\n\t\t\t// No need to transform if nothing's changed\n\t\t\tif( translate || scale ) {\n\n\t\t\t\tlet transform = [];\n\n\t\t\t\tif( translate ) transform.push( `translate(${delta.x}px, ${delta.y}px)` );\n\t\t\t\tif( scale ) transform.push( `scale(${delta.scaleX}, ${delta.scaleY})` );\n\n\t\t\t\tfromProps.styles['transform'] = transform.join( ' ' );\n\t\t\t\tfromProps.styles['transform-origin'] = 'top left';\n\n\t\t\t\ttoProps.styles['transform'] = 'none';\n\n\t\t\t}\n\n\t\t}\n\n\t\t// Delete all unchanged 'to' styles\n\t\tfor( let propertyName in toProps.styles ) {\n\t\t\tconst toValue = toProps.styles[propertyName];\n\t\t\tconst fromValue = fromProps.styles[propertyName];\n\n\t\t\tif( toValue === fromValue ) {\n\t\t\t\tdelete toProps.styles[propertyName];\n\t\t\t}\n\t\t\telse {\n\t\t\t\t// If these property values were set via a custom matcher providing\n\t\t\t\t// an explicit 'from' and/or 'to' value, we always inject those values.\n\t\t\t\tif( toValue.explicitValue === true ) {\n\t\t\t\t\ttoProps.styles[propertyName] = toValue.value;\n\t\t\t\t}\n\n\t\t\t\tif( fromValue.explicitValue === true ) {\n\t\t\t\t\tfromProps.styles[propertyName] = fromValue.value;\n\t\t\t\t}\n\t\t\t}\n\t\t}\n\n\t\tlet css = '';\n\n\t\tlet toStyleProperties = Object.keys( toProps.styles );\n\n\t\t// Only create animate this element IF at least one style\n\t\t// property has changed\n\t\tif( toStyleProperties.length > 0 ) {\n\n\t\t\t// Instantly move to the 'from' state\n\t\t\tfromProps.styles['transition'] = 'none';\n\n\t\t\t// Animate towards the 'to' state\n\t\t\ttoProps.styles['transition'] = `all ${options.duration}s ${options.easing} ${options.delay}s`;\n\t\t\ttoProps.styles['transition-property'] = toStyleProperties.join( ', ' );\n\t\t\ttoProps.styles['will-change'] = toStyleProperties.join( ', ' );\n\n\t\t\t// Build up our custom CSS. We need to override inline styles\n\t\t\t// so we need to make our styles vErY IMPORTANT!1!!\n\t\t\tlet fromCSS = Object.keys( fromProps.styles ).map( propertyName => {\n\t\t\t\treturn propertyName + ': ' + fromProps.styles[propertyName] + ' !important;';\n\t\t\t} ).join( '' );\n\n\t\t\tlet toCSS = Object.keys( toProps.styles ).map( propertyName => {\n\t\t\t\treturn propertyName + ': ' + toProps.styles[propertyName] + ' !important;';\n\t\t\t} ).join( '' );\n\n\t\t\tcss = \t'[data-auto-animate-target=\"'+ id +'\"] {'+ fromCSS +'}' +\n\t\t\t\t\t'[data-auto-animate=\"running\"] [data-auto-animate-target=\"'+ id +'\"] {'+ toCSS +'}';\n\n\t\t}\n\n\t\treturn css;\n\n\t}\n\n\t/**\n\t * Returns the auto-animate options for the given element.\n\t *\n\t * @param {HTMLElement} element Element to pick up options\n\t * from, either a slide or an animation target\n\t * @param {Object} [inheritedOptions] Optional set of existing\n\t * options\n\t */\n\tgetAutoAnimateOptions( element, inheritedOptions ) {\n\n\t\tlet options = {\n\t\t\teasing: this.Reveal.getConfig().autoAnimateEasing,\n\t\t\tduration: this.Reveal.getConfig().autoAnimateDuration,\n\t\t\tdelay: 0\n\t\t};\n\n\t\toptions = extend( options, inheritedOptions );\n\n\t\t// Inherit options from parent elements\n\t\tif( element.parentNode ) {\n\t\t\tlet autoAnimatedParent = closest( element.parentNode, '[data-auto-animate-target]' );\n\t\t\tif( autoAnimatedParent ) {\n\t\t\t\toptions = this.getAutoAnimateOptions( autoAnimatedParent, options );\n\t\t\t}\n\t\t}\n\n\t\tif( element.dataset.autoAnimateEasing ) {\n\t\t\toptions.easing = element.dataset.autoAnimateEasing;\n\t\t}\n\n\t\tif( element.dataset.autoAnimateDuration ) {\n\t\t\toptions.duration = parseFloat( element.dataset.autoAnimateDuration );\n\t\t}\n\n\t\tif( element.dataset.autoAnimateDelay ) {\n\t\t\toptions.delay = parseFloat( element.dataset.autoAnimateDelay );\n\t\t}\n\n\t\treturn options;\n\n\t}\n\n\t/**\n\t * Returns an object containing all of the properties\n\t * that can be auto-animated for the given element and\n\t * their current computed values.\n\t *\n\t * @param {String} direction 'from' or 'to'\n\t */\n\tgetAutoAnimatableProperties( direction, element, elementOptions ) {\n\n\t\tlet config = this.Reveal.getConfig();\n\n\t\tlet properties = { styles: [] };\n\n\t\t// Position and size\n\t\tif( elementOptions.translate !== false || elementOptions.scale !== false ) {\n\t\t\tlet bounds;\n\n\t\t\t// Custom auto-animate may optionally return a custom tailored\n\t\t\t// measurement function\n\t\t\tif( typeof elementOptions.measure === 'function' ) {\n\t\t\t\tbounds = elementOptions.measure( element );\n\t\t\t}\n\t\t\telse {\n\t\t\t\tif( config.center ) {\n\t\t\t\t\t// More precise, but breaks when used in combination\n\t\t\t\t\t// with zoom for scaling the deck ¯\\_(ツ)_/¯\n\t\t\t\t\tbounds = element.getBoundingClientRect();\n\t\t\t\t}\n\t\t\t\telse {\n\t\t\t\t\tlet scale = this.Reveal.getScale();\n\t\t\t\t\tbounds = {\n\t\t\t\t\t\tx: element.offsetLeft * scale,\n\t\t\t\t\t\ty: element.offsetTop * scale,\n\t\t\t\t\t\twidth: element.offsetWidth * scale,\n\t\t\t\t\t\theight: element.offsetHeight * scale\n\t\t\t\t\t};\n\t\t\t\t}\n\t\t\t}\n\n\t\t\tproperties.x = bounds.x;\n\t\t\tproperties.y = bounds.y;\n\t\t\tproperties.width = bounds.width;\n\t\t\tproperties.height = bounds.height;\n\t\t}\n\n\t\tconst computedStyles = getComputedStyle( element );\n\n\t\t// CSS styles\n\t\t( elementOptions.styles || config.autoAnimateStyles ).forEach( style => {\n\t\t\tlet value;\n\n\t\t\t// `style` is either the property name directly, or an object\n\t\t\t// definition of a style property\n\t\t\tif( typeof style === 'string' ) style = { property: style };\n\n\t\t\tif( typeof style.from !== 'undefined' && direction === 'from' ) {\n\t\t\t\tvalue = { value: style.from, explicitValue: true };\n\t\t\t}\n\t\t\telse if( typeof style.to !== 'undefined' && direction === 'to' ) {\n\t\t\t\tvalue = { value: style.to, explicitValue: true };\n\t\t\t}\n\t\t\telse {\n\t\t\t\tvalue = computedStyles[style.property];\n\t\t\t}\n\n\t\t\tif( value !== '' ) {\n\t\t\t\tproperties.styles[style.property] = value;\n\t\t\t}\n\t\t} );\n\n\t\treturn properties;\n\n\t}\n\n\t/**\n\t * Get a list of all element pairs that we can animate\n\t * between the given slides.\n\t *\n\t * @param {HTMLElement} fromSlide\n\t * @param {HTMLElement} toSlide\n\t *\n\t * @return {Array} Each value is an array where [0] is\n\t * the element we're animating from and [1] is the\n\t * element we're animating to\n\t */\n\tgetAutoAnimatableElements( fromSlide, toSlide ) {\n\n\t\tlet matcher = typeof this.Reveal.getConfig().autoAnimateMatcher === 'function' ? this.Reveal.getConfig().autoAnimateMatcher : this.getAutoAnimatePairs;\n\n\t\tlet pairs = matcher.call( this, fromSlide, toSlide );\n\n\t\tlet reserved = [];\n\n\t\t// Remove duplicate pairs\n\t\treturn pairs.filter( ( pair, index ) => {\n\t\t\tif( reserved.indexOf( pair.to ) === -1 ) {\n\t\t\t\treserved.push( pair.to );\n\t\t\t\treturn true;\n\t\t\t}\n\t\t} );\n\n\t}\n\n\t/**\n\t * Identifies matching elements between slides.\n\t *\n\t * You can specify a custom matcher function by using\n\t * the `autoAnimateMatcher` config option.\n\t */\n\tgetAutoAnimatePairs( fromSlide, toSlide ) {\n\n\t\tlet pairs = [];\n\n\t\tconst codeNodes = 'pre';\n\t\tconst textNodes = 'h1, h2, h3, h4, h5, h6, p, li';\n\t\tconst mediaNodes = 'img, video, iframe';\n\n\t\t// Eplicit matches via data-id\n\t\tthis.findAutoAnimateMatches( pairs, fromSlide, toSlide, '[data-id]', node => {\n\t\t\treturn node.nodeName + ':::' + node.getAttribute( 'data-id' );\n\t\t} );\n\n\t\t// Text\n\t\tthis.findAutoAnimateMatches( pairs, fromSlide, toSlide, textNodes, node => {\n\t\t\treturn node.nodeName + ':::' + node.innerText;\n\t\t} );\n\n\t\t// Media\n\t\tthis.findAutoAnimateMatches( pairs, fromSlide, toSlide, mediaNodes, node => {\n\t\t\treturn node.nodeName + ':::' + ( node.getAttribute( 'src' ) || node.getAttribute( 'data-src' ) );\n\t\t} );\n\n\t\t// Code\n\t\tthis.findAutoAnimateMatches( pairs, fromSlide, toSlide, codeNodes, node => {\n\t\t\treturn node.nodeName + ':::' + node.innerText;\n\t\t} );\n\n\t\tpairs.forEach( pair => {\n\n\t\t\t// Disable scale transformations on text nodes, we transition\n\t\t\t// each individual text property instead\n\t\t\tif( matches( pair.from, textNodes ) ) {\n\t\t\t\tpair.options = { scale: false };\n\t\t\t}\n\t\t\t// Animate individual lines of code\n\t\t\telse if( matches( pair.from, codeNodes ) ) {\n\n\t\t\t\t// Transition the code block's width and height instead of scaling\n\t\t\t\t// to prevent its content from being squished\n\t\t\t\tpair.options = { scale: false, styles: [ 'width', 'height' ] };\n\n\t\t\t\t// Lines of code\n\t\t\t\tthis.findAutoAnimateMatches( pairs, pair.from, pair.to, '.hljs .hljs-ln-code', node => {\n\t\t\t\t\treturn node.textContent;\n\t\t\t\t}, {\n\t\t\t\t\tscale: false,\n\t\t\t\t\tstyles: [],\n\t\t\t\t\tmeasure: this.getLocalBoundingBox.bind( this )\n\t\t\t\t} );\n\n\t\t\t\t// Line numbers\n\t\t\t\tthis.findAutoAnimateMatches( pairs, pair.from, pair.to, '.hljs .hljs-ln-line[data-line-number]', node => {\n\t\t\t\t\treturn node.getAttribute( 'data-line-number' );\n\t\t\t\t}, {\n\t\t\t\t\tscale: false,\n\t\t\t\t\tstyles: [ 'width' ],\n\t\t\t\t\tmeasure: this.getLocalBoundingBox.bind( this )\n\t\t\t\t} );\n\n\t\t\t}\n\n\t\t}, this );\n\n\t\treturn pairs;\n\n\t}\n\n\t/**\n\t * Helper method which returns a bounding box based on\n\t * the given elements offset coordinates.\n\t *\n\t * @param {HTMLElement} element\n\t * @return {Object} x, y, width, height\n\t */\n\tgetLocalBoundingBox( element ) {\n\n\t\tconst presentationScale = this.Reveal.getScale();\n\n\t\treturn {\n\t\t\tx: Math.round( ( element.offsetLeft * presentationScale ) * 100 ) / 100,\n\t\t\ty: Math.round( ( element.offsetTop * presentationScale ) * 100 ) / 100,\n\t\t\twidth: Math.round( ( element.offsetWidth * presentationScale ) * 100 ) / 100,\n\t\t\theight: Math.round( ( element.offsetHeight * presentationScale ) * 100 ) / 100\n\t\t};\n\n\t}\n\n\t/**\n\t * Finds matching elements between two slides.\n\t *\n\t * @param {Array} pairs \tList of pairs to push matches to\n\t * @param {HTMLElement} fromScope Scope within the from element exists\n\t * @param {HTMLElement} toScope Scope within the to element exists\n\t * @param {String} selector CSS selector of the element to match\n\t * @param {Function} serializer A function that accepts an element and returns\n\t * a stringified ID based on its contents\n\t * @param {Object} animationOptions Optional config options for this pair\n\t */\n\tfindAutoAnimateMatches( pairs, fromScope, toScope, selector, serializer, animationOptions ) {\n\n\t\tlet fromMatches = {};\n\t\tlet toMatches = {};\n\n\t\t[].slice.call( fromScope.querySelectorAll( selector ) ).forEach( ( element, i ) => {\n\t\t\tconst key = serializer( element );\n\t\t\tif( typeof key === 'string' && key.length ) {\n\t\t\t\tfromMatches[key] = fromMatches[key] || [];\n\t\t\t\tfromMatches[key].push( element );\n\t\t\t}\n\t\t} );\n\n\t\t[].slice.call( toScope.querySelectorAll( selector ) ).forEach( ( element, i ) => {\n\t\t\tconst key = serializer( element );\n\t\t\ttoMatches[key] = toMatches[key] || [];\n\t\t\ttoMatches[key].push( element );\n\n\t\t\tlet fromElement;\n\n\t\t\t// Retrieve the 'from' element\n\t\t\tif( fromMatches[key] ) {\n\t\t\t\tconst pimaryIndex = toMatches[key].length - 1;\n\t\t\t\tconst secondaryIndex = fromMatches[key].length - 1;\n\n\t\t\t\t// If there are multiple identical from elements, retrieve\n\t\t\t\t// the one at the same index as our to-element.\n\t\t\t\tif( fromMatches[key][ pimaryIndex ] ) {\n\t\t\t\t\tfromElement = fromMatches[key][ pimaryIndex ];\n\t\t\t\t\tfromMatches[key][ pimaryIndex ] = null;\n\t\t\t\t}\n\t\t\t\t// If there are no matching from-elements at the same index,\n\t\t\t\t// use the last one.\n\t\t\t\telse if( fromMatches[key][ secondaryIndex ] ) {\n\t\t\t\t\tfromElement = fromMatches[key][ secondaryIndex ];\n\t\t\t\t\tfromMatches[key][ secondaryIndex ] = null;\n\t\t\t\t}\n\t\t\t}\n\n\t\t\t// If we've got a matching pair, push it to the list of pairs\n\t\t\tif( fromElement ) {\n\t\t\t\tpairs.push({\n\t\t\t\t\tfrom: fromElement,\n\t\t\t\t\tto: element,\n\t\t\t\t\toptions: animationOptions\n\t\t\t\t});\n\t\t\t}\n\t\t} );\n\n\t}\n\n\t/**\n\t * Returns a all elements within the given scope that should\n\t * be considered unmatched in an auto-animate transition. If\n\t * fading of unmatched elements is turned on, these elements\n\t * will fade when going between auto-animate slides.\n\t *\n\t * Note that parents of auto-animate targets are NOT considerd\n\t * unmatched since fading them would break the auto-animation.\n\t *\n\t * @param {HTMLElement} rootElement\n\t * @return {Array}\n\t */\n\tgetUnmatchedAutoAnimateElements( rootElement ) {\n\n\t\treturn [].slice.call( rootElement.children ).reduce( ( result, element ) => {\n\n\t\t\tconst containsAnimatedElements = element.querySelector( '[data-auto-animate-target]' );\n\n\t\t\t// The element is unmatched if\n\t\t\t// - It is not an auto-animate target\n\t\t\t// - It does not contain any auto-animate targets\n\t\t\tif( !element.hasAttribute( 'data-auto-animate-target' ) && !containsAnimatedElements ) {\n\t\t\t\tresult.push( element );\n\t\t\t}\n\n\t\t\tif( element.querySelector( '[data-auto-animate-target]' ) ) {\n\t\t\t\tresult = result.concat( this.getUnmatchedAutoAnimateElements( element ) );\n\t\t\t}\n\n\t\t\treturn result;\n\n\t\t}, [] );\n\n\t}\n\n}\n","import { extend, queryAll } from '../utils/util.js'\n\n/**\n * Handles sorting and navigation of slide fragments.\n * Fragments are elements within a slide that are\n * revealed/animated incrementally.\n */\nexport default class Fragments {\n\n\tconstructor( Reveal ) {\n\n\t\tthis.Reveal = Reveal;\n\n\t}\n\n\t/**\n\t * Called when the reveal.js config is updated.\n\t */\n\tconfigure( config, oldConfig ) {\n\n\t\tif( config.fragments === false ) {\n\t\t\tthis.disable();\n\t\t}\n\t\telse if( oldConfig.fragments === false ) {\n\t\t\tthis.enable();\n\t\t}\n\n\t}\n\n\t/**\n\t * If fragments are disabled in the deck, they should all be\n\t * visible rather than stepped through.\n\t */\n\tdisable() {\n\n\t\tqueryAll( this.Reveal.getSlidesElement(), '.fragment' ).forEach( element => {\n\t\t\telement.classList.add( 'visible' );\n\t\t\telement.classList.remove( 'current-fragment' );\n\t\t} );\n\n\t}\n\n\t/**\n\t * Reverse of #disable(). Only called if fragments have\n\t * previously been disabled.\n\t */\n\tenable() {\n\n\t\tqueryAll( this.Reveal.getSlidesElement(), '.fragment' ).forEach( element => {\n\t\t\telement.classList.remove( 'visible' );\n\t\t\telement.classList.remove( 'current-fragment' );\n\t\t} );\n\n\t}\n\n\t/**\n\t * Returns an object describing the available fragment\n\t * directions.\n\t *\n\t * @return {{prev: boolean, next: boolean}}\n\t */\n\tavailableRoutes() {\n\n\t\tlet currentSlide = this.Reveal.getCurrentSlide();\n\t\tif( currentSlide && this.Reveal.getConfig().fragments ) {\n\t\t\tlet fragments = currentSlide.querySelectorAll( '.fragment:not(.disabled)' );\n\t\t\tlet hiddenFragments = currentSlide.querySelectorAll( '.fragment:not(.disabled):not(.visible)' );\n\n\t\t\treturn {\n\t\t\t\tprev: fragments.length - hiddenFragments.length > 0,\n\t\t\t\tnext: !!hiddenFragments.length\n\t\t\t};\n\t\t}\n\t\telse {\n\t\t\treturn { prev: false, next: false };\n\t\t}\n\n\t}\n\n\t/**\n\t * Return a sorted fragments list, ordered by an increasing\n\t * \"data-fragment-index\" attribute.\n\t *\n\t * Fragments will be revealed in the order that they are returned by\n\t * this function, so you can use the index attributes to control the\n\t * order of fragment appearance.\n\t *\n\t * To maintain a sensible default fragment order, fragments are presumed\n\t * to be passed in document order. This function adds a \"fragment-index\"\n\t * attribute to each node if such an attribute is not already present,\n\t * and sets that attribute to an integer value which is the position of\n\t * the fragment within the fragments list.\n\t *\n\t * @param {object[]|*} fragments\n\t * @param {boolean} grouped If true the returned array will contain\n\t * nested arrays for all fragments with the same index\n\t * @return {object[]} sorted Sorted array of fragments\n\t */\n\tsort( fragments, grouped = false ) {\n\n\t\tfragments = Array.from( fragments );\n\n\t\tlet ordered = [],\n\t\t\tunordered = [],\n\t\t\tsorted = [];\n\n\t\t// Group ordered and unordered elements\n\t\tfragments.forEach( fragment => {\n\t\t\tif( fragment.hasAttribute( 'data-fragment-index' ) ) {\n\t\t\t\tlet index = parseInt( fragment.getAttribute( 'data-fragment-index' ), 10 );\n\n\t\t\t\tif( !ordered[index] ) {\n\t\t\t\t\tordered[index] = [];\n\t\t\t\t}\n\n\t\t\t\tordered[index].push( fragment );\n\t\t\t}\n\t\t\telse {\n\t\t\t\tunordered.push( [ fragment ] );\n\t\t\t}\n\t\t} );\n\n\t\t// Append fragments without explicit indices in their\n\t\t// DOM order\n\t\tordered = ordered.concat( unordered );\n\n\t\t// Manually count the index up per group to ensure there\n\t\t// are no gaps\n\t\tlet index = 0;\n\n\t\t// Push all fragments in their sorted order to an array,\n\t\t// this flattens the groups\n\t\tordered.forEach( group => {\n\t\t\tgroup.forEach( fragment => {\n\t\t\t\tsorted.push( fragment );\n\t\t\t\tfragment.setAttribute( 'data-fragment-index', index );\n\t\t\t} );\n\n\t\t\tindex ++;\n\t\t} );\n\n\t\treturn grouped === true ? ordered : sorted;\n\n\t}\n\n\t/**\n\t * Sorts and formats all of fragments in the\n\t * presentation.\n\t */\n\tsortAll() {\n\n\t\tthis.Reveal.getHorizontalSlides().forEach( horizontalSlide => {\n\n\t\t\tlet verticalSlides = queryAll( horizontalSlide, 'section' );\n\t\t\tverticalSlides.forEach( ( verticalSlide, y ) => {\n\n\t\t\t\tthis.sort( verticalSlide.querySelectorAll( '.fragment' ) );\n\n\t\t\t}, this );\n\n\t\t\tif( verticalSlides.length === 0 ) this.sort( horizontalSlide.querySelectorAll( '.fragment' ) );\n\n\t\t} );\n\n\t}\n\n\t/**\n\t * Refreshes the fragments on the current slide so that they\n\t * have the appropriate classes (.visible + .current-fragment).\n\t *\n\t * @param {number} [index] The index of the current fragment\n\t * @param {array} [fragments] Array containing all fragments\n\t * in the current slide\n\t *\n\t * @return {{shown: array, hidden: array}}\n\t */\n\tupdate( index, fragments ) {\n\n\t\tlet changedFragments = {\n\t\t\tshown: [],\n\t\t\thidden: []\n\t\t};\n\n\t\tlet currentSlide = this.Reveal.getCurrentSlide();\n\t\tif( currentSlide && this.Reveal.getConfig().fragments ) {\n\n\t\t\tfragments = fragments || this.sort( currentSlide.querySelectorAll( '.fragment' ) );\n\n\t\t\tif( fragments.length ) {\n\n\t\t\t\tlet maxIndex = 0;\n\n\t\t\t\tif( typeof index !== 'number' ) {\n\t\t\t\t\tlet currentFragment = this.sort( currentSlide.querySelectorAll( '.fragment.visible' ) ).pop();\n\t\t\t\t\tif( currentFragment ) {\n\t\t\t\t\t\tindex = parseInt( currentFragment.getAttribute( 'data-fragment-index' ) || 0, 10 );\n\t\t\t\t\t}\n\t\t\t\t}\n\n\t\t\t\tArray.from( fragments ).forEach( ( el, i ) => {\n\n\t\t\t\t\tif( el.hasAttribute( 'data-fragment-index' ) ) {\n\t\t\t\t\t\ti = parseInt( el.getAttribute( 'data-fragment-index' ), 10 );\n\t\t\t\t\t}\n\n\t\t\t\t\tmaxIndex = Math.max( maxIndex, i );\n\n\t\t\t\t\t// Visible fragments\n\t\t\t\t\tif( i <= index ) {\n\t\t\t\t\t\tlet wasVisible = el.classList.contains( 'visible' )\n\t\t\t\t\t\tel.classList.add( 'visible' );\n\t\t\t\t\t\tel.classList.remove( 'current-fragment' );\n\n\t\t\t\t\t\tif( i === index ) {\n\t\t\t\t\t\t\t// Announce the fragments one by one to the Screen Reader\n\t\t\t\t\t\t\tthis.Reveal.announceStatus( this.Reveal.getStatusText( el ) );\n\n\t\t\t\t\t\t\tel.classList.add( 'current-fragment' );\n\t\t\t\t\t\t\tthis.Reveal.slideContent.startEmbeddedContent( el );\n\t\t\t\t\t\t}\n\n\t\t\t\t\t\tif( !wasVisible ) {\n\t\t\t\t\t\t\tchangedFragments.shown.push( el )\n\t\t\t\t\t\t\tthis.Reveal.dispatchEvent({\n\t\t\t\t\t\t\t\ttarget: el,\n\t\t\t\t\t\t\t\ttype: 'visible',\n\t\t\t\t\t\t\t\tbubbles: false\n\t\t\t\t\t\t\t});\n\t\t\t\t\t\t}\n\t\t\t\t\t}\n\t\t\t\t\t// Hidden fragments\n\t\t\t\t\telse {\n\t\t\t\t\t\tlet wasVisible = el.classList.contains( 'visible' )\n\t\t\t\t\t\tel.classList.remove( 'visible' );\n\t\t\t\t\t\tel.classList.remove( 'current-fragment' );\n\n\t\t\t\t\t\tif( wasVisible ) {\n\t\t\t\t\t\t\tthis.Reveal.slideContent.stopEmbeddedContent( el );\n\t\t\t\t\t\t\tchangedFragments.hidden.push( el );\n\t\t\t\t\t\t\tthis.Reveal.dispatchEvent({\n\t\t\t\t\t\t\t\ttarget: el,\n\t\t\t\t\t\t\t\ttype: 'hidden',\n\t\t\t\t\t\t\t\tbubbles: false\n\t\t\t\t\t\t\t});\n\t\t\t\t\t\t}\n\t\t\t\t\t}\n\n\t\t\t\t} );\n\n\t\t\t\t// Write the current fragment index to the slide
    .\n\t\t\t\t// This can be used by end users to apply styles based on\n\t\t\t\t// the current fragment index.\n\t\t\t\tindex = typeof index === 'number' ? index : -1;\n\t\t\t\tindex = Math.max( Math.min( index, maxIndex ), -1 );\n\t\t\t\tcurrentSlide.setAttribute( 'data-fragment', index );\n\n\t\t\t}\n\n\t\t}\n\n\t\treturn changedFragments;\n\n\t}\n\n\t/**\n\t * Formats the fragments on the given slide so that they have\n\t * valid indices. Call this if fragments are changed in the DOM\n\t * after reveal.js has already initialized.\n\t *\n\t * @param {HTMLElement} slide\n\t * @return {Array} a list of the HTML fragments that were synced\n\t */\n\tsync( slide = this.Reveal.getCurrentSlide() ) {\n\n\t\treturn this.sort( slide.querySelectorAll( '.fragment' ) );\n\n\t}\n\n\t/**\n\t * Navigate to the specified slide fragment.\n\t *\n\t * @param {?number} index The index of the fragment that\n\t * should be shown, -1 means all are invisible\n\t * @param {number} offset Integer offset to apply to the\n\t * fragment index\n\t *\n\t * @return {boolean} true if a change was made in any\n\t * fragments visibility as part of this call\n\t */\n\tgoto( index, offset = 0 ) {\n\n\t\tlet currentSlide = this.Reveal.getCurrentSlide();\n\t\tif( currentSlide && this.Reveal.getConfig().fragments ) {\n\n\t\t\tlet fragments = this.sort( currentSlide.querySelectorAll( '.fragment:not(.disabled)' ) );\n\t\t\tif( fragments.length ) {\n\n\t\t\t\t// If no index is specified, find the current\n\t\t\t\tif( typeof index !== 'number' ) {\n\t\t\t\t\tlet lastVisibleFragment = this.sort( currentSlide.querySelectorAll( '.fragment:not(.disabled).visible' ) ).pop();\n\n\t\t\t\t\tif( lastVisibleFragment ) {\n\t\t\t\t\t\tindex = parseInt( lastVisibleFragment.getAttribute( 'data-fragment-index' ) || 0, 10 );\n\t\t\t\t\t}\n\t\t\t\t\telse {\n\t\t\t\t\t\tindex = -1;\n\t\t\t\t\t}\n\t\t\t\t}\n\n\t\t\t\t// Apply the offset if there is one\n\t\t\t\tindex += offset;\n\n\t\t\t\tlet changedFragments = this.update( index, fragments );\n\n\t\t\t\tif( changedFragments.hidden.length ) {\n\t\t\t\t\tthis.Reveal.dispatchEvent({\n\t\t\t\t\t\ttype: 'fragmenthidden',\n\t\t\t\t\t\tdata: {\n\t\t\t\t\t\t\tfragment: changedFragments.hidden[0],\n\t\t\t\t\t\t\tfragments: changedFragments.hidden\n\t\t\t\t\t\t}\n\t\t\t\t\t});\n\t\t\t\t}\n\n\t\t\t\tif( changedFragments.shown.length ) {\n\t\t\t\t\tthis.Reveal.dispatchEvent({\n\t\t\t\t\t\ttype: 'fragmentshown',\n\t\t\t\t\t\tdata: {\n\t\t\t\t\t\t\tfragment: changedFragments.shown[0],\n\t\t\t\t\t\t\tfragments: changedFragments.shown\n\t\t\t\t\t\t}\n\t\t\t\t\t});\n\t\t\t\t}\n\n\t\t\t\tthis.Reveal.controls.update();\n\t\t\t\tthis.Reveal.progress.update();\n\n\t\t\t\tif( this.Reveal.getConfig().fragmentInURL ) {\n\t\t\t\t\tthis.Reveal.location.writeURL();\n\t\t\t\t}\n\n\t\t\t\treturn !!( changedFragments.shown.length || changedFragments.hidden.length );\n\n\t\t\t}\n\n\t\t}\n\n\t\treturn false;\n\n\t}\n\n\t/**\n\t * Navigate to the next slide fragment.\n\t *\n\t * @return {boolean} true if there was a next fragment,\n\t * false otherwise\n\t */\n\tnext() {\n\n\t\treturn this.goto( null, 1 );\n\n\t}\n\n\t/**\n\t * Navigate to the previous slide fragment.\n\t *\n\t * @return {boolean} true if there was a previous fragment,\n\t * false otherwise\n\t */\n\tprev() {\n\n\t\treturn this.goto( null, -1 );\n\n\t}\n\n}","import { SLIDES_SELECTOR } from '../utils/constants.js'\nimport { extend, queryAll, transformElement } from '../utils/util.js'\n\n/**\n * Handles all logic related to the overview mode\n * (birds-eye view of all slides).\n */\nexport default class Overview {\n\n\tconstructor( Reveal ) {\n\n\t\tthis.Reveal = Reveal;\n\n\t\tthis.active = false;\n\n\t\tthis.onSlideClicked = this.onSlideClicked.bind( this );\n\n\t}\n\n\t/**\n\t * Displays the overview of slides (quick nav) by scaling\n\t * down and arranging all slide elements.\n\t */\n\tactivate() {\n\n\t\t// Only proceed if enabled in config\n\t\tif( this.Reveal.getConfig().overview && !this.isActive() ) {\n\n\t\t\tthis.active = true;\n\n\t\t\tthis.Reveal.getRevealElement().classList.add( 'overview' );\n\n\t\t\t// Don't auto-slide while in overview mode\n\t\t\tthis.Reveal.cancelAutoSlide();\n\n\t\t\t// Move the backgrounds element into the slide container to\n\t\t\t// that the same scaling is applied\n\t\t\tthis.Reveal.getSlidesElement().appendChild( this.Reveal.getBackgroundsElement() );\n\n\t\t\t// Clicking on an overview slide navigates to it\n\t\t\tqueryAll( this.Reveal.getRevealElement(), SLIDES_SELECTOR ).forEach( slide => {\n\t\t\t\tif( !slide.classList.contains( 'stack' ) ) {\n\t\t\t\t\tslide.addEventListener( 'click', this.onSlideClicked, true );\n\t\t\t\t}\n\t\t\t} );\n\n\t\t\t// Calculate slide sizes\n\t\t\tconst margin = 70;\n\t\t\tconst slideSize = this.Reveal.getComputedSlideSize();\n\t\t\tthis.overviewSlideWidth = slideSize.width + margin;\n\t\t\tthis.overviewSlideHeight = slideSize.height + margin;\n\n\t\t\t// Reverse in RTL mode\n\t\t\tif( this.Reveal.getConfig().rtl ) {\n\t\t\t\tthis.overviewSlideWidth = -this.overviewSlideWidth;\n\t\t\t}\n\n\t\t\tthis.Reveal.updateSlidesVisibility();\n\n\t\t\tthis.layout();\n\t\t\tthis.update();\n\n\t\t\tthis.Reveal.layout();\n\n\t\t\tconst indices = this.Reveal.getIndices();\n\n\t\t\t// Notify observers of the overview showing\n\t\t\tthis.Reveal.dispatchEvent({\n\t\t\t\ttype: 'overviewshown',\n\t\t\t\tdata: {\n\t\t\t\t\t'indexh': indices.h,\n\t\t\t\t\t'indexv': indices.v,\n\t\t\t\t\t'currentSlide': this.Reveal.getCurrentSlide()\n\t\t\t\t}\n\t\t\t});\n\n\t\t}\n\n\t}\n\n\t/**\n\t * Uses CSS transforms to position all slides in a grid for\n\t * display inside of the overview mode.\n\t */\n\tlayout() {\n\n\t\t// Layout slides\n\t\tthis.Reveal.getHorizontalSlides().forEach( ( hslide, h ) => {\n\t\t\thslide.setAttribute( 'data-index-h', h );\n\t\t\ttransformElement( hslide, 'translate3d(' + ( h * this.overviewSlideWidth ) + 'px, 0, 0)' );\n\n\t\t\tif( hslide.classList.contains( 'stack' ) ) {\n\n\t\t\t\tqueryAll( hslide, 'section' ).forEach( ( vslide, v ) => {\n\t\t\t\t\tvslide.setAttribute( 'data-index-h', h );\n\t\t\t\t\tvslide.setAttribute( 'data-index-v', v );\n\n\t\t\t\t\ttransformElement( vslide, 'translate3d(0, ' + ( v * this.overviewSlideHeight ) + 'px, 0)' );\n\t\t\t\t} );\n\n\t\t\t}\n\t\t} );\n\n\t\t// Layout slide backgrounds\n\t\tArray.from( this.Reveal.getBackgroundsElement().childNodes ).forEach( ( hbackground, h ) => {\n\t\t\ttransformElement( hbackground, 'translate3d(' + ( h * this.overviewSlideWidth ) + 'px, 0, 0)' );\n\n\t\t\tqueryAll( hbackground, '.slide-background' ).forEach( ( vbackground, v ) => {\n\t\t\t\ttransformElement( vbackground, 'translate3d(0, ' + ( v * this.overviewSlideHeight ) + 'px, 0)' );\n\t\t\t} );\n\t\t} );\n\n\t}\n\n\t/**\n\t * Moves the overview viewport to the current slides.\n\t * Called each time the current slide changes.\n\t */\n\tupdate() {\n\n\t\tconst vmin = Math.min( window.innerWidth, window.innerHeight );\n\t\tconst scale = Math.max( vmin / 5, 150 ) / vmin;\n\t\tconst indices = this.Reveal.getIndices();\n\n\t\tthis.Reveal.transformSlides( {\n\t\t\toverview: [\n\t\t\t\t'scale('+ scale +')',\n\t\t\t\t'translateX('+ ( -indices.h * this.overviewSlideWidth ) +'px)',\n\t\t\t\t'translateY('+ ( -indices.v * this.overviewSlideHeight ) +'px)'\n\t\t\t].join( ' ' )\n\t\t} );\n\n\t}\n\n\t/**\n\t * Exits the slide overview and enters the currently\n\t * active slide.\n\t */\n\tdeactivate() {\n\n\t\t// Only proceed if enabled in config\n\t\tif( this.Reveal.getConfig().overview ) {\n\n\t\t\tthis.active = false;\n\n\t\t\tthis.Reveal.getRevealElement().classList.remove( 'overview' );\n\n\t\t\t// Temporarily add a class so that transitions can do different things\n\t\t\t// depending on whether they are exiting/entering overview, or just\n\t\t\t// moving from slide to slide\n\t\t\tthis.Reveal.getRevealElement().classList.add( 'overview-deactivating' );\n\n\t\t\tsetTimeout( () => {\n\t\t\t\tthis.Reveal.getRevealElement().classList.remove( 'overview-deactivating' );\n\t\t\t}, 1 );\n\n\t\t\t// Move the background element back out\n\t\t\tthis.Reveal.getRevealElement().appendChild( this.Reveal.getBackgroundsElement() );\n\n\t\t\t// Clean up changes made to slides\n\t\t\tqueryAll( this.Reveal.getRevealElement(), SLIDES_SELECTOR ).forEach( slide => {\n\t\t\t\ttransformElement( slide, '' );\n\n\t\t\t\tslide.removeEventListener( 'click', this.onSlideClicked, true );\n\t\t\t} );\n\n\t\t\t// Clean up changes made to backgrounds\n\t\t\tqueryAll( this.Reveal.getBackgroundsElement(), '.slide-background' ).forEach( background => {\n\t\t\t\ttransformElement( background, '' );\n\t\t\t} );\n\n\t\t\tthis.Reveal.transformSlides( { overview: '' } );\n\n\t\t\tconst indices = this.Reveal.getIndices();\n\n\t\t\tthis.Reveal.slide( indices.h, indices.v );\n\t\t\tthis.Reveal.layout();\n\t\t\tthis.Reveal.cueAutoSlide();\n\n\t\t\t// Notify observers of the overview hiding\n\t\t\tthis.Reveal.dispatchEvent({\n\t\t\t\ttype: 'overviewhidden',\n\t\t\t\tdata: {\n\t\t\t\t\t'indexh': indices.h,\n\t\t\t\t\t'indexv': indices.v,\n\t\t\t\t\t'currentSlide': this.Reveal.getCurrentSlide()\n\t\t\t\t}\n\t\t\t});\n\n\t\t}\n\t}\n\n\t/**\n\t * Toggles the slide overview mode on and off.\n\t *\n\t * @param {Boolean} [override] Flag which overrides the\n\t * toggle logic and forcibly sets the desired state. True means\n\t * overview is open, false means it's closed.\n\t */\n\ttoggle( override ) {\n\n\t\tif( typeof override === 'boolean' ) {\n\t\t\toverride ? this.activate() : this.deactivate();\n\t\t}\n\t\telse {\n\t\t\tthis.isActive() ? this.deactivate() : this.activate();\n\t\t}\n\n\t}\n\n\t/**\n\t * Checks if the overview is currently active.\n\t *\n\t * @return {Boolean} true if the overview is active,\n\t * false otherwise\n\t */\n\tisActive() {\n\n\t\treturn this.active;\n\n\t}\n\n\t/**\n\t * Invoked when a slide is and we're in the overview.\n\t *\n\t * @param {object} event\n\t */\n\tonSlideClicked( event ) {\n\n\t\tif( this.isActive() ) {\n\t\t\tevent.preventDefault();\n\n\t\t\tlet element = event.target;\n\n\t\t\twhile( element && !element.nodeName.match( /section/gi ) ) {\n\t\t\t\telement = element.parentNode;\n\t\t\t}\n\n\t\t\tif( element && !element.classList.contains( 'disabled' ) ) {\n\n\t\t\t\tthis.deactivate();\n\n\t\t\t\tif( element.nodeName.match( /section/gi ) ) {\n\t\t\t\t\tlet h = parseInt( element.getAttribute( 'data-index-h' ), 10 ),\n\t\t\t\t\t\tv = parseInt( element.getAttribute( 'data-index-v' ), 10 );\n\n\t\t\t\t\tthis.Reveal.slide( h, v );\n\t\t\t\t}\n\n\t\t\t}\n\t\t}\n\n\t}\n\n}","import { enterFullscreen } from '../utils/util.js'\n\n/**\n * Handles all reveal.js keyboard interactions.\n */\nexport default class Keyboard {\n\n\tconstructor( Reveal ) {\n\n\t\tthis.Reveal = Reveal;\n\n\t\t// A key:value map of keyboard keys and descriptions of\n\t\t// the actions they trigger\n\t\tthis.shortcuts = {};\n\n\t\t// Holds custom key code mappings\n\t\tthis.bindings = {};\n\n\t\tthis.onDocumentKeyDown = this.onDocumentKeyDown.bind( this );\n\t\tthis.onDocumentKeyPress = this.onDocumentKeyPress.bind( this );\n\n\t}\n\n\t/**\n\t * Called when the reveal.js config is updated.\n\t */\n\tconfigure( config, oldConfig ) {\n\n\t\tif( config.navigationMode === 'linear' ) {\n\t\t\tthis.shortcuts['→ , ↓ , SPACE , N , L , J'] = 'Next slide';\n\t\t\tthis.shortcuts['← , ↑ , P , H , K'] = 'Previous slide';\n\t\t}\n\t\telse {\n\t\t\tthis.shortcuts['N , SPACE'] = 'Next slide';\n\t\t\tthis.shortcuts['P , Shift SPACE'] = 'Previous slide';\n\t\t\tthis.shortcuts['← , H'] = 'Navigate left';\n\t\t\tthis.shortcuts['→ , L'] = 'Navigate right';\n\t\t\tthis.shortcuts['↑ , K'] = 'Navigate up';\n\t\t\tthis.shortcuts['↓ , J'] = 'Navigate down';\n\t\t}\n\n\t\tthis.shortcuts['Alt + ←/↑/→/↓'] = 'Navigate without fragments';\n\t\tthis.shortcuts['Shift + ←/↑/→/↓'] = 'Jump to first/last slide';\n\t\tthis.shortcuts['B , .'] = 'Pause';\n\t\tthis.shortcuts['F'] = 'Fullscreen';\n\t\tthis.shortcuts['ESC, O'] = 'Slide overview';\n\n\t}\n\n\t/**\n\t * Starts listening for keyboard events.\n\t */\n\tbind() {\n\n\t\tdocument.addEventListener( 'keydown', this.onDocumentKeyDown, false );\n\t\tdocument.addEventListener( 'keypress', this.onDocumentKeyPress, false );\n\n\t}\n\n\t/**\n\t * Stops listening for keyboard events.\n\t */\n\tunbind() {\n\n\t\tdocument.removeEventListener( 'keydown', this.onDocumentKeyDown, false );\n\t\tdocument.removeEventListener( 'keypress', this.onDocumentKeyPress, false );\n\n\t}\n\n\t/**\n\t * Add a custom key binding with optional description to\n\t * be added to the help screen.\n\t */\n\taddKeyBinding( binding, callback ) {\n\n\t\tif( typeof binding === 'object' && binding.keyCode ) {\n\t\t\tthis.bindings[binding.keyCode] = {\n\t\t\t\tcallback: callback,\n\t\t\t\tkey: binding.key,\n\t\t\t\tdescription: binding.description\n\t\t\t};\n\t\t}\n\t\telse {\n\t\t\tthis.bindings[binding] = {\n\t\t\t\tcallback: callback,\n\t\t\t\tkey: null,\n\t\t\t\tdescription: null\n\t\t\t};\n\t\t}\n\n\t}\n\n\t/**\n\t * Removes the specified custom key binding.\n\t */\n\tremoveKeyBinding( keyCode ) {\n\n\t\tdelete this.bindings[keyCode];\n\n\t}\n\n\t/**\n\t * Programmatically triggers a keyboard event\n\t *\n\t * @param {int} keyCode\n\t */\n\ttriggerKey( keyCode ) {\n\n\t\tthis.onDocumentKeyDown( { keyCode } );\n\n\t}\n\n\t/**\n\t * Registers a new shortcut to include in the help overlay\n\t *\n\t * @param {String} key\n\t * @param {String} value\n\t */\n\tregisterKeyboardShortcut( key, value ) {\n\n\t\tthis.shortcuts[key] = value;\n\n\t}\n\n\tgetShortcuts() {\n\n\t\treturn this.shortcuts;\n\n\t}\n\n\tgetBindings() {\n\n\t\treturn this.bindings;\n\n\t}\n\n\t/**\n\t * Handler for the document level 'keypress' event.\n\t *\n\t * @param {object} event\n\t */\n\tonDocumentKeyPress( event ) {\n\n\t\t// Check if the pressed key is question mark\n\t\tif( event.shiftKey && event.charCode === 63 ) {\n\t\t\tthis.Reveal.toggleHelp();\n\t\t}\n\n\t}\n\n\t/**\n\t * Handler for the document level 'keydown' event.\n\t *\n\t * @param {object} event\n\t */\n\tonDocumentKeyDown( event ) {\n\n\t\tlet config = this.Reveal.getConfig();\n\n\t\t// If there's a condition specified and it returns false,\n\t\t// ignore this event\n\t\tif( typeof config.keyboardCondition === 'function' && config.keyboardCondition(event) === false ) {\n\t\t\treturn true;\n\t\t}\n\n\t\t// If keyboardCondition is set, only capture keyboard events\n\t\t// for embedded decks when they are focused\n\t\tif( config.keyboardCondition === 'focused' && !this.Reveal.isFocused() ) {\n\t\t\treturn true;\n\t\t}\n\n\t\t// Shorthand\n\t\tlet keyCode = event.keyCode;\n\n\t\t// Remember if auto-sliding was paused so we can toggle it\n\t\tlet autoSlideWasPaused = !this.Reveal.isAutoSliding();\n\n\t\tthis.Reveal.onUserInput( event );\n\n\t\t// Is there a focused element that could be using the keyboard?\n\t\tlet activeElementIsCE = document.activeElement && document.activeElement.isContentEditable === true;\n\t\tlet activeElementIsInput = document.activeElement && document.activeElement.tagName && /input|textarea/i.test( document.activeElement.tagName );\n\t\tlet activeElementIsNotes = document.activeElement && document.activeElement.className && /speaker-notes/i.test( document.activeElement.className);\n\n\t\t// Whitelist certain modifiers for slide navigation shortcuts\n\t\tlet isNavigationKey = [32, 37, 38, 39, 40, 78, 80].indexOf( event.keyCode ) !== -1;\n\n\t\t// Prevent all other events when a modifier is pressed\n\t\tlet unusedModifier = \t!( isNavigationKey && event.shiftKey || event.altKey ) &&\n\t\t\t\t\t\t\t\t( event.shiftKey || event.altKey || event.ctrlKey || event.metaKey );\n\n\t\t// Disregard the event if there's a focused element or a\n\t\t// keyboard modifier key is present\n\t\tif( activeElementIsCE || activeElementIsInput || activeElementIsNotes || unusedModifier ) return;\n\n\t\t// While paused only allow resume keyboard events; 'b', 'v', '.'\n\t\tlet resumeKeyCodes = [66,86,190,191];\n\t\tlet key;\n\n\t\t// Custom key bindings for togglePause should be able to resume\n\t\tif( typeof config.keyboard === 'object' ) {\n\t\t\tfor( key in config.keyboard ) {\n\t\t\t\tif( config.keyboard[key] === 'togglePause' ) {\n\t\t\t\t\tresumeKeyCodes.push( parseInt( key, 10 ) );\n\t\t\t\t}\n\t\t\t}\n\t\t}\n\n\t\tif( this.Reveal.isPaused() && resumeKeyCodes.indexOf( keyCode ) === -1 ) {\n\t\t\treturn false;\n\t\t}\n\n\t\t// Use linear navigation if we're configured to OR if\n\t\t// the presentation is one-dimensional\n\t\tlet useLinearMode = config.navigationMode === 'linear' || !this.Reveal.hasHorizontalSlides() || !this.Reveal.hasVerticalSlides();\n\n\t\tlet triggered = false;\n\n\t\t// 1. User defined key bindings\n\t\tif( typeof config.keyboard === 'object' ) {\n\n\t\t\tfor( key in config.keyboard ) {\n\n\t\t\t\t// Check if this binding matches the pressed key\n\t\t\t\tif( parseInt( key, 10 ) === keyCode ) {\n\n\t\t\t\t\tlet value = config.keyboard[ key ];\n\n\t\t\t\t\t// Callback function\n\t\t\t\t\tif( typeof value === 'function' ) {\n\t\t\t\t\t\tvalue.apply( null, [ event ] );\n\t\t\t\t\t}\n\t\t\t\t\t// String shortcuts to reveal.js API\n\t\t\t\t\telse if( typeof value === 'string' && typeof this.Reveal[ value ] === 'function' ) {\n\t\t\t\t\t\tthis.Reveal[ value ].call();\n\t\t\t\t\t}\n\n\t\t\t\t\ttriggered = true;\n\n\t\t\t\t}\n\n\t\t\t}\n\n\t\t}\n\n\t\t// 2. Registered custom key bindings\n\t\tif( triggered === false ) {\n\n\t\t\tfor( key in this.bindings ) {\n\n\t\t\t\t// Check if this binding matches the pressed key\n\t\t\t\tif( parseInt( key, 10 ) === keyCode ) {\n\n\t\t\t\t\tlet action = this.bindings[ key ].callback;\n\n\t\t\t\t\t// Callback function\n\t\t\t\t\tif( typeof action === 'function' ) {\n\t\t\t\t\t\taction.apply( null, [ event ] );\n\t\t\t\t\t}\n\t\t\t\t\t// String shortcuts to reveal.js API\n\t\t\t\t\telse if( typeof action === 'string' && typeof this.Reveal[ action ] === 'function' ) {\n\t\t\t\t\t\tthis.Reveal[ action ].call();\n\t\t\t\t\t}\n\n\t\t\t\t\ttriggered = true;\n\t\t\t\t}\n\t\t\t}\n\t\t}\n\n\t\t// 3. System defined key bindings\n\t\tif( triggered === false ) {\n\n\t\t\t// Assume true and try to prove false\n\t\t\ttriggered = true;\n\n\t\t\t// P, PAGE UP\n\t\t\tif( keyCode === 80 || keyCode === 33 ) {\n\t\t\t\tthis.Reveal.prev({skipFragments: event.altKey});\n\t\t\t}\n\t\t\t// N, PAGE DOWN\n\t\t\telse if( keyCode === 78 || keyCode === 34 ) {\n\t\t\t\tthis.Reveal.next({skipFragments: event.altKey});\n\t\t\t}\n\t\t\t// H, LEFT\n\t\t\telse if( keyCode === 72 || keyCode === 37 ) {\n\t\t\t\tif( event.shiftKey ) {\n\t\t\t\t\tthis.Reveal.slide( 0 );\n\t\t\t\t}\n\t\t\t\telse if( !this.Reveal.overview.isActive() && useLinearMode ) {\n\t\t\t\t\tthis.Reveal.prev({skipFragments: event.altKey});\n\t\t\t\t}\n\t\t\t\telse {\n\t\t\t\t\tthis.Reveal.left({skipFragments: event.altKey});\n\t\t\t\t}\n\t\t\t}\n\t\t\t// L, RIGHT\n\t\t\telse if( keyCode === 76 || keyCode === 39 ) {\n\t\t\t\tif( event.shiftKey ) {\n\t\t\t\t\tthis.Reveal.slide( this.Reveal.getHorizontalSlides().length - 1 );\n\t\t\t\t}\n\t\t\t\telse if( !this.Reveal.overview.isActive() && useLinearMode ) {\n\t\t\t\t\tthis.Reveal.next({skipFragments: event.altKey});\n\t\t\t\t}\n\t\t\t\telse {\n\t\t\t\t\tthis.Reveal.right({skipFragments: event.altKey});\n\t\t\t\t}\n\t\t\t}\n\t\t\t// K, UP\n\t\t\telse if( keyCode === 75 || keyCode === 38 ) {\n\t\t\t\tif( event.shiftKey ) {\n\t\t\t\t\tthis.Reveal.slide( undefined, 0 );\n\t\t\t\t}\n\t\t\t\telse if( !this.Reveal.overview.isActive() && useLinearMode ) {\n\t\t\t\t\tthis.Reveal.prev({skipFragments: event.altKey});\n\t\t\t\t}\n\t\t\t\telse {\n\t\t\t\t\tthis.Reveal.up({skipFragments: event.altKey});\n\t\t\t\t}\n\t\t\t}\n\t\t\t// J, DOWN\n\t\t\telse if( keyCode === 74 || keyCode === 40 ) {\n\t\t\t\tif( event.shiftKey ) {\n\t\t\t\t\tthis.Reveal.slide( undefined, Number.MAX_VALUE );\n\t\t\t\t}\n\t\t\t\telse if( !this.Reveal.overview.isActive() && useLinearMode ) {\n\t\t\t\t\tthis.Reveal.next({skipFragments: event.altKey});\n\t\t\t\t}\n\t\t\t\telse {\n\t\t\t\t\tthis.Reveal.down({skipFragments: event.altKey});\n\t\t\t\t}\n\t\t\t}\n\t\t\t// HOME\n\t\t\telse if( keyCode === 36 ) {\n\t\t\t\tthis.Reveal.slide( 0 );\n\t\t\t}\n\t\t\t// END\n\t\t\telse if( keyCode === 35 ) {\n\t\t\t\tthis.Reveal.slide( this.Reveal.getHorizontalSlides().length - 1 );\n\t\t\t}\n\t\t\t// SPACE\n\t\t\telse if( keyCode === 32 ) {\n\t\t\t\tif( this.Reveal.overview.isActive() ) {\n\t\t\t\t\tthis.Reveal.overview.deactivate();\n\t\t\t\t}\n\t\t\t\tif( event.shiftKey ) {\n\t\t\t\t\tthis.Reveal.prev({skipFragments: event.altKey});\n\t\t\t\t}\n\t\t\t\telse {\n\t\t\t\t\tthis.Reveal.next({skipFragments: event.altKey});\n\t\t\t\t}\n\t\t\t}\n\t\t\t// TWO-SPOT, SEMICOLON, B, V, PERIOD, LOGITECH PRESENTER TOOLS \"BLACK SCREEN\" BUTTON\n\t\t\telse if( keyCode === 58 || keyCode === 59 || keyCode === 66 || keyCode === 86 || keyCode === 190 || keyCode === 191 ) {\n\t\t\t\tthis.Reveal.togglePause();\n\t\t\t}\n\t\t\t// F\n\t\t\telse if( keyCode === 70 ) {\n\t\t\t\tenterFullscreen( config.embedded ? this.Reveal.getViewportElement() : document.documentElement );\n\t\t\t}\n\t\t\t// A\n\t\t\telse if( keyCode === 65 ) {\n\t\t\t\tif ( config.autoSlideStoppable ) {\n\t\t\t\t\tthis.Reveal.toggleAutoSlide( autoSlideWasPaused );\n\t\t\t\t}\n\t\t\t}\n\t\t\telse {\n\t\t\t\ttriggered = false;\n\t\t\t}\n\n\t\t}\n\n\t\t// If the input resulted in a triggered action we should prevent\n\t\t// the browsers default behavior\n\t\tif( triggered ) {\n\t\t\tevent.preventDefault && event.preventDefault();\n\t\t}\n\t\t// ESC or O key\n\t\telse if( keyCode === 27 || keyCode === 79 ) {\n\t\t\tif( this.Reveal.closeOverlay() === false ) {\n\t\t\t\tthis.Reveal.overview.toggle();\n\t\t\t}\n\n\t\t\tevent.preventDefault && event.preventDefault();\n\t\t}\n\n\t\t// If auto-sliding is enabled we need to cue up\n\t\t// another timeout\n\t\tthis.Reveal.cueAutoSlide();\n\n\t}\n\n}","/**\n * Reads and writes the URL based on reveal.js' current state.\n */\nexport default class Location {\n\n\t// The minimum number of milliseconds that must pass between\n\t// calls to history.replaceState\n\tMAX_REPLACE_STATE_FREQUENCY = 1000\n\n\tconstructor( Reveal ) {\n\n\t\tthis.Reveal = Reveal;\n\n\t\t// Delays updates to the URL due to a Chrome thumbnailer bug\n\t\tthis.writeURLTimeout = 0;\n\n\t\tthis.replaceStateTimestamp = 0;\n\n\t\tthis.onWindowHashChange = this.onWindowHashChange.bind( this );\n\n\t}\n\n\tbind() {\n\n\t\twindow.addEventListener( 'hashchange', this.onWindowHashChange, false );\n\n\t}\n\n\tunbind() {\n\n\t\twindow.removeEventListener( 'hashchange', this.onWindowHashChange, false );\n\n\t}\n\n\t/**\n\t * Returns the slide indices for the given hash link.\n\t *\n\t * @param {string} [hash] the hash string that we want to\n\t * find the indices for\n\t *\n\t * @returns slide indices or null\n\t */\n\tgetIndicesFromHash( hash=window.location.hash ) {\n\n\t\t// Attempt to parse the hash as either an index or name\n\t\tlet name = hash.replace( /^#\\/?/, '' );\n\t\tlet bits = name.split( '/' );\n\n\t\t// If the first bit is not fully numeric and there is a name we\n\t\t// can assume that this is a named link\n\t\tif( !/^[0-9]*$/.test( bits[0] ) && name.length ) {\n\t\t\tlet element;\n\n\t\t\tlet f;\n\n\t\t\t// Parse named links with fragments (#/named-link/2)\n\t\t\tif( /\\/[-\\d]+$/g.test( name ) ) {\n\t\t\t\tf = parseInt( name.split( '/' ).pop(), 10 );\n\t\t\t\tf = isNaN(f) ? undefined : f;\n\t\t\t\tname = name.split( '/' ).shift();\n\t\t\t}\n\n\t\t\t// Ensure the named link is a valid HTML ID attribute\n\t\t\ttry {\n\t\t\t\telement = document.getElementById( decodeURIComponent( name ) );\n\t\t\t}\n\t\t\tcatch ( error ) { }\n\n\t\t\tif( element ) {\n\t\t\t\treturn { ...this.Reveal.getIndices( element ), f };\n\t\t\t}\n\t\t}\n\t\telse {\n\t\t\tconst config = this.Reveal.getConfig();\n\t\t\tlet hashIndexBase = config.hashOneBasedIndex ? 1 : 0;\n\n\t\t\t// Read the index components of the hash\n\t\t\tlet h = ( parseInt( bits[0], 10 ) - hashIndexBase ) || 0,\n\t\t\t\tv = ( parseInt( bits[1], 10 ) - hashIndexBase ) || 0,\n\t\t\t\tf;\n\n\t\t\tif( config.fragmentInURL ) {\n\t\t\t\tf = parseInt( bits[2], 10 );\n\t\t\t\tif( isNaN( f ) ) {\n\t\t\t\t\tf = undefined;\n\t\t\t\t}\n\t\t\t}\n\n\t\t\treturn { h, v, f };\n\t\t}\n\n\t\t// The hash couldn't be parsed or no matching named link was found\n\t\treturn null\n\n\t}\n\n\t/**\n\t * Reads the current URL (hash) and navigates accordingly.\n\t */\n\treadURL() {\n\n\t\tconst currentIndices = this.Reveal.getIndices();\n\t\tconst newIndices = this.getIndicesFromHash();\n\n\t\tif( newIndices ) {\n\t\t\tif( ( newIndices.h !== currentIndices.h || newIndices.v !== currentIndices.v || newIndices.f !== undefined ) ) {\n\t\t\t\t\tthis.Reveal.slide( newIndices.h, newIndices.v, newIndices.f );\n\t\t\t}\n\t\t}\n\t\t// If no new indices are available, we're trying to navigate to\n\t\t// a slide hash that does not exist\n\t\telse {\n\t\t\tthis.Reveal.slide( currentIndices.h || 0, currentIndices.v || 0 );\n\t\t}\n\n\t}\n\n\t/**\n\t * Updates the page URL (hash) to reflect the current\n\t * state.\n\t *\n\t * @param {number} delay The time in ms to wait before\n\t * writing the hash\n\t */\n\twriteURL( delay ) {\n\n\t\tlet config = this.Reveal.getConfig();\n\t\tlet currentSlide = this.Reveal.getCurrentSlide();\n\n\t\t// Make sure there's never more than one timeout running\n\t\tclearTimeout( this.writeURLTimeout );\n\n\t\t// If a delay is specified, timeout this call\n\t\tif( typeof delay === 'number' ) {\n\t\t\tthis.writeURLTimeout = setTimeout( this.writeURL, delay );\n\t\t}\n\t\telse if( currentSlide ) {\n\n\t\t\tlet hash = this.getHash();\n\n\t\t\t// If we're configured to push to history OR the history\n\t\t\t// API is not avaialble.\n\t\t\tif( config.history ) {\n\t\t\t\twindow.location.hash = hash;\n\t\t\t}\n\t\t\t// If we're configured to reflect the current slide in the\n\t\t\t// URL without pushing to history.\n\t\t\telse if( config.hash ) {\n\t\t\t\t// If the hash is empty, don't add it to the URL\n\t\t\t\tif( hash === '/' ) {\n\t\t\t\t\tthis.debouncedReplaceState( window.location.pathname + window.location.search );\n\t\t\t\t}\n\t\t\t\telse {\n\t\t\t\t\tthis.debouncedReplaceState( '#' + hash );\n\t\t\t\t}\n\t\t\t}\n\t\t\t// UPDATE: The below nuking of all hash changes breaks\n\t\t\t// anchors on pages where reveal.js is running. Removed\n\t\t\t// in 4.0. Why was it here in the first place? ¯\\_(ツ)_/¯\n\t\t\t//\n\t\t\t// If history and hash are both disabled, a hash may still\n\t\t\t// be added to the URL by clicking on a href with a hash\n\t\t\t// target. Counter this by always removing the hash.\n\t\t\t// else {\n\t\t\t// \twindow.history.replaceState( null, null, window.location.pathname + window.location.search );\n\t\t\t// }\n\n\t\t}\n\n\t}\n\n\treplaceState( url ) {\n\n\t\twindow.history.replaceState( null, null, url );\n\t\tthis.replaceStateTimestamp = Date.now();\n\n\t}\n\n\tdebouncedReplaceState( url ) {\n\n\t\tclearTimeout( this.replaceStateTimeout );\n\n\t\tif( Date.now() - this.replaceStateTimestamp > this.MAX_REPLACE_STATE_FREQUENCY ) {\n\t\t\tthis.replaceState( url );\n\t\t}\n\t\telse {\n\t\t\tthis.replaceStateTimeout = setTimeout( () => this.replaceState( url ), this.MAX_REPLACE_STATE_FREQUENCY );\n\t\t}\n\n\t}\n\n\t/**\n\t * Return a hash URL that will resolve to the given slide location.\n\t *\n\t * @param {HTMLElement} [slide=currentSlide] The slide to link to\n\t */\n\tgetHash( slide ) {\n\n\t\tlet url = '/';\n\n\t\t// Attempt to create a named link based on the slide's ID\n\t\tlet s = slide || this.Reveal.getCurrentSlide();\n\t\tlet id = s ? s.getAttribute( 'id' ) : null;\n\t\tif( id ) {\n\t\t\tid = encodeURIComponent( id );\n\t\t}\n\n\t\tlet index = this.Reveal.getIndices( slide );\n\t\tif( !this.Reveal.getConfig().fragmentInURL ) {\n\t\t\tindex.f = undefined;\n\t\t}\n\n\t\t// If the current slide has an ID, use that as a named link,\n\t\t// but we don't support named links with a fragment index\n\t\tif( typeof id === 'string' && id.length ) {\n\t\t\turl = '/' + id;\n\n\t\t\t// If there is also a fragment, append that at the end\n\t\t\t// of the named link, like: #/named-link/2\n\t\t\tif( index.f >= 0 ) url += '/' + index.f;\n\t\t}\n\t\t// Otherwise use the /h/v index\n\t\telse {\n\t\t\tlet hashIndexBase = this.Reveal.getConfig().hashOneBasedIndex ? 1 : 0;\n\t\t\tif( index.h > 0 || index.v > 0 || index.f >= 0 ) url += index.h + hashIndexBase;\n\t\t\tif( index.v > 0 || index.f >= 0 ) url += '/' + (index.v + hashIndexBase );\n\t\t\tif( index.f >= 0 ) url += '/' + index.f;\n\t\t}\n\n\t\treturn url;\n\n\t}\n\n\t/**\n\t * Handler for the window level 'hashchange' event.\n\t *\n\t * @param {object} [event]\n\t */\n\tonWindowHashChange( event ) {\n\n\t\tthis.readURL();\n\n\t}\n\n}","import { queryAll } from '../utils/util.js'\nimport { isAndroid } from '../utils/device.js'\n\n/**\n * Manages our presentation controls. This includes both\n * the built-in control arrows as well as event monitoring\n * of any elements within the presentation with either of the\n * following helper classes:\n * - .navigate-up\n * - .navigate-right\n * - .navigate-down\n * - .navigate-left\n * - .navigate-next\n * - .navigate-prev\n */\nexport default class Controls {\n\n\tconstructor( Reveal ) {\n\n\t\tthis.Reveal = Reveal;\n\n\t\tthis.onNavigateLeftClicked = this.onNavigateLeftClicked.bind( this );\n\t\tthis.onNavigateRightClicked = this.onNavigateRightClicked.bind( this );\n\t\tthis.onNavigateUpClicked = this.onNavigateUpClicked.bind( this );\n\t\tthis.onNavigateDownClicked = this.onNavigateDownClicked.bind( this );\n\t\tthis.onNavigatePrevClicked = this.onNavigatePrevClicked.bind( this );\n\t\tthis.onNavigateNextClicked = this.onNavigateNextClicked.bind( this );\n\n\t}\n\n\trender() {\n\n\t\tconst rtl = this.Reveal.getConfig().rtl;\n\t\tconst revealElement = this.Reveal.getRevealElement();\n\n\t\tthis.element = document.createElement( 'aside' );\n\t\tthis.element.className = 'controls';\n\t\tthis.element.innerHTML =\n\t\t\t`\n\t\t\t\n\t\t\t\n\t\t\t`;\n\n\t\tthis.Reveal.getRevealElement().appendChild( this.element );\n\n\t\t// There can be multiple instances of controls throughout the page\n\t\tthis.controlsLeft = queryAll( revealElement, '.navigate-left' );\n\t\tthis.controlsRight = queryAll( revealElement, '.navigate-right' );\n\t\tthis.controlsUp = queryAll( revealElement, '.navigate-up' );\n\t\tthis.controlsDown = queryAll( revealElement, '.navigate-down' );\n\t\tthis.controlsPrev = queryAll( revealElement, '.navigate-prev' );\n\t\tthis.controlsNext = queryAll( revealElement, '.navigate-next' );\n\n\t\t// The left, right and down arrows in the standard reveal.js controls\n\t\tthis.controlsRightArrow = this.element.querySelector( '.navigate-right' );\n\t\tthis.controlsLeftArrow = this.element.querySelector( '.navigate-left' );\n\t\tthis.controlsDownArrow = this.element.querySelector( '.navigate-down' );\n\n\t}\n\n\t/**\n\t * Called when the reveal.js config is updated.\n\t */\n\tconfigure( config, oldConfig ) {\n\n\t\tthis.element.style.display = config.controls ? 'block' : 'none';\n\n\t\tthis.element.setAttribute( 'data-controls-layout', config.controlsLayout );\n\t\tthis.element.setAttribute( 'data-controls-back-arrows', config.controlsBackArrows );\n\n\t}\n\n\tbind() {\n\n\t\t// Listen to both touch and click events, in case the device\n\t\t// supports both\n\t\tlet pointerEvents = [ 'touchstart', 'click' ];\n\n\t\t// Only support touch for Android, fixes double navigations in\n\t\t// stock browser\n\t\tif( isAndroid ) {\n\t\t\tpointerEvents = [ 'touchstart' ];\n\t\t}\n\n\t\tpointerEvents.forEach( eventName => {\n\t\t\tthis.controlsLeft.forEach( el => el.addEventListener( eventName, this.onNavigateLeftClicked, false ) );\n\t\t\tthis.controlsRight.forEach( el => el.addEventListener( eventName, this.onNavigateRightClicked, false ) );\n\t\t\tthis.controlsUp.forEach( el => el.addEventListener( eventName, this.onNavigateUpClicked, false ) );\n\t\t\tthis.controlsDown.forEach( el => el.addEventListener( eventName, this.onNavigateDownClicked, false ) );\n\t\t\tthis.controlsPrev.forEach( el => el.addEventListener( eventName, this.onNavigatePrevClicked, false ) );\n\t\t\tthis.controlsNext.forEach( el => el.addEventListener( eventName, this.onNavigateNextClicked, false ) );\n\t\t} );\n\n\t}\n\n\tunbind() {\n\n\t\t[ 'touchstart', 'click' ].forEach( eventName => {\n\t\t\tthis.controlsLeft.forEach( el => el.removeEventListener( eventName, this.onNavigateLeftClicked, false ) );\n\t\t\tthis.controlsRight.forEach( el => el.removeEventListener( eventName, this.onNavigateRightClicked, false ) );\n\t\t\tthis.controlsUp.forEach( el => el.removeEventListener( eventName, this.onNavigateUpClicked, false ) );\n\t\t\tthis.controlsDown.forEach( el => el.removeEventListener( eventName, this.onNavigateDownClicked, false ) );\n\t\t\tthis.controlsPrev.forEach( el => el.removeEventListener( eventName, this.onNavigatePrevClicked, false ) );\n\t\t\tthis.controlsNext.forEach( el => el.removeEventListener( eventName, this.onNavigateNextClicked, false ) );\n\t\t} );\n\n\t}\n\n\t/**\n\t * Updates the state of all control/navigation arrows.\n\t */\n\tupdate() {\n\n\t\tlet routes = this.Reveal.availableRoutes();\n\n\t\t// Remove the 'enabled' class from all directions\n\t\t[...this.controlsLeft, ...this.controlsRight, ...this.controlsUp, ...this.controlsDown, ...this.controlsPrev, ...this.controlsNext].forEach( node => {\n\t\t\tnode.classList.remove( 'enabled', 'fragmented' );\n\n\t\t\t// Set 'disabled' attribute on all directions\n\t\t\tnode.setAttribute( 'disabled', 'disabled' );\n\t\t} );\n\n\t\t// Add the 'enabled' class to the available routes; remove 'disabled' attribute to enable buttons\n\t\tif( routes.left ) this.controlsLeft.forEach( el => { el.classList.add( 'enabled' ); el.removeAttribute( 'disabled' ); } );\n\t\tif( routes.right ) this.controlsRight.forEach( el => { el.classList.add( 'enabled' ); el.removeAttribute( 'disabled' ); } );\n\t\tif( routes.up ) this.controlsUp.forEach( el => { el.classList.add( 'enabled' ); el.removeAttribute( 'disabled' ); } );\n\t\tif( routes.down ) this.controlsDown.forEach( el => { el.classList.add( 'enabled' ); el.removeAttribute( 'disabled' ); } );\n\n\t\t// Prev/next buttons\n\t\tif( routes.left || routes.up ) this.controlsPrev.forEach( el => { el.classList.add( 'enabled' ); el.removeAttribute( 'disabled' ); } );\n\t\tif( routes.right || routes.down ) this.controlsNext.forEach( el => { el.classList.add( 'enabled' ); el.removeAttribute( 'disabled' ); } );\n\n\t\t// Highlight fragment directions\n\t\tlet currentSlide = this.Reveal.getCurrentSlide();\n\t\tif( currentSlide ) {\n\n\t\t\tlet fragmentsRoutes = this.Reveal.fragments.availableRoutes();\n\n\t\t\t// Always apply fragment decorator to prev/next buttons\n\t\t\tif( fragmentsRoutes.prev ) this.controlsPrev.forEach( el => { el.classList.add( 'fragmented', 'enabled' ); el.removeAttribute( 'disabled' ); } );\n\t\t\tif( fragmentsRoutes.next ) this.controlsNext.forEach( el => { el.classList.add( 'fragmented', 'enabled' ); el.removeAttribute( 'disabled' ); } );\n\n\t\t\t// Apply fragment decorators to directional buttons based on\n\t\t\t// what slide axis they are in\n\t\t\tif( this.Reveal.isVerticalSlide( currentSlide ) ) {\n\t\t\t\tif( fragmentsRoutes.prev ) this.controlsUp.forEach( el => { el.classList.add( 'fragmented', 'enabled' ); el.removeAttribute( 'disabled' ); } );\n\t\t\t\tif( fragmentsRoutes.next ) this.controlsDown.forEach( el => { el.classList.add( 'fragmented', 'enabled' ); el.removeAttribute( 'disabled' ); } );\n\t\t\t}\n\t\t\telse {\n\t\t\t\tif( fragmentsRoutes.prev ) this.controlsLeft.forEach( el => { el.classList.add( 'fragmented', 'enabled' ); el.removeAttribute( 'disabled' ); } );\n\t\t\t\tif( fragmentsRoutes.next ) this.controlsRight.forEach( el => { el.classList.add( 'fragmented', 'enabled' ); el.removeAttribute( 'disabled' ); } );\n\t\t\t}\n\n\t\t}\n\n\t\tif( this.Reveal.getConfig().controlsTutorial ) {\n\n\t\t\tlet indices = this.Reveal.getIndices();\n\n\t\t\t// Highlight control arrows with an animation to ensure\n\t\t\t// that the viewer knows how to navigate\n\t\t\tif( !this.Reveal.hasNavigatedVertically() && routes.down ) {\n\t\t\t\tthis.controlsDownArrow.classList.add( 'highlight' );\n\t\t\t}\n\t\t\telse {\n\t\t\t\tthis.controlsDownArrow.classList.remove( 'highlight' );\n\n\t\t\t\tif( this.Reveal.getConfig().rtl ) {\n\n\t\t\t\t\tif( !this.Reveal.hasNavigatedHorizontally() && routes.left && indices.v === 0 ) {\n\t\t\t\t\t\tthis.controlsLeftArrow.classList.add( 'highlight' );\n\t\t\t\t\t}\n\t\t\t\t\telse {\n\t\t\t\t\t\tthis.controlsLeftArrow.classList.remove( 'highlight' );\n\t\t\t\t\t}\n\n\t\t\t\t} else {\n\n\t\t\t\t\tif( !this.Reveal.hasNavigatedHorizontally() && routes.right && indices.v === 0 ) {\n\t\t\t\t\t\tthis.controlsRightArrow.classList.add( 'highlight' );\n\t\t\t\t\t}\n\t\t\t\t\telse {\n\t\t\t\t\t\tthis.controlsRightArrow.classList.remove( 'highlight' );\n\t\t\t\t\t}\n\t\t\t\t}\n\t\t\t}\n\t\t}\n\t}\n\n\tdestroy() {\n\n\t\tthis.unbind();\n\t\tthis.element.remove();\n\n\t}\n\n\t/**\n\t * Event handlers for navigation control buttons.\n\t */\n\tonNavigateLeftClicked( event ) {\n\n\t\tevent.preventDefault();\n\t\tthis.Reveal.onUserInput();\n\n\t\tif( this.Reveal.getConfig().navigationMode === 'linear' ) {\n\t\t\tthis.Reveal.prev();\n\t\t}\n\t\telse {\n\t\t\tthis.Reveal.left();\n\t\t}\n\n\t}\n\n\tonNavigateRightClicked( event ) {\n\n\t\tevent.preventDefault();\n\t\tthis.Reveal.onUserInput();\n\n\t\tif( this.Reveal.getConfig().navigationMode === 'linear' ) {\n\t\t\tthis.Reveal.next();\n\t\t}\n\t\telse {\n\t\t\tthis.Reveal.right();\n\t\t}\n\n\t}\n\n\tonNavigateUpClicked( event ) {\n\n\t\tevent.preventDefault();\n\t\tthis.Reveal.onUserInput();\n\n\t\tthis.Reveal.up();\n\n\t}\n\n\tonNavigateDownClicked( event ) {\n\n\t\tevent.preventDefault();\n\t\tthis.Reveal.onUserInput();\n\n\t\tthis.Reveal.down();\n\n\t}\n\n\tonNavigatePrevClicked( event ) {\n\n\t\tevent.preventDefault();\n\t\tthis.Reveal.onUserInput();\n\n\t\tthis.Reveal.prev();\n\n\t}\n\n\tonNavigateNextClicked( event ) {\n\n\t\tevent.preventDefault();\n\t\tthis.Reveal.onUserInput();\n\n\t\tthis.Reveal.next();\n\n\t}\n\n\n}","/**\n * Creates a visual progress bar for the presentation.\n */\nexport default class Progress {\n\n\tconstructor( Reveal ) {\n\n\t\tthis.Reveal = Reveal;\n\n\t\tthis.onProgressClicked = this.onProgressClicked.bind( this );\n\n\t}\n\n\trender() {\n\n\t\tthis.element = document.createElement( 'div' );\n\t\tthis.element.className = 'progress';\n\t\tthis.Reveal.getRevealElement().appendChild( this.element );\n\n\t\tthis.bar = document.createElement( 'span' );\n\t\tthis.element.appendChild( this.bar );\n\n\t}\n\n\t/**\n\t * Called when the reveal.js config is updated.\n\t */\n\tconfigure( config, oldConfig ) {\n\n\t\tthis.element.style.display = config.progress ? 'block' : 'none';\n\n\t}\n\n\tbind() {\n\n\t\tif( this.Reveal.getConfig().progress && this.element ) {\n\t\t\tthis.element.addEventListener( 'click', this.onProgressClicked, false );\n\t\t}\n\n\t}\n\n\tunbind() {\n\n\t\tif ( this.Reveal.getConfig().progress && this.element ) {\n\t\t\tthis.element.removeEventListener( 'click', this.onProgressClicked, false );\n\t\t}\n\n\t}\n\n\t/**\n\t * Updates the progress bar to reflect the current slide.\n\t */\n\tupdate() {\n\n\t\t// Update progress if enabled\n\t\tif( this.Reveal.getConfig().progress && this.bar ) {\n\n\t\t\tlet scale = this.Reveal.getProgress();\n\n\t\t\t// Don't fill the progress bar if there's only one slide\n\t\t\tif( this.Reveal.getTotalSlides() < 2 ) {\n\t\t\t\tscale = 0;\n\t\t\t}\n\n\t\t\tthis.bar.style.transform = 'scaleX('+ scale +')';\n\n\t\t}\n\n\t}\n\n\tgetMaxWidth() {\n\n\t\treturn this.Reveal.getRevealElement().offsetWidth;\n\n\t}\n\n\t/**\n\t * Clicking on the progress bar results in a navigation to the\n\t * closest approximate horizontal slide using this equation:\n\t *\n\t * ( clickX / presentationWidth ) * numberOfSlides\n\t *\n\t * @param {object} event\n\t */\n\tonProgressClicked( event ) {\n\n\t\tthis.Reveal.onUserInput( event );\n\n\t\tevent.preventDefault();\n\n\t\tlet slides = this.Reveal.getSlides();\n\t\tlet slidesTotal = slides.length;\n\t\tlet slideIndex = Math.floor( ( event.clientX / this.getMaxWidth() ) * slidesTotal );\n\n\t\tif( this.Reveal.getConfig().rtl ) {\n\t\t\tslideIndex = slidesTotal - slideIndex;\n\t\t}\n\n\t\tlet targetIndices = this.Reveal.getIndices(slides[slideIndex]);\n\t\tthis.Reveal.slide( targetIndices.h, targetIndices.v );\n\n\t}\n\n\tdestroy() {\n\n\t\tthis.element.remove();\n\n\t}\n\n}","/**\n * Handles hiding of the pointer/cursor when inactive.\n */\nexport default class Pointer {\n\n\tconstructor( Reveal ) {\n\n\t\tthis.Reveal = Reveal;\n\n\t\t// Throttles mouse wheel navigation\n\t\tthis.lastMouseWheelStep = 0;\n\n\t\t// Is the mouse pointer currently hidden from view\n\t\tthis.cursorHidden = false;\n\n\t\t// Timeout used to determine when the cursor is inactive\n\t\tthis.cursorInactiveTimeout = 0;\n\n\t\tthis.onDocumentCursorActive = this.onDocumentCursorActive.bind( this );\n\t\tthis.onDocumentMouseScroll = this.onDocumentMouseScroll.bind( this );\n\n\t}\n\n\t/**\n\t * Called when the reveal.js config is updated.\n\t */\n\tconfigure( config, oldConfig ) {\n\n\t\tif( config.mouseWheel ) {\n\t\t\tdocument.addEventListener( 'DOMMouseScroll', this.onDocumentMouseScroll, false ); // FF\n\t\t\tdocument.addEventListener( 'mousewheel', this.onDocumentMouseScroll, false );\n\t\t}\n\t\telse {\n\t\t\tdocument.removeEventListener( 'DOMMouseScroll', this.onDocumentMouseScroll, false ); // FF\n\t\t\tdocument.removeEventListener( 'mousewheel', this.onDocumentMouseScroll, false );\n\t\t}\n\n\t\t// Auto-hide the mouse pointer when its inactive\n\t\tif( config.hideInactiveCursor ) {\n\t\t\tdocument.addEventListener( 'mousemove', this.onDocumentCursorActive, false );\n\t\t\tdocument.addEventListener( 'mousedown', this.onDocumentCursorActive, false );\n\t\t}\n\t\telse {\n\t\t\tthis.showCursor();\n\n\t\t\tdocument.removeEventListener( 'mousemove', this.onDocumentCursorActive, false );\n\t\t\tdocument.removeEventListener( 'mousedown', this.onDocumentCursorActive, false );\n\t\t}\n\n\t}\n\n\t/**\n\t * Shows the mouse pointer after it has been hidden with\n\t * #hideCursor.\n\t */\n\tshowCursor() {\n\n\t\tif( this.cursorHidden ) {\n\t\t\tthis.cursorHidden = false;\n\t\t\tthis.Reveal.getRevealElement().style.cursor = '';\n\t\t}\n\n\t}\n\n\t/**\n\t * Hides the mouse pointer when it's on top of the .reveal\n\t * container.\n\t */\n\thideCursor() {\n\n\t\tif( this.cursorHidden === false ) {\n\t\t\tthis.cursorHidden = true;\n\t\t\tthis.Reveal.getRevealElement().style.cursor = 'none';\n\t\t}\n\n\t}\n\n\tdestroy() {\n\n\t\tthis.showCursor();\n\n\t\tdocument.removeEventListener( 'DOMMouseScroll', this.onDocumentMouseScroll, false );\n\t\tdocument.removeEventListener( 'mousewheel', this.onDocumentMouseScroll, false );\n\t\tdocument.removeEventListener( 'mousemove', this.onDocumentCursorActive, false );\n\t\tdocument.removeEventListener( 'mousedown', this.onDocumentCursorActive, false );\n\n\t}\n\n\t/**\n\t * Called whenever there is mouse input at the document level\n\t * to determine if the cursor is active or not.\n\t *\n\t * @param {object} event\n\t */\n\tonDocumentCursorActive( event ) {\n\n\t\tthis.showCursor();\n\n\t\tclearTimeout( this.cursorInactiveTimeout );\n\n\t\tthis.cursorInactiveTimeout = setTimeout( this.hideCursor.bind( this ), this.Reveal.getConfig().hideCursorTime );\n\n\t}\n\n\t/**\n\t * Handles mouse wheel scrolling, throttled to avoid skipping\n\t * multiple slides.\n\t *\n\t * @param {object} event\n\t */\n\tonDocumentMouseScroll( event ) {\n\n\t\tif( Date.now() - this.lastMouseWheelStep > 1000 ) {\n\n\t\t\tthis.lastMouseWheelStep = Date.now();\n\n\t\t\tlet delta = event.detail || -event.wheelDelta;\n\t\t\tif( delta > 0 ) {\n\t\t\t\tthis.Reveal.next();\n\t\t\t}\n\t\t\telse if( delta < 0 ) {\n\t\t\t\tthis.Reveal.prev();\n\t\t\t}\n\n\t\t}\n\n\t}\n\n}","/**\n * Loads a JavaScript file from the given URL and executes it.\n *\n * @param {string} url Address of the .js file to load\n * @param {function} callback Method to invoke when the script\n * has loaded and executed\n */\nexport const loadScript = ( url, callback ) => {\n\n\tconst script = document.createElement( 'script' );\n\tscript.type = 'text/javascript';\n\tscript.async = false;\n\tscript.defer = false;\n\tscript.src = url;\n\n\tif( typeof callback === 'function' ) {\n\n\t\t// Success callback\n\t\tscript.onload = script.onreadystatechange = event => {\n\t\t\tif( event.type === 'load' || /loaded|complete/.test( script.readyState ) ) {\n\n\t\t\t\t// Kill event listeners\n\t\t\t\tscript.onload = script.onreadystatechange = script.onerror = null;\n\n\t\t\t\tcallback();\n\n\t\t\t}\n\t\t};\n\n\t\t// Error callback\n\t\tscript.onerror = err => {\n\n\t\t\t// Kill event listeners\n\t\t\tscript.onload = script.onreadystatechange = script.onerror = null;\n\n\t\t\tcallback( new Error( 'Failed loading script: ' + script.src + '\\n' + err ) );\n\n\t\t};\n\n\t}\n\n\t// Append the script at the end of \n\tconst head = document.querySelector( 'head' );\n\thead.insertBefore( script, head.lastChild );\n\n}","import { loadScript } from '../utils/loader.js'\n\n/**\n * Manages loading and registering of reveal.js plugins.\n */\nexport default class Plugins {\n\n\tconstructor( reveal ) {\n\n\t\tthis.Reveal = reveal;\n\n\t\t// Flags our current state (idle -> loading -> loaded)\n\t\tthis.state = 'idle';\n\n\t\t// An id:instance map of currently registed plugins\n\t\tthis.registeredPlugins = {};\n\n\t\tthis.asyncDependencies = [];\n\n\t}\n\n\t/**\n\t * Loads reveal.js dependencies, registers and\n\t * initializes plugins.\n\t *\n\t * Plugins are direct references to a reveal.js plugin\n\t * object that we register and initialize after any\n\t * synchronous dependencies have loaded.\n\t *\n\t * Dependencies are defined via the 'dependencies' config\n\t * option and will be loaded prior to starting reveal.js.\n\t * Some dependencies may have an 'async' flag, if so they\n\t * will load after reveal.js has been started up.\n\t */\n\tload( plugins, dependencies ) {\n\n\t\tthis.state = 'loading';\n\n\t\tplugins.forEach( this.registerPlugin.bind( this ) );\n\n\t\treturn new Promise( resolve => {\n\n\t\t\tlet scripts = [],\n\t\t\t\tscriptsToLoad = 0;\n\n\t\t\tdependencies.forEach( s => {\n\t\t\t\t// Load if there's no condition or the condition is truthy\n\t\t\t\tif( !s.condition || s.condition() ) {\n\t\t\t\t\tif( s.async ) {\n\t\t\t\t\t\tthis.asyncDependencies.push( s );\n\t\t\t\t\t}\n\t\t\t\t\telse {\n\t\t\t\t\t\tscripts.push( s );\n\t\t\t\t\t}\n\t\t\t\t}\n\t\t\t} );\n\n\t\t\tif( scripts.length ) {\n\t\t\t\tscriptsToLoad = scripts.length;\n\n\t\t\t\tconst scriptLoadedCallback = (s) => {\n\t\t\t\t\tif( s && typeof s.callback === 'function' ) s.callback();\n\n\t\t\t\t\tif( --scriptsToLoad === 0 ) {\n\t\t\t\t\t\tthis.initPlugins().then( resolve );\n\t\t\t\t\t}\n\t\t\t\t};\n\n\t\t\t\t// Load synchronous scripts\n\t\t\t\tscripts.forEach( s => {\n\t\t\t\t\tif( typeof s.id === 'string' ) {\n\t\t\t\t\t\tthis.registerPlugin( s );\n\t\t\t\t\t\tscriptLoadedCallback( s );\n\t\t\t\t\t}\n\t\t\t\t\telse if( typeof s.src === 'string' ) {\n\t\t\t\t\t\tloadScript( s.src, () => scriptLoadedCallback(s) );\n\t\t\t\t\t}\n\t\t\t\t\telse {\n\t\t\t\t\t\tconsole.warn( 'Unrecognized plugin format', s );\n\t\t\t\t\t\tscriptLoadedCallback();\n\t\t\t\t\t}\n\t\t\t\t} );\n\t\t\t}\n\t\t\telse {\n\t\t\t\tthis.initPlugins().then( resolve );\n\t\t\t}\n\n\t\t} );\n\n\t}\n\n\t/**\n\t * Initializes our plugins and waits for them to be ready\n\t * before proceeding.\n\t */\n\tinitPlugins() {\n\n\t\treturn new Promise( resolve => {\n\n\t\t\tlet pluginValues = Object.values( this.registeredPlugins );\n\t\t\tlet pluginsToInitialize = pluginValues.length;\n\n\t\t\t// If there are no plugins, skip this step\n\t\t\tif( pluginsToInitialize === 0 ) {\n\t\t\t\tthis.loadAsync().then( resolve );\n\t\t\t}\n\t\t\t// ... otherwise initialize plugins\n\t\t\telse {\n\n\t\t\t\tlet initNextPlugin;\n\n\t\t\t\tlet afterPlugInitialized = () => {\n\t\t\t\t\tif( --pluginsToInitialize === 0 ) {\n\t\t\t\t\t\tthis.loadAsync().then( resolve );\n\t\t\t\t\t}\n\t\t\t\t\telse {\n\t\t\t\t\t\tinitNextPlugin();\n\t\t\t\t\t}\n\t\t\t\t};\n\n\t\t\t\tlet i = 0;\n\n\t\t\t\t// Initialize plugins serially\n\t\t\t\tinitNextPlugin = () => {\n\n\t\t\t\t\tlet plugin = pluginValues[i++];\n\n\t\t\t\t\t// If the plugin has an 'init' method, invoke it\n\t\t\t\t\tif( typeof plugin.init === 'function' ) {\n\t\t\t\t\t\tlet promise = plugin.init( this.Reveal );\n\n\t\t\t\t\t\t// If the plugin returned a Promise, wait for it\n\t\t\t\t\t\tif( promise && typeof promise.then === 'function' ) {\n\t\t\t\t\t\t\tpromise.then( afterPlugInitialized );\n\t\t\t\t\t\t}\n\t\t\t\t\t\telse {\n\t\t\t\t\t\t\tafterPlugInitialized();\n\t\t\t\t\t\t}\n\t\t\t\t\t}\n\t\t\t\t\telse {\n\t\t\t\t\t\tafterPlugInitialized();\n\t\t\t\t\t}\n\n\t\t\t\t}\n\n\t\t\t\tinitNextPlugin();\n\n\t\t\t}\n\n\t\t} )\n\n\t}\n\n\t/**\n\t * Loads all async reveal.js dependencies.\n\t */\n\tloadAsync() {\n\n\t\tthis.state = 'loaded';\n\n\t\tif( this.asyncDependencies.length ) {\n\t\t\tthis.asyncDependencies.forEach( s => {\n\t\t\t\tloadScript( s.src, s.callback );\n\t\t\t} );\n\t\t}\n\n\t\treturn Promise.resolve();\n\n\t}\n\n\t/**\n\t * Registers a new plugin with this reveal.js instance.\n\t *\n\t * reveal.js waits for all regisered plugins to initialize\n\t * before considering itself ready, as long as the plugin\n\t * is registered before calling `Reveal.initialize()`.\n\t */\n\tregisterPlugin( plugin ) {\n\n\t\t// Backwards compatibility to make reveal.js ~3.9.0\n\t\t// plugins work with reveal.js 4.0.0\n\t\tif( arguments.length === 2 && typeof arguments[0] === 'string' ) {\n\t\t\tplugin = arguments[1];\n\t\t\tplugin.id = arguments[0];\n\t\t}\n\t\t// Plugin can optionally be a function which we call\n\t\t// to create an instance of the plugin\n\t\telse if( typeof plugin === 'function' ) {\n\t\t\tplugin = plugin();\n\t\t}\n\n\t\tlet id = plugin.id;\n\n\t\tif( typeof id !== 'string' ) {\n\t\t\tconsole.warn( 'Unrecognized plugin format; can\\'t find plugin.id', plugin );\n\t\t}\n\t\telse if( this.registeredPlugins[id] === undefined ) {\n\t\t\tthis.registeredPlugins[id] = plugin;\n\n\t\t\t// If a plugin is registered after reveal.js is loaded,\n\t\t\t// initialize it right away\n\t\t\tif( this.state === 'loaded' && typeof plugin.init === 'function' ) {\n\t\t\t\tplugin.init( this.Reveal );\n\t\t\t}\n\t\t}\n\t\telse {\n\t\t\tconsole.warn( 'reveal.js: \"'+ id +'\" plugin has already been registered' );\n\t\t}\n\n\t}\n\n\t/**\n\t * Checks if a specific plugin has been registered.\n\t *\n\t * @param {String} id Unique plugin identifier\n\t */\n\thasPlugin( id ) {\n\n\t\treturn !!this.registeredPlugins[id];\n\n\t}\n\n\t/**\n\t * Returns the specific plugin instance, if a plugin\n\t * with the given ID has been registered.\n\t *\n\t * @param {String} id Unique plugin identifier\n\t */\n\tgetPlugin( id ) {\n\n\t\treturn this.registeredPlugins[id];\n\n\t}\n\n\tgetRegisteredPlugins() {\n\n\t\treturn this.registeredPlugins;\n\n\t}\n\n\tdestroy() {\n\n\t\tObject.values( this.registeredPlugins ).forEach( plugin => {\n\t\t\tif( typeof plugin.destroy === 'function' ) {\n\t\t\t\tplugin.destroy();\n\t\t\t}\n\t\t} );\n\n\t\tthis.registeredPlugins = {};\n\t\tthis.asyncDependencies = [];\n\n\t}\n\n}\n","import { SLIDES_SELECTOR } from '../utils/constants.js'\nimport { queryAll, createStyleSheet } from '../utils/util.js'\n\n/**\n * Setups up our presentation for printing/exporting to PDF.\n */\nexport default class Print {\n\n\tconstructor( Reveal ) {\n\n\t\tthis.Reveal = Reveal;\n\n\t}\n\n\t/**\n\t * Configures the presentation for printing to a static\n\t * PDF.\n\t */\n\tasync setupPDF() {\n\n\t\tconst config = this.Reveal.getConfig();\n\t\tconst slides = queryAll( this.Reveal.getRevealElement(), SLIDES_SELECTOR )\n\n\t\t// Compute slide numbers now, before we start duplicating slides\n\t\tconst doingSlideNumbers = config.slideNumber && /all|print/i.test( config.showSlideNumber );\n\n\t\tconst slideSize = this.Reveal.getComputedSlideSize( window.innerWidth, window.innerHeight );\n\n\t\t// Dimensions of the PDF pages\n\t\tconst pageWidth = Math.floor( slideSize.width * ( 1 + config.margin ) ),\n\t\t\tpageHeight = Math.floor( slideSize.height * ( 1 + config.margin ) );\n\n\t\t// Dimensions of slides within the pages\n\t\tconst slideWidth = slideSize.width,\n\t\t\tslideHeight = slideSize.height;\n\n\t\tawait new Promise( requestAnimationFrame );\n\n\t\t// Let the browser know what page size we want to print\n\t\tcreateStyleSheet( '@page{size:'+ pageWidth +'px '+ pageHeight +'px; margin: 0px;}' );\n\n\t\t// Limit the size of certain elements to the dimensions of the slide\n\t\tcreateStyleSheet( '.reveal section>img, .reveal section>video, .reveal section>iframe{max-width: '+ slideWidth +'px; max-height:'+ slideHeight +'px}' );\n\n\t\tdocument.documentElement.classList.add( 'print-pdf' );\n\t\tdocument.body.style.width = pageWidth + 'px';\n\t\tdocument.body.style.height = pageHeight + 'px';\n\n\t\tconst viewportElement = document.querySelector( '.reveal-viewport' );\n\t\tlet presentationBackground;\n\t\tif( viewportElement ) {\n\t\t\tconst viewportStyles = window.getComputedStyle( viewportElement );\n\t\t\tif( viewportStyles && viewportStyles.background ) {\n\t\t\t\tpresentationBackground = viewportStyles.background;\n\t\t\t}\n\t\t}\n\n\t\t// Make sure stretch elements fit on slide\n\t\tawait new Promise( requestAnimationFrame );\n\t\tthis.Reveal.layoutSlideContents( slideWidth, slideHeight );\n\n\t\t// Batch scrollHeight access to prevent layout thrashing\n\t\tawait new Promise( requestAnimationFrame );\n\n\t\tconst slideScrollHeights = slides.map( slide => slide.scrollHeight );\n\n\t\tconst pages = [];\n\t\tconst pageContainer = slides[0].parentNode;\n\n\t\t// Slide and slide background layout\n\t\tslides.forEach( function( slide, index ) {\n\n\t\t\t// Vertical stacks are not centred since their section\n\t\t\t// children will be\n\t\t\tif( slide.classList.contains( 'stack' ) === false ) {\n\t\t\t\t// Center the slide inside of the page, giving the slide some margin\n\t\t\t\tlet left = ( pageWidth - slideWidth ) / 2;\n\t\t\t\tlet top = ( pageHeight - slideHeight ) / 2;\n\n\t\t\t\tconst contentHeight = slideScrollHeights[ index ];\n\t\t\t\tlet numberOfPages = Math.max( Math.ceil( contentHeight / pageHeight ), 1 );\n\n\t\t\t\t// Adhere to configured pages per slide limit\n\t\t\t\tnumberOfPages = Math.min( numberOfPages, config.pdfMaxPagesPerSlide );\n\n\t\t\t\t// Center slides vertically\n\t\t\t\tif( numberOfPages === 1 && config.center || slide.classList.contains( 'center' ) ) {\n\t\t\t\t\ttop = Math.max( ( pageHeight - contentHeight ) / 2, 0 );\n\t\t\t\t}\n\n\t\t\t\t// Wrap the slide in a page element and hide its overflow\n\t\t\t\t// so that no page ever flows onto another\n\t\t\t\tconst page = document.createElement( 'div' );\n\t\t\t\tpages.push( page );\n\n\t\t\t\tpage.className = 'pdf-page';\n\t\t\t\tpage.style.height = ( ( pageHeight + config.pdfPageHeightOffset ) * numberOfPages ) + 'px';\n\n\t\t\t\t// Copy the presentation-wide background to each individual\n\t\t\t\t// page when printing\n\t\t\t\tif( presentationBackground ) {\n\t\t\t\t\tpage.style.background = presentationBackground;\n\t\t\t\t}\n\n\t\t\t\tpage.appendChild( slide );\n\n\t\t\t\t// Position the slide inside of the page\n\t\t\t\tslide.style.left = left + 'px';\n\t\t\t\tslide.style.top = top + 'px';\n\t\t\t\tslide.style.width = slideWidth + 'px';\n\n\t\t\t\t// Re-run the slide layout so that r-fit-text is applied based on\n\t\t\t\t// the printed slide size\n\t\t\t\tthis.Reveal.slideContent.layout( slide )\n\n\t\t\t\tif( slide.slideBackgroundElement ) {\n\t\t\t\t\tpage.insertBefore( slide.slideBackgroundElement, slide );\n\t\t\t\t}\n\n\t\t\t\t// Inject notes if `showNotes` is enabled\n\t\t\t\tif( config.showNotes ) {\n\n\t\t\t\t\t// Are there notes for this slide?\n\t\t\t\t\tconst notes = this.Reveal.getSlideNotes( slide );\n\t\t\t\t\tif( notes ) {\n\n\t\t\t\t\t\tconst notesSpacing = 8;\n\t\t\t\t\t\tconst notesLayout = typeof config.showNotes === 'string' ? config.showNotes : 'inline';\n\t\t\t\t\t\tconst notesElement = document.createElement( 'div' );\n\t\t\t\t\t\tnotesElement.classList.add( 'speaker-notes' );\n\t\t\t\t\t\tnotesElement.classList.add( 'speaker-notes-pdf' );\n\t\t\t\t\t\tnotesElement.setAttribute( 'data-layout', notesLayout );\n\t\t\t\t\t\tnotesElement.innerHTML = notes;\n\n\t\t\t\t\t\tif( notesLayout === 'separate-page' ) {\n\t\t\t\t\t\t\tpages.push( notesElement );\n\t\t\t\t\t\t}\n\t\t\t\t\t\telse {\n\t\t\t\t\t\t\tnotesElement.style.left = notesSpacing + 'px';\n\t\t\t\t\t\t\tnotesElement.style.bottom = notesSpacing + 'px';\n\t\t\t\t\t\t\tnotesElement.style.width = ( pageWidth - notesSpacing*2 ) + 'px';\n\t\t\t\t\t\t\tpage.appendChild( notesElement );\n\t\t\t\t\t\t}\n\n\t\t\t\t\t}\n\n\t\t\t\t}\n\n\t\t\t\t// Inject slide numbers if `slideNumbers` are enabled\n\t\t\t\tif( doingSlideNumbers ) {\n\t\t\t\t\tconst slideNumber = index + 1;\n\t\t\t\t\tconst numberElement = document.createElement( 'div' );\n\t\t\t\t\tnumberElement.classList.add( 'slide-number' );\n\t\t\t\t\tnumberElement.classList.add( 'slide-number-pdf' );\n\t\t\t\t\tnumberElement.innerHTML = slideNumber;\n\t\t\t\t\tpage.appendChild( numberElement );\n\t\t\t\t}\n\n\t\t\t\t// Copy page and show fragments one after another\n\t\t\t\tif( config.pdfSeparateFragments ) {\n\n\t\t\t\t\t// Each fragment 'group' is an array containing one or more\n\t\t\t\t\t// fragments. Multiple fragments that appear at the same time\n\t\t\t\t\t// are part of the same group.\n\t\t\t\t\tconst fragmentGroups = this.Reveal.fragments.sort( page.querySelectorAll( '.fragment' ), true );\n\n\t\t\t\t\tlet previousFragmentStep;\n\n\t\t\t\t\tfragmentGroups.forEach( function( fragments ) {\n\n\t\t\t\t\t\t// Remove 'current-fragment' from the previous group\n\t\t\t\t\t\tif( previousFragmentStep ) {\n\t\t\t\t\t\t\tpreviousFragmentStep.forEach( function( fragment ) {\n\t\t\t\t\t\t\t\tfragment.classList.remove( 'current-fragment' );\n\t\t\t\t\t\t\t} );\n\t\t\t\t\t\t}\n\n\t\t\t\t\t\t// Show the fragments for the current index\n\t\t\t\t\t\tfragments.forEach( function( fragment ) {\n\t\t\t\t\t\t\tfragment.classList.add( 'visible', 'current-fragment' );\n\t\t\t\t\t\t}, this );\n\n\t\t\t\t\t\t// Create a separate page for the current fragment state\n\t\t\t\t\t\tconst clonedPage = page.cloneNode( true );\n\t\t\t\t\t\tpages.push( clonedPage );\n\n\t\t\t\t\t\tpreviousFragmentStep = fragments;\n\n\t\t\t\t\t}, this );\n\n\t\t\t\t\t// Reset the first/original page so that all fragments are hidden\n\t\t\t\t\tfragmentGroups.forEach( function( fragments ) {\n\t\t\t\t\t\tfragments.forEach( function( fragment ) {\n\t\t\t\t\t\t\tfragment.classList.remove( 'visible', 'current-fragment' );\n\t\t\t\t\t\t} );\n\t\t\t\t\t} );\n\n\t\t\t\t}\n\t\t\t\t// Show all fragments\n\t\t\t\telse {\n\t\t\t\t\tqueryAll( page, '.fragment:not(.fade-out)' ).forEach( function( fragment ) {\n\t\t\t\t\t\tfragment.classList.add( 'visible' );\n\t\t\t\t\t} );\n\t\t\t\t}\n\n\t\t\t}\n\n\t\t}, this );\n\n\t\tawait new Promise( requestAnimationFrame );\n\n\t\tpages.forEach( page => pageContainer.appendChild( page ) );\n\n\t\t// Notify subscribers that the PDF layout is good to go\n\t\tthis.Reveal.dispatchEvent({ type: 'pdf-ready' });\n\n\t}\n\n\t/**\n\t * Checks if this instance is being used to print a PDF.\n\t */\n\tisPrintingPDF() {\n\n\t\treturn ( /print-pdf/gi ).test( window.location.search );\n\n\t}\n\n}\n","import { isAndroid } from '../utils/device.js'\nimport { matches } from '../utils/util.js'\n\nconst SWIPE_THRESHOLD = 40;\n\n/**\n * Controls all touch interactions and navigations for\n * a presentation.\n */\nexport default class Touch {\n\n\tconstructor( Reveal ) {\n\n\t\tthis.Reveal = Reveal;\n\n\t\t// Holds information about the currently ongoing touch interaction\n\t\tthis.touchStartX = 0;\n\t\tthis.touchStartY = 0;\n\t\tthis.touchStartCount = 0;\n\t\tthis.touchCaptured = false;\n\n\t\tthis.onPointerDown = this.onPointerDown.bind( this );\n\t\tthis.onPointerMove = this.onPointerMove.bind( this );\n\t\tthis.onPointerUp = this.onPointerUp.bind( this );\n\t\tthis.onTouchStart = this.onTouchStart.bind( this );\n\t\tthis.onTouchMove = this.onTouchMove.bind( this );\n\t\tthis.onTouchEnd = this.onTouchEnd.bind( this );\n\n\t}\n\n\t/**\n\t *\n\t */\n\tbind() {\n\n\t\tlet revealElement = this.Reveal.getRevealElement();\n\n\t\tif( 'onpointerdown' in window ) {\n\t\t\t// Use W3C pointer events\n\t\t\trevealElement.addEventListener( 'pointerdown', this.onPointerDown, false );\n\t\t\trevealElement.addEventListener( 'pointermove', this.onPointerMove, false );\n\t\t\trevealElement.addEventListener( 'pointerup', this.onPointerUp, false );\n\t\t}\n\t\telse if( window.navigator.msPointerEnabled ) {\n\t\t\t// IE 10 uses prefixed version of pointer events\n\t\t\trevealElement.addEventListener( 'MSPointerDown', this.onPointerDown, false );\n\t\t\trevealElement.addEventListener( 'MSPointerMove', this.onPointerMove, false );\n\t\t\trevealElement.addEventListener( 'MSPointerUp', this.onPointerUp, false );\n\t\t}\n\t\telse {\n\t\t\t// Fall back to touch events\n\t\t\trevealElement.addEventListener( 'touchstart', this.onTouchStart, false );\n\t\t\trevealElement.addEventListener( 'touchmove', this.onTouchMove, false );\n\t\t\trevealElement.addEventListener( 'touchend', this.onTouchEnd, false );\n\t\t}\n\n\t}\n\n\t/**\n\t *\n\t */\n\tunbind() {\n\n\t\tlet revealElement = this.Reveal.getRevealElement();\n\n\t\trevealElement.removeEventListener( 'pointerdown', this.onPointerDown, false );\n\t\trevealElement.removeEventListener( 'pointermove', this.onPointerMove, false );\n\t\trevealElement.removeEventListener( 'pointerup', this.onPointerUp, false );\n\n\t\trevealElement.removeEventListener( 'MSPointerDown', this.onPointerDown, false );\n\t\trevealElement.removeEventListener( 'MSPointerMove', this.onPointerMove, false );\n\t\trevealElement.removeEventListener( 'MSPointerUp', this.onPointerUp, false );\n\n\t\trevealElement.removeEventListener( 'touchstart', this.onTouchStart, false );\n\t\trevealElement.removeEventListener( 'touchmove', this.onTouchMove, false );\n\t\trevealElement.removeEventListener( 'touchend', this.onTouchEnd, false );\n\n\t}\n\n\t/**\n\t * Checks if the target element prevents the triggering of\n\t * swipe navigation.\n\t */\n\tisSwipePrevented( target ) {\n\n\t\t// Prevent accidental swipes when scrubbing timelines\n\t\tif( matches( target, 'video, audio' ) ) return true;\n\n\t\twhile( target && typeof target.hasAttribute === 'function' ) {\n\t\t\tif( target.hasAttribute( 'data-prevent-swipe' ) ) return true;\n\t\t\ttarget = target.parentNode;\n\t\t}\n\n\t\treturn false;\n\n\t}\n\n\t/**\n\t * Handler for the 'touchstart' event, enables support for\n\t * swipe and pinch gestures.\n\t *\n\t * @param {object} event\n\t */\n\tonTouchStart( event ) {\n\n\t\tif( this.isSwipePrevented( event.target ) ) return true;\n\n\t\tthis.touchStartX = event.touches[0].clientX;\n\t\tthis.touchStartY = event.touches[0].clientY;\n\t\tthis.touchStartCount = event.touches.length;\n\n\t}\n\n\t/**\n\t * Handler for the 'touchmove' event.\n\t *\n\t * @param {object} event\n\t */\n\tonTouchMove( event ) {\n\n\t\tif( this.isSwipePrevented( event.target ) ) return true;\n\n\t\tlet config = this.Reveal.getConfig();\n\n\t\t// Each touch should only trigger one action\n\t\tif( !this.touchCaptured ) {\n\t\t\tthis.Reveal.onUserInput( event );\n\n\t\t\tlet currentX = event.touches[0].clientX;\n\t\t\tlet currentY = event.touches[0].clientY;\n\n\t\t\t// There was only one touch point, look for a swipe\n\t\t\tif( event.touches.length === 1 && this.touchStartCount !== 2 ) {\n\n\t\t\t\tlet availableRoutes = this.Reveal.availableRoutes({ includeFragments: true });\n\n\t\t\t\tlet deltaX = currentX - this.touchStartX,\n\t\t\t\t\tdeltaY = currentY - this.touchStartY;\n\n\t\t\t\tif( deltaX > SWIPE_THRESHOLD && Math.abs( deltaX ) > Math.abs( deltaY ) ) {\n\t\t\t\t\tthis.touchCaptured = true;\n\t\t\t\t\tif( config.navigationMode === 'linear' ) {\n\t\t\t\t\t\tif( config.rtl ) {\n\t\t\t\t\t\t\tthis.Reveal.next();\n\t\t\t\t\t\t}\n\t\t\t\t\t\telse {\n\t\t\t\t\t\t\tthis.Reveal.prev();\n\t\t\t\t\t\t}\n\t\t\t\t\t}\n\t\t\t\t\telse {\n\t\t\t\t\t\tthis.Reveal.left();\n\t\t\t\t\t}\n\t\t\t\t}\n\t\t\t\telse if( deltaX < -SWIPE_THRESHOLD && Math.abs( deltaX ) > Math.abs( deltaY ) ) {\n\t\t\t\t\tthis.touchCaptured = true;\n\t\t\t\t\tif( config.navigationMode === 'linear' ) {\n\t\t\t\t\t\tif( config.rtl ) {\n\t\t\t\t\t\t\tthis.Reveal.prev();\n\t\t\t\t\t\t}\n\t\t\t\t\t\telse {\n\t\t\t\t\t\t\tthis.Reveal.next();\n\t\t\t\t\t\t}\n\t\t\t\t\t}\n\t\t\t\t\telse {\n\t\t\t\t\t\tthis.Reveal.right();\n\t\t\t\t\t}\n\t\t\t\t}\n\t\t\t\telse if( deltaY > SWIPE_THRESHOLD && availableRoutes.up ) {\n\t\t\t\t\tthis.touchCaptured = true;\n\t\t\t\t\tif( config.navigationMode === 'linear' ) {\n\t\t\t\t\t\tthis.Reveal.prev();\n\t\t\t\t\t}\n\t\t\t\t\telse {\n\t\t\t\t\t\tthis.Reveal.up();\n\t\t\t\t\t}\n\t\t\t\t}\n\t\t\t\telse if( deltaY < -SWIPE_THRESHOLD && availableRoutes.down ) {\n\t\t\t\t\tthis.touchCaptured = true;\n\t\t\t\t\tif( config.navigationMode === 'linear' ) {\n\t\t\t\t\t\tthis.Reveal.next();\n\t\t\t\t\t}\n\t\t\t\t\telse {\n\t\t\t\t\t\tthis.Reveal.down();\n\t\t\t\t\t}\n\t\t\t\t}\n\n\t\t\t\t// If we're embedded, only block touch events if they have\n\t\t\t\t// triggered an action\n\t\t\t\tif( config.embedded ) {\n\t\t\t\t\tif( this.touchCaptured || this.Reveal.isVerticalSlide() ) {\n\t\t\t\t\t\tevent.preventDefault();\n\t\t\t\t\t}\n\t\t\t\t}\n\t\t\t\t// Not embedded? Block them all to avoid needless tossing\n\t\t\t\t// around of the viewport in iOS\n\t\t\t\telse {\n\t\t\t\t\tevent.preventDefault();\n\t\t\t\t}\n\n\t\t\t}\n\t\t}\n\t\t// There's a bug with swiping on some Android devices unless\n\t\t// the default action is always prevented\n\t\telse if( isAndroid ) {\n\t\t\tevent.preventDefault();\n\t\t}\n\n\t}\n\n\t/**\n\t * Handler for the 'touchend' event.\n\t *\n\t * @param {object} event\n\t */\n\tonTouchEnd( event ) {\n\n\t\tthis.touchCaptured = false;\n\n\t}\n\n\t/**\n\t * Convert pointer down to touch start.\n\t *\n\t * @param {object} event\n\t */\n\tonPointerDown( event ) {\n\n\t\tif( event.pointerType === event.MSPOINTER_TYPE_TOUCH || event.pointerType === \"touch\" ) {\n\t\t\tevent.touches = [{ clientX: event.clientX, clientY: event.clientY }];\n\t\t\tthis.onTouchStart( event );\n\t\t}\n\n\t}\n\n\t/**\n\t * Convert pointer move to touch move.\n\t *\n\t * @param {object} event\n\t */\n\tonPointerMove( event ) {\n\n\t\tif( event.pointerType === event.MSPOINTER_TYPE_TOUCH || event.pointerType === \"touch\" ) {\n\t\t\tevent.touches = [{ clientX: event.clientX, clientY: event.clientY }];\n\t\t\tthis.onTouchMove( event );\n\t\t}\n\n\t}\n\n\t/**\n\t * Convert pointer up to touch end.\n\t *\n\t * @param {object} event\n\t */\n\tonPointerUp( event ) {\n\n\t\tif( event.pointerType === event.MSPOINTER_TYPE_TOUCH || event.pointerType === \"touch\" ) {\n\t\t\tevent.touches = [{ clientX: event.clientX, clientY: event.clientY }];\n\t\t\tthis.onTouchEnd( event );\n\t\t}\n\n\t}\n\n}","import { closest } from '../utils/util.js'\n\n/**\n * Manages focus when a presentation is embedded. This\n * helps us only capture keyboard from the presentation\n * a user is currently interacting with in a page where\n * multiple presentations are embedded.\n */\n\nconst STATE_FOCUS = 'focus';\nconst STATE_BLUR = 'blur';\n\nexport default class Focus {\n\n\tconstructor( Reveal ) {\n\n\t\tthis.Reveal = Reveal;\n\n\t\tthis.onRevealPointerDown = this.onRevealPointerDown.bind( this );\n\t\tthis.onDocumentPointerDown = this.onDocumentPointerDown.bind( this );\n\n\t}\n\n\t/**\n\t * Called when the reveal.js config is updated.\n\t */\n\tconfigure( config, oldConfig ) {\n\n\t\tif( config.embedded ) {\n\t\t\tthis.blur();\n\t\t}\n\t\telse {\n\t\t\tthis.focus();\n\t\t\tthis.unbind();\n\t\t}\n\n\t}\n\n\tbind() {\n\n\t\tif( this.Reveal.getConfig().embedded ) {\n\t\t\tthis.Reveal.getRevealElement().addEventListener( 'pointerdown', this.onRevealPointerDown, false );\n\t\t}\n\n\t}\n\n\tunbind() {\n\n\t\tthis.Reveal.getRevealElement().removeEventListener( 'pointerdown', this.onRevealPointerDown, false );\n\t\tdocument.removeEventListener( 'pointerdown', this.onDocumentPointerDown, false );\n\n\t}\n\n\tfocus() {\n\n\t\tif( this.state !== STATE_FOCUS ) {\n\t\t\tthis.Reveal.getRevealElement().classList.add( 'focused' );\n\t\t\tdocument.addEventListener( 'pointerdown', this.onDocumentPointerDown, false );\n\t\t}\n\n\t\tthis.state = STATE_FOCUS;\n\n\t}\n\n\tblur() {\n\n\t\tif( this.state !== STATE_BLUR ) {\n\t\t\tthis.Reveal.getRevealElement().classList.remove( 'focused' );\n\t\t\tdocument.removeEventListener( 'pointerdown', this.onDocumentPointerDown, false );\n\t\t}\n\n\t\tthis.state = STATE_BLUR;\n\n\t}\n\n\tisFocused() {\n\n\t\treturn this.state === STATE_FOCUS;\n\n\t}\n\n\tdestroy() {\n\n\t\tthis.Reveal.getRevealElement().classList.remove( 'focused' );\n\n\t}\n\n\tonRevealPointerDown( event ) {\n\n\t\tthis.focus();\n\n\t}\n\n\tonDocumentPointerDown( event ) {\n\n\t\tlet revealElement = closest( event.target, '.reveal' );\n\t\tif( !revealElement || revealElement !== this.Reveal.getRevealElement() ) {\n\t\t\tthis.blur();\n\t\t}\n\n\t}\n\n}","/**\n * Handles the showing and \n */\nexport default class Notes {\n\n\tconstructor( Reveal ) {\n\n\t\tthis.Reveal = Reveal;\n\n\t}\n\n\trender() {\n\n\t\tthis.element = document.createElement( 'div' );\n\t\tthis.element.className = 'speaker-notes';\n\t\tthis.element.setAttribute( 'data-prevent-swipe', '' );\n\t\tthis.element.setAttribute( 'tabindex', '0' );\n\t\tthis.Reveal.getRevealElement().appendChild( this.element );\n\n\t}\n\n\t/**\n\t * Called when the reveal.js config is updated.\n\t */\n\tconfigure( config, oldConfig ) {\n\n\t\tif( config.showNotes ) {\n\t\t\tthis.element.setAttribute( 'data-layout', typeof config.showNotes === 'string' ? config.showNotes : 'inline' );\n\t\t}\n\n\t}\n\n\t/**\n\t * Pick up notes from the current slide and display them\n\t * to the viewer.\n\t *\n\t * @see {@link config.showNotes}\n\t */\n\tupdate() {\n\n\t\tif( this.Reveal.getConfig().showNotes && this.element && this.Reveal.getCurrentSlide() && !this.Reveal.print.isPrintingPDF() ) {\n\n\t\t\tthis.element.innerHTML = this.getSlideNotes() || 'No notes on this slide.';\n\n\t\t}\n\n\t}\n\n\t/**\n\t * Updates the visibility of the speaker notes sidebar that\n\t * is used to share annotated slides. The notes sidebar is\n\t * only visible if showNotes is true and there are notes on\n\t * one or more slides in the deck.\n\t */\n\tupdateVisibility() {\n\n\t\tif( this.Reveal.getConfig().showNotes && this.hasNotes() && !this.Reveal.print.isPrintingPDF() ) {\n\t\t\tthis.Reveal.getRevealElement().classList.add( 'show-notes' );\n\t\t}\n\t\telse {\n\t\t\tthis.Reveal.getRevealElement().classList.remove( 'show-notes' );\n\t\t}\n\n\t}\n\n\t/**\n\t * Checks if there are speaker notes for ANY slide in the\n\t * presentation.\n\t */\n\thasNotes() {\n\n\t\treturn this.Reveal.getSlidesElement().querySelectorAll( '[data-notes], aside.notes' ).length > 0;\n\n\t}\n\n\t/**\n\t * Checks if this presentation is running inside of the\n\t * speaker notes window.\n\t *\n\t * @return {boolean}\n\t */\n\tisSpeakerNotesWindow() {\n\n\t\treturn !!window.location.search.match( /receiver/gi );\n\n\t}\n\n\t/**\n\t * Retrieves the speaker notes from a slide. Notes can be\n\t * defined in two ways:\n\t * 1. As a data-notes attribute on the slide
    \n\t * 2. As an

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-0,0 +1,142 @@ +.pagedtable { + overflow: auto; + padding-left: 8px; + padding-right: 8px; +} + +.pagedtable-wrapper { + border: 1px solid #ccc; + border-radius: 4px; + margin-bottom: 10px; +} + +.pagedtable table { + width: 100%; + max-width: 100%; + margin: 0; +} + +.pagedtable th { + padding: 0 5px 0 5px; + border: none; + border-bottom: 2px solid #dddddd; + + min-width: 45px; +} + +.pagedtable-empty th { + display: none; +} + +.pagedtable td { + padding: 0 4px 0 4px; +} + +.pagedtable .even { + background-color: rgba(140, 140, 140, 0.1); +} + +.pagedtable-padding-col { + display: none; +} + +.pagedtable a { + -webkit-touch-callout: none; + -webkit-user-select: none; + -khtml-user-select: none; + -moz-user-select: none; + -ms-user-select: none; + user-select: none; +} + +.pagedtable-index-nav { + cursor: pointer; + padding: 0 5px 0 5px; + float: right; + border: 0; +} + +.pagedtable-index-nav-disabled { + cursor: default; + text-decoration: none; + color: #999; +} + +a.pagedtable-index-nav-disabled:hover { + text-decoration: none; + color: #999; +} + +.pagedtable-indexes { + cursor: pointer; + float: right; + border: 0; +} + +.pagedtable-index-current { + cursor: default; + text-decoration: none; + font-weight: bold; + color: #333; + border: 0; +} + +a.pagedtable-index-current:hover { + text-decoration: none; + font-weight: bold; + color: #333; +} + +.pagedtable-index { + width: 30px; + display: inline-block; + text-align: center; + border: 0; +} + +.pagedtable-index-separator-left { + display: inline-block; + color: #333; + font-size: 9px; + padding: 0 0 0 0; + cursor: default; +} + +.pagedtable-index-separator-right { + display: inline-block; + color: #333; + font-size: 9px; + padding: 0 4px 0 0; + cursor: default; +} + +.pagedtable-footer { + padding-top: 4px; + padding-bottom: 5px; +} + +.pagedtable-not-empty .pagedtable-footer { + border-top: 2px solid #dddddd; +} + +.pagedtable-info { + overflow: hidden; + color: #999; + white-space: nowrap; + text-overflow: ellipsis; +} + +.pagedtable-header-name { + overflow: hidden; + text-overflow: ellipsis; +} + +.pagedtable-header-type { + color: #999; + font-weight: 400; +} + +.pagedtable-na-cell { + font-style: italic; + opacity: 0.3; +} diff --git a/probability_files/libs/pagedtable-1.1/js/pagedtable.js b/probability_files/libs/pagedtable-1.1/js/pagedtable.js new file mode 100644 index 000000000..52ada2bc5 --- /dev/null +++ b/probability_files/libs/pagedtable-1.1/js/pagedtable.js @@ -0,0 +1,1151 @@ +// Production steps of ECMA-262, Edition 5, 15.4.4.18 +// Reference: http://es5.github.io/#x15.4.4.18 +if (!Array.prototype.forEach) { + + Array.prototype.forEach = function(callback, thisArg) { + + var T, k; + + if (this === null) { + throw new TypeError(' this is null or not defined'); + } + + // 1. Let O be the result of calling toObject() passing the + // |this| value as the argument. + var O = Object(this); + + // 2. Let lenValue be the result of calling the Get() internal + // method of O with the argument "length". + // 3. Let len be toUint32(lenValue). + var len = O.length >>> 0; + + // 4. If isCallable(callback) is false, throw a TypeError exception. + // See: http://es5.github.com/#x9.11 + if (typeof callback !== "function") { + throw new TypeError(callback + ' is not a function'); + } + + // 5. If thisArg was supplied, let T be thisArg; else let + // T be undefined. + if (arguments.length > 1) { + T = thisArg; + } + + // 6. Let k be 0 + k = 0; + + // 7. Repeat, while k < len + while (k < len) { + + var kValue; + + // a. Let Pk be ToString(k). + // This is implicit for LHS operands of the in operator + // b. Let kPresent be the result of calling the HasProperty + // internal method of O with argument Pk. + // This step can be combined with c + // c. If kPresent is true, then + if (k in O) { + + // i. Let kValue be the result of calling the Get internal + // method of O with argument Pk. + kValue = O[k]; + + // ii. Call the Call internal method of callback with T as + // the this value and argument list containing kValue, k, and O. + callback.call(T, kValue, k, O); + } + // d. Increase k by 1. + k++; + } + // 8. return undefined + }; +} + +// Production steps of ECMA-262, Edition 5, 15.4.4.19 +// Reference: http://es5.github.io/#x15.4.4.19 +if (!Array.prototype.map) { + + Array.prototype.map = function(callback, thisArg) { + + var T, A, k; + + if (this == null) { + throw new TypeError(' this is null or not defined'); + } + + // 1. Let O be the result of calling ToObject passing the |this| + // value as the argument. + var O = Object(this); + + // 2. Let lenValue be the result of calling the Get internal + // method of O with the argument "length". + // 3. Let len be ToUint32(lenValue). + var len = O.length >>> 0; + + // 4. If IsCallable(callback) is false, throw a TypeError exception. + // See: http://es5.github.com/#x9.11 + if (typeof callback !== 'function') { + throw new TypeError(callback + ' is not a function'); + } + + // 5. If thisArg was supplied, let T be thisArg; else let T be undefined. + if (arguments.length > 1) { + T = thisArg; + } + + // 6. Let A be a new array created as if by the expression new Array(len) + // where Array is the standard built-in constructor with that name and + // len is the value of len. + A = new Array(len); + + // 7. Let k be 0 + k = 0; + + // 8. Repeat, while k < len + while (k < len) { + + var kValue, mappedValue; + + // a. Let Pk be ToString(k). + // This is implicit for LHS operands of the in operator + // b. Let kPresent be the result of calling the HasProperty internal + // method of O with argument Pk. + // This step can be combined with c + // c. If kPresent is true, then + if (k in O) { + + // i. Let kValue be the result of calling the Get internal + // method of O with argument Pk. + kValue = O[k]; + + // ii. Let mappedValue be the result of calling the Call internal + // method of callback with T as the this value and argument + // list containing kValue, k, and O. + mappedValue = callback.call(T, kValue, k, O); + + // iii. Call the DefineOwnProperty internal method of A with arguments + // Pk, Property Descriptor + // { Value: mappedValue, + // Writable: true, + // Enumerable: true, + // Configurable: true }, + // and false. + + // In browsers that support Object.defineProperty, use the following: + // Object.defineProperty(A, k, { + // value: mappedValue, + // writable: true, + // enumerable: true, + // configurable: true + // }); + + // For best browser support, use the following: + A[k] = mappedValue; + } + // d. Increase k by 1. + k++; + } + + // 9. return A + return A; + }; +} + +var PagedTable = function (pagedTable) { + var me = this; + + var source = function(pagedTable) { + var sourceElems = [].slice.call(pagedTable.children).filter(function(e) { + return e.hasAttribute("data-pagedtable-source"); + }); + + if (sourceElems === null || sourceElems.length !== 1) { + throw("A single data-pagedtable-source was not found"); + } + + return JSON.parse(sourceElems[0].innerHTML); + }(pagedTable); + + var options = function(source) { + var options = typeof(source.options) !== "undefined" && + source.options !== null ? source.options : {}; + + var columns = typeof(options.columns) !== "undefined" ? options.columns : {}; + var rows = typeof(options.rows) !== "undefined" ? options.rows : {}; + + var positiveIntOrNull = function(value) { + return parseInt(value) >= 0 ? parseInt(value) : null; + }; + + return { + pages: positiveIntOrNull(options.pages), + rows: { + min: positiveIntOrNull(rows.min), + max: positiveIntOrNull(rows.max), + total: positiveIntOrNull(rows.total) + }, + columns: { + min: positiveIntOrNull(columns.min), + max: positiveIntOrNull(columns.max), + total: positiveIntOrNull(columns.total) + } + }; + }(source); + + var Measurer = function() { + + // set some default initial values that will get adjusted in runtime + me.measures = { + padding: 12, + character: 8, + height: 15, + defaults: true + }; + + me.calculate = function(measuresCell) { + if (!me.measures.defaults) + return; + + var measuresCellStyle = window.getComputedStyle(measuresCell, null); + + var newPadding = parsePadding(measuresCellStyle.paddingLeft) + + parsePadding(measuresCellStyle.paddingRight); + + var sampleString = "ABCDEFGHIJ0123456789"; + var newCharacter = Math.ceil(measuresCell.clientWidth / sampleString.length); + + if (newPadding <= 0 || newCharacter <= 0) + return; + + me.measures.padding = newPadding; + me.measures.character = newCharacter; + me.measures.height = measuresCell.clientHeight; + me.measures.defaults = false; + }; + + return me; + }; + + var Page = function(data, options) { + var me = this; + + var defaults = { + max: 7, + rows: 10 + }; + + var totalPages = function() { + return Math.ceil(data.length / me.rows); + }; + + me.number = 0; + me.max = options.pages !== null ? options.pages : defaults.max; + me.visible = me.max; + me.rows = options.rows.min !== null ? options.rows.min : defaults.rows; + me.total = totalPages(); + + me.setRows = function(newRows) { + me.rows = newRows; + me.total = totalPages(); + }; + + me.setPageNumber = function(newPageNumber) { + if (newPageNumber < 0) newPageNumber = 0; + if (newPageNumber >= me.total) newPageNumber = me.total - 1; + + me.number = newPageNumber; + }; + + me.setVisiblePages = function(visiblePages) { + me.visible = Math.min(me.max, visiblePages); + me.setPageNumber(me.number); + }; + + me.getVisiblePageRange = function() { + var start = me.number - Math.max(Math.floor((me.visible - 1) / 2), 0); + var end = me.number + Math.floor(me.visible / 2) + 1; + var pageCount = me.total; + + if (start < 0) { + var diffToStart = 0 - start; + start += diffToStart; + end += diffToStart; + } + + if (end > pageCount) { + var diffToEnd = end - pageCount; + start -= diffToEnd; + end -= diffToEnd; + } + + start = start < 0 ? 0 : start; + end = end >= pageCount ? pageCount : end; + + var first = false; + var last = false; + + if (start > 0 && me.visible > 1) { + start = start + 1; + first = true; + } + + if (end < pageCount && me.visible > 2) { + end = end - 1; + last = true; + } + + return { + first: first, + start: start, + end: end, + last: last + }; + }; + + me.getRowStart = function() { + var rowStart = page.number * page.rows; + if (rowStart < 0) + rowStart = 0; + + return rowStart; + }; + + me.getRowEnd = function() { + var rowStart = me.getRowStart(); + return Math.min(rowStart + me.rows, data.length); + }; + + me.getPaddingRows = function() { + var rowStart = me.getRowStart(); + var rowEnd = me.getRowEnd(); + return data.length > me.rows ? me.rows - (rowEnd - rowStart) : 0; + }; + }; + + var Columns = function(data, columns, options) { + var me = this; + + me.defaults = { + min: 5 + }; + + me.number = 0; + me.visible = 0; + me.total = columns.length; + me.subset = []; + me.padding = 0; + me.min = options.columns.min !== null ? options.columns.min : me.defaults.min; + me.max = options.columns.max !== null ? options.columns.max : null; + me.widths = {}; + + var widthsLookAhead = Math.max(100, options.rows.min); + var paddingColChars = 10; + + me.emptyNames = function() { + columns.forEach(function(column) { + if (columns.label !== null && columns.label !== "") + return false; + }); + + return true; + }; + + var parsePadding = function(value) { + return parseInt(value) >= 0 ? parseInt(value) : 0; + }; + + me.calculateWidths = function(measures) { + columns.forEach(function(column) { + var maxChars = Math.max( + column.label.toString().length, + column.type.toString().length + ); + + for (var idxRow = 0; idxRow < Math.min(widthsLookAhead, data.length); idxRow++) { + maxChars = Math.max(maxChars, data[idxRow][column.name.toString()].length); + } + + me.widths[column.name] = { + // width in characters + chars: maxChars, + // width for the inner html columns + inner: maxChars * measures.character, + // width adding outer styles like padding + outer: maxChars * measures.character + measures.padding + }; + }); + }; + + me.getWidth = function() { + var widthOuter = 0; + for (var idxCol = 0; idxCol < me.subset.length; idxCol++) { + var columnName = me.subset[idxCol].name; + widthOuter = widthOuter + me.widths[columnName].outer; + } + + widthOuter = widthOuter + me.padding * paddingColChars * measurer.measures.character; + + if (me.hasMoreLeftColumns()) { + widthOuter = widthOuter + columnNavigationWidthPX + measurer.measures.padding; + } + + if (me.hasMoreRightColumns()) { + widthOuter = widthOuter + columnNavigationWidthPX + measurer.measures.padding; + } + + return widthOuter; + }; + + me.updateSlice = function() { + if (me.number + me.visible >= me.total) + me.number = me.total - me.visible; + + if (me.number < 0) me.number = 0; + + me.subset = columns.slice(me.number, Math.min(me.number + me.visible, me.total)); + + me.subset = me.subset.map(function(column) { + Object.keys(column).forEach(function(colKey) { + column[colKey] = column[colKey] === null ? "" : column[colKey].toString(); + }); + + column.width = null; + return column; + }); + }; + + me.setVisibleColumns = function(columnNumber, newVisibleColumns, paddingCount) { + me.number = columnNumber; + me.visible = newVisibleColumns; + me.padding = paddingCount; + + me.updateSlice(); + }; + + me.incColumnNumber = function(increment) { + me.number = me.number + increment; + }; + + me.setColumnNumber = function(newNumber) { + me.number = newNumber; + }; + + me.setPaddingCount = function(newPadding) { + me.padding = newPadding; + }; + + me.getPaddingCount = function() { + return me.padding; + }; + + me.hasMoreLeftColumns = function() { + return me.number > 0; + }; + + me.hasMoreRightColumns = function() { + return me.number + me.visible < me.total; + }; + + me.updateSlice(0); + return me; + }; + + var data = source.data; + var page = new Page(data, options); + var measurer = new Measurer(data, options); + var columns = new Columns(data, source.columns, options); + + var table = null; + var tableDiv = null; + var header = null; + var footer = null; + var tbody = null; + + // Caches pagedTable.clientWidth, specially for webkit + var cachedPagedTableClientWidth = null; + + var onChangeCallbacks = []; + + var clearSelection = function() { + if(document.selection && document.selection.empty) { + document.selection.empty(); + } else if(window.getSelection) { + var sel = window.getSelection(); + sel.removeAllRanges(); + } + }; + + var columnNavigationWidthPX = 5; + + var renderColumnNavigation = function(increment, backwards) { + var arrow = document.createElement("div"); + arrow.setAttribute("style", + "border-top: " + columnNavigationWidthPX + "px solid transparent;" + + "border-bottom: " + columnNavigationWidthPX + "px solid transparent;" + + "border-" + (backwards ? "right" : "left") + ": " + columnNavigationWidthPX + "px solid;"); + + var header = document.createElement("th"); + header.appendChild(arrow); + header.setAttribute("style", + "cursor: pointer;" + + "vertical-align: middle;" + + "min-width: " + columnNavigationWidthPX + "px;" + + "width: " + columnNavigationWidthPX + "px;"); + + header.onclick = function() { + columns.incColumnNumber(backwards ? -1 : increment); + + me.animateColumns(backwards); + renderFooter(); + + clearSelection(); + triggerOnChange(); + }; + + return header; + }; + + var maxColumnWidth = function(width) { + var padding = 80; + var columnMax = Math.max(cachedPagedTableClientWidth - padding, 0); + + return parseInt(width) > 0 ? + Math.min(columnMax, parseInt(width)) + "px" : + columnMax + "px"; + }; + + var clearHeader = function() { + var thead = pagedTable.querySelectorAll("thead")[0]; + thead.innerHTML = ""; + }; + + var renderHeader = function(clear) { + cachedPagedTableClientWidth = pagedTable.clientWidth; + + var fragment = document.createDocumentFragment(); + + header = document.createElement("tr"); + fragment.appendChild(header); + + if (columns.number > 0) + header.appendChild(renderColumnNavigation(-columns.visible, true)); + + columns.subset = columns.subset.map(function(columnData) { + var column = document.createElement("th"); + column.setAttribute("align", columnData.align); + column.style.textAlign = columnData.align; + + column.style.maxWidth = maxColumnWidth(null); + if (columnData.width) { + column.style.minWidth = + column.style.maxWidth = maxColumnWidth(columnData.width); + } + + var columnName = document.createElement("div"); + columnName.setAttribute("class", "pagedtable-header-name"); + if (columnData.label === "") { + columnName.innerHTML = " "; + } + else { + columnName.appendChild(document.createTextNode(columnData.label)); + } + column.appendChild(columnName); + + var columnType = document.createElement("div"); + columnType.setAttribute("class", "pagedtable-header-type"); + if (columnData.type === "") { + columnType.innerHTML = " "; + } + else { + columnType.appendChild(document.createTextNode("<" + columnData.type + ">")); + } + column.appendChild(columnType); + + header.appendChild(column); + + columnData.element = column; + + return columnData; + }); + + for (var idx = 0; idx < columns.getPaddingCount(); idx++) { + var paddingCol = document.createElement("th"); + paddingCol.setAttribute("class", "pagedtable-padding-col"); + header.appendChild(paddingCol); + } + + if (columns.number + columns.visible < columns.total) + header.appendChild(renderColumnNavigation(columns.visible, false)); + + if (typeof(clear) == "undefined" || clear) clearHeader(); + var thead = pagedTable.querySelectorAll("thead")[0]; + thead.appendChild(fragment); + }; + + me.animateColumns = function(backwards) { + var thead = pagedTable.querySelectorAll("thead")[0]; + + var headerOld = thead.querySelectorAll("tr")[0]; + var tbodyOld = table.querySelectorAll("tbody")[0]; + + me.fitColumns(backwards); + + renderHeader(false); + + header.style.opacity = "0"; + header.style.transform = backwards ? "translateX(-30px)" : "translateX(30px)"; + header.style.transition = "transform 200ms linear, opacity 200ms"; + header.style.transitionDelay = "0"; + + renderBody(false); + + if (headerOld) { + headerOld.style.position = "absolute"; + headerOld.style.transform = "translateX(0px)"; + headerOld.style.opacity = "1"; + headerOld.style.transition = "transform 100ms linear, opacity 100ms"; + headerOld.setAttribute("class", "pagedtable-remove-head"); + if (headerOld.style.transitionEnd) { + headerOld.addEventListener("transitionend", function() { + var headerOldByClass = thead.querySelector(".pagedtable-remove-head"); + if (headerOldByClass) thead.removeChild(headerOldByClass); + }); + } + else { + thead.removeChild(headerOld); + } + } + + if (tbodyOld) table.removeChild(tbodyOld); + + tbody.style.opacity = "0"; + tbody.style.transition = "transform 200ms linear, opacity 200ms"; + tbody.style.transitionDelay = "0ms"; + + // force relayout + window.getComputedStyle(header).opacity; + window.getComputedStyle(tbody).opacity; + + if (headerOld) { + headerOld.style.transform = backwards ? "translateX(20px)" : "translateX(-30px)"; + headerOld.style.opacity = "0"; + } + + header.style.transform = "translateX(0px)"; + header.style.opacity = "1"; + + tbody.style.opacity = "1"; + } + + me.onChange = function(callback) { + onChangeCallbacks.push(callback); + }; + + var triggerOnChange = function() { + onChangeCallbacks.forEach(function(onChange) { + onChange(); + }); + }; + + var clearBody = function() { + if (tbody) { + table.removeChild(tbody); + tbody = null; + } + }; + + var renderBody = function(clear) { + cachedPagedTableClientWidth = pagedTable.clientWidth + + var fragment = document.createDocumentFragment(); + + var pageData = data.slice(page.getRowStart(), page.getRowEnd()); + + pageData.forEach(function(dataRow, idxRow) { + var htmlRow = document.createElement("tr"); + htmlRow.setAttribute("class", (idxRow % 2 !==0) ? "even" : "odd"); + + if (columns.hasMoreLeftColumns()) + htmlRow.appendChild(document.createElement("td")); + + columns.subset.forEach(function(columnData) { + var cellName = columnData.name; + var dataCell = dataRow[cellName]; + var htmlCell = document.createElement("td"); + + if (dataCell === "NA") htmlCell.setAttribute("class", "pagedtable-na-cell"); + if (dataCell === "__NA__") dataCell = "NA"; + + var cellText = document.createTextNode(dataCell); + htmlCell.appendChild(cellText); + if (dataCell.length > 50) { + htmlCell.setAttribute("title", dataCell); + } + htmlCell.setAttribute("align", columnData.align); + htmlCell.style.textAlign = columnData.align; + htmlCell.style.maxWidth = maxColumnWidth(null); + if (columnData.width) { + htmlCell.style.minWidth = htmlCell.style.maxWidth = maxColumnWidth(columnData.width); + } + htmlRow.appendChild(htmlCell); + }); + + for (var idx = 0; idx < columns.getPaddingCount(); idx++) { + var paddingCol = document.createElement("td"); + paddingCol.setAttribute("class", "pagedtable-padding-col"); + htmlRow.appendChild(paddingCol); + } + + if (columns.hasMoreRightColumns()) + htmlRow.appendChild(document.createElement("td")); + + fragment.appendChild(htmlRow); + }); + + for (var idxPadding = 0; idxPadding < page.getPaddingRows(); idxPadding++) { + var paddingRow = document.createElement("tr"); + + var paddingCellRow = document.createElement("td"); + paddingCellRow.innerHTML = " "; + paddingCellRow.setAttribute("colspan", "100%"); + paddingRow.appendChild(paddingCellRow); + + fragment.appendChild(paddingRow); + } + + if (typeof(clear) == "undefined" || clear) clearBody(); + tbody = document.createElement("tbody"); + tbody.appendChild(fragment); + + table.appendChild(tbody); + }; + + var getLabelInfo = function() { + var pageStart = page.getRowStart(); + var pageEnd = page.getRowEnd(); + var totalRows = data.length; + + var totalRowsLabel = options.rows.total ? options.rows.total : totalRows; + var totalRowsLabelFormat = totalRowsLabel.toString().replace(/(\d)(?=(\d\d\d)+(?!\d))/g, '$1,'); + + var infoText = (pageStart + 1) + "-" + pageEnd + " of " + totalRowsLabelFormat + " rows"; + if (totalRows < page.rows) { + infoText = totalRowsLabel + " row" + (totalRows != 1 ? "s" : ""); + } + if (columns.total > columns.visible) { + var totalColumnsLabel = options.columns.total ? options.columns.total : columns.total; + + infoText = infoText + " | " + (columns.number + 1) + "-" + + (Math.min(columns.number + columns.visible, columns.total)) + + " of " + totalColumnsLabel + " columns"; + } + + return infoText; + }; + + var clearFooter = function() { + footer = pagedTable.querySelectorAll("div.pagedtable-footer")[0]; + footer.innerHTML = ""; + + return footer; + }; + + var createPageLink = function(idxPage) { + var pageLink = document.createElement("a"); + pageLinkClass = idxPage === page.number ? 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The choice of a reference level is arbitrary and does not limit what we can do with the resulting model; it only makes it more computationally convenient to make inferences about comparisons involving that reference group.\n\n\n\n2.2.2 Parallel lines regression\nWe don’t have enough data to model the distribution of birth weight separately for each combination of gestational age and sex, so let’s instead consider a (relatively) simple model for how that distribution varies with gestational age and sex:\n\\[p(Y=y|A=a,S=s) \\ \\sim_{\\text{iid}}\\ N(\\mu(a,s), \\sigma^2)\\]\n\\[\n\\begin{aligned}\n\\mu(a,s)\n&\\stackrel{\\text{def}}{=}\\mathbb{E}\\left[Y|A=a, S=s\\right] \\\\\n&= \\beta_0 + \\beta_A a+ \\beta_M m\n\\end{aligned}\n\\tag{2.1}\\]\n\nTable 2.3 shows the parameter estimates from R. Figure 2.2 shows the estimated model, superimposed on the data.\n\n\nShow R codebw_lm1 = lm(\n formula = weight ~ sex + age, \n data = bw)\n\nbw_lm1 |> \n parameters() |>\n print_md(\n include_reference = TRUE,\n # show_sigma = TRUE,\n select = \"{estimate}\")\n\n\nTable 2.3: Estimate of Model 2.1 for birthweight data\n\n\n\n\nParameter\nEstimate\n\n\n\n(Intercept)\n-1773.32\n\n\nsex (female)\n0.00\n\n\nsex (male)\n163.04\n\n\nage\n120.89\n\n\n\n\n\n\n\n\n\n\n\n\nShow R codebw = \n bw |> \n mutate(`E[Y|X=x]` = fitted(bw_lm1)) |> \n arrange(sex, age)\n\nplot2 = \n plot1 %+% bw +\n geom_line(aes(y = `E[Y|X=x]`))\n\nprint(plot2)\n\n\n\nFigure 2.2: Parallel-slopes model of birthweight\n\n\n\n\n\n\n\n\n\nModel assumptions and predictions\n\nTo learn what this model is assuming, let’s plug in a few values.\n\n\nExercise 2.1 According to this model, what’s the mean birthweight for a female born at 36 weeks?\n\n\nTable 2.4: Estimated coefficients for model 2.1\n\nShow R codecoef(bw_lm1)\n#> (Intercept) sexmale age \n#> -1773.32184 163.03930 120.89433\n\n\n\n\n\n\n\nSolution.  \n\nShow R codepred_female = coef(bw_lm1)[\"(Intercept)\"] + coef(bw_lm1)[\"age\"]*36\ncoef(bw_lm1)\n#> (Intercept) sexmale age \n#> -1773.3 163.0 120.9\n# print(pred_female)\n### built-in prediction: \n# predict(bw_lm1, newdata = tibble(sex = \"female\", age = 36))\n\n\n\\[\n\\begin{aligned}\nE[Y|A = 0, A = 36]\n&= \\beta_0 + \\beta_M \\cdot 0+ \\beta_A \\cdot 36 \\\\\n&= 2578.873934\n\\end{aligned}\n\\]\n\n\n\nExercise 2.2 What’s the mean birthweight for a male born at 36 weeks?\n\nShow R codecoef(bw_lm1)\n#> (Intercept) sexmale age \n#> -1773.3 163.0 120.9\n\n\n\n\n\nSolution.  \n\nShow R codepred_male = \n coef(bw_lm1)[\"(Intercept)\"] + \n coef(bw_lm1)[\"sexmale\"] + \n coef(bw_lm1)[\"age\"]*36\ncoef(bw_lm1)\n#> (Intercept) sexmale age \n#> -1773.3 163.0 120.9\n\n\n\\[\n\\begin{aligned}\nE[Y|M = 1, A = 36]\n&= \\beta_0 + \\beta_M \\cdot 1+ \\beta_A \\cdot 36 \\\\\n&= 2741.91323693\n\\end{aligned}\n\\]\n\n\n\nExercise 2.3 What’s the difference in mean birthweights between males born at 36 weeks and females born at 36 weeks?\n\n\nShow R codecoef(bw_lm1)\n#> (Intercept) sexmale age \n#> -1773.3 163.0 120.9\n\n\n\n\nSolution. \\[\n\\begin{aligned}\n& E[Y|M = 1, A = 36] - E[Y|M = 0, A = 36]\\\\\n&=\n2741.91323693 - 2578.873934\\\\\n&=\n163.03930293\n\\end{aligned}\n\\]\nShortcut:\n\\[\n\\begin{aligned}\n& E[Y|M = 1, A = 36] - E[Y|M = 0, A = 36]\\\\\n&= (\\beta_0 + \\beta_M \\cdot 1+ \\beta_A \\cdot 36) -\n(\\beta_0 + \\beta_M \\cdot 0+ \\beta_A \\cdot 36) \\\\\n&= \\beta_M \\\\\n&= 163.03930293\n\\end{aligned}\n\\]\n\n\nNote that age doesn’t show up in this difference: in other words, according to this model, the difference between females and males with the same gestational age is the same for every age.\nThat’s an assumption of the model; it’s built-in to the parametric structure, even before we plug in the estimated values of those parameters.\nThat’s why the lines are parallel.\n\n\n2.2.3 Interactions\n\nWhat if we don’t like that parallel lines assumption?\nThen we need to allow an “interaction” between age \\(A\\) and sex \\(S\\):\n\n\\[\nE[Y|A=a, S=s] = \\beta_0 + \\beta_A a+ \\beta_M m + \\beta_{AM} (a \\cdot m)\n\\tag{2.2}\\]\n\nNow, the slope of mean birthweight \\(E[Y|A,S]\\) with respect to gestational age \\(A\\) depends on the value of sex \\(S\\).\n\n\n\nShow R codebw_lm2 = lm(weight ~ sex + age + sex:age, data = bw)\nbw_lm2 |> \n parameters() |>\n print_md(\n include_reference = TRUE,\n # show_sigma = TRUE,\n select = \"{estimate}\")\n\n\nTable 2.5: Birthweight model with interaction term\n\n\n\n\nParameter\nEstimate\n\n\n\n(Intercept)\n-2141.67\n\n\nsex (female)\n0.00\n\n\nsex (male)\n872.99\n\n\nage\n130.40\n\n\nsex (male) × age\n-18.42\n\n\n\n\n\n\n\n\n\n\n\n\nShow R codebw = \n bw |> \n mutate(\n predlm2 = predict(bw_lm2)\n ) |> \n arrange(sex, age)\n\nplot1_interact = \n plot1 %+% bw +\n geom_line(aes(y = predlm2))\n\nprint(plot1_interact)\n\n\n\nFigure 2.3: Birthweight model with interaction term\n\n\n\n\n\n\n\n\n\nNow we can see that the lines aren’t parallel.\n\n\nHere’s another way we could rewrite this model (by collecting terms involving \\(S\\)):\n\\[\nE[Y|A, M] = \\beta_0 + \\beta_M M+ (\\beta_A + \\beta_{AM} M) A\n\\]\n\n\n\n\n\n\nNote\n\n\n\nIf you want to understand a coefficient in a model with interactions, collect terms for the corresponding variable, and you will see what other variables are interacting with the variable you are interested in.\n\n\n\nIn this case, the coefficient \\(S\\) is interacting with \\(A\\). So the slope of \\(Y\\) with respect to \\(A\\) depends on the value of \\(M\\).\nAccording to this model, there is no such thing as “the slope of birthweight with respect to age”. There are two slopes, one for each sex.3 We can only talk about “the slope of birthweight with respect to age among males” and “the slope of birthweight with respect to age among females”.\nThen: that coefficient is the difference in means per unit change in its corresponding coefficient, when the other collected variables are set to 0.\n\n\n\nTo learn what this model is assuming, let’s plug in a few values.\n\n\nExercise 2.4 According to this model, what’s the mean birthweight for a female born at 36 weeks?\n\n\n\nSolution.  \n\nShow R codepred_female = coef(bw_lm2)[\"(Intercept)\"] + coef(bw_lm2)[\"age\"]*36\n\n\n\\[\nE[Y|A = 0, X_2 = 36] =\n\\beta_0 + \\beta_M \\cdot 0+ \\beta_A \\cdot 36 + \\beta_{AM} \\cdot (0 * 36)\n= 2552.73333333\n\\]\n\n\n\nExercise 2.5 What’s the mean birthweight for a male born at 36 weeks?\n\n\n\nSolution.  \n\nShow R codepred_male = \n coef(bw_lm2)[\"(Intercept)\"] + \n coef(bw_lm2)[\"sexmale\"] + \n coef(bw_lm2)[\"age\"]*36 + \n coef(bw_lm2)[\"sexmale:age\"] * 36\n\n\n\\[\n\\begin{aligned}\nE[Y|A = 0, X_2 = 36]\n&= \\beta_0 + \\beta_M \\cdot 1+ \\beta_A \\cdot 36 + \\beta_{AM} \\cdot 1 \\cdot 36\\\\\n&= 2762.70689655\n\\end{aligned}\n\\]\n\n\n\nExercise 2.6 What’s the difference in mean birthweights between males born at 36 weeks and females born at 36 weeks?\n\n\n\nSolution. \\[\n\\begin{aligned}\n& E[Y|M = 1, A = 36] - E[Y|M = 0, A = 36]\\\\\n&= (\\beta_0 + \\beta_M \\cdot 1+ \\beta_A \\cdot 36 + \\beta_{AM} \\cdot 1 \\cdot 36)\\\\\n&\\ \\ \\ \\ \\ -(\\beta_0 + \\beta_M \\cdot 0+ \\beta_A \\cdot 36 + \\beta_{AM} \\cdot 0 \\cdot 36) \\\\\n&= \\beta_{S} + \\beta_{AM}\\cdot 36\\\\\n&= 209.97356322\n\\end{aligned}\n\\]\n\n\nNote that age now does show up in the difference: in other words, according to this model, the difference in mean birthweights between females and males with the same gestational age can vary by gestational age.\nThat’s how the lines in the graph ended up non-parallel.\n\n\n2.2.4 Stratified regression\n\nWe could re-write the interaction model as a stratified model, with a slope and intercept for each sex:\n\n\\[\n\\mathbb{E}\\left[Y|A=a, S=s\\right] =\n\\beta_M m + \\beta_{AM} (a \\cdot m) +\n\\beta_F f + \\beta_{AF} (a \\cdot f)\n\\tag{2.3}\\]\nCompare this stratified model with our interaction model, Equation 2.2:\n\\[\n\\mathbb{E}\\left[Y|A=a, S=s\\right] =\n\\beta_0 + \\beta_A a + \\beta_M m + \\beta_{AM} (a \\cdot m)\n\\]\n\nIn the stratified model, the intercept term \\(\\beta_0\\) has been relabeled as \\(\\beta_F\\).\n\n\nShow R codebw_lm2 = lm(weight ~ sex + age + sex:age, data = bw)\nbw_lm2 |> \n parameters() |>\n print_md(\n include_reference = TRUE,\n # show_sigma = TRUE,\n select = \"{estimate}\")\n\n\nTable 2.6: Birthweight model with interaction term\n\n\n\n\nParameter\nEstimate\n\n\n\n(Intercept)\n-2141.67\n\n\nsex (female)\n0.00\n\n\nsex (male)\n872.99\n\n\nage\n130.40\n\n\nsex (male) × age\n-18.42\n\n\n\n\n\n\n\n\n\n\n\n\nShow R codebw_lm_strat = \n bw |> \n lm(\n formula = weight ~ sex + sex:age - 1, \n data = _)\n\nbw_lm_strat |> \n parameters() |>\n print_md(\n # show_sigma = TRUE,\n select = \"{estimate}\")\n\n\nTable 2.7: Birthweight model - stratified betas\n\n\n\n\nParameter\nEstimate\n\n\n\nsex (female)\n-2141.67\n\n\nsex (male)\n-1268.67\n\n\nsex (female) × age\n130.40\n\n\nsex (male) × age\n111.98\n\n\n\n\n\n\n\n\n\n\n2.2.5 Curved-line regression\n\nIf we transform some of our covariates (\\(X\\)s) and plot the resulting model on the original covariate scale, we end up with curved regression lines:\n\n\nShow R codebw_lm3 = lm(weight ~ sex:log(age) - 1, data = bw)\nlibrary(palmerpenguins)\n\nggpenguins <- \n palmerpenguins::penguins |> \n dplyr::filter(species == \"Adelie\") |> \n ggplot(\n aes(x = bill_length_mm , y = body_mass_g)) +\n geom_point() + \n xlab(\"Bill length (mm)\") + \n ylab(\"Body mass (g)\")\n\nggpenguins2 = ggpenguins +\n stat_smooth(\n method = \"lm\",\n formula = y ~ log(x),\n geom = \"smooth\") +\n xlab(\"Bill length (mm)\") + \n ylab(\"Body mass (g)\")\n\n\nggpenguins2 |> print()\n\n\n\nFigure 2.4: palmerpenguins model with bill_length entering on log scale", + "text": "2.2 Understanding Gaussian Linear Regression Models\n\n2.2.1 Motivating example: birthweights and gestational age\nSuppose we want to learn about the distributions of birthweights (outcome \\(Y\\)) for (human) babies born at different gestational ages (covariate \\(A\\)) and with different chromosomal sexes (covariate \\(S\\)) (Dobson and Barnett (2018) Example 2.2.2).\n\n\nData as table\nReshape data for graphing\nData as graph\n\n\n\n\nShow R codelibrary(dobson)\ndata(\"birthweight\", package = \"dobson\")\nbirthweight |> knitr::kable()\n\n\nTable 2.1: birthweight data (Dobson and Barnett (2018) Example 2.2.2)\n\n\n\n\n\n\n\n\n\n\nboys gestational age\nboys weight\ngirls gestational age\ngirls weight\n\n\n\n40\n2968\n40\n3317\n\n\n38\n2795\n36\n2729\n\n\n40\n3163\n40\n2935\n\n\n35\n2925\n38\n2754\n\n\n36\n2625\n42\n3210\n\n\n37\n2847\n39\n2817\n\n\n41\n3292\n40\n3126\n\n\n40\n3473\n37\n2539\n\n\n37\n2628\n36\n2412\n\n\n38\n3176\n38\n2991\n\n\n40\n3421\n39\n2875\n\n\n38\n2975\n40\n3231\n\n\n\n\n\n\n\n\n\n\n\nShow R codebw = \n birthweight |> \n pivot_longer(\n cols = everything(),\n names_to = c(\"sex\", \".value\"),\n names_sep = \"s \"\n ) |> \n rename(age = `gestational age`) |> \n mutate(\n sex = sex |> \n case_match(\n \"boy\" ~ \"male\",\n \"girl\" ~ \"female\") |> \n factor(levels = c(\"female\", \"male\")))\n\nbw\n\n\nTable 2.2: birthweight data reshaped\n\n\n\n \n\n\n\n\n\n\n\n\n\nShow R codeplot1 = bw |> \n ggplot(aes(\n x = age, \n y = weight,\n linetype = sex,\n shape = sex,\n col = sex)) +\n theme_bw() +\n xlab(\"Gestational age (weeks)\") +\n ylab(\"Birthweight (grams)\") +\n theme(legend.position = \"bottom\") +\n # expand_limits(y = 0, x = 0) +\n geom_point(alpha = .7)\nprint(plot1 + facet_wrap(~ sex))\n\n\n\nFigure 2.1: birthweight data (Dobson and Barnett (2018) Example 2.2.2)\n\n\n\n\n\n\n\n\n\n\n\nData notation\nLet’s define some notation to represent this data.\n\n\\(Y\\): birthweight (measured in grams)\n\\(S\\): chromosomal sex: “male” (XY) or “female” (XX)\n\\(M\\): indicator variable for \\(S\\) = “male”1\n\\(M = 0\\) if female (XX)\n\\(M = 1\\) if male (XY)\n\\(F\\): indicator variable for \\(S\\) = “female”2\n\\(F = 1\\) if female (XX)\n\\(F = 0\\) if male (XY)\n\\(A\\): estimated gestational age at birth (measured in weeks).\n\n\n\n\n\n\n\nNote\n\n\n\nFemale is the reference level for the categorical variable \\(S\\) (chromosomal sex) and corresponding indicator variable \\(M\\) . The choice of a reference level is arbitrary and does not limit what we can do with the resulting model; it only makes it more computationally convenient to make inferences about comparisons involving that reference group.\n\n\n\n2.2.2 Parallel lines regression\nWe don’t have enough data to model the distribution of birth weight separately for each combination of gestational age and sex, so let’s instead consider a (relatively) simple model for how that distribution varies with gestational age and sex:\n\\[p(Y=y|A=a,S=s) \\ \\sim_{\\text{iid}}\\ N(\\mu(a,s), \\sigma^2)\\]\n\\[\n\\begin{aligned}\n\\mu(a,s)\n&\\stackrel{\\text{def}}{=}\\mathbb{E}\\left[Y|A=a, S=s\\right] \\\\\n&= \\beta_0 + \\beta_A a+ \\beta_M m\n\\end{aligned}\n\\tag{2.1}\\]\n\nTable 2.3 shows the parameter estimates from R. Figure 2.2 shows the estimated model, superimposed on the data.\n\n\nShow R codebw_lm1 = lm(\n formula = weight ~ sex + age, \n data = bw)\n\nbw_lm1 |> \n parameters() |>\n print_md(\n include_reference = TRUE,\n # show_sigma = TRUE,\n select = \"{estimate}\")\n\n\nTable 2.3: Estimate of Model 2.1 for birthweight data\n\n\n\n\nParameter\nEstimate\n\n\n\n(Intercept)\n-1773.32\n\n\nsex (female)\n0.00\n\n\nsex (male)\n163.04\n\n\nage\n120.89\n\n\n\n\n\n\n\n\n\n\n\n\nShow R codebw = \n bw |> \n mutate(`E[Y|X=x]` = fitted(bw_lm1)) |> \n arrange(sex, age)\n\nplot2 = \n plot1 %+% bw +\n geom_line(aes(y = `E[Y|X=x]`))\n\nprint(plot2)\n\n\n\nFigure 2.2: Parallel-slopes model of birthweight\n\n\n\n\n\n\n\n\n\nModel assumptions and predictions\n\nTo learn what this model is assuming, let’s plug in a few values.\n\n\nExercise 2.1 According to this model, what’s the mean birthweight for a female born at 36 weeks?\n\n\nTable 2.4: Estimated coefficients for model 2.1\n\nShow R codecoef(bw_lm1)\n#> (Intercept) sexmale age \n#> -1773.3 163.0 120.9\n\n\n\n\n\n\n\nSolution.  \n\nShow R codepred_female = coef(bw_lm1)[\"(Intercept)\"] + coef(bw_lm1)[\"age\"]*36\ncoef(bw_lm1)\n#> (Intercept) sexmale age \n#> -1773.3 163.0 120.9\n# print(pred_female)\n### built-in prediction: \n# predict(bw_lm1, newdata = tibble(sex = \"female\", age = 36))\n\n\n\\[\n\\begin{aligned}\nE[Y|A = 0, A = 36]\n&= \\beta_0 + \\beta_M \\cdot 0+ \\beta_A \\cdot 36 \\\\\n&= 2578.8739\n\\end{aligned}\n\\]\n\n\n\nExercise 2.2 What’s the mean birthweight for a male born at 36 weeks?\n\nShow R codecoef(bw_lm1)\n#> (Intercept) sexmale age \n#> -1773.3 163.0 120.9\n\n\n\n\n\nSolution.  \n\nShow R codepred_male = \n coef(bw_lm1)[\"(Intercept)\"] + \n coef(bw_lm1)[\"sexmale\"] + \n coef(bw_lm1)[\"age\"]*36\ncoef(bw_lm1)\n#> (Intercept) sexmale age \n#> -1773.3 163.0 120.9\n\n\n\\[\n\\begin{aligned}\nE[Y|M = 1, A = 36]\n&= \\beta_0 + \\beta_M \\cdot 1+ \\beta_A \\cdot 36 \\\\\n&= 2741.9132\n\\end{aligned}\n\\]\n\n\n\nExercise 2.3 What’s the difference in mean birthweights between males born at 36 weeks and females born at 36 weeks?\n\n\nShow R codecoef(bw_lm1)\n#> (Intercept) sexmale age \n#> -1773.3 163.0 120.9\n\n\n\n\nSolution. \\[\n\\begin{aligned}\n& E[Y|M = 1, A = 36] - E[Y|M = 0, A = 36]\\\\\n&=\n2741.9132 - 2578.8739\\\\\n&=\n163.0393\n\\end{aligned}\n\\]\nShortcut:\n\\[\n\\begin{aligned}\n& E[Y|M = 1, A = 36] - E[Y|M = 0, A = 36]\\\\\n&= (\\beta_0 + \\beta_M \\cdot 1+ \\beta_A \\cdot 36) -\n(\\beta_0 + \\beta_M \\cdot 0+ \\beta_A \\cdot 36) \\\\\n&= \\beta_M \\\\\n&= 163.0393\n\\end{aligned}\n\\]\n\n\nNote that age doesn’t show up in this difference: in other words, according to this model, the difference between females and males with the same gestational age is the same for every age.\nThat’s an assumption of the model; it’s built-in to the parametric structure, even before we plug in the estimated values of those parameters.\nThat’s why the lines are parallel.\n\n\n2.2.3 Interactions\n\nWhat if we don’t like that parallel lines assumption?\nThen we need to allow an “interaction” between age \\(A\\) and sex \\(S\\):\n\n\\[\nE[Y|A=a, S=s] = \\beta_0 + \\beta_A a+ \\beta_M m + \\beta_{AM} (a \\cdot m)\n\\tag{2.2}\\]\n\nNow, the slope of mean birthweight \\(E[Y|A,S]\\) with respect to gestational age \\(A\\) depends on the value of sex \\(S\\).\n\n\n\nShow R codebw_lm2 = lm(weight ~ sex + age + sex:age, data = bw)\nbw_lm2 |> \n parameters() |>\n print_md(\n include_reference = TRUE,\n # show_sigma = TRUE,\n select = \"{estimate}\")\n\n\nTable 2.5: Birthweight model with interaction term\n\n\n\n\nParameter\nEstimate\n\n\n\n(Intercept)\n-2141.67\n\n\nsex (female)\n0.00\n\n\nsex (male)\n872.99\n\n\nage\n130.40\n\n\nsex (male) × age\n-18.42\n\n\n\n\n\n\n\n\n\n\n\n\nShow R codebw = \n bw |> \n mutate(\n predlm2 = predict(bw_lm2)\n ) |> \n arrange(sex, age)\n\nplot1_interact = \n plot1 %+% bw +\n geom_line(aes(y = predlm2))\n\nprint(plot1_interact)\n\n\n\nFigure 2.3: Birthweight model with interaction term\n\n\n\n\n\n\n\n\n\nNow we can see that the lines aren’t parallel.\n\n\nHere’s another way we could rewrite this model (by collecting terms involving \\(S\\)):\n\\[\nE[Y|A, M] = \\beta_0 + \\beta_M M+ (\\beta_A + \\beta_{AM} M) A\n\\]\n\n\n\n\n\n\nNote\n\n\n\nIf you want to understand a coefficient in a model with interactions, collect terms for the corresponding variable, and you will see what other variables are interacting with the variable you are interested in.\n\n\n\nIn this case, the coefficient \\(S\\) is interacting with \\(A\\). So the slope of \\(Y\\) with respect to \\(A\\) depends on the value of \\(M\\).\nAccording to this model, there is no such thing as “the slope of birthweight with respect to age”. There are two slopes, one for each sex.3 We can only talk about “the slope of birthweight with respect to age among males” and “the slope of birthweight with respect to age among females”.\nThen: that coefficient is the difference in means per unit change in its corresponding coefficient, when the other collected variables are set to 0.\n\n\n\nTo learn what this model is assuming, let’s plug in a few values.\n\n\nExercise 2.4 According to this model, what’s the mean birthweight for a female born at 36 weeks?\n\n\n\nSolution.  \n\nShow R codepred_female = coef(bw_lm2)[\"(Intercept)\"] + coef(bw_lm2)[\"age\"]*36\n\n\n\\[\nE[Y|A = 0, X_2 = 36] =\n\\beta_0 + \\beta_M \\cdot 0+ \\beta_A \\cdot 36 + \\beta_{AM} \\cdot (0 * 36)\n= 2552.7333\n\\]\n\n\n\nExercise 2.5 What’s the mean birthweight for a male born at 36 weeks?\n\n\n\nSolution.  \n\nShow R codepred_male = \n coef(bw_lm2)[\"(Intercept)\"] + \n coef(bw_lm2)[\"sexmale\"] + \n coef(bw_lm2)[\"age\"]*36 + \n coef(bw_lm2)[\"sexmale:age\"] * 36\n\n\n\\[\n\\begin{aligned}\nE[Y|A = 0, X_2 = 36]\n&= \\beta_0 + \\beta_M \\cdot 1+ \\beta_A \\cdot 36 + \\beta_{AM} \\cdot 1 \\cdot 36\\\\\n&= 2762.7069\n\\end{aligned}\n\\]\n\n\n\nExercise 2.6 What’s the difference in mean birthweights between males born at 36 weeks and females born at 36 weeks?\n\n\n\nSolution. \\[\n\\begin{aligned}\n& E[Y|M = 1, A = 36] - E[Y|M = 0, A = 36]\\\\\n&= (\\beta_0 + \\beta_M \\cdot 1+ \\beta_A \\cdot 36 + \\beta_{AM} \\cdot 1 \\cdot 36)\\\\\n&\\ \\ \\ \\ \\ -(\\beta_0 + \\beta_M \\cdot 0+ \\beta_A \\cdot 36 + \\beta_{AM} \\cdot 0 \\cdot 36) \\\\\n&= \\beta_{S} + \\beta_{AM}\\cdot 36\\\\\n&= 209.9736\n\\end{aligned}\n\\]\n\n\nNote that age now does show up in the difference: in other words, according to this model, the difference in mean birthweights between females and males with the same gestational age can vary by gestational age.\nThat’s how the lines in the graph ended up non-parallel.\n\n\n2.2.4 Stratified regression\n\nWe could re-write the interaction model as a stratified model, with a slope and intercept for each sex:\n\n\\[\n\\mathbb{E}\\left[Y|A=a, S=s\\right] =\n\\beta_M m + \\beta_{AM} (a \\cdot m) +\n\\beta_F f + \\beta_{AF} (a \\cdot f)\n\\tag{2.3}\\]\nCompare this stratified model with our interaction model, Equation 2.2:\n\\[\n\\mathbb{E}\\left[Y|A=a, S=s\\right] =\n\\beta_0 + \\beta_A a + \\beta_M m + \\beta_{AM} (a \\cdot m)\n\\]\n\nIn the stratified model, the intercept term \\(\\beta_0\\) has been relabeled as \\(\\beta_F\\).\n\n\nShow R codebw_lm2 = lm(weight ~ sex + age + sex:age, data = bw)\nbw_lm2 |> \n parameters() |>\n print_md(\n include_reference = TRUE,\n # show_sigma = TRUE,\n select = \"{estimate}\")\n\n\nTable 2.6: Birthweight model with interaction term\n\n\n\n\nParameter\nEstimate\n\n\n\n(Intercept)\n-2141.67\n\n\nsex (female)\n0.00\n\n\nsex (male)\n872.99\n\n\nage\n130.40\n\n\nsex (male) × age\n-18.42\n\n\n\n\n\n\n\n\n\n\n\n\nShow R codebw_lm_strat = \n bw |> \n lm(\n formula = weight ~ sex + sex:age - 1, \n data = _)\n\nbw_lm_strat |> \n parameters() |>\n print_md(\n # show_sigma = TRUE,\n select = \"{estimate}\")\n\n\nTable 2.7: Birthweight model - stratified betas\n\n\n\n\nParameter\nEstimate\n\n\n\nsex (female)\n-2141.67\n\n\nsex (male)\n-1268.67\n\n\nsex (female) × age\n130.40\n\n\nsex (male) × age\n111.98\n\n\n\n\n\n\n\n\n\n\n2.2.5 Curved-line regression\n\nIf we transform some of our covariates (\\(X\\)s) and plot the resulting model on the original covariate scale, we end up with curved regression lines:\n\n\nShow R codebw_lm3 = lm(weight ~ sex:log(age) - 1, data = bw)\nlibrary(palmerpenguins)\n\nggpenguins <- \n palmerpenguins::penguins |> \n dplyr::filter(species == \"Adelie\") |> \n ggplot(\n aes(x = bill_length_mm , y = body_mass_g)) +\n geom_point() + \n xlab(\"Bill length (mm)\") + \n ylab(\"Body mass (g)\")\n\nggpenguins2 = ggpenguins +\n stat_smooth(\n method = \"lm\",\n formula = y ~ log(x),\n geom = \"smooth\") +\n xlab(\"Bill length (mm)\") + \n ylab(\"Body mass (g)\")\n\n\nggpenguins2 |> print()\n\n\n\nFigure 2.4: palmerpenguins model with bill_length entering on log scale", "crumbs": [ "Generalized Linear Models", "2  Linear (Gaussian) Models" @@ -138,7 +138,7 @@ "href": "Linear-models-overview.html#sec-infer-LMs", "title": "\n2  Linear (Gaussian) Models\n", "section": "\n2.4 Inference about Gaussian Linear Regression Models", - "text": "2.4 Inference about Gaussian Linear Regression Models\n\n2.4.1 Motivating example: birthweight data\nResearch question: is there really an interaction between sex and age?\n\\(H_0: \\beta_{AM} = 0\\)\n\\(H_A: \\beta_{AM} \\neq 0\\)\n\\(P(|\\hat\\beta_{AM}| > |-18.41724138| \\mid H_0)\\) = ?\n\n2.4.2 Wald tests and CIs\nR can give you Wald tests for single coefficients and corresponding CIs:\n\nShow R code\nbw_lm2 |> \n parameters() |>\n print_md(\n include_reference = TRUE)\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(20)\np\n\n\n\n(Intercept)\n-2141.67\n1163.60\n(-4568.90, 285.56)\n-1.84\n0.081\n\n\nsex (female)\n0.00\n\n\n\n\n\n\nsex (male)\n872.99\n1611.33\n(-2488.18, 4234.17)\n0.54\n0.594\n\n\nage\n130.40\n30.00\n(67.82, 192.98)\n4.35\n< .001\n\n\nsex (male) × age\n-18.42\n41.76\n(-105.52, 68.68)\n-0.44\n0.664\n\n\n\n\n\nTo understand what’s happening, let’s replicate these results by hand for the interaction term.\n\n2.4.3 P-values\n\nShow R codebw_lm2 |> \n parameters(keep = \"sexmale:age\") |>\n print_md(\n include_reference = TRUE)\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(20)\np\n\n\nsex (male) × age\n-18.42\n41.76\n(-105.52, 68.68)\n-0.44\n0.664\n\n\n\n\n\nShow R codebeta_hat = coef(summary(bw_lm2))[\"sexmale:age\", \"Estimate\"]\nse_hat = coef(summary(bw_lm2))[\"sexmale:age\", \"Std. Error\"]\ndfresid = bw_lm2$df.residual\nt_stat = abs(beta_hat)/se_hat\npval_t = \n pt(-t_stat, df = dfresid, lower.tail = TRUE) +\n pt(t_stat, df = dfresid, lower.tail = FALSE)\n\n\n\\[\n\\begin{aligned}\n&P\\left(\n| \\hat \\beta_{AM} | >\n| -18.41724138| \\middle| H_0\n\\right)\n\\\\\n&= \\Pr \\left(\n\\left| \\frac{\\hat\\beta_{AM}}{\\hat{SE}(\\hat\\beta_{AM})} \\right| >\n\\left| \\frac{-18.41724138}{41.75581745} \\right| \\middle| H_0\n\\right)\\\\\n&= \\Pr \\left(\n\\left| T_{20} \\right| > 0.44107007 | H_0\n\\right)\\\\\n&= 0.66389342\n\\end{aligned}\n\\]\n\nThis matches the result in the table above.\n\n\n2.4.4 Confidence intervals\n\nShow R codebw_lm2 |> \n parameters(keep = \"sexmale:age\") |>\n print_md(\n include_reference = TRUE)\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(20)\np\n\n\nsex (male) × age\n-18.42\n41.76\n(-105.52, 68.68)\n-0.44\n0.664\n\n\n\n\n\nShow R codeq_t = qt(\n p = 0.975, \n df = dfresid, \n lower.tail = TRUE)\n\nq_t = qt(\n p = 0.025, \n df = dfresid, \n lower.tail = TRUE)\n\n\nconfint_radius_t = \n se_hat * q_t\n\nconfint_t = beta_hat + c(-1,1) * confint_radius_t\n\nprint(confint_t)\n#> [1] 68.68 -105.52\n\n\n\nThis also matches.\n\n\n2.4.5 Gaussian approximations\nHere are the asymptotic (Gaussian approximation) equivalents:\n\n2.4.6 P-values\n\nShow R codebw_lm2 |> \n parameters(keep = \"sexmale:age\") |>\n print_md(\n include_reference = TRUE)\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(20)\np\n\n\nsex (male) × age\n-18.42\n41.76\n(-105.52, 68.68)\n-0.44\n0.664\n\n\n\n\n\nShow R codepval_z = pnorm(abs(t_stat), lower = FALSE) * 2\n\nprint(pval_z)\n#> [1] 0.6592\n\n\n\n2.4.7 Confidence intervals\n\nShow R codebw_lm2 |> \n parameters(keep = \"sexmale:age\") |>\n print_md(\n include_reference = TRUE)\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(20)\np\n\n\nsex (male) × age\n-18.42\n41.76\n(-105.52, 68.68)\n-0.44\n0.664\n\n\n\n\n\nShow R codeconfint_radius_z = se_hat * qnorm(0.975, lower = TRUE)\nconfint_z = \n beta_hat + c(-1,1) * confint_radius_z\nprint(confint_z)\n#> [1] -100.26 63.42\n\n\n\n2.4.8 Likelihood ratio statistics\n\nShow R code\nlogLik(bw_lm2)\n#> 'log Lik.' -156.6 (df=5)\nlogLik(bw_lm1)\n#> 'log Lik.' -156.7 (df=4)\n\nlLR = (logLik(bw_lm2) - logLik(bw_lm1)) |> as.numeric()\ndelta_df = (bw_lm1$df.residual - df.residual(bw_lm2))\n\n\nx_max = 1\n\n\n\n\nShow R coded_lLR = function(x, df = delta_df) dchisq(x, df = df)\n\nchisq_plot = \n ggplot() + \n geom_function(fun = d_lLR) +\n stat_function( fun = d_lLR, xlim = c(lLR, x_max), geom = \"area\", fill = \"gray\") +\n geom_segment(aes(x = lLR, xend = lLR, y = 0, yend = d_lLR(lLR)), col = \"red\") + \n xlim(0.0001,x_max) + \n ylim(0,4) + \n ylab(\"p(X=x)\") + \n xlab(\"log(likelihood ratio) statistic [x]\") +\n theme_classic()\nchisq_plot |> print()\n\n\n\nFigure 2.5: Chi-square distribution\n\n\n\n\n\n\n\n\nNow we can get the p-value:\n\nShow R codepchisq(\n q = 2*lLR, \n df = delta_df, \n lower = FALSE) |> \n print()\n#> [1] 0.6298\n\n\n\nIn practice you don’t have to do this by hand; there are functions to do it for you:\n\nShow R code\n# built in\nlibrary(lmtest)\nlrtest(bw_lm2, bw_lm1)\n\n\n\n#Df\nLogLik\nDf\nChisq\nPr(>Chisq)\n\n\n\n5\n-156.6\nNA\nNA\nNA\n\n\n4\n-156.7\n-1\n0.2323\n0.6298", + "text": "2.4 Inference about Gaussian Linear Regression Models\n\n2.4.1 Motivating example: birthweight data\nResearch question: is there really an interaction between sex and age?\n\\(H_0: \\beta_{AM} = 0\\)\n\\(H_A: \\beta_{AM} \\neq 0\\)\n\\(P(|\\hat\\beta_{AM}| > |-18.4172| \\mid H_0)\\) = ?\n\n2.4.2 Wald tests and CIs\nR can give you Wald tests for single coefficients and corresponding CIs:\n\nShow R code\nbw_lm2 |> \n parameters() |>\n print_md(\n include_reference = TRUE)\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(20)\np\n\n\n\n(Intercept)\n-2141.67\n1163.60\n(-4568.90, 285.56)\n-1.84\n0.081\n\n\nsex (female)\n0.00\n\n\n\n\n\n\nsex (male)\n872.99\n1611.33\n(-2488.18, 4234.17)\n0.54\n0.594\n\n\nage\n130.40\n30.00\n(67.82, 192.98)\n4.35\n< .001\n\n\nsex (male) × age\n-18.42\n41.76\n(-105.52, 68.68)\n-0.44\n0.664\n\n\n\n\n\nTo understand what’s happening, let’s replicate these results by hand for the interaction term.\n\n2.4.3 P-values\n\nShow R codebw_lm2 |> \n parameters(keep = \"sexmale:age\") |>\n print_md(\n include_reference = TRUE)\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(20)\np\n\n\nsex (male) × age\n-18.42\n41.76\n(-105.52, 68.68)\n-0.44\n0.664\n\n\n\n\n\nShow R codebeta_hat = coef(summary(bw_lm2))[\"sexmale:age\", \"Estimate\"]\nse_hat = coef(summary(bw_lm2))[\"sexmale:age\", \"Std. Error\"]\ndfresid = bw_lm2$df.residual\nt_stat = abs(beta_hat)/se_hat\npval_t = \n pt(-t_stat, df = dfresid, lower.tail = TRUE) +\n pt(t_stat, df = dfresid, lower.tail = FALSE)\n\n\n\\[\n\\begin{aligned}\n&P\\left(\n| \\hat \\beta_{AM} | >\n| -18.4172| \\middle| H_0\n\\right)\n\\\\\n&= \\Pr \\left(\n\\left| \\frac{\\hat\\beta_{AM}}{\\hat{SE}(\\hat\\beta_{AM})} \\right| >\n\\left| \\frac{-18.4172}{41.7558} \\right| \\middle| H_0\n\\right)\\\\\n&= \\Pr \\left(\n\\left| T_{20} \\right| > 0.4411 | H_0\n\\right)\\\\\n&= 0.6639\n\\end{aligned}\n\\]\n\nThis matches the result in the table above.\n\n\n2.4.4 Confidence intervals\n\nShow R codebw_lm2 |> \n parameters(keep = \"sexmale:age\") |>\n print_md(\n include_reference = TRUE)\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(20)\np\n\n\nsex (male) × age\n-18.42\n41.76\n(-105.52, 68.68)\n-0.44\n0.664\n\n\n\n\n\nShow R codeq_t = qt(\n p = 0.975, \n df = dfresid, \n lower.tail = TRUE)\n\nq_t = qt(\n p = 0.025, \n df = dfresid, \n lower.tail = TRUE)\n\n\nconfint_radius_t = \n se_hat * q_t\n\nconfint_t = beta_hat + c(-1,1) * confint_radius_t\n\nprint(confint_t)\n#> [1] 68.68 -105.52\n\n\n\nThis also matches.\n\n\n2.4.5 Gaussian approximations\nHere are the asymptotic (Gaussian approximation) equivalents:\n\n2.4.6 P-values\n\nShow R codebw_lm2 |> \n parameters(keep = \"sexmale:age\") |>\n print_md(\n include_reference = TRUE)\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(20)\np\n\n\nsex (male) × age\n-18.42\n41.76\n(-105.52, 68.68)\n-0.44\n0.664\n\n\n\n\n\nShow R codepval_z = pnorm(abs(t_stat), lower = FALSE) * 2\n\nprint(pval_z)\n#> [1] 0.6592\n\n\n\n2.4.7 Confidence intervals\n\nShow R codebw_lm2 |> \n parameters(keep = \"sexmale:age\") |>\n print_md(\n include_reference = TRUE)\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(20)\np\n\n\nsex (male) × age\n-18.42\n41.76\n(-105.52, 68.68)\n-0.44\n0.664\n\n\n\n\n\nShow R codeconfint_radius_z = se_hat * qnorm(0.975, lower = TRUE)\nconfint_z = \n beta_hat + c(-1,1) * confint_radius_z\nprint(confint_z)\n#> [1] -100.26 63.42\n\n\n\n2.4.8 Likelihood ratio statistics\n\nShow R code\nlogLik(bw_lm2)\n#> 'log Lik.' -156.6 (df=5)\nlogLik(bw_lm1)\n#> 'log Lik.' -156.7 (df=4)\n\nlLR = (logLik(bw_lm2) - logLik(bw_lm1)) |> as.numeric()\ndelta_df = (bw_lm1$df.residual - df.residual(bw_lm2))\n\n\nx_max = 1\n\n\n\n\nShow R coded_lLR = function(x, df = delta_df) dchisq(x, df = df)\n\nchisq_plot = \n ggplot() + \n geom_function(fun = d_lLR) +\n stat_function( fun = d_lLR, xlim = c(lLR, x_max), geom = \"area\", fill = \"gray\") +\n geom_segment(aes(x = lLR, xend = lLR, y = 0, yend = d_lLR(lLR)), col = \"red\") + \n xlim(0.0001,x_max) + \n ylim(0,4) + \n ylab(\"p(X=x)\") + \n xlab(\"log(likelihood ratio) statistic [x]\") +\n theme_classic()\nchisq_plot |> print()\n\n\n\nFigure 2.5: Chi-square distribution\n\n\n\n\n\n\n\n\nNow we can get the p-value:\n\nShow R codepchisq(\n q = 2*lLR, \n df = delta_df, \n lower = FALSE) |> \n print()\n#> [1] 0.6298\n\n\n\nIn practice you don’t have to do this by hand; there are functions to do it for you:\n\nShow R code\n# built in\nlibrary(lmtest)\nlrtest(bw_lm2, bw_lm1)\n\n\n\n#Df\nLogLik\nDf\nChisq\nPr(>Chisq)\n\n\n\n5\n-156.6\nNA\nNA\nNA\n\n\n4\n-156.7\n-1\n0.2323\n0.6298", "crumbs": [ "Generalized Linear Models", "2  Linear (Gaussian) Models" @@ -149,7 +149,7 @@ "href": "Linear-models-overview.html#goodness-of-fit", "title": "\n2  Linear (Gaussian) Models\n", "section": "\n2.5 Goodness of fit", - "text": "2.5 Goodness of fit\n\n2.5.1 AIC and BIC\n\nWhen we use likelihood ratio tests, we are comparing how well different models fit the data.\nLikelihood ratio tests require “nested” models: one must be a special case of the other.\nIf we have non-nested models, we can instead use the Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC):\n\n\nAIC = \\(-2 * \\ell(\\hat\\theta) + 2 * p\\)\nBIC = \\(-2 * \\ell(\\hat\\theta) + p * \\text{log}(n)\\)\n\nwhere \\(\\ell\\) is the log-likelihood of the data evaluated using the parameter estimates \\(\\hat\\theta\\), \\(p\\) is the number of estimated parameters in the model (including \\(\\hat\\sigma^2\\)), and \\(n\\) is the number of observations.\nYou can calculate these criteria using the logLik() function, or use the built-in R functions:\nAIC in R\n\nShow R code\n-2 * logLik(bw_lm2) |> as.numeric() + \n 2*(length(coef(bw_lm2))+1) # sigma counts as a parameter here\n#> [1] 323.2\n\nAIC(bw_lm2)\n#> [1] 323.2\n\n\nBIC in R\n\nShow R code\n-2 * logLik(bw_lm2) |> as.numeric() + \n (length(coef(bw_lm2))+1) * log(nobs(bw_lm2))\n#> [1] 329\n\nBIC(bw_lm2)\n#> [1] 329\n\n\nLarge values of AIC and BIC are worse than small values. There are no hypothesis tests or p-values associated with these criteria.\n\n2.5.2 (Residual) Deviance\nLet \\(q\\) be the number of distinct covariate combinations in a data set.\n\nShow R codebw.X.unique = \n bw |> \n count(sex, age)\n\nn_unique.bw = nrow(bw.X.unique)\n\n\nFor example, in the birthweight data, there are \\(q = 12\\) unique patterns (Table 2.9).\n\nShow R codebw.X.unique\n\n\nTable 2.9: Unique covariate combinations in the birthweight data, with replicate counts\n\n\n\n\nsex\nage\nn\n\n\n\nfemale\n36\n2\n\n\nfemale\n37\n1\n\n\nfemale\n38\n2\n\n\nfemale\n39\n2\n\n\nfemale\n40\n4\n\n\nfemale\n42\n1\n\n\nmale\n35\n1\n\n\nmale\n36\n1\n\n\nmale\n37\n2\n\n\nmale\n38\n3\n\n\nmale\n40\n4\n\n\nmale\n41\n1\n\n\n\n\n\n\n\n\n\n\nDefinition 2.1 (Replicates) If a given covariate pattern has more than one observation in a dataset, those observations are called replicates.\n\n\n\nExample 2.1 (Replicates in the birthweight data) In the birthweight dataset, there are 2 replicates of the combination “female, age 36” (Table 2.9).\n\n\n\nExercise 2.7 (Replicates in the birthweight data) Which covariate pattern(s) in the birthweight data has the most replicates?\n\n\n\nSolution 2.1 (Replicates in the birthweight data). Two covariate patterns are tied for most replicates: males at age 40 weeks and females at age 40 weeks. 40 weeks is the usual length for human pregnancy (Polin, Fox, and Abman (2011)), so this result makes sense.\n\nShow R codebw.X.unique |> dplyr::filter(n == max(n))\n\n\n\nsex\nage\nn\n\n\n\nfemale\n40\n4\n\n\nmale\n40\n4\n\n\n\n\n\n\n\nSaturated models\nThe most complicated model we could fit would have one parameter (a mean) for each covariate pattern, plus a variance parameter:\n\nShow R codelm_max = \n bw |> \n mutate(age = factor(age)) |> \n lm(\n formula = weight ~ sex:age - 1, \n data = _)\n\nlm_max |> \n parameters() |> \n print_md()\n\n\nTable 2.10: Saturated model for the birthweight data\n\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(12)\np\n\n\n\nsex (male) × age35\n2925.00\n187.92\n(2515.55, 3334.45)\n15.56\n< .001\n\n\nsex (female) × age36\n2570.50\n132.88\n(2280.98, 2860.02)\n19.34\n< .001\n\n\nsex (male) × age36\n2625.00\n187.92\n(2215.55, 3034.45)\n13.97\n< .001\n\n\nsex (female) × age37\n2539.00\n187.92\n(2129.55, 2948.45)\n13.51\n< .001\n\n\nsex (male) × age37\n2737.50\n132.88\n(2447.98, 3027.02)\n20.60\n< .001\n\n\nsex (female) × age38\n2872.50\n132.88\n(2582.98, 3162.02)\n21.62\n< .001\n\n\nsex (male) × age38\n2982.00\n108.50\n(2745.60, 3218.40)\n27.48\n< .001\n\n\nsex (female) × age39\n2846.00\n132.88\n(2556.48, 3135.52)\n21.42\n< .001\n\n\nsex (female) × age40\n3152.25\n93.96\n(2947.52, 3356.98)\n33.55\n< .001\n\n\nsex (male) × age40\n3256.25\n93.96\n(3051.52, 3460.98)\n34.66\n< .001\n\n\nsex (male) × age41\n3292.00\n187.92\n(2882.55, 3701.45)\n17.52\n< .001\n\n\nsex (female) × age42\n3210.00\n187.92\n(2800.55, 3619.45)\n17.08\n< .001\n\n\n\n\n\n\n\n\nWe call this model the full, maximal, or saturated model for this dataset.\nWe can calculate the log-likelihood of this model as usual:\n\nShow R codelogLik(lm_max)\n#> 'log Lik.' -151.4 (df=13)\n\n\nWe can compare this model to our other models using chi-square tests, as usual:\n\nShow R codelrtest(lm_max, bw_lm2)\n\n\n\n#Df\nLogLik\nDf\nChisq\nPr(>Chisq)\n\n\n\n13\n-151.4\nNA\nNA\nNA\n\n\n5\n-156.6\n-8\n10.36\n0.241\n\n\n\n\n\nThe likelihood ratio statistic for this test is \\[\\lambda = 2 * (\\ell_{\\text{full}} - \\ell) = 10.35537421\\] where:\n\n\n\\(\\ell_{\\text{max}}\\) is the log-likelihood of the full model: -151.40160056\n\n\\(\\ell\\) is the log-likelihood of our comparison model (two slopes, two intercepts): -156.57928767\n\nThis statistic is called the deviance or residual deviance for our two-slopes and two-intercepts model; it tells us how much the likelihood of that model deviates from the likelihood of the maximal model.\nThe corresponding p-value tells us whether there we have enough evidence to detect that our two-slopes, two-intercepts model is a worse fit for the data than the maximal model; in other words, it tells us if there’s evidence that we missed any important patterns. (Remember, a nonsignificant p-value could mean that we didn’t miss anything and a more complicated model is unnecessary, or it could mean we just don’t have enough data to tell the difference between these models.)\n\n2.5.3 Null Deviance\nSimilarly, the least complicated model we could fit would have only one mean parameter, an intercept:\n\\[\\text E[Y|X=x] = \\beta_0\\] We can fit this model in R like so:\n\nShow R codelm0 = lm(weight ~ 1, data = bw)\n\nlm0 |> parameters() |> print_md()\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(23)\np\n\n\n(Intercept)\n2967.67\n57.58\n(2848.56, 3086.77)\n51.54\n< .001\n\n\n\n\nThis model also has a likelihood:\n\nShow R codelogLik(lm0)\n#> 'log Lik.' -169 (df=2)\n\n\nAnd we can compare it to more complicated models using a likelihood ratio test:\n\nShow R code\nlrtest(bw_lm2, lm0)\n\n\n\n#Df\nLogLik\nDf\nChisq\nPr(>Chisq)\n\n\n\n5\n-156.6\nNA\nNA\nNA\n\n\n2\n-169.0\n-3\n24.75\n0\n\n\n\n\n\nThe likelihood ratio statistic for the test comparing the null model to the maximal model is \\[\\lambda = 2 * (\\ell_{\\text{full}} - \\ell_{0}) = 35.10673188\\] where:\n\n\n\\(\\ell_{\\text{0}}\\) is the log-likelihood of the null model: -168.9549665\n\n\\(\\ell_{\\text{full}}\\) is the log-likelihood of the maximal model: -151.40160056\n\nIn R, this test is:\n\nShow R codelrtest(lm_max, lm0)\n\n\n\n#Df\nLogLik\nDf\nChisq\nPr(>Chisq)\n\n\n\n13\n-151.4\nNA\nNA\nNA\n\n\n2\n-169.0\n-11\n35.11\n2e-04\n\n\n\n\n\nThis log-likelihood ratio statistic is called the null deviance. It tells us whether we have enough data to detect a difference between the null and full models.", + "text": "2.5 Goodness of fit\n\n2.5.1 AIC and BIC\n\nWhen we use likelihood ratio tests, we are comparing how well different models fit the data.\nLikelihood ratio tests require “nested” models: one must be a special case of the other.\nIf we have non-nested models, we can instead use the Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC):\n\n\nAIC = \\(-2 * \\ell(\\hat\\theta) + 2 * p\\)\nBIC = \\(-2 * \\ell(\\hat\\theta) + p * \\text{log}(n)\\)\n\nwhere \\(\\ell\\) is the log-likelihood of the data evaluated using the parameter estimates \\(\\hat\\theta\\), \\(p\\) is the number of estimated parameters in the model (including \\(\\hat\\sigma^2\\)), and \\(n\\) is the number of observations.\nYou can calculate these criteria using the logLik() function, or use the built-in R functions:\nAIC in R\n\nShow R code\n-2 * logLik(bw_lm2) |> as.numeric() + \n 2*(length(coef(bw_lm2))+1) # sigma counts as a parameter here\n#> [1] 323.2\n\nAIC(bw_lm2)\n#> [1] 323.2\n\n\nBIC in R\n\nShow R code\n-2 * logLik(bw_lm2) |> as.numeric() + \n (length(coef(bw_lm2))+1) * log(nobs(bw_lm2))\n#> [1] 329\n\nBIC(bw_lm2)\n#> [1] 329\n\n\nLarge values of AIC and BIC are worse than small values. There are no hypothesis tests or p-values associated with these criteria.\n\n2.5.2 (Residual) Deviance\nLet \\(q\\) be the number of distinct covariate combinations in a data set.\n\nShow R codebw.X.unique = \n bw |> \n count(sex, age)\n\nn_unique.bw = nrow(bw.X.unique)\n\n\nFor example, in the birthweight data, there are \\(q = 12\\) unique patterns (Table 2.9).\n\nShow R codebw.X.unique\n\n\nTable 2.9: Unique covariate combinations in the birthweight data, with replicate counts\n\n\n\n\nsex\nage\nn\n\n\n\nfemale\n36\n2\n\n\nfemale\n37\n1\n\n\nfemale\n38\n2\n\n\nfemale\n39\n2\n\n\nfemale\n40\n4\n\n\nfemale\n42\n1\n\n\nmale\n35\n1\n\n\nmale\n36\n1\n\n\nmale\n37\n2\n\n\nmale\n38\n3\n\n\nmale\n40\n4\n\n\nmale\n41\n1\n\n\n\n\n\n\n\n\n\n\nDefinition 2.1 (Replicates) If a given covariate pattern has more than one observation in a dataset, those observations are called replicates.\n\n\n\nExample 2.1 (Replicates in the birthweight data) In the birthweight dataset, there are 2 replicates of the combination “female, age 36” (Table 2.9).\n\n\n\nExercise 2.7 (Replicates in the birthweight data) Which covariate pattern(s) in the birthweight data has the most replicates?\n\n\n\nSolution 2.1 (Replicates in the birthweight data). Two covariate patterns are tied for most replicates: males at age 40 weeks and females at age 40 weeks. 40 weeks is the usual length for human pregnancy (Polin, Fox, and Abman (2011)), so this result makes sense.\n\nShow R codebw.X.unique |> dplyr::filter(n == max(n))\n\n\n\nsex\nage\nn\n\n\n\nfemale\n40\n4\n\n\nmale\n40\n4\n\n\n\n\n\n\n\nSaturated models\nThe most complicated model we could fit would have one parameter (a mean) for each covariate pattern, plus a variance parameter:\n\nShow R codelm_max = \n bw |> \n mutate(age = factor(age)) |> \n lm(\n formula = weight ~ sex:age - 1, \n data = _)\n\nlm_max |> \n parameters() |> \n print_md()\n\n\nTable 2.10: Saturated model for the birthweight data\n\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(12)\np\n\n\n\nsex (male) × age35\n2925.00\n187.92\n(2515.55, 3334.45)\n15.56\n< .001\n\n\nsex (female) × age36\n2570.50\n132.88\n(2280.98, 2860.02)\n19.34\n< .001\n\n\nsex (male) × age36\n2625.00\n187.92\n(2215.55, 3034.45)\n13.97\n< .001\n\n\nsex (female) × age37\n2539.00\n187.92\n(2129.55, 2948.45)\n13.51\n< .001\n\n\nsex (male) × age37\n2737.50\n132.88\n(2447.98, 3027.02)\n20.60\n< .001\n\n\nsex (female) × age38\n2872.50\n132.88\n(2582.98, 3162.02)\n21.62\n< .001\n\n\nsex (male) × age38\n2982.00\n108.50\n(2745.60, 3218.40)\n27.48\n< .001\n\n\nsex (female) × age39\n2846.00\n132.88\n(2556.48, 3135.52)\n21.42\n< .001\n\n\nsex (female) × age40\n3152.25\n93.96\n(2947.52, 3356.98)\n33.55\n< .001\n\n\nsex (male) × age40\n3256.25\n93.96\n(3051.52, 3460.98)\n34.66\n< .001\n\n\nsex (male) × age41\n3292.00\n187.92\n(2882.55, 3701.45)\n17.52\n< .001\n\n\nsex (female) × age42\n3210.00\n187.92\n(2800.55, 3619.45)\n17.08\n< .001\n\n\n\n\n\n\n\n\nWe call this model the full, maximal, or saturated model for this dataset.\nWe can calculate the log-likelihood of this model as usual:\n\nShow R codelogLik(lm_max)\n#> 'log Lik.' -151.4 (df=13)\n\n\nWe can compare this model to our other models using chi-square tests, as usual:\n\nShow R codelrtest(lm_max, bw_lm2)\n\n\n\n#Df\nLogLik\nDf\nChisq\nPr(>Chisq)\n\n\n\n13\n-151.4\nNA\nNA\nNA\n\n\n5\n-156.6\n-8\n10.36\n0.241\n\n\n\n\n\nThe likelihood ratio statistic for this test is \\[\\lambda = 2 * (\\ell_{\\text{full}} - \\ell) = 10.3554\\] where:\n\n\n\\(\\ell_{\\text{max}}\\) is the log-likelihood of the full model: -151.4016\n\n\\(\\ell\\) is the log-likelihood of our comparison model (two slopes, two intercepts): -156.5793\n\nThis statistic is called the deviance or residual deviance for our two-slopes and two-intercepts model; it tells us how much the likelihood of that model deviates from the likelihood of the maximal model.\nThe corresponding p-value tells us whether there we have enough evidence to detect that our two-slopes, two-intercepts model is a worse fit for the data than the maximal model; in other words, it tells us if there’s evidence that we missed any important patterns. (Remember, a nonsignificant p-value could mean that we didn’t miss anything and a more complicated model is unnecessary, or it could mean we just don’t have enough data to tell the difference between these models.)\n\n2.5.3 Null Deviance\nSimilarly, the least complicated model we could fit would have only one mean parameter, an intercept:\n\\[\\text E[Y|X=x] = \\beta_0\\] We can fit this model in R like so:\n\nShow R codelm0 = lm(weight ~ 1, data = bw)\n\nlm0 |> parameters() |> print_md()\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(23)\np\n\n\n(Intercept)\n2967.67\n57.58\n(2848.56, 3086.77)\n51.54\n< .001\n\n\n\n\nThis model also has a likelihood:\n\nShow R codelogLik(lm0)\n#> 'log Lik.' -169 (df=2)\n\n\nAnd we can compare it to more complicated models using a likelihood ratio test:\n\nShow R code\nlrtest(bw_lm2, lm0)\n\n\n\n#Df\nLogLik\nDf\nChisq\nPr(>Chisq)\n\n\n\n5\n-156.6\nNA\nNA\nNA\n\n\n2\n-169.0\n-3\n24.75\n0\n\n\n\n\n\nThe likelihood ratio statistic for the test comparing the null model to the maximal model is \\[\\lambda = 2 * (\\ell_{\\text{full}} - \\ell_{0}) = 35.1067\\] where:\n\n\n\\(\\ell_{\\text{0}}\\) is the log-likelihood of the null model: -168.955\n\n\\(\\ell_{\\text{full}}\\) is the log-likelihood of the maximal model: -151.4016\n\nIn R, this test is:\n\nShow R codelrtest(lm_max, lm0)\n\n\n\n#Df\nLogLik\nDf\nChisq\nPr(>Chisq)\n\n\n\n13\n-151.4\nNA\nNA\nNA\n\n\n2\n-169.0\n-11\n35.11\n2e-04\n\n\n\n\n\nThis log-likelihood ratio statistic is called the null deviance. It tells us whether we have enough data to detect a difference between the null and full models.", "crumbs": [ "Generalized Linear Models", "2  Linear (Gaussian) Models" @@ -193,7 +193,7 @@ "href": "Linear-models-overview.html#model-selection-1", "title": "\n2  Linear (Gaussian) Models\n", "section": "\n2.9 Model selection", - "text": "2.9 Model selection\n(adapted from Dobson and Barnett (2018) §6.3.3; for more information on prediction, see James et al. (2013) and Harrell (2015)).\n\nIf we have a lot of covariates in our dataset, we might want to choose a small subset to use in our model.\nThere are a few possible metrics to consider for choosing a “best” model.\n\n\n2.9.1 Mean squared error\nWe might want to minimize the mean squared error, \\(\\text E[(y-\\hat y)^2]\\), for new observations that weren’t in our data set when we fit the model.\nUnfortunately, \\[\\frac{1}{n}\\sum_{i=1}^n (y_i-\\hat y_i)^2\\] gives a biased estimate of \\(\\text E[(y-\\hat y)^2]\\) for new data. If we want an unbiased estimate, we will have to be clever.\n\nCross-validation\n\nShow R codedata(\"carbohydrate\", package = \"dobson\")\nlibrary(cvTools)\nfull_model <- lm(carbohydrate ~ ., data = carbohydrate)\ncv_full = \n full_model |> cvFit(\n data = carbohydrate, K = 5, R = 10,\n y = carbohydrate$carbohydrate)\n\nreduced_model = update(full_model, \n formula = ~ . - age)\n\ncv_reduced = \n reduced_model |> cvFit(\n data = carbohydrate, K = 5, R = 10,\n y = carbohydrate$carbohydrate)\n\n\n\n\nShow R coderesults_reduced = \n tibble(\n model = \"wgt+protein\",\n errs = cv_reduced$reps[])\nresults_full = \n tibble(model = \"wgt+age+protein\",\n errs = cv_full$reps[])\n\ncv_results = \n bind_rows(results_reduced, results_full)\n\ncv_results |> \n ggplot(aes(y = model, x = errs)) +\n geom_boxplot()\n\n\n\n\n\n\n\n\ncomparing metrics\n\nShow R code\ncompare_results = tribble(\n ~ model, ~ cvRMSE, ~ r.squared, ~adj.r.squared, ~ trainRMSE, ~loglik,\n \"full\", cv_full$cv, summary(full_model)$r.squared, summary(full_model)$adj.r.squared, sigma(full_model), logLik(full_model) |> as.numeric(),\n \"reduced\", cv_reduced$cv, summary(reduced_model)$r.squared, summary(reduced_model)$adj.r.squared, sigma(reduced_model), logLik(reduced_model) |> as.numeric())\n\ncompare_results\n\n\n\nmodel\ncvRMSE\nr.squared\nadj.r.squared\ntrainRMSE\nloglik\n\n\n\nfull\n6.946\n0.4805\n0.3831\n5.956\n-61.84\n\n\nreduced\n6.627\n0.4454\n0.3802\n5.971\n-62.49\n\n\n\n\n\n\n\nShow R codeanova(full_model, reduced_model)\n\n\n\nRes.Df\nRSS\nDf\nSum of Sq\nF\nPr(>F)\n\n\n\n16\n567.7\nNA\nNA\nNA\nNA\n\n\n17\n606.0\n-1\n-38.36\n1.081\n0.3139\n\n\n\n\n\nstepwise regression\n\nShow R codelibrary(olsrr)\nolsrr:::ols_step_both_aic(full_model)\n#> \n#> \n#> Stepwise Summary \n#> -------------------------------------------------------------------------\n#> Step Variable AIC SBC SBIC R2 Adj. R2 \n#> -------------------------------------------------------------------------\n#> 0 Base Model 140.773 142.764 83.068 0.00000 0.00000 \n#> 1 protein (+) 137.950 140.937 80.438 0.21427 0.17061 \n#> 2 weight (+) 132.981 136.964 77.191 0.44544 0.38020 \n#> -------------------------------------------------------------------------\n#> \n#> Final Model Output \n#> ------------------\n#> \n#> Model Summary \n#> ---------------------------------------------------------------\n#> R 0.667 RMSE 5.505 \n#> R-Squared 0.445 MSE 35.648 \n#> Adj. R-Squared 0.380 Coef. Var 15.879 \n#> Pred R-Squared 0.236 AIC 132.981 \n#> MAE 4.593 SBC 136.964 \n#> ---------------------------------------------------------------\n#> RMSE: Root Mean Square Error \n#> MSE: Mean Square Error \n#> MAE: Mean Absolute Error \n#> AIC: Akaike Information Criteria \n#> SBC: Schwarz Bayesian Criteria \n#> \n#> ANOVA \n#> -------------------------------------------------------------------\n#> Sum of \n#> Squares DF Mean Square F Sig. \n#> -------------------------------------------------------------------\n#> Regression 486.778 2 243.389 6.827 0.0067 \n#> Residual 606.022 17 35.648 \n#> Total 1092.800 19 \n#> -------------------------------------------------------------------\n#> \n#> Parameter Estimates \n#> ----------------------------------------------------------------------------------------\n#> model Beta Std. Error Std. Beta t Sig lower upper \n#> ----------------------------------------------------------------------------------------\n#> (Intercept) 33.130 12.572 2.635 0.017 6.607 59.654 \n#> protein 1.824 0.623 0.534 2.927 0.009 0.509 3.139 \n#> weight -0.222 0.083 -0.486 -2.662 0.016 -0.397 -0.046 \n#> ----------------------------------------------------------------------------------------\n\n\nLasso\n\\[\\arg min_{\\theta} \\ell(\\theta) + \\lambda \\sum_{j=1}^p|\\beta_j|\\]\n\nShow R codelibrary(glmnet)\ny = carbohydrate$carbohydrate\nx = carbohydrate |> \n select(age, weight, protein) |> \n as.matrix()\nfit = glmnet(x,y)\n\n\n\n\nShow R codeautoplot(fit, xvar = 'lambda')\n\n\n\nFigure 2.19: Lasso selection\n\n\n\n\n\n\n\n\n\nShow R codecvfit = cv.glmnet(x,y)\nplot(cvfit)\n\n\n\n\n\n\n\n\n\nShow R codecoef(cvfit, s = \"lambda.1se\")\n#> 4 x 1 sparse Matrix of class \"dgCMatrix\"\n#> s1\n#> (Intercept) 34.4241\n#> age . \n#> weight -0.0662\n#> protein 0.6607", + "text": "2.9 Model selection\n(adapted from Dobson and Barnett (2018) §6.3.3; for more information on prediction, see James et al. (2013) and Harrell (2015)).\n\nIf we have a lot of covariates in our dataset, we might want to choose a small subset to use in our model.\nThere are a few possible metrics to consider for choosing a “best” model.\n\n\n2.9.1 Mean squared error\nWe might want to minimize the mean squared error, \\(\\text E[(y-\\hat y)^2]\\), for new observations that weren’t in our data set when we fit the model.\nUnfortunately, \\[\\frac{1}{n}\\sum_{i=1}^n (y_i-\\hat y_i)^2\\] gives a biased estimate of \\(\\text E[(y-\\hat y)^2]\\) for new data. If we want an unbiased estimate, we will have to be clever.\n\nCross-validation\n\nShow R codedata(\"carbohydrate\", package = \"dobson\")\nlibrary(cvTools)\nfull_model <- lm(carbohydrate ~ ., data = carbohydrate)\ncv_full = \n full_model |> cvFit(\n data = carbohydrate, K = 5, R = 10,\n y = carbohydrate$carbohydrate)\n\nreduced_model = update(full_model, \n formula = ~ . - age)\n\ncv_reduced = \n reduced_model |> cvFit(\n data = carbohydrate, K = 5, R = 10,\n y = carbohydrate$carbohydrate)\n\n\n\n\nShow R coderesults_reduced = \n tibble(\n model = \"wgt+protein\",\n errs = cv_reduced$reps[])\nresults_full = \n tibble(model = \"wgt+age+protein\",\n errs = cv_full$reps[])\n\ncv_results = \n bind_rows(results_reduced, results_full)\n\ncv_results |> \n ggplot(aes(y = model, x = errs)) +\n geom_boxplot()\n\n\n\n\n\n\n\n\ncomparing metrics\n\nShow R code\ncompare_results = tribble(\n ~ model, ~ cvRMSE, ~ r.squared, ~adj.r.squared, ~ trainRMSE, ~loglik,\n \"full\", cv_full$cv, summary(full_model)$r.squared, summary(full_model)$adj.r.squared, sigma(full_model), logLik(full_model) |> as.numeric(),\n \"reduced\", cv_reduced$cv, summary(reduced_model)$r.squared, summary(reduced_model)$adj.r.squared, sigma(reduced_model), logLik(reduced_model) |> as.numeric())\n\ncompare_results\n\n\n\nmodel\ncvRMSE\nr.squared\nadj.r.squared\ntrainRMSE\nloglik\n\n\n\nfull\n6.958\n0.4805\n0.3831\n5.956\n-61.84\n\n\nreduced\n6.778\n0.4454\n0.3802\n5.971\n-62.49\n\n\n\n\n\n\n\nShow R codeanova(full_model, reduced_model)\n\n\n\nRes.Df\nRSS\nDf\nSum of Sq\nF\nPr(>F)\n\n\n\n16\n567.7\nNA\nNA\nNA\nNA\n\n\n17\n606.0\n-1\n-38.36\n1.081\n0.3139\n\n\n\n\n\nstepwise regression\n\nShow R codelibrary(olsrr)\nolsrr:::ols_step_both_aic(full_model)\n#> \n#> \n#> Stepwise Summary \n#> -------------------------------------------------------------------------\n#> Step Variable AIC SBC SBIC R2 Adj. R2 \n#> -------------------------------------------------------------------------\n#> 0 Base Model 140.773 142.764 83.068 0.00000 0.00000 \n#> 1 protein (+) 137.950 140.937 80.438 0.21427 0.17061 \n#> 2 weight (+) 132.981 136.964 77.191 0.44544 0.38020 \n#> -------------------------------------------------------------------------\n#> \n#> Final Model Output \n#> ------------------\n#> \n#> Model Summary \n#> ---------------------------------------------------------------\n#> R 0.667 RMSE 5.505 \n#> R-Squared 0.445 MSE 35.648 \n#> Adj. R-Squared 0.380 Coef. Var 15.879 \n#> Pred R-Squared 0.236 AIC 132.981 \n#> MAE 4.593 SBC 136.964 \n#> ---------------------------------------------------------------\n#> RMSE: Root Mean Square Error \n#> MSE: Mean Square Error \n#> MAE: Mean Absolute Error \n#> AIC: Akaike Information Criteria \n#> SBC: Schwarz Bayesian Criteria \n#> \n#> ANOVA \n#> -------------------------------------------------------------------\n#> Sum of \n#> Squares DF Mean Square F Sig. \n#> -------------------------------------------------------------------\n#> Regression 486.778 2 243.389 6.827 0.0067 \n#> Residual 606.022 17 35.648 \n#> Total 1092.800 19 \n#> -------------------------------------------------------------------\n#> \n#> Parameter Estimates \n#> ----------------------------------------------------------------------------------------\n#> model Beta Std. Error Std. Beta t Sig lower upper \n#> ----------------------------------------------------------------------------------------\n#> (Intercept) 33.130 12.572 2.635 0.017 6.607 59.654 \n#> protein 1.824 0.623 0.534 2.927 0.009 0.509 3.139 \n#> weight -0.222 0.083 -0.486 -2.662 0.016 -0.397 -0.046 \n#> ----------------------------------------------------------------------------------------\n\n\nLasso\n\\[\\arg min_{\\theta} \\ell(\\theta) + \\lambda \\sum_{j=1}^p|\\beta_j|\\]\n\nShow R codelibrary(glmnet)\ny = carbohydrate$carbohydrate\nx = carbohydrate |> \n select(age, weight, protein) |> \n as.matrix()\nfit = glmnet(x,y)\n\n\n\n\nShow R codeautoplot(fit, xvar = 'lambda')\n\n\n\nFigure 2.19: Lasso selection\n\n\n\n\n\n\n\n\n\nShow R codecvfit = cv.glmnet(x,y)\nplot(cvfit)\n\n\n\n\n\n\n\n\n\nShow R codecoef(cvfit, s = \"lambda.1se\")\n#> 4 x 1 sparse Matrix of class \"dgCMatrix\"\n#> s1\n#> (Intercept) 34.1090\n#> age . \n#> weight -0.1041\n#> protein 0.9441", "crumbs": [ "Generalized Linear Models", "2  Linear (Gaussian) Models" @@ -270,7 +270,7 @@ "href": "logistic-regression.html#comparing-probabilities", "title": "3  Models for Binary Outcomes", "section": "3.3 Comparing Probabilities", - "text": "3.3 Comparing Probabilities\n\n3.3.1 Risk differences\n\nThe simplest comparison of two probabilities, \\(\\pi_1\\), and \\(\\pi_2\\), is the difference of their values:\n\n\nDefinition 3.3 (Risk difference) The risk difference of two probabilities, \\(\\pi_1\\), and \\(\\pi_2\\), is the difference of their values: \\[\\delta(\\pi_1,\\pi_2) \\stackrel{\\text{def}}{=}\\pi_1 - \\pi_2\\]\n\n\n\nExample 3.2 (Difference in MI risk) In Example 3.1, the maximum likelihood estimate of the difference in MI risk between OC users and OC non-users is:\n\\[\n\\begin{aligned}\n\\hat\\delta(\\pi(OC), \\pi(\\neg OC))\n&= \\delta(\\hat\\pi(OC), \\hat\\pi(\\neg OC))\\\\\n&= \\hat\\pi(OC) - \\hat\\pi(\\neg OC)\\\\\n&= 0.0026 - 7\\times 10^{-4}\\\\\n&= 0.0019\n\\end{aligned}\n\\]\n\n\n\n\n3.3.2 Risk ratios\n\nDefinition 3.4 (Relative risk ratios)  \n\nThe relative risk of probability \\(\\pi_1\\) compared to another probability \\(\\pi_2\\), also called the risk ratio, relative risk ratio, probability ratio, or rate ratio, is the ratio of those probabilities:\n\n\\[\\rho(\\pi_1,\\pi_2) = \\frac{\\pi_1}{\\pi_2}\\]\n\n\n\nExample 3.3  \n\nAbove, we estimated that:\n\n\\[\\hat{\\text{p}}(MI|OC) = 0.0026\\]\n\\[\\hat{\\text{p}}(MI|\\neg OC) = 7\\times 10^{-4}\\]\n\nSo we might estimate that the relative risk of MI for OC versus non-OC is:\n\n\n\nShow R code\nrr = (13/5000)/(7/10000)\n\n\n\\[\n\\begin{aligned}\n\\hat\\rho(OC, \\neg OC)\n&=\\frac{\\hat{\\text{p}}(MI|OC)}{\\hat{\\text{p}}(MI|\\neg OC)}\\\\\n&= \\frac{0.0026}{7\\times 10^{-4}}\\\\\n&= 3.71428571\n\\end{aligned}\n\\]\n\nWe might summarize this result by saying that “the estimated probability of MI among OC users was 3.71428571 as high as the estimated probability among OC non-users.\n\n\n\n\n\n3.3.3 Relative risk differences\n\nDefinition 3.5 (Relative risk difference)  \n\nSometimes, we divide the risk difference by the comparison probability; the result is called the relative risk difference:\n\n\\[\\xi(\\pi_1,\\pi_2) \\stackrel{\\text{def}}{=}\\frac{\\delta(\\pi_1,\\pi_2)}{\\pi_2}\\]\n\n\n\nTheorem 3.1 (Relative risk difference equals risk ratio minus 1) \\[\\xi(\\pi_1,\\pi_2) = \\rho(\\pi_1,\\pi_2) - 1\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n\\xi(\\pi_1,\\pi_2)\n&\\stackrel{\\text{def}}{=}\\frac{\\delta(\\pi_1,\\pi_2)}{\\pi_2}\n\\\\&= \\frac{\\pi_1-\\pi_2}{\\pi_2}\n\\\\&= \\frac{\\pi_1}{\\pi_2} - 1\n\\\\&= \\rho(\\pi_1,\\pi_2) - 1\n\\end{aligned}\n\\]\n\n\n\n\n3.3.4 Changing reference groups in risk comparisons\n\nRisk differences, risk ratios, and relative risk differences are defined by two probabilities, plus a choice of which probability is the baseline or reference probability (i.e., which probability is the subtrahend of the risk difference or the denominator of the risk ratio).\n\n\\[\\delta(\\pi_2,\\pi_1) = -\\delta(\\pi_1, \\pi_2)\\]\n\\[\\rho(\\pi_2,\\pi_1) = \\left(\\rho(\\pi_1,\\pi_2)\\right)^{-1}\\] \\[\\xi(\\pi_2,\\pi_1) = \\left(\\xi(\\pi_2,\\pi_1) + 1\\right)^{-1} - 1\\]\n\nExercise 3.5 Prove the relationships above.\n\n\n\nExample 3.4 (Switching the reference group in a risk ratio) Above, we estimated that the risk ratio of OC versus non-OC is:\n\\[\n\\begin{aligned}\n\\rho(OC, \\neg OC)\n&= 3.71428571\n\\end{aligned}\n\\]\nIn comparison, the risk ratio for non-OC versus OC is:\n\\[\n\\begin{aligned}\n\\rho(\\neg OC, OC)\n&=\\frac{\\hat{\\text{p}}(MI|\\neg OC)}{\\hat{\\text{p}}(MI|OC)}\\\\\n&= \\frac{7\\times 10^{-4}}{0.0026}\\\\\n&= 0.26923077\\\\\n&= \\frac{1}{\\rho(OC, \\neg OC)}\n\\end{aligned}\n\\]", + "text": "3.3 Comparing Probabilities\n\n3.3.1 Risk differences\n\nThe simplest comparison of two probabilities, \\(\\pi_1\\), and \\(\\pi_2\\), is the difference of their values:\n\n\nDefinition 3.3 (Risk difference) The risk difference of two probabilities, \\(\\pi_1\\), and \\(\\pi_2\\), is the difference of their values: \\[\\delta(\\pi_1,\\pi_2) \\stackrel{\\text{def}}{=}\\pi_1 - \\pi_2\\]\n\n\n\nExample 3.2 (Difference in MI risk) In Example 3.1, the maximum likelihood estimate of the difference in MI risk between OC users and OC non-users is:\n\\[\n\\begin{aligned}\n\\hat\\delta(\\pi(OC), \\pi(\\neg OC))\n&= \\delta(\\hat\\pi(OC), \\hat\\pi(\\neg OC))\\\\\n&= \\hat\\pi(OC) - \\hat\\pi(\\neg OC)\\\\\n&= 0.0026 - 7\\times 10^{-4}\\\\\n&= 0.0019\n\\end{aligned}\n\\]\n\n\n\n\n3.3.2 Risk ratios\n\nDefinition 3.4 (Relative risk ratios)  \n\nThe relative risk of probability \\(\\pi_1\\) compared to another probability \\(\\pi_2\\), also called the risk ratio, relative risk ratio, probability ratio, or rate ratio, is the ratio of those probabilities:\n\n\\[\\rho(\\pi_1,\\pi_2) = \\frac{\\pi_1}{\\pi_2}\\]\n\n\n\nExample 3.3  \n\nAbove, we estimated that:\n\n\\[\\hat{\\text{p}}(MI|OC) = 0.0026\\]\n\\[\\hat{\\text{p}}(MI|\\neg OC) = 7\\times 10^{-4}\\]\n\nSo we might estimate that the relative risk of MI for OC versus non-OC is:\n\n\n\nShow R code\nrr = (13/5000)/(7/10000)\n\n\n\\[\n\\begin{aligned}\n\\hat\\rho(OC, \\neg OC)\n&=\\frac{\\hat{\\text{p}}(MI|OC)}{\\hat{\\text{p}}(MI|\\neg OC)}\\\\\n&= \\frac{0.0026}{7\\times 10^{-4}}\\\\\n&= 3.7143\n\\end{aligned}\n\\]\n\nWe might summarize this result by saying that “the estimated probability of MI among OC users was 3.7143 as high as the estimated probability among OC non-users.\n\n\n\n\n\n3.3.3 Relative risk differences\n\nDefinition 3.5 (Relative risk difference)  \n\nSometimes, we divide the risk difference by the comparison probability; the result is called the relative risk difference:\n\n\\[\\xi(\\pi_1,\\pi_2) \\stackrel{\\text{def}}{=}\\frac{\\delta(\\pi_1,\\pi_2)}{\\pi_2}\\]\n\n\n\nTheorem 3.1 (Relative risk difference equals risk ratio minus 1) \\[\\xi(\\pi_1,\\pi_2) = \\rho(\\pi_1,\\pi_2) - 1\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n\\xi(\\pi_1,\\pi_2)\n&\\stackrel{\\text{def}}{=}\\frac{\\delta(\\pi_1,\\pi_2)}{\\pi_2}\n\\\\&= \\frac{\\pi_1-\\pi_2}{\\pi_2}\n\\\\&= \\frac{\\pi_1}{\\pi_2} - 1\n\\\\&= \\rho(\\pi_1,\\pi_2) - 1\n\\end{aligned}\n\\]\n\n\n\n\n3.3.4 Changing reference groups in risk comparisons\n\nRisk differences, risk ratios, and relative risk differences are defined by two probabilities, plus a choice of which probability is the baseline or reference probability (i.e., which probability is the subtrahend of the risk difference or the denominator of the risk ratio).\n\n\\[\\delta(\\pi_2,\\pi_1) = -\\delta(\\pi_1, \\pi_2)\\]\n\\[\\rho(\\pi_2,\\pi_1) = \\left(\\rho(\\pi_1,\\pi_2)\\right)^{-1}\\] \\[\\xi(\\pi_2,\\pi_1) = \\left(\\xi(\\pi_2,\\pi_1) + 1\\right)^{-1} - 1\\]\n\nExercise 3.5 Prove the relationships above.\n\n\n\nExample 3.4 (Switching the reference group in a risk ratio) Above, we estimated that the risk ratio of OC versus non-OC is:\n\\[\n\\begin{aligned}\n\\rho(OC, \\neg OC)\n&= 3.7143\n\\end{aligned}\n\\]\nIn comparison, the risk ratio for non-OC versus OC is:\n\\[\n\\begin{aligned}\n\\rho(\\neg OC, OC)\n&=\\frac{\\hat{\\text{p}}(MI|\\neg OC)}{\\hat{\\text{p}}(MI|OC)}\\\\\n&= \\frac{7\\times 10^{-4}}{0.0026}\\\\\n&= 0.2692\\\\\n&= \\frac{1}{\\rho(OC, \\neg OC)}\n\\end{aligned}\n\\]", "crumbs": [ "Generalized Linear Models", "3  Models for Binary Outcomes" @@ -281,7 +281,7 @@ "href": "logistic-regression.html#odds-and-odds-ratios", "title": "3  Models for Binary Outcomes", "section": "3.4 Odds and Odds Ratios", - "text": "3.4 Odds and Odds Ratios\n\n3.4.1 Odds and probabilities\n\nIn logistic regression, we will make use of a mathematically-convenient transformation of probability, called odds.\n\n\nDefinition 3.6 (Odds) The odds of an outcome \\(A\\), which we will represent using \\(\\omega\\) (“omega”), is the probability that the outcome occurs, divided by the probability that it doesn’t occur:\n\\[\\omega(A) \\stackrel{\\text{def}}{=}\\frac{\\Pr(A)}{\\Pr(\\neg A)}\\]\n\n\n\nTheorem 3.2 If the probability of an outcome \\(A\\) is \\(\\Pr(A)=\\pi\\), then the corresponding odds of \\(A\\) is:\n\\[\\text{odds}\\left\\{\\pi\\right\\} = \\frac{\\pi}{1-\\pi} \\tag{3.1}\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n\\Pr(\\neg A) &= 1 - \\Pr(A)\n\\\\ &= 1 - \\pi\n\\end{aligned}\n\\]\n\\[\n\\begin{aligned}\n\\therefore\\omega(A) &\\stackrel{\\text{def}}{=}\\frac{\\Pr(A)}{\\Pr(\\neg A)}\n\\\\ &= \\frac{\\pi}{1-\\pi}\n\\end{aligned}\n\\]\n\n\n\nFunction 3.1, which transforms probabilities into odds, can be called the odds function. Figure 3.1 graphs the shape of this function.\n\n\n\nShow R code\nodds = function(pi) pi / (1 - pi)\nlibrary(ggplot2)\nggplot() + \n geom_function(fun = odds, \n mapping = aes(col = \"odds function\")) +\n xlim(0, .5) +\n xlab(\"Probability\") +\n ylab(\"Odds\") +\n geom_abline(aes(intercept = 0, slope = 1, col = \"y=x\")) + \n theme_bw() +\n labs(colour = \"\") +\n theme(legend.position = \"bottom\")\n\n\n\n\n\nFigure 3.1: Odds versus probability\n\n\n\n\n\n\n\n\n\n\nExample 3.5 (Computing odds from probabilities) In Exercise 3.4, we estimated that the probability of MI, given OC use, is \\(\\pi(OC) \\stackrel{\\text{def}}{=}\\Pr(MI|OC) = 0.0026\\). If this estimate is correct, then the odds of MI, given OC use, is:\n\\[\n\\begin{aligned}\n\\omega(OC)\n&\\stackrel{\\text{def}}{=}\\frac{\\Pr(MI|OC)}{\\Pr(\\neg MI|OC)}\\\\\n&=\\frac{\\Pr(MI|OC)}{1-\\Pr(MI|OC)}\\\\\n&=\\frac{\\pi(OC)}{1-\\pi(OC)}\\\\\n&=\\frac{0.0026}{1-0.0026}\\\\\n&\\approx 0.002607\n\\end{aligned}\n\\]\n\n\n\nExercise 3.6 (Computing odds from probabilities) Estimate the odds of MI, for non-OC users.\n\nSolution. \\[\n\\omega(\\neg OC) = 7.00490343\\times 10^{-4}\n\\]\n\n\n\n\nTheorem 3.3 (One-sample MLE for odds) Let \\(X_1,...X_n\\) be a set of \\(n\\) \\(\\text{iid}\\) Bernoulli trials, and let \\(X = \\sum_{i=1}^nX_i\\) be their sum.\nThen the maximum likelihood estimate of the odds of \\(X=1\\), \\(\\omega\\), is:\n\\[\n\\hat{\\omega}= \\frac{x}{n-x}\n\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n1-\\hat\\pi\n&= 1-\\frac{x}{n}\\\\\n&= \\frac{n}{n} - \\frac{x}{n}\\\\\n&= \\frac{n - x}{n}\n\\end{aligned}\n\\]\nThus, the estimated odds is:\n\\[\n\\begin{aligned}\n\\frac{\\hat\\pi}{1-\\hat\\pi}\n&= \\frac{\\left(\\frac{x}{n}\\right)}{\\left(\\frac{n-x}{n}\\right)}\\\\\n&= \\frac{x}{n-x}\n\\end{aligned}\n\\]\n\nThat is, odds can be calculated directly as “# events” divided by “# nonevents” (without needing to calculate \\(\\hat\\pi\\) and \\(1-\\hat\\pi\\) first).\n\n\n\n\nExample 3.6 (calculating odds using the shortcut) In Example 3.5, we calculated \\[\n\\begin{aligned}\n\\omega(OC)\n&=0.00260678\n\\end{aligned}\n\\]\nLet’s recalculate this result using our shortcut.\n\n\n\nSolution 3.1. \\[\n\\begin{aligned}\n\\omega(OC)\n&=\\frac{13}{5000-13}\\\\\n&=0.00260678\n\\end{aligned}\n\\]\nSame answer as in Example 3.5!\n\n\n\nTheorem 3.4 (Simplified expression for odds function)  \n\nAn equivalent expression for the odds function is\n\n\\[\n\\text{odds}\\left\\{\\pi\\right\\} = \\left(\\left(\\pi\\right)^{-1}-1)\\right)^{-1}\n\\tag{3.2}\\]\n\n\n\nExercise 3.7 Prove that Equation 3.2 is equivalent to Definition 3.6.\n\n\n\nTheorem 3.5 (Derivative of odds function) \\[{\\text{odds}}'\\left\\{\\pi\\right\\} = \\frac{1}{\\left(1-\\pi\\right)^2}\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n{\\text{odds}}'\\left\\{\\pi\\right\\}\n &= \\frac{\\partial}{\\partial \\pi}\\left(\\frac{\\pi}{1-\\pi}\\right)\n\\\\ &= \\frac {\\frac{\\partial}{\\partial \\pi}\\pi} {1-\\pi} -\n \\left(\\frac{\\pi}{\\left(1-\\pi\\right)^2} \\cdot\\frac{\\partial}{\\partial \\pi}\\left(1-\\pi\\right)\\right)\n\\\\ &= \\frac{1}{1-\\pi} - \\frac{\\pi}{\\left(1-\\pi\\right)^2} \\cdot(-1)\n\\\\ &= \\frac{1}{1-\\pi} + \\frac{\\pi}{\\left(1-\\pi\\right)^2}\n\\\\ &= \\frac{1-\\pi}{\\left(1-\\pi\\right)^2} + \\frac{\\pi}{\\left(1-\\pi\\right)^2}\n\\\\ &= \\frac{1-\\pi + \\pi}{\\left(1-\\pi\\right)^2}\n\\\\ &= \\frac{1}{\\left(1-\\pi\\right)^2}\n\\end{aligned}\n\\]\n\n\n\nOdds of rare events\nFor rare events (small \\(\\pi\\)), odds and probabilities are nearly equal, because \\(1-\\pi \\approx 1\\) (see Figure 3.1).\nFor example, in Example 3.5, the probability and odds differ by \\(6.77762182\\times 10^{-6}\\).\n\n\nExercise 3.8 What odds value corresponds to the probability \\(\\pi = 0.2\\), and what is the numerical difference between these two values?\n\n\n\nSolution. \\[\n\\omega= \\frac{\\pi}{1-\\pi}\n=\\frac{.2}{.8}\n= .25\n\\]\n\n\n\nTheorem 3.6 Let \\(\\omega= \\frac{\\pi}{1-\\pi}\\). Then:\n\\[\\omega- \\pi = \\omega\\cdot\\pi\\]\n\n\n\nExercise 3.9 Prove Theorem 3.6.\n\n\n\nSolution. \\[\n\\begin{aligned}\n\\omega- \\pi\n&= \\frac{\\pi}{1-\\pi} - \\pi\n\\\\ &= \\frac{\\pi}{1-\\pi} - \\frac{\\pi(1-\\pi)}{1-\\pi}\n\\\\ &= \\frac{\\pi}{1-\\pi} - \\frac{\\pi - \\pi^2}{1-\\pi}\n\\\\ &= \\frac{\\pi - (\\pi - \\pi^2)}{1-\\pi}\n\\\\ &= \\frac{\\pi - \\pi + \\pi^2}{1-\\pi}\n\\\\ &= \\frac{\\pi^2}{1-\\pi}\n\\\\ &= \\frac{\\pi}{1-\\pi} \\pi\n\\\\ &= \\omega\\pi\n\\end{aligned}\n\\]\n\n\n\nLemma 3.1 (Odds of a non-event) If \\(\\pi\\) is the odds of event \\(A\\) and \\(\\omega\\) is the corresponding odds of \\(A\\), then the odds of \\(\\neg A\\) are:\n\\[\n\\omega(\\neg A) = \\frac{1-\\pi}{\\pi}\n\\]\n\n\nProof. Left to the reader.\n\n\n\n\n\n3.4.2 The inverse odds function\n\nDefinition 3.7 (inverse odds function) The inverse odds function,\n\\[\\text{invodds}\\left\\{\\omega\\right\\} \\stackrel{\\text{def}}{=}\\frac{\\omega}{1 + \\omega}\\] converts odds into their corresponding probabilities (Figure 3.2).\n\n\nThe inverse-odds function takes an odds as input and produces a probability as output. Its domain of inputs is \\([0,\\infty)\\) and its range of outputs is \\([0,1]\\).\nI haven’t seen anyone give the inverse-odds function a concise name; maybe \\(\\text{prob}()\\)?\n\n\n\n\nShow R code\nodds_inv = function(omega) (1 + omega^-1)^-1\nggplot() +\n geom_function(fun = odds_inv, aes(col = \"inverse-odds\")) +\n xlab(\"Odds\") +\n ylab(\"Probability\") +\n xlim(0,5) +\n ylim(0,1) +\n geom_abline(aes(intercept = 0, slope = 1, col = \"x=y\"))\n\n\n\n\n\nFigure 3.2: The inverse odds function, \\(\\text{invodds}\\left\\{\\omega\\right\\}\\)\n\n\n\n\n\n\n\n\n\n\nExercise 3.10 What probability corresponds to an odds of \\(\\omega= 1\\), and what is the numerical difference between these two values?\n\n\n\nSolution. \\[\n\\pi(1) = \\frac{1}{1+1}\n=\\frac{1}{2}\n= .5\n\\] \\[\n1 - \\pi(1) = 1 - .5 = .5\n\\]\n\n\n\nLemma 3.2 (Simplified expression for inverse odds function)  \n\nAn equivalent expression for the inverse odds function is\n\n\\[\n\\pi(\\omega) = (1+\\omega^{-1})^{-1}\n\\tag{3.3}\\]\n\n\n\nExercise 3.11 Prove that Equation 3.3 is equivalent to Definition 3.7.\n\n\n\nLemma 3.3 (One minus inverse-odds) \\[1 - \\text{invodds}\\left\\{\\omega\\right\\} = \\frac{1}{1+\\omega}\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n1 - \\text{invodds}\\left\\{\\omega\\right\\} &= 1 - \\frac{\\omega}{1 + \\omega}\n\\\\ &= \\frac{{\\color{red}1+\\omega}}{1 + \\omega} - \\frac{{\\color{blue}\\omega}}{1 + \\omega}\n\\\\ &= \\frac{{\\color{red}(1+\\omega)} - {\\color{blue}\\omega}}{1 + \\omega}\n\\\\ &= \\frac{1 + \\omega- \\omega}{1 + \\omega}\n\\\\ &= \\frac{1}{1 + \\omega}\n\\end{aligned}\n\\]\n\n\n\nTheorem 3.7 If \\(\\omega\\) is the odds of event \\(A\\), then the probability that \\(A\\) does not occur is:\n\\[\\Pr(\\neg A) = \\frac{1}{1+\\omega}\\]\n\n\n\nProof. \n\nUse Lemma 3.3:\n\n\\[\n\\begin{aligned}\n\\Pr(\\neg A)\n&= 1 - \\Pr(A)\n\\\\ &= 1 - \\text{invodds}\\left\\{\\omega\\right\\}\n\\\\ &= \\frac{1}{1 + \\omega}\n\\end{aligned}\n\\]\n\n\n\nTheorem 3.8 (Derivative of inverse odds function) \\[{\\text{invodds}}'\\left\\{\\omega\\right\\} = \\frac{1}{\\left(1+\\omega\\right)^2}\\]\n\n\n\nProof. \n\nUse the quotient rule:\n\n\\[\n\\begin{aligned}\n{\\text{invodds}}'(\\omega)\n&= \\frac{\\partial}{\\partial \\omega} \\text{invodds}\\left\\{\\omega\\right\\}\n\\\\ &= \\frac{\\partial}{\\partial \\omega} \\frac{\\omega}{1+\\omega}\n\\\\ &= \\frac{\\frac{\\partial}{\\partial \\omega}\\omega}{1+\\omega} - \\frac{\\omega}{\\left(1+\\omega\\right)^2} \\cdot\\frac{\\partial}{\\partial \\omega}(1+\\omega)\n\\\\ &= \\frac{1}{1+\\omega} - \\frac{\\omega}{\\left(1+\\omega\\right)^2} \\cdot 1\n\\\\ &= \\frac{1}{1+\\omega} - \\frac{\\omega}{\\left(1+\\omega\\right)^2}\n\\\\ &= \\frac{1+\\omega}{\\left(1+\\omega\\right)^2} - \\frac{\\omega}{\\left(1+\\omega\\right)^2}\n\\\\ &= \\frac{1+\\omega- \\omega}{\\left(1+\\omega\\right)^2}\n\\\\ &= \\frac{1}{\\left(1+\\omega\\right)^2}\n\\end{aligned}\n\\]\n\n\n\nCorollary 3.1 \\[{\\text{invodds}}'\\left\\{\\omega\\right\\} = \\left(1 - \\text{invodds}\\left\\{\\omega\\right\\}\\right)^2\\]\n\n\n\n\n3.4.3 Odds ratios\n\nNow that we have defined odds, we can introduce another way of comparing event probabilities: odds ratios.\n\n\nDefinition 3.8 (Odds ratio) The odds ratio for two odds \\(\\omega_1\\), \\(\\omega_2\\) is their ratio:\n\\[\\theta(\\omega_1, \\omega_2) = \\frac{\\omega_1}{\\omega_2}\\]\n\n\n\nExample 3.7 (Calculating odds ratios) In Example 3.1, the odds ratio for OC users versus OC-non-users is:\n\\[\n\\begin{aligned}\n\\theta(\\omega(OC), \\omega(\\neg OC))\n&= \\frac{\\omega(OC)}{\\omega(\\neg OC)}\\\\\n&= \\frac{0.0026}{7\\times 10^{-4}}\\\\\n&= 3.71428571\\\\\n\\end{aligned}\n\\]\n\n\n\nA shortcut for calculating odds ratio estimates\n\nThe general form of a two-by-two table is shown in Table 3.2.\n\n\n\nTable 3.2: A generic 2x2 table\n\n\n\n\n\n\nEvent\nNon-Event\nTotal\n\n\n\n\nExposed\na\nb\na+b\n\n\nNon-exposed\nc\nd\nc+d\n\n\nTotal\na+c\nb+d\na+b+c+d\n\n\n\n\n\n\n\n\nFrom this table, we have:\n\n\n\\(\\hat\\pi(Event|Exposed) = a/(a+b)\\)\n\\(\\hat\\pi(\\neg Event|Exposed) = b/(a+b)\\)\n\\(\\hat\\omega(Event|Exposed) = \\frac{\\left(\\frac{a}{a+b}\\right)}{\\left(\\frac{b}{a+b}\\right)}=\\frac{a}{b}\\)\n\\(\\hat\\omega(Event|\\neg Exposed) = \\frac{c}{d}\\) (see Exercise 3.12)\n\\(\\theta(Exposed,\\neg Exposed) = \\frac{\\frac{a}{b}}{\\frac{c}{d}} = \\frac{ad}{bc}\\)\n\n\n\nExercise 3.12 Given Table 3.2, show that \\(\\hat\\omega(Event|\\neg Exposed) = \\frac{c}{d}\\).\n\n\n\n\nProperties of odds ratios\n\nOdds ratios have a special property: we can swap a covariate with the outcome, and the odds ratio remains the same.\n\n\nTheorem 3.9 (Odds ratios are reversible) For any two events \\(A\\), \\(B\\):\n\\[\\theta(A|B) = \\theta(B|A)\\]\n\n\n\nProof. \n\\[\n\\begin{aligned}\n\\theta(A|B) &\\stackrel{\\text{def}}{=}\\frac{\\omega(A|B)}{\\omega(A|\\neg B)}\n\\\\ &= \\frac\n{\\left(\\frac{\\text{p}(A|B)}{\\text{p}(\\neg A|B)}\\right)}\n{\\left(\\frac{\\text{p}(A|\\neg B)}{\\text{p}(\\neg A| \\neg B)}\\right)}\n\\\\ &=\n\\left(\\frac{\\text{p}(A|B)}{\\text{p}(\\neg A|B)}\\right)\n\\left(\\frac{\\text{p}(A|\\neg B)}{\\text{p}(\\neg A| \\neg B)}\\right)^{-1}\n\\\\ &=\n\\left(\\frac{\\text{p}(A|B)}{\\text{p}(\\neg A|B)}\\right)\n\\left(\\frac{\\text{p}(\\neg A| \\neg B)}{\\text{p}(A|\\neg B)}\\right)\n\\\\ &=\n\\left(\\frac{\\text{p}(A|B)}{\\text{p}(\\neg A|B)} \\cdot \\frac{\\text{p}(B)}{\\text{p}(B)}\\right)\n\\left(\\frac{\\text{p}(\\neg A| \\neg B)}{\\text{p}(A|\\neg B)} \\cdot \\frac{\\text{p}(\\neg B)}{\\text{p}(\\neg B)}\\right)\n\\\\ &=\n\\left(\\frac{\\text{p}(A,B)}{\\text{p}(\\neg A,B)}\\right)\n\\left(\\frac{\\text{p}(\\neg A, \\neg B)}{\\text{p}(A, \\neg B)}\\right)\n\\\\ &=\n\\left(\\frac{\\text{p}(B,A)}{{\\color{red}\\text{p}(B,\\neg A)}}\\right)\n\\left(\\frac{\\text{p}(\\neg B, \\neg A)}{{\\color{blue}\\text{p}(\\neg B, A)}}\\right)\n\\\\ &=\n\\left(\\frac{\\text{p}(B,A)}{{\\color{blue}\\text{p}(\\neg B, A)}}\\right)\n\\left(\\frac{\\text{p}(\\neg B, \\neg A)}{{\\color{red}\\text{p}(B,\\neg A)}}\\right)\n\\\\ &= \\text{[reverse the preceding steps]}\n\\\\ &= \\theta(B|A)\n\\end{aligned}\n\\]\n\n\n\nExample 3.8 In Example 3.1, we have:\n\\[\n\\begin{aligned}\n\\theta(MI; OC)\n&\\stackrel{\\text{def}}{=}\n\\frac{\\omega(MI|OC)}{\\omega(MI|\\neg OC)}\\\\\n&\\stackrel{\\text{def}}{=}\\frac\n{\\left(\\frac{\\Pr(MI|OC)}{\\Pr(\\neg MI|OC)}\\right)}\n{\\left(\\frac{\\Pr(MI|\\neg OC)}{\\Pr(\\neg MI|\\neg OC)}\\right)}\\\\\n&= \\frac\n{\\left(\\frac{\\Pr(MI,OC)}{\\Pr(\\neg MI,OC)}\\right)}\n{\\left(\\frac{\\Pr(MI,\\neg OC)}{\\Pr(\\neg MI,\\neg OC)}\\right)}\\\\\n&= \\left(\\frac{\\Pr(MI,OC)}{\\Pr(\\neg MI,OC)}\\right)\n\\left(\\frac{\\Pr(\\neg MI,\\neg OC)}{\\Pr(MI,\\neg OC)}\\right)\\\\\n&= \\left(\\frac{\\Pr(MI,OC)}{\\Pr(MI,\\neg OC)}\\right)\n\\left(\\frac{\\Pr(\\neg MI,\\neg OC)}{\\Pr(\\neg MI,OC)}\\right)\\\\\n&= \\left(\\frac{\\Pr(OC,MI)}{\\Pr(\\neg OC,MI)}\\right)\n\\left(\\frac{\\Pr(\\neg OC,\\neg MI)}{\\Pr(OC,\\neg MI)}\\right)\\\\\n&= \\left(\\frac{\\Pr(OC|MI)}{\\Pr(\\neg OC|MI)}\\right)\n\\left(\\frac{\\Pr(\\neg OC|\\neg MI)}{\\Pr(OC|\\neg MI)}\\right)\\\\\n&= \\frac{\\left(\\frac{\\Pr(OC|MI)}{\\Pr(\\neg OC|MI)}\\right)}\n{\\left(\\frac{\\Pr(OC|\\neg MI)}{\\Pr(\\neg OC|\\neg MI)}\\right)}\\\\\n&\\stackrel{\\text{def}}{=}\\frac{\\omega(OC|MI)}\n{\\omega(OC|\\neg MI)}\\\\\n&\\stackrel{\\text{def}}{=}\\theta(OC; MI)\n\\end{aligned}\n\\]\n\n\n\nExercise 3.13 For Table 3.2, show that \\(\\hat\\theta(Exposed, Unexposed) = \\hat\\theta(Event, \\neg Event)\\).\n\n\n\nConditional odds ratios have the same reversibility property:\n\n\nTheorem 3.10 (Conditional odds ratios are reversible) For any three events \\(A\\), \\(B\\), \\(C\\):\n\\[\\theta(A|B,C) = \\theta(B|A,C)\\]\n\n\n\nProof. Apply the same steps as for Theorem 3.9, inserting \\(C\\) into the conditions (RHS of \\(|\\)) of every expression.\n\n\n\n\nOdds Ratios vs Probability (Risk) Ratios\n\nWhen the outcome is rare (i.e., its probability is small) for both groups being compared in an odds ratio, the odds of the outcome will be similar to the probability of the outcome, and thus the risk ratio will be similar to the odds ratio.\n\n\nCase 1: rare events\nFor rare events, odds ratios and probability (a.k.a. risk, a.k.a. prevalence) ratios will be close:\n\\(\\pi_1 = .01\\) \\(\\pi_2 = .02\\)\n\n\nShow R code\npi1 = .01\npi2 = .02\npi2/pi1\n#> [1] 2\nodds(pi2)/odds(pi1)\n#> [1] 2.0204082\n\n\n\n\nExample 3.9 In Example 3.1, the outcome is rare for both OC and non-OC participants, so the odds for both groups are similar to the corresponding probabilities, and the odds ratio is similar the risk ratio.\n\n\n\n\nCase 2: frequent events\n\\(\\pi_1 = .4\\) \\(\\pi_2 = .5\\)\nFor more frequently-occurring outcomes, this won’t be the case:\n\n\nShow R code\npi1 = .4\npi2 = .5\npi2/pi1\n#> [1] 1.25\nodds(pi2)/odds(pi1)\n#> [1] 1.5\n\n\n\n\n\n\nOdds Ratios in Case-Control Studies\n\nTable 3.1 simulates a follow-up study in which two populations were followed and the number of MI’s was observed. The risks are \\(P(MI|OC)\\) and \\(P(MI|\\neg OC)\\) and we can estimate these risks from the data.\nBut suppose we had a case-control study in which we had 100 women with MI and selected a comparison group of 100 women without MI (matched as groups on age, etc.). Then MI is not random, and we cannot compute P(MI|OC) and we cannot compute the risk ratio. However, the odds ratio however can be computed.\nThe disease odds ratio is the odds for the disease in the exposed group divided by the odds for the disease in the unexposed group, and we cannot validly compute and use these separate parts.\nWe can still validly compute and use the exposure odds ratio, which is the odds for exposure in the disease group divided by the odds for exposure in the non-diseased group (because exposure can be treated as random):\n\n\\[\n\\hat\\theta(OC|MI) =\n\\frac{\\hat{\\omega}(OC|MI)}{\\hat{\\omega}(OC|\\neg MI)}\n\\]\n\nAnd these two odds ratios, \\(\\hat\\theta(MI|OC)\\) and \\(\\hat\\theta(OC|MI)\\), are mathematically equivalent, as we saw in Section 3.4.3.2:\n\n\\[\\hat\\theta(MI|OC) = \\hat\\theta(OC|MI)\\]\n\n\nExercise 3.14 Calculate the odds ratio of MI with respect to OC use, assuming that Table 3.1 comes from a case-control study. Confirm that the result is the same as in Example 3.7.\n\n\nSolution. \n\n\\(\\omega(OC|MI) = P(OC|MI)/(1 – P(OC|MI) = \\frac{13}{7} = 1.85714286\\)\n\\(\\omega(OC|\\neg MI) = P(OC|\\neg MI)/(1 – P(OC|\\neg MI) = \\frac{4987}{9993} = 0.49904933\\)\n\\(\\theta(OC,MI) = \\frac{\\omega(OC|MI)}{\\omega(OC|\\neg MI)} = \\frac{13/7}{4987/9993} = 3.72136125\\)\n\n\nThis is the same estimate we calculated in Example 3.7.\n\n\n\n\n\n\nOdds Ratios in Cross-Sectional Studies\n\nIf a cross-sectional study is a probability sample of a population (which it rarely is) then we can estimate risks.\nIf it is a sample, but not an unbiased probability sample, then we need to treat it in the same way as a case-control study.\nWe can validly estimate odds ratios in either case.\nBut we can usually not validly estimate risks and risk ratios.", + "text": "3.4 Odds and Odds Ratios\n\n3.4.1 Odds and probabilities\n\nIn logistic regression, we will make use of a mathematically-convenient transformation of probability, called odds.\n\n\nDefinition 3.6 (Odds) The odds of an outcome \\(A\\), which we will represent using \\(\\omega\\) (“omega”), is the probability that the outcome occurs, divided by the probability that it doesn’t occur:\n\\[\\omega(A) \\stackrel{\\text{def}}{=}\\frac{\\Pr(A)}{\\Pr(\\neg A)}\\]\n\n\n\nTheorem 3.2 If the probability of an outcome \\(A\\) is \\(\\Pr(A)=\\pi\\), then the corresponding odds of \\(A\\) is:\n\\[\\text{odds}\\left\\{\\pi\\right\\} = \\frac{\\pi}{1-\\pi} \\tag{3.1}\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n\\Pr(\\neg A) &= 1 - \\Pr(A)\n\\\\ &= 1 - \\pi\n\\end{aligned}\n\\]\n\\[\n\\begin{aligned}\n\\therefore\\omega(A) &\\stackrel{\\text{def}}{=}\\frac{\\Pr(A)}{\\Pr(\\neg A)}\n\\\\ &= \\frac{\\pi}{1-\\pi}\n\\end{aligned}\n\\]\n\n\n\nFunction 3.1, which transforms probabilities into odds, can be called the odds function. Figure 3.1 graphs the shape of this function.\n\n\n\nShow R code\nodds = function(pi) pi / (1 - pi)\nlibrary(ggplot2)\nggplot() + \n geom_function(fun = odds, \n mapping = aes(col = \"odds function\")) +\n xlim(0, .5) +\n xlab(\"Probability\") +\n ylab(\"Odds\") +\n geom_abline(aes(intercept = 0, slope = 1, col = \"y=x\")) + \n theme_bw() +\n labs(colour = \"\") +\n theme(legend.position = \"bottom\")\n\n\n\n\n\nFigure 3.1: Odds versus probability\n\n\n\n\n\n\n\n\n\n\nExample 3.5 (Computing odds from probabilities) In Exercise 3.4, we estimated that the probability of MI, given OC use, is \\(\\pi(OC) \\stackrel{\\text{def}}{=}\\Pr(MI|OC) = 0.0026\\). If this estimate is correct, then the odds of MI, given OC use, is:\n\\[\n\\begin{aligned}\n\\omega(OC)\n&\\stackrel{\\text{def}}{=}\\frac{\\Pr(MI|OC)}{\\Pr(\\neg MI|OC)}\\\\\n&=\\frac{\\Pr(MI|OC)}{1-\\Pr(MI|OC)}\\\\\n&=\\frac{\\pi(OC)}{1-\\pi(OC)}\\\\\n&=\\frac{0.0026}{1-0.0026}\\\\\n&\\approx 0.002607\n\\end{aligned}\n\\]\n\n\n\nExercise 3.6 (Computing odds from probabilities) Estimate the odds of MI, for non-OC users.\n\nSolution. \\[\n\\omega(\\neg OC) = 7.0049\\times 10^{-4}\n\\]\n\n\n\n\nTheorem 3.3 (One-sample MLE for odds) Let \\(X_1,...X_n\\) be a set of \\(n\\) \\(\\text{iid}\\) Bernoulli trials, and let \\(X = \\sum_{i=1}^nX_i\\) be their sum.\nThen the maximum likelihood estimate of the odds of \\(X=1\\), \\(\\omega\\), is:\n\\[\n\\hat{\\omega}= \\frac{x}{n-x}\n\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n1-\\hat\\pi\n&= 1-\\frac{x}{n}\\\\\n&= \\frac{n}{n} - \\frac{x}{n}\\\\\n&= \\frac{n - x}{n}\n\\end{aligned}\n\\]\nThus, the estimated odds is:\n\\[\n\\begin{aligned}\n\\frac{\\hat\\pi}{1-\\hat\\pi}\n&= \\frac{\\left(\\frac{x}{n}\\right)}{\\left(\\frac{n-x}{n}\\right)}\\\\\n&= \\frac{x}{n-x}\n\\end{aligned}\n\\]\n\nThat is, odds can be calculated directly as “# events” divided by “# nonevents” (without needing to calculate \\(\\hat\\pi\\) and \\(1-\\hat\\pi\\) first).\n\n\n\n\nExample 3.6 (calculating odds using the shortcut) In Example 3.5, we calculated \\[\n\\begin{aligned}\n\\omega(OC)\n&=0.0026\n\\end{aligned}\n\\]\nLet’s recalculate this result using our shortcut.\n\n\n\nSolution 3.1. \\[\n\\begin{aligned}\n\\omega(OC)\n&=\\frac{13}{5000-13}\\\\\n&=0.0026\n\\end{aligned}\n\\]\nSame answer as in Example 3.5!\n\n\n\nTheorem 3.4 (Simplified expression for odds function)  \n\nAn equivalent expression for the odds function is\n\n\\[\n\\text{odds}\\left\\{\\pi\\right\\} = \\left(\\left(\\pi\\right)^{-1}-1)\\right)^{-1}\n\\tag{3.2}\\]\n\n\n\nExercise 3.7 Prove that Equation 3.2 is equivalent to Definition 3.6.\n\n\n\nTheorem 3.5 (Derivative of odds function) \\[{\\text{odds}}'\\left\\{\\pi\\right\\} = \\frac{1}{\\left(1-\\pi\\right)^2}\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n{\\text{odds}}'\\left\\{\\pi\\right\\}\n &= \\frac{\\partial}{\\partial \\pi}\\left(\\frac{\\pi}{1-\\pi}\\right)\n\\\\ &= \\frac {\\frac{\\partial}{\\partial \\pi}\\pi} {1-\\pi} -\n \\left(\\frac{\\pi}{\\left(1-\\pi\\right)^2} \\cdot\\frac{\\partial}{\\partial \\pi}\\left(1-\\pi\\right)\\right)\n\\\\ &= \\frac{1}{1-\\pi} - \\frac{\\pi}{\\left(1-\\pi\\right)^2} \\cdot(-1)\n\\\\ &= \\frac{1}{1-\\pi} + \\frac{\\pi}{\\left(1-\\pi\\right)^2}\n\\\\ &= \\frac{1-\\pi}{\\left(1-\\pi\\right)^2} + \\frac{\\pi}{\\left(1-\\pi\\right)^2}\n\\\\ &= \\frac{1-\\pi + \\pi}{\\left(1-\\pi\\right)^2}\n\\\\ &= \\frac{1}{\\left(1-\\pi\\right)^2}\n\\end{aligned}\n\\]\n\n\n\nOdds of rare events\nFor rare events (small \\(\\pi\\)), odds and probabilities are nearly equal, because \\(1-\\pi \\approx 1\\) (see Figure 3.1).\nFor example, in Example 3.5, the probability and odds differ by \\(6.7776\\times 10^{-6}\\).\n\n\nExercise 3.8 What odds value corresponds to the probability \\(\\pi = 0.2\\), and what is the numerical difference between these two values?\n\n\n\nSolution. \\[\n\\omega= \\frac{\\pi}{1-\\pi}\n=\\frac{.2}{.8}\n= .25\n\\]\n\n\n\nTheorem 3.6 Let \\(\\omega= \\frac{\\pi}{1-\\pi}\\). Then:\n\\[\\omega- \\pi = \\omega\\cdot\\pi\\]\n\n\n\nExercise 3.9 Prove Theorem 3.6.\n\n\n\nSolution. \\[\n\\begin{aligned}\n\\omega- \\pi\n&= \\frac{\\pi}{1-\\pi} - \\pi\n\\\\ &= \\frac{\\pi}{1-\\pi} - \\frac{\\pi(1-\\pi)}{1-\\pi}\n\\\\ &= \\frac{\\pi}{1-\\pi} - \\frac{\\pi - \\pi^2}{1-\\pi}\n\\\\ &= \\frac{\\pi - (\\pi - \\pi^2)}{1-\\pi}\n\\\\ &= \\frac{\\pi - \\pi + \\pi^2}{1-\\pi}\n\\\\ &= \\frac{\\pi^2}{1-\\pi}\n\\\\ &= \\frac{\\pi}{1-\\pi} \\pi\n\\\\ &= \\omega\\pi\n\\end{aligned}\n\\]\n\n\n\nLemma 3.1 (Odds of a non-event) If \\(\\pi\\) is the odds of event \\(A\\) and \\(\\omega\\) is the corresponding odds of \\(A\\), then the odds of \\(\\neg A\\) are:\n\\[\n\\omega(\\neg A) = \\frac{1-\\pi}{\\pi}\n\\]\n\n\nProof. Left to the reader.\n\n\n\n\n\n3.4.2 The inverse odds function\n\nDefinition 3.7 (inverse odds function) The inverse odds function,\n\\[\\text{invodds}\\left\\{\\omega\\right\\} \\stackrel{\\text{def}}{=}\\frac{\\omega}{1 + \\omega}\\] converts odds into their corresponding probabilities (Figure 3.2).\n\n\nThe inverse-odds function takes an odds as input and produces a probability as output. Its domain of inputs is \\([0,\\infty)\\) and its range of outputs is \\([0,1]\\).\nI haven’t seen anyone give the inverse-odds function a concise name; maybe \\(\\text{prob}()\\)?\n\n\n\n\nShow R code\nodds_inv = function(omega) (1 + omega^-1)^-1\nggplot() +\n geom_function(fun = odds_inv, aes(col = \"inverse-odds\")) +\n xlab(\"Odds\") +\n ylab(\"Probability\") +\n xlim(0,5) +\n ylim(0,1) +\n geom_abline(aes(intercept = 0, slope = 1, col = \"x=y\"))\n\n\n\n\n\nFigure 3.2: The inverse odds function, \\(\\text{invodds}\\left\\{\\omega\\right\\}\\)\n\n\n\n\n\n\n\n\n\n\nExercise 3.10 What probability corresponds to an odds of \\(\\omega= 1\\), and what is the numerical difference between these two values?\n\n\n\nSolution. \\[\n\\pi(1) = \\frac{1}{1+1}\n=\\frac{1}{2}\n= .5\n\\] \\[\n1 - \\pi(1) = 1 - .5 = .5\n\\]\n\n\n\nLemma 3.2 (Simplified expression for inverse odds function)  \n\nAn equivalent expression for the inverse odds function is\n\n\\[\n\\pi(\\omega) = (1+\\omega^{-1})^{-1}\n\\tag{3.3}\\]\n\n\n\nExercise 3.11 Prove that Equation 3.3 is equivalent to Definition 3.7.\n\n\n\nLemma 3.3 (One minus inverse-odds) \\[1 - \\text{invodds}\\left\\{\\omega\\right\\} = \\frac{1}{1+\\omega}\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n1 - \\text{invodds}\\left\\{\\omega\\right\\} &= 1 - \\frac{\\omega}{1 + \\omega}\n\\\\ &= \\frac{{\\color{red}1+\\omega}}{1 + \\omega} - \\frac{{\\color{blue}\\omega}}{1 + \\omega}\n\\\\ &= \\frac{{\\color{red}(1+\\omega)} - {\\color{blue}\\omega}}{1 + \\omega}\n\\\\ &= \\frac{1 + \\omega- \\omega}{1 + \\omega}\n\\\\ &= \\frac{1}{1 + \\omega}\n\\end{aligned}\n\\]\n\n\n\nTheorem 3.7 If \\(\\omega\\) is the odds of event \\(A\\), then the probability that \\(A\\) does not occur is:\n\\[\\Pr(\\neg A) = \\frac{1}{1+\\omega}\\]\n\n\n\nProof. \n\nUse Lemma 3.3:\n\n\\[\n\\begin{aligned}\n\\Pr(\\neg A)\n&= 1 - \\Pr(A)\n\\\\ &= 1 - \\text{invodds}\\left\\{\\omega\\right\\}\n\\\\ &= \\frac{1}{1 + \\omega}\n\\end{aligned}\n\\]\n\n\n\nTheorem 3.8 (Derivative of inverse odds function) \\[{\\text{invodds}}'\\left\\{\\omega\\right\\} = \\frac{1}{\\left(1+\\omega\\right)^2}\\]\n\n\n\nProof. \n\nUse the quotient rule:\n\n\\[\n\\begin{aligned}\n{\\text{invodds}}'(\\omega)\n&= \\frac{\\partial}{\\partial \\omega} \\text{invodds}\\left\\{\\omega\\right\\}\n\\\\ &= \\frac{\\partial}{\\partial \\omega} \\frac{\\omega}{1+\\omega}\n\\\\ &= \\frac{\\frac{\\partial}{\\partial \\omega}\\omega}{1+\\omega} - \\frac{\\omega}{\\left(1+\\omega\\right)^2} \\cdot\\frac{\\partial}{\\partial \\omega}(1+\\omega)\n\\\\ &= \\frac{1}{1+\\omega} - \\frac{\\omega}{\\left(1+\\omega\\right)^2} \\cdot 1\n\\\\ &= \\frac{1}{1+\\omega} - \\frac{\\omega}{\\left(1+\\omega\\right)^2}\n\\\\ &= \\frac{1+\\omega}{\\left(1+\\omega\\right)^2} - \\frac{\\omega}{\\left(1+\\omega\\right)^2}\n\\\\ &= \\frac{1+\\omega- \\omega}{\\left(1+\\omega\\right)^2}\n\\\\ &= \\frac{1}{\\left(1+\\omega\\right)^2}\n\\end{aligned}\n\\]\n\n\n\nCorollary 3.1 \\[{\\text{invodds}}'\\left\\{\\omega\\right\\} = \\left(1 - \\text{invodds}\\left\\{\\omega\\right\\}\\right)^2\\]\n\n\n\n\n3.4.3 Odds ratios\n\nNow that we have defined odds, we can introduce another way of comparing event probabilities: odds ratios.\n\n\nDefinition 3.8 (Odds ratio) The odds ratio for two odds \\(\\omega_1\\), \\(\\omega_2\\) is their ratio:\n\\[\\theta(\\omega_1, \\omega_2) = \\frac{\\omega_1}{\\omega_2}\\]\n\n\n\nExample 3.7 (Calculating odds ratios) In Example 3.1, the odds ratio for OC users versus OC-non-users is:\n\\[\n\\begin{aligned}\n\\theta(\\omega(OC), \\omega(\\neg OC))\n&= \\frac{\\omega(OC)}{\\omega(\\neg OC)}\\\\\n&= \\frac{0.0026}{7\\times 10^{-4}}\\\\\n&= 3.7143\\\\\n\\end{aligned}\n\\]\n\n\n\nA shortcut for calculating odds ratio estimates\n\nThe general form of a two-by-two table is shown in Table 3.2.\n\n\n\nTable 3.2: A generic 2x2 table\n\n\n\n\n\n\nEvent\nNon-Event\nTotal\n\n\n\n\nExposed\na\nb\na+b\n\n\nNon-exposed\nc\nd\nc+d\n\n\nTotal\na+c\nb+d\na+b+c+d\n\n\n\n\n\n\n\n\nFrom this table, we have:\n\n\n\\(\\hat\\pi(Event|Exposed) = a/(a+b)\\)\n\\(\\hat\\pi(\\neg Event|Exposed) = b/(a+b)\\)\n\\(\\hat\\omega(Event|Exposed) = \\frac{\\left(\\frac{a}{a+b}\\right)}{\\left(\\frac{b}{a+b}\\right)}=\\frac{a}{b}\\)\n\\(\\hat\\omega(Event|\\neg Exposed) = \\frac{c}{d}\\) (see Exercise 3.12)\n\\(\\theta(Exposed,\\neg Exposed) = \\frac{\\frac{a}{b}}{\\frac{c}{d}} = \\frac{ad}{bc}\\)\n\n\n\nExercise 3.12 Given Table 3.2, show that \\(\\hat\\omega(Event|\\neg Exposed) = \\frac{c}{d}\\).\n\n\n\n\nProperties of odds ratios\n\nOdds ratios have a special property: we can swap a covariate with the outcome, and the odds ratio remains the same.\n\n\nTheorem 3.9 (Odds ratios are reversible) For any two events \\(A\\), \\(B\\):\n\\[\\theta(A|B) = \\theta(B|A)\\]\n\n\n\nProof. \n\\[\n\\begin{aligned}\n\\theta(A|B) &\\stackrel{\\text{def}}{=}\\frac{\\omega(A|B)}{\\omega(A|\\neg B)}\n\\\\ &= \\frac\n{\\left(\\frac{\\text{p}(A|B)}{\\text{p}(\\neg A|B)}\\right)}\n{\\left(\\frac{\\text{p}(A|\\neg B)}{\\text{p}(\\neg A| \\neg B)}\\right)}\n\\\\ &=\n\\left(\\frac{\\text{p}(A|B)}{\\text{p}(\\neg A|B)}\\right)\n\\left(\\frac{\\text{p}(A|\\neg B)}{\\text{p}(\\neg A| \\neg B)}\\right)^{-1}\n\\\\ &=\n\\left(\\frac{\\text{p}(A|B)}{\\text{p}(\\neg A|B)}\\right)\n\\left(\\frac{\\text{p}(\\neg A| \\neg B)}{\\text{p}(A|\\neg B)}\\right)\n\\\\ &=\n\\left(\\frac{\\text{p}(A|B)}{\\text{p}(\\neg A|B)} \\cdot \\frac{\\text{p}(B)}{\\text{p}(B)}\\right)\n\\left(\\frac{\\text{p}(\\neg A| \\neg B)}{\\text{p}(A|\\neg B)} \\cdot \\frac{\\text{p}(\\neg B)}{\\text{p}(\\neg B)}\\right)\n\\\\ &=\n\\left(\\frac{\\text{p}(A,B)}{\\text{p}(\\neg A,B)}\\right)\n\\left(\\frac{\\text{p}(\\neg A, \\neg B)}{\\text{p}(A, \\neg B)}\\right)\n\\\\ &=\n\\left(\\frac{\\text{p}(B,A)}{{\\color{red}\\text{p}(B,\\neg A)}}\\right)\n\\left(\\frac{\\text{p}(\\neg B, \\neg A)}{{\\color{blue}\\text{p}(\\neg B, A)}}\\right)\n\\\\ &=\n\\left(\\frac{\\text{p}(B,A)}{{\\color{blue}\\text{p}(\\neg B, A)}}\\right)\n\\left(\\frac{\\text{p}(\\neg B, \\neg A)}{{\\color{red}\\text{p}(B,\\neg A)}}\\right)\n\\\\ &= \\text{[reverse the preceding steps]}\n\\\\ &= \\theta(B|A)\n\\end{aligned}\n\\]\n\n\n\nExample 3.8 In Example 3.1, we have:\n\\[\n\\begin{aligned}\n\\theta(MI; OC)\n&\\stackrel{\\text{def}}{=}\n\\frac{\\omega(MI|OC)}{\\omega(MI|\\neg OC)}\\\\\n&\\stackrel{\\text{def}}{=}\\frac\n{\\left(\\frac{\\Pr(MI|OC)}{\\Pr(\\neg MI|OC)}\\right)}\n{\\left(\\frac{\\Pr(MI|\\neg OC)}{\\Pr(\\neg MI|\\neg OC)}\\right)}\\\\\n&= \\frac\n{\\left(\\frac{\\Pr(MI,OC)}{\\Pr(\\neg MI,OC)}\\right)}\n{\\left(\\frac{\\Pr(MI,\\neg OC)}{\\Pr(\\neg MI,\\neg OC)}\\right)}\\\\\n&= \\left(\\frac{\\Pr(MI,OC)}{\\Pr(\\neg MI,OC)}\\right)\n\\left(\\frac{\\Pr(\\neg MI,\\neg OC)}{\\Pr(MI,\\neg OC)}\\right)\\\\\n&= \\left(\\frac{\\Pr(MI,OC)}{\\Pr(MI,\\neg OC)}\\right)\n\\left(\\frac{\\Pr(\\neg MI,\\neg OC)}{\\Pr(\\neg MI,OC)}\\right)\\\\\n&= \\left(\\frac{\\Pr(OC,MI)}{\\Pr(\\neg OC,MI)}\\right)\n\\left(\\frac{\\Pr(\\neg OC,\\neg MI)}{\\Pr(OC,\\neg MI)}\\right)\\\\\n&= \\left(\\frac{\\Pr(OC|MI)}{\\Pr(\\neg OC|MI)}\\right)\n\\left(\\frac{\\Pr(\\neg OC|\\neg MI)}{\\Pr(OC|\\neg MI)}\\right)\\\\\n&= \\frac{\\left(\\frac{\\Pr(OC|MI)}{\\Pr(\\neg OC|MI)}\\right)}\n{\\left(\\frac{\\Pr(OC|\\neg MI)}{\\Pr(\\neg OC|\\neg MI)}\\right)}\\\\\n&\\stackrel{\\text{def}}{=}\\frac{\\omega(OC|MI)}\n{\\omega(OC|\\neg MI)}\\\\\n&\\stackrel{\\text{def}}{=}\\theta(OC; MI)\n\\end{aligned}\n\\]\n\n\n\nExercise 3.13 For Table 3.2, show that \\(\\hat\\theta(Exposed, Unexposed) = \\hat\\theta(Event, \\neg Event)\\).\n\n\n\nConditional odds ratios have the same reversibility property:\n\n\nTheorem 3.10 (Conditional odds ratios are reversible) For any three events \\(A\\), \\(B\\), \\(C\\):\n\\[\\theta(A|B,C) = \\theta(B|A,C)\\]\n\n\n\nProof. Apply the same steps as for Theorem 3.9, inserting \\(C\\) into the conditions (RHS of \\(|\\)) of every expression.\n\n\n\n\nOdds Ratios vs Probability (Risk) Ratios\n\nWhen the outcome is rare (i.e., its probability is small) for both groups being compared in an odds ratio, the odds of the outcome will be similar to the probability of the outcome, and thus the risk ratio will be similar to the odds ratio.\n\n\nCase 1: rare events\nFor rare events, odds ratios and probability (a.k.a. risk, a.k.a. prevalence) ratios will be close:\n\\(\\pi_1 = .01\\) \\(\\pi_2 = .02\\)\n\n\nShow R code\npi1 = .01\npi2 = .02\npi2/pi1\n#> [1] 2\nodds(pi2)/odds(pi1)\n#> [1] 2.02\n\n\n\n\nExample 3.9 In Example 3.1, the outcome is rare for both OC and non-OC participants, so the odds for both groups are similar to the corresponding probabilities, and the odds ratio is similar the risk ratio.\n\n\n\n\nCase 2: frequent events\n\\(\\pi_1 = .4\\) \\(\\pi_2 = .5\\)\nFor more frequently-occurring outcomes, this won’t be the case:\n\n\nShow R code\npi1 = .4\npi2 = .5\npi2/pi1\n#> [1] 1.25\nodds(pi2)/odds(pi1)\n#> [1] 1.5\n\n\n\n\n\n\nOdds Ratios in Case-Control Studies\n\nTable 3.1 simulates a follow-up study in which two populations were followed and the number of MI’s was observed. The risks are \\(P(MI|OC)\\) and \\(P(MI|\\neg OC)\\) and we can estimate these risks from the data.\nBut suppose we had a case-control study in which we had 100 women with MI and selected a comparison group of 100 women without MI (matched as groups on age, etc.). Then MI is not random, and we cannot compute P(MI|OC) and we cannot compute the risk ratio. However, the odds ratio however can be computed.\nThe disease odds ratio is the odds for the disease in the exposed group divided by the odds for the disease in the unexposed group, and we cannot validly compute and use these separate parts.\nWe can still validly compute and use the exposure odds ratio, which is the odds for exposure in the disease group divided by the odds for exposure in the non-diseased group (because exposure can be treated as random):\n\n\\[\n\\hat\\theta(OC|MI) =\n\\frac{\\hat{\\omega}(OC|MI)}{\\hat{\\omega}(OC|\\neg MI)}\n\\]\n\nAnd these two odds ratios, \\(\\hat\\theta(MI|OC)\\) and \\(\\hat\\theta(OC|MI)\\), are mathematically equivalent, as we saw in Section 3.4.3.2:\n\n\\[\\hat\\theta(MI|OC) = \\hat\\theta(OC|MI)\\]\n\n\nExercise 3.14 Calculate the odds ratio of MI with respect to OC use, assuming that Table 3.1 comes from a case-control study. Confirm that the result is the same as in Example 3.7.\n\n\nSolution. \n\n\\(\\omega(OC|MI) = P(OC|MI)/(1 – P(OC|MI) = \\frac{13}{7} = 1.8571\\)\n\\(\\omega(OC|\\neg MI) = P(OC|\\neg MI)/(1 – P(OC|\\neg MI) = \\frac{4987}{9993} = 0.499\\)\n\\(\\theta(OC,MI) = \\frac{\\omega(OC|MI)}{\\omega(OC|\\neg MI)} = \\frac{13/7}{4987/9993} = 3.7214\\)\n\n\nThis is the same estimate we calculated in Example 3.7.\n\n\n\n\n\n\nOdds Ratios in Cross-Sectional Studies\n\nIf a cross-sectional study is a probability sample of a population (which it rarely is) then we can estimate risks.\nIf it is a sample, but not an unbiased probability sample, then we need to treat it in the same way as a case-control study.\nWe can validly estimate odds ratios in either case.\nBut we can usually not validly estimate risks and risk ratios.", "crumbs": [ "Generalized Linear Models", "3  Models for Binary Outcomes" @@ -303,7 +303,7 @@ "href": "logistic-regression.html#introduction-to-logistic-regression", "title": "3  Models for Binary Outcomes", "section": "3.6 Introduction to logistic regression", - "text": "3.6 Introduction to logistic regression\n\n\nIn Example 3.1, we estimated the risk and the odds of MI for two groups, defined by oral contraceptive use.\nIf the predictor is quantitative (dose) or there is more than one predictor, the task becomes more difficult.\nIn this case, we will use logistic regression, which is a generalization of the linear regression models you have been using that can account for a binary response instead of a continuous one.\n\n\n\n3.6.1 Binary outcomes models - one group, no covariates\n\\[\n\\begin{aligned}\n\\text{P}(Y=1) &= \\pi\\\\\n\\text{P}(Y=0) &= 1-\\pi\\\\\n\\text{P}(Y=y) &= \\pi^y (1-\\pi)^{1-y}\\\\\n\\mathbf y &= (y_1, ..., y_n)\\\\\n\\mathcal L(\\pi;\\mathbf y) &= \\pi^{\\sum y_i} (1-\\pi)^{n - \\sum y_i}\\\\\n\\ell(\\pi, \\mathbf y) &= \\left({\\sum y_i}\\right) \\text{log}\\left\\{\\pi\\right\\} + \\left(n - \\sum y_i\\right) \\text{log}\\left\\{1-\\pi\\right\\}\\\\\n&= \\left({\\sum y_i}\\right) \\left(\\text{log}\\left\\{\\pi\\right\\} - \\text{log}\\left\\{1-\\pi\\right\\}\\right) + n \\cdot \\text{log}\\left\\{1-\\pi\\right\\}\\\\\n&= \\left({\\sum y_i}\\right) \\text{log}\\left\\{\\frac{\\pi}{ 1-\\pi}\\right\\} + n \\cdot \\text{log}\\left\\{1-\\pi\\right\\}\n\\\\ &= \\left({\\sum y_i}\\right) \\text{logit}(\\pi) + n \\cdot \\text{log}\\left\\{1-\\pi\\right\\}\n\\end{aligned}\n\\]\n\n\n3.6.2 Binary outcomes - general\n\\[\n\\begin{aligned}\n\\text{P}(Y_i=1) &= \\pi_i\n\\\\ \\text{P}(Y_i=0) &= 1-\\pi_i\n\\end{aligned}\n\\]\n\\[\\text{P}(Y_i=y_i) = (\\pi_i)^y_i (1-\\pi_i)^{1-y_i}\\]\n\\[\\mathcal{L}_i(\\pi_i) = \\text{P}(Y_i=y_i)\\]\n\\[\n\\begin{aligned}\n\\ell_i(\\pi_i)\n &= \\text{log}\\left\\{\\mathcal{L}_i(\\pi_i)\\right\\}\n\\\\ &= y_i \\text{log}\\left\\{\\pi_i\\right\\} + (1-y_i) \\text{log}\\left\\{1-\\pi_i\\right\\}\n\\end{aligned}\n\\]\n\nFor \\(\\text{iid}\\) data \\(\\tilde{y} = (y_1, ..., y_n)\\):\n\\[\n\\begin{aligned}\n\\mathcal{L}(\\pi;\\tilde{y}) &= \\text{P}(Y_1=y_1, \\ldots, Y_n = y_n)\n\\\\ &= \\prod_{i=1}^n \\text{P}(Y_i=y_i)\n\\\\ &= \\prod_{i=1}^n \\mathcal{L}_i(\\pi_i)\n\\end{aligned}\n\\]\n\n\n3.6.3 Modeling \\(\\pi_i\\) as a function of \\(X_i\\)\nIf there are only a few distinct \\(X_i\\) values, we can model \\(\\pi_i\\) separately for each value of \\(X_i\\).\nOtherwise, we need regression.\n\\[\n\\begin{aligned}\n\\pi(x) &\\equiv \\text{E}(Y=1|X=x)\\\\\n&= f(x^\\top\\beta)\n\\end{aligned}\n\\]\nTypically, we use the \\(\\text{expit}\\) inverse-link:\n\\[\\pi(\\tilde{x}) = \\text{expit}(\\tilde{x}'\\beta) \\tag{3.4}\\]\n\n\n3.6.4 Meet the beetles\n\n\nShow R code\nlibrary(glmx)\n\ndata(BeetleMortality, package = \"glmx\")\nbeetles = BeetleMortality |>\n mutate(\n pct = died/n,\n survived = n - died\n )\n\nplot1 = \n beetles |> \n ggplot(aes(x = dose, y = pct)) +\n geom_point(aes(size = n)) +\n xlab(\"Dose (log mg/L)\") +\n ylab(\"Mortality rate (%)\") +\n scale_y_continuous(labels = scales::percent) +\n scale_size(range = c(1,2)) +\n theme_bw(base_size = 18)\n\nprint(plot1)\n\n\n\n\n\nFigure 3.5: Mortality rates of adult flour beetles after five hours’ exposure to gaseous carbon disulphide (Bliss 1935)\n\n\n\n\n\n\n\n\n\n\n3.6.5 Why don’t we use linear regression?\n\n\nShow R code\nbeetles_long = beetles |> \n reframe(\n .by = everything(),\n outcome = c(\n rep(1, times = died), \n rep(0, times = survived))) |> \n as_tibble()\n\nlm1 = beetles_long |> lm(formula = outcome ~ dose)\nf.linear = function(x) predict(lm1, newdata = data.frame(dose = x))\n\nrange1 = range(beetles$dose) + c(-.2, .2)\n\nplot2 = \n plot1 + \n geom_function(\n fun = f.linear, \n aes(col = \"Straight line\")) +\n labs(colour=\"Model\", size = \"\")\n\nplot2 |> print()\n\n\n\n\n\nFigure 3.6: Mortality rates of adult flour beetles after five hours’ exposure to gaseous carbon disulphide (Bliss 1935)\n\n\n\n\n\n\n\n\n\n\n3.6.6 Zoom out\n\n\nShow R code\n(plot2 + expand_limits(x = c(1.6, 2))) |> print()\n\n\n\n\n\nFigure 3.7: Mortality rates of adult flour beetles after five hours’ exposure to gaseous carbon disulphide (Bliss 1935)\n\n\n\n\n\n\n\n\n\n\n3.6.7 log transformation of dose?\n\n\nShow R code\nlm2 = beetles_long |> lm(formula = outcome ~ log(dose))\nf.linearlog = function(x) predict(lm2, newdata = data.frame(dose = x))\n\nplot3 = plot2 + \n expand_limits(x = c(1.6, 2)) +\n geom_function(fun = f.linearlog, aes(col = \"Log-transform dose\"))\n(plot3 + expand_limits(x = c(1.6, 2))) |> print()\n\n\n\n\n\nFigure 3.8: Mortality rates of adult flour beetles after five hours’ exposure to gaseous carbon disulphide (Bliss 1935)\n\n\n\n\n\n\n\n\n\n\n3.6.8 Logistic regression\n\n\nShow R code\n\n#| label: fig-beetles_5\n#| fig-cap: \"Mortality rates of adult flour beetles after five hours' exposure to gaseous carbon disulphide (Bliss 1935)\"\n\nbeetles_glm_grouped = beetles |> \n glm(formula = cbind(died, survived) ~ dose, family = \"binomial\")\nf = function(x) \n beetles_glm_grouped |> \n predict(newdata = data.frame(dose = x), type = \"response\")\n\nplot4 = plot3 + geom_function(fun = f, aes(col = \"Logistic regression\"))\nplot4 |> print()\n\n\n\n\n\n\n\n\n\n\n\n\n3.6.9 Three parts to regression models\n\nWhat distribution does the outcome have for a specific subpopulation defined by covariates? (outcome model)\nHow does the combination of covariates relate to the mean? (link function)\nHow do the covariates combine? (linear predictor, interactions)\n\n\n\n\n3.6.10 Logistic regression in R\n\n\nShow R code\n\nbeetles_glm_grouped = \n beetles |> \n glm(\n formula = cbind(died, survived) ~ dose, \n family = \"binomial\")\n\nlibrary(parameters)\nbeetles_glm_grouped |> \n parameters() |> \n print_md()\n\n\n\n\n\nParameter\nLog-Odds\nSE\n95% CI\nz\np\n\n\n\n\n(Intercept)\n-60.72\n5.18\n(-71.44, -51.08)\n-11.72\n< .001\n\n\ndose\n34.27\n2.91\n(28.85, 40.30)\n11.77\n< .001\n\n\n\n\n\n\nFitted values:\n\n\nShow R code\nfitted.values(beetles_glm_grouped)\n#> 1 2 3 4 5 6 \n#> 0.058601026 0.164027869 0.362119006 0.605314906 0.795171772 0.903235819 \n#> 7 8 \n#> 0.955196107 0.979049344\npredict(beetles_glm_grouped, type = \"response\")\n#> 1 2 3 4 5 6 \n#> 0.058601026 0.164027869 0.362119006 0.605314906 0.795171772 0.903235819 \n#> 7 8 \n#> 0.955196107 0.979049344\npredict(beetles_glm_grouped, type = \"link\")\n#> 1 2 3 4 5 6 \n#> -2.77661484 -1.62855893 -0.56617883 0.42766061 1.35638644 2.23370678 \n#> 7 8 \n#> 3.05962163 3.84441209\n\nfit_y = beetles$n * fitted.values(beetles_glm_grouped)\n\n\n\n\n\n3.6.11 Individual observations\n\n\nShow R code\nbeetles_long\n\n\n\n\nTable 3.3: beetles data in long format\n\n\n\n\n \n\n\n\n\n\n\n\n\n\nShow R code\nbeetles_glm_ungrouped = \n beetles_long |> \n glm(\n formula = outcome ~ dose, \n family = \"binomial\")\n\nbeetles_glm_ungrouped |> parameters() |> print_md()\n\n\n\n\nTable 3.4: logistic regression model for beetles data with individual Bernoulli data\n\n\n\n\n\n\nParameter\nLog-Odds\nSE\n95% CI\nz\np\n\n\n\n\n(Intercept)\n-60.72\n5.18\n(-71.44, -51.08)\n-11.72\n< .001\n\n\ndose\n34.27\n2.91\n(28.85, 40.30)\n11.77\n< .001\n\n\n\n\n\n\n\n\n\nHere’s the previous version again:\n\n\n\nShow R code\nbeetles_glm_grouped |> parameters() |> print_md()\n\n\n\n\nTable 3.5: logistic regression model for beetles data with grouped (binomial) data\n\n\n\n\n\n\nParameter\nLog-Odds\nSE\n95% CI\nz\np\n\n\n\n\n(Intercept)\n-60.72\n5.18\n(-71.44, -51.08)\n-11.72\n< .001\n\n\ndose\n34.27\n2.91\n(28.85, 40.30)\n11.77\n< .001\n\n\n\n\n\n\n\n\n\n\nThey seem the same! But not quite:\n\n\n\nShow R code\n\nlogLik(beetles_glm_grouped)\n#> 'log Lik.' -18.715135 (df=2)\nlogLik(beetles_glm_ungrouped)\n#> 'log Lik.' -186.2354 (df=2)\n\n\n\nThe difference is due to the binomial coefficient \\(\\left(n\\atop x \\right)\\) which isn’t included in the individual-observations (Bernoulli) version of the model.", + "text": "3.6 Introduction to logistic regression\n\n\nIn Example 3.1, we estimated the risk and the odds of MI for two groups, defined by oral contraceptive use.\nIf the predictor is quantitative (dose) or there is more than one predictor, the task becomes more difficult.\nIn this case, we will use logistic regression, which is a generalization of the linear regression models you have been using that can account for a binary response instead of a continuous one.\n\n\n\n3.6.1 Binary outcomes models - one group, no covariates\n\\[\n\\begin{aligned}\n\\text{P}(Y=1) &= \\pi\\\\\n\\text{P}(Y=0) &= 1-\\pi\\\\\n\\text{P}(Y=y) &= \\pi^y (1-\\pi)^{1-y}\\\\\n\\mathbf y &= (y_1, ..., y_n)\\\\\n\\mathcal L(\\pi;\\mathbf y) &= \\pi^{\\sum y_i} (1-\\pi)^{n - \\sum y_i}\\\\\n\\ell(\\pi, \\mathbf y) &= \\left({\\sum y_i}\\right) \\text{log}\\left\\{\\pi\\right\\} + \\left(n - \\sum y_i\\right) \\text{log}\\left\\{1-\\pi\\right\\}\\\\\n&= \\left({\\sum y_i}\\right) \\left(\\text{log}\\left\\{\\pi\\right\\} - \\text{log}\\left\\{1-\\pi\\right\\}\\right) + n \\cdot \\text{log}\\left\\{1-\\pi\\right\\}\\\\\n&= \\left({\\sum y_i}\\right) \\text{log}\\left\\{\\frac{\\pi}{ 1-\\pi}\\right\\} + n \\cdot \\text{log}\\left\\{1-\\pi\\right\\}\n\\\\ &= \\left({\\sum y_i}\\right) \\text{logit}(\\pi) + n \\cdot \\text{log}\\left\\{1-\\pi\\right\\}\n\\end{aligned}\n\\]\n\n\n3.6.2 Binary outcomes - general\n\\[\n\\begin{aligned}\n\\text{P}(Y_i=1) &= \\pi_i\n\\\\ \\text{P}(Y_i=0) &= 1-\\pi_i\n\\end{aligned}\n\\]\n\\[\\text{P}(Y_i=y_i) = (\\pi_i)^y_i (1-\\pi_i)^{1-y_i}\\]\n\\[\\mathcal{L}_i(\\pi_i) = \\text{P}(Y_i=y_i)\\]\n\\[\n\\begin{aligned}\n\\ell_i(\\pi_i)\n &= \\text{log}\\left\\{\\mathcal{L}_i(\\pi_i)\\right\\}\n\\\\ &= y_i \\text{log}\\left\\{\\pi_i\\right\\} + (1-y_i) \\text{log}\\left\\{1-\\pi_i\\right\\}\n\\end{aligned}\n\\]\n\nFor \\(\\text{iid}\\) data \\(\\tilde{y} = (y_1, ..., y_n)\\):\n\\[\n\\begin{aligned}\n\\mathcal{L}(\\pi;\\tilde{y}) &= \\text{P}(Y_1=y_1, \\ldots, Y_n = y_n)\n\\\\ &= \\prod_{i=1}^n \\text{P}(Y_i=y_i)\n\\\\ &= \\prod_{i=1}^n \\mathcal{L}_i(\\pi_i)\n\\end{aligned}\n\\]\n\n\n3.6.3 Modeling \\(\\pi_i\\) as a function of \\(X_i\\)\nIf there are only a few distinct \\(X_i\\) values, we can model \\(\\pi_i\\) separately for each value of \\(X_i\\).\nOtherwise, we need regression.\n\\[\n\\begin{aligned}\n\\pi(x) &\\equiv \\text{E}(Y=1|X=x)\\\\\n&= f(x^\\top\\beta)\n\\end{aligned}\n\\]\nTypically, we use the \\(\\text{expit}\\) inverse-link:\n\\[\\pi(\\tilde{x}) = \\text{expit}(\\tilde{x}'\\beta) \\tag{3.4}\\]\n\n\n3.6.4 Meet the beetles\n\n\nShow R code\nlibrary(glmx)\n\ndata(BeetleMortality, package = \"glmx\")\nbeetles = BeetleMortality |>\n mutate(\n pct = died/n,\n survived = n - died\n )\n\nplot1 = \n beetles |> \n ggplot(aes(x = dose, y = pct)) +\n geom_point(aes(size = n)) +\n xlab(\"Dose (log mg/L)\") +\n ylab(\"Mortality rate (%)\") +\n scale_y_continuous(labels = scales::percent) +\n scale_size(range = c(1,2)) +\n theme_bw(base_size = 18)\n\nprint(plot1)\n\n\n\n\n\nFigure 3.5: Mortality rates of adult flour beetles after five hours’ exposure to gaseous carbon disulphide (Bliss 1935)\n\n\n\n\n\n\n\n\n\n\n3.6.5 Why don’t we use linear regression?\n\n\nShow R code\nbeetles_long = beetles |> \n reframe(\n .by = everything(),\n outcome = c(\n rep(1, times = died), \n rep(0, times = survived))) |> \n as_tibble()\n\nlm1 = beetles_long |> lm(formula = outcome ~ dose)\nf.linear = function(x) predict(lm1, newdata = data.frame(dose = x))\n\nrange1 = range(beetles$dose) + c(-.2, .2)\n\nplot2 = \n plot1 + \n geom_function(\n fun = f.linear, \n aes(col = \"Straight line\")) +\n labs(colour=\"Model\", size = \"\")\n\nplot2 |> print()\n\n\n\n\n\nFigure 3.6: Mortality rates of adult flour beetles after five hours’ exposure to gaseous carbon disulphide (Bliss 1935)\n\n\n\n\n\n\n\n\n\n\n3.6.6 Zoom out\n\n\nShow R code\n(plot2 + expand_limits(x = c(1.6, 2))) |> print()\n\n\n\n\n\nFigure 3.7: Mortality rates of adult flour beetles after five hours’ exposure to gaseous carbon disulphide (Bliss 1935)\n\n\n\n\n\n\n\n\n\n\n3.6.7 log transformation of dose?\n\n\nShow R code\nlm2 = beetles_long |> lm(formula = outcome ~ log(dose))\nf.linearlog = function(x) predict(lm2, newdata = data.frame(dose = x))\n\nplot3 = plot2 + \n expand_limits(x = c(1.6, 2)) +\n geom_function(fun = f.linearlog, aes(col = \"Log-transform dose\"))\n(plot3 + expand_limits(x = c(1.6, 2))) |> print()\n\n\n\n\n\nFigure 3.8: Mortality rates of adult flour beetles after five hours’ exposure to gaseous carbon disulphide (Bliss 1935)\n\n\n\n\n\n\n\n\n\n\n3.6.8 Logistic regression\n\n\nShow R code\n\n#| label: fig-beetles_5\n#| fig-cap: \"Mortality rates of adult flour beetles after five hours' exposure to gaseous carbon disulphide (Bliss 1935)\"\n\nbeetles_glm_grouped = beetles |> \n glm(formula = cbind(died, survived) ~ dose, family = \"binomial\")\nf = function(x) \n beetles_glm_grouped |> \n predict(newdata = data.frame(dose = x), type = \"response\")\n\nplot4 = plot3 + geom_function(fun = f, aes(col = \"Logistic regression\"))\nplot4 |> print()\n\n\n\n\n\n\n\n\n\n\n\n\n3.6.9 Three parts to regression models\n\nWhat distribution does the outcome have for a specific subpopulation defined by covariates? (outcome model)\nHow does the combination of covariates relate to the mean? (link function)\nHow do the covariates combine? (linear predictor, interactions)\n\n\n\n\n3.6.10 Logistic regression in R\n\n\nShow R code\n\nbeetles_glm_grouped = \n beetles |> \n glm(\n formula = cbind(died, survived) ~ dose, \n family = \"binomial\")\n\nlibrary(parameters)\nbeetles_glm_grouped |> \n parameters() |> \n print_md()\n\n\n\n\n\nParameter\nLog-Odds\nSE\n95% CI\nz\np\n\n\n\n\n(Intercept)\n-60.72\n5.18\n(-71.44, -51.08)\n-11.72\n< .001\n\n\ndose\n34.27\n2.91\n(28.85, 40.30)\n11.77\n< .001\n\n\n\n\n\n\nFitted values:\n\n\nShow R code\nfitted.values(beetles_glm_grouped)\n#> 1 2 3 4 5 6 7 8 \n#> 0.0586 0.1640 0.3621 0.6053 0.7952 0.9032 0.9552 0.9790\npredict(beetles_glm_grouped, type = \"response\")\n#> 1 2 3 4 5 6 7 8 \n#> 0.0586 0.1640 0.3621 0.6053 0.7952 0.9032 0.9552 0.9790\npredict(beetles_glm_grouped, type = \"link\")\n#> 1 2 3 4 5 6 7 8 \n#> -2.7766 -1.6286 -0.5662 0.4277 1.3564 2.2337 3.0596 3.8444\n\nfit_y = beetles$n * fitted.values(beetles_glm_grouped)\n\n\n\n\n\n3.6.11 Individual observations\n\n\nShow R code\nbeetles_long\n\n\n\n\nTable 3.3: beetles data in long format\n\n\n\n\n \n\n\n\n\n\n\n\n\n\nShow R code\nbeetles_glm_ungrouped = \n beetles_long |> \n glm(\n formula = outcome ~ dose, \n family = \"binomial\")\n\nbeetles_glm_ungrouped |> parameters() |> print_md()\n\n\n\n\nTable 3.4: logistic regression model for beetles data with individual Bernoulli data\n\n\n\n\n\n\nParameter\nLog-Odds\nSE\n95% CI\nz\np\n\n\n\n\n(Intercept)\n-60.72\n5.18\n(-71.44, -51.08)\n-11.72\n< .001\n\n\ndose\n34.27\n2.91\n(28.85, 40.30)\n11.77\n< .001\n\n\n\n\n\n\n\n\n\nHere’s the previous version again:\n\n\n\nShow R code\nbeetles_glm_grouped |> parameters() |> print_md()\n\n\n\n\nTable 3.5: logistic regression model for beetles data with grouped (binomial) data\n\n\n\n\n\n\nParameter\nLog-Odds\nSE\n95% CI\nz\np\n\n\n\n\n(Intercept)\n-60.72\n5.18\n(-71.44, -51.08)\n-11.72\n< .001\n\n\ndose\n34.27\n2.91\n(28.85, 40.30)\n11.77\n< .001\n\n\n\n\n\n\n\n\n\n\nThey seem the same! But not quite:\n\n\n\nShow R code\n\nlogLik(beetles_glm_grouped)\n#> 'log Lik.' -18.72 (df=2)\nlogLik(beetles_glm_ungrouped)\n#> 'log Lik.' -186.2 (df=2)\n\n\n\nThe difference is due to the binomial coefficient \\(\\left(n\\atop x \\right)\\) which isn’t included in the individual-observations (Bernoulli) version of the model.", "crumbs": [ "Generalized Linear Models", "3  Models for Binary Outcomes" @@ -336,7 +336,7 @@ "href": "logistic-regression.html#sec-gof", "title": "3  Models for Binary Outcomes", "section": "3.9 Model comparisons for logistic models", - "text": "3.9 Model comparisons for logistic models\n\n3.9.1 Deviance test\nWe can compare the maximized log-likelihood of our model, \\(\\ell(\\hat\\beta; \\mathbf x)\\), versus the log-likelihood of the full model (aka saturated model aka maximal model), \\(\\ell_{\\text{full}}\\), which has one parameter per covariate pattern. With enough data, \\(2(\\ell_{\\text{full}} - \\ell(\\hat\\beta; \\mathbf x)) \\dot \\sim \\chi^2(N - p)\\), where \\(N\\) is the number of distinct covariate patterns and \\(p\\) is the number of \\(\\beta\\) parameters in our model. A significant p-value for this deviance statistic indicates that there’s some detectable pattern in the data that our model isn’t flexible enough to catch.\n\n\n\n\n\n\nCaution\n\n\n\nThe deviance statistic needs to have a large amount of data for each covariate pattern for the \\(\\chi^2\\) approximation to hold. A guideline from Dobson is that if there are \\(q\\) distinct covariate patterns \\(x_1...,x_q\\), with \\(n_1,...,n_q\\) observations per pattern, then the expected frequencies \\(n_k \\cdot \\pi(x_k)\\) should be at least 1 for every pattern \\(k\\in 1:q\\).\n\n\nIf you have covariates measured on a continuous scale, you may not be able to use the deviance tests to assess goodness of fit.\n\n\n3.9.2 Hosmer-Lemeshow test\nIf our covariate patterns produce groups that are too small, a reasonable solution is to make bigger groups by merging some of the covariate-pattern groups together.\nHosmer and Lemeshow (1980) proposed that we group the patterns by their predicted probabilities according to the model of interest. For example, you could group all of the observations with predicted probabilities of 10% or less together, then group the observations with 11%-20% probability together, and so on; \\(g=10\\) categories in all.\nThen we can construct a statistic \\[X^2 = \\sum_{c=1}^g \\frac{(o_c - e_c)^2}{e_c}\\] where \\(o_c\\) is the number of events observed in group \\(c\\), and \\(e_c\\) is the number of events expected in group \\(c\\) (based on the sum of the fitted values \\(\\hat\\pi_i\\) for observations in group \\(c\\)).\nIf each group has enough observations in it, you can compare \\(X^2\\) to a \\(\\chi^2\\) distribution; by simulation, the degrees of freedom has been found to be approximately \\(g-2\\).\nFor our CHD model, this procedure would be:\n\n\nShow R code\nwcgs = \n wcgs |> \n mutate(\n pred_probs_glm1 = chd_glm_strat |> fitted(),\n pred_prob_cats1 = \n pred_probs_glm1 |> \n cut(breaks = seq(0, 1, by = .1), \n include.lowest = TRUE))\n\nHL_table = \n wcgs |> \n summarize(\n .by = pred_prob_cats1,\n n = n(),\n o = sum(chd69 == \"Yes\"),\n e = sum(pred_probs_glm1)\n )\n\nlibrary(pander)\nHL_table |> pander()\n\n\n\n\n\n\n\n\n\n\n\npred_prob_cats1\nn\no\ne\n\n\n\n\n(0.1,0.2]\n785\n116\n108\n\n\n(0.2,0.3]\n64\n12\n13.77\n\n\n[0,0.1]\n2305\n129\n135.2\n\n\n\n\n\nShow R code\n\nX2 = HL_table |> \n summarize(\n `X^2` = sum((o-e)^2/e)\n ) |> \n pull(`X^2`)\nprint(X2)\n#> [1] 1.1102871\n\npval1 = pchisq(X2, lower = FALSE, df = nrow(HL_table) - 2)\n\n\nOur statistic is \\(X^2 = 1.11028711\\); \\(p(\\chi^2(1) > 1.11028711) = 0.29201955\\), which is our p-value for detecting a lack of goodness of fit.\nUnfortunately that grouping plan left us with just three categories with any observations, so instead of grouping by 10% increments of predicted probability, typically analysts use deciles of the predicted probabilities:\n\n\nShow R code\nwcgs = \n wcgs |> \n mutate(\n pred_probs_glm1 = chd_glm_strat |> fitted(),\n pred_prob_cats1 = \n pred_probs_glm1 |> \n cut(breaks = quantile(pred_probs_glm1, seq(0, 1, by = .1)), \n include.lowest = TRUE))\n\nHL_table = \n wcgs |> \n summarize(\n .by = pred_prob_cats1,\n n = n(),\n o = sum(chd69 == \"Yes\"),\n e = sum(pred_probs_glm1)\n )\n\nHL_table |> pander()\n\n\n\n\n\n\n\n\n\n\n\npred_prob_cats1\nn\no\ne\n\n\n\n\n(0.114,0.147]\n275\n48\n36.81\n\n\n(0.147,0.222]\n314\n51\n57.19\n\n\n(0.0774,0.0942]\n371\n27\n32.56\n\n\n(0.0942,0.114]\n282\n30\n29.89\n\n\n(0.0633,0.069]\n237\n17\n15.97\n\n\n(0.069,0.0774]\n306\n20\n22.95\n\n\n(0.0487,0.0633]\n413\n27\n24.1\n\n\n(0.0409,0.0487]\n310\n14\n14.15\n\n\n[0.0322,0.0363]\n407\n16\n13.91\n\n\n(0.0363,0.0409]\n239\n7\n9.48\n\n\n\n\n\nShow R code\n\nX2 = HL_table |> \n summarize(\n `X^2` = sum((o-e)^2/e)\n ) |> \n pull(`X^2`)\n\nprint(X2)\n#> [1] 6.7811383\n\npval1 = pchisq(X2, lower = FALSE, df = nrow(HL_table) - 2)\n\n\nNow we have more evenly split categories. The p-value is \\(0.56041994\\), still not significant.\nGraphically, we have compared:\n\n\nShow R code\n\nHL_plot = \n HL_table |> \n ggplot(aes(x = pred_prob_cats1)) + \n geom_line(aes(y = e, x = pred_prob_cats1, group = \"Expected\", col = \"Expected\")) +\n geom_point(aes(y = e, size = n, col = \"Expected\")) +\n geom_point(aes(y = o, size = n, col = \"Observed\")) +\n geom_line(aes(y = o, col = \"Observed\", group = \"Observed\")) +\n scale_size(range = c(1,4)) +\n theme_bw() +\n ylab(\"number of CHD events\") +\n theme(axis.text.x = element_text(angle = 45))\n\n\n\n\nShow R code\nggplotly(HL_plot)\n\n\n\n\n\n\n\n\n3.9.3 Comparing models\n\nAIC = \\(-2 * \\ell(\\hat\\theta) + 2 * p\\) [lower is better]\nBIC = \\(-2 * \\ell(\\hat\\theta) + p * \\text{log}(n)\\) [lower is better]\nlikelihood ratio [higher is better]", + "text": "3.9 Model comparisons for logistic models\n\n3.9.1 Deviance test\nWe can compare the maximized log-likelihood of our model, \\(\\ell(\\hat\\beta; \\mathbf x)\\), versus the log-likelihood of the full model (aka saturated model aka maximal model), \\(\\ell_{\\text{full}}\\), which has one parameter per covariate pattern. With enough data, \\(2(\\ell_{\\text{full}} - \\ell(\\hat\\beta; \\mathbf x)) \\dot \\sim \\chi^2(N - p)\\), where \\(N\\) is the number of distinct covariate patterns and \\(p\\) is the number of \\(\\beta\\) parameters in our model. A significant p-value for this deviance statistic indicates that there’s some detectable pattern in the data that our model isn’t flexible enough to catch.\n\n\n\n\n\n\nCaution\n\n\n\nThe deviance statistic needs to have a large amount of data for each covariate pattern for the \\(\\chi^2\\) approximation to hold. A guideline from Dobson is that if there are \\(q\\) distinct covariate patterns \\(x_1...,x_q\\), with \\(n_1,...,n_q\\) observations per pattern, then the expected frequencies \\(n_k \\cdot \\pi(x_k)\\) should be at least 1 for every pattern \\(k\\in 1:q\\).\n\n\nIf you have covariates measured on a continuous scale, you may not be able to use the deviance tests to assess goodness of fit.\n\n\n3.9.2 Hosmer-Lemeshow test\nIf our covariate patterns produce groups that are too small, a reasonable solution is to make bigger groups by merging some of the covariate-pattern groups together.\nHosmer and Lemeshow (1980) proposed that we group the patterns by their predicted probabilities according to the model of interest. For example, you could group all of the observations with predicted probabilities of 10% or less together, then group the observations with 11%-20% probability together, and so on; \\(g=10\\) categories in all.\nThen we can construct a statistic \\[X^2 = \\sum_{c=1}^g \\frac{(o_c - e_c)^2}{e_c}\\] where \\(o_c\\) is the number of events observed in group \\(c\\), and \\(e_c\\) is the number of events expected in group \\(c\\) (based on the sum of the fitted values \\(\\hat\\pi_i\\) for observations in group \\(c\\)).\nIf each group has enough observations in it, you can compare \\(X^2\\) to a \\(\\chi^2\\) distribution; by simulation, the degrees of freedom has been found to be approximately \\(g-2\\).\nFor our CHD model, this procedure would be:\n\n\nShow R code\nwcgs = \n wcgs |> \n mutate(\n pred_probs_glm1 = chd_glm_strat |> fitted(),\n pred_prob_cats1 = \n pred_probs_glm1 |> \n cut(breaks = seq(0, 1, by = .1), \n include.lowest = TRUE))\n\nHL_table = \n wcgs |> \n summarize(\n .by = pred_prob_cats1,\n n = n(),\n o = sum(chd69 == \"Yes\"),\n e = sum(pred_probs_glm1)\n )\n\nlibrary(pander)\nHL_table |> pander()\n\n\n\n\n\n\n\n\n\n\n\npred_prob_cats1\nn\no\ne\n\n\n\n\n(0.1,0.2]\n785\n116\n108\n\n\n(0.2,0.3]\n64\n12\n13.77\n\n\n[0,0.1]\n2,305\n129\n135.2\n\n\n\n\n\nShow R code\n\nX2 = HL_table |> \n summarize(\n `X^2` = sum((o-e)^2/e)\n ) |> \n pull(`X^2`)\nprint(X2)\n#> [1] 1.1102871\n\npval1 = pchisq(X2, lower = FALSE, df = nrow(HL_table) - 2)\n\n\nOur statistic is \\(X^2 = 1.11028711\\); \\(p(\\chi^2(1) > 1.11028711) = 0.29201955\\), which is our p-value for detecting a lack of goodness of fit.\nUnfortunately that grouping plan left us with just three categories with any observations, so instead of grouping by 10% increments of predicted probability, typically analysts use deciles of the predicted probabilities:\n\n\nShow R code\nwcgs = \n wcgs |> \n mutate(\n pred_probs_glm1 = chd_glm_strat |> fitted(),\n pred_prob_cats1 = \n pred_probs_glm1 |> \n cut(breaks = quantile(pred_probs_glm1, seq(0, 1, by = .1)), \n include.lowest = TRUE))\n\nHL_table = \n wcgs |> \n summarize(\n .by = pred_prob_cats1,\n n = n(),\n o = sum(chd69 == \"Yes\"),\n e = sum(pred_probs_glm1)\n )\n\nHL_table |> pander()\n\n\n\n\n\n\n\n\n\n\n\npred_prob_cats1\nn\no\ne\n\n\n\n\n(0.114,0.147]\n275\n48\n36.81\n\n\n(0.147,0.222]\n314\n51\n57.19\n\n\n(0.0774,0.0942]\n371\n27\n32.56\n\n\n(0.0942,0.114]\n282\n30\n29.89\n\n\n(0.0633,0.069]\n237\n17\n15.97\n\n\n(0.069,0.0774]\n306\n20\n22.95\n\n\n(0.0487,0.0633]\n413\n27\n24.1\n\n\n(0.0409,0.0487]\n310\n14\n14.15\n\n\n[0.0322,0.0363]\n407\n16\n13.91\n\n\n(0.0363,0.0409]\n239\n7\n9.48\n\n\n\n\n\nShow R code\n\nX2 = HL_table |> \n summarize(\n `X^2` = sum((o-e)^2/e)\n ) |> \n pull(`X^2`)\n\nprint(X2)\n#> [1] 6.7811383\n\npval1 = pchisq(X2, lower = FALSE, df = nrow(HL_table) - 2)\n\n\nNow we have more evenly split categories. The p-value is \\(0.56041994\\), still not significant.\nGraphically, we have compared:\n\n\nShow R code\n\nHL_plot = \n HL_table |> \n ggplot(aes(x = pred_prob_cats1)) + \n geom_line(aes(y = e, x = pred_prob_cats1, group = \"Expected\", col = \"Expected\")) +\n geom_point(aes(y = e, size = n, col = \"Expected\")) +\n geom_point(aes(y = o, size = n, col = \"Observed\")) +\n geom_line(aes(y = o, col = \"Observed\", group = \"Observed\")) +\n scale_size(range = c(1,4)) +\n theme_bw() +\n ylab(\"number of CHD events\") +\n theme(axis.text.x = element_text(angle = 45))\n\n\n\n\nShow R code\nggplotly(HL_plot)\n\n\n\n\n\n\n\n\n3.9.3 Comparing models\n\nAIC = \\(-2 * \\ell(\\hat\\theta) + 2 * p\\) [lower is better]\nBIC = \\(-2 * \\ell(\\hat\\theta) + p * \\text{log}(n)\\) [lower is better]\nlikelihood ratio [higher is better]", "crumbs": [ "Generalized Linear Models", "3  Models for Binary Outcomes" @@ -479,7 +479,18 @@ "href": "count-regression.html#example-needle-sharing", "title": "\n4  Models for Count Outcomes\n", "section": "\n4.7 Example: needle-sharing", - "text": "4.7 Example: needle-sharing\n(adapted from Vittinghoff et al. (2012), §8)\n\nShow R codelibrary(tidyverse)\nlibrary(haven)\nneedles = read_dta(\"inst/extdata/needle_sharing.dta\") |> \n as_tibble() |> \n mutate(polydrug = \n ifelse(polydrug, \"multiple drugs used\", \"one drug used\") |> \n factor() |> \n relevel(ref = \"one drug used\"),\n homeless = ifelse(homeless, \"homeless\", \"not homeless\") |> \n factor() |> relevel(ref = \"not homeless\"))\nneedles\n\n\nTable 4.1: Needle-sharing data\n\n\n\n \n\n\n\n\n\n\n\n\nShow R codelibrary(ggplot2)\n\nneedles |> \n ggplot(\n aes(\n x = age,\n y = shared_syr,\n col = sex,\n shape = ethn\n )\n ) + \n geom_point(\n size = 4, \n alpha = .5) +\n facet_grid(\n cols = vars(polydrug), \n rows = vars(homeless)) +\n theme(legend.position = \"bottom\")\n\n\n\nFigure 4.1: Rates of needle sharing", + "text": "4.7 Example: needle-sharing\n(adapted from Vittinghoff et al. (2012), §8)\n\nShow R codelibrary(tidyverse)\nlibrary(haven)\nneedles = read_dta(\"inst/extdata/needle_sharing.dta\") |> \n as_tibble() |> \n mutate(polydrug = \n ifelse(polydrug, \"multiple drugs used\", \"one drug used\") |> \n factor() |> \n relevel(ref = \"one drug used\"),\n homeless = ifelse(homeless, \"homeless\", \"not homeless\") |> \n factor() |> relevel(ref = \"not homeless\")) |> \n mutate(sex = factor(sex) |> relevel(ref = \"M\"))\nneedles\n\n\nTable 4.1: Needle-sharing data\n\n\n\n \n\n\n\n\n\n\n\n\nShow R codelibrary(ggplot2)\n\nneedles |> \n ggplot(\n aes(\n x = age,\n y = shared_syr,\n shape = sex,\n col = ethn\n )\n ) + \n geom_point(\n size = 3, \n alpha = .5) +\n facet_grid(\n cols = vars(polydrug), \n rows = vars(homeless)) +\n theme(legend.position = \"bottom\")\n\n\n\nFigure 4.1: Rates of needle sharing", + "crumbs": [ + "Generalized Linear Models", + "4  Models for Count Outcomes" + ] + }, + { + "objectID": "count-regression.html#covariate-counts", + "href": "count-regression.html#covariate-counts", + "title": "\n4  Models for Count Outcomes\n", + "section": "\n4.8 Covariate counts:", + "text": "4.8 Covariate counts:\n\nShow R codeneedles |> \n dplyr::select(sex, homeless, polydrug) |> \n summary()\n#> sex homeless polydrug \n#> M :97 not homeless:63 one drug used :109 \n#> F :30 homeless :61 multiple drugs used: 19 \n#> Trans: 1 NA's : 4\n\n\nThere’s only one individual with sex = Trans, which unfortunately isn’t enough data to analyze. We will remove that individual:\n\nShow R code\nneedles = needles |> filter(sex != \"Trans\")", "crumbs": [ "Generalized Linear Models", "4  Models for Count Outcomes" @@ -489,8 +500,8 @@ "objectID": "count-regression.html#model", "href": "count-regression.html#model", "title": "\n4  Models for Count Outcomes\n", - "section": "\n4.8 model", - "text": "4.8 model\n\nShow R codeglm1 = glm(\n data = needles,\n family = stats::poisson,\n shared_syr ~ age + sex + homeless*polydrug\n)\n\nlibrary(parameters)\nglm1 |> parameters(exponentiate = TRUE) |> \n print_md()\n\n\nTable 4.2: Poisson model for needle-sharing data\n\n\n\n\n\n\n\n\n\n\n\n\nParameter\nIRR\nSE\n95% CI\nz\np\n\n\n\n(Intercept)\n8.95\n2.26\n(5.46, 14.68)\n8.69\n< .001\n\n\nage\n0.97\n5.58e-03\n(0.96, 0.98)\n-5.41\n< .001\n\n\nsex (M)\n0.50\n0.06\n(0.40, 0.63)\n-5.88\n< .001\n\n\nsex (Trans)\n7.99e-08\n1.02e-04\n(0.00, Inf)\n-0.01\n0.990\n\n\nhomeless (homeless)\n3.58\n0.45\n(2.79, 4.59)\n10.06\n< .001\n\n\npolydrug (multiple drugs used)\n1.45e-07\n5.82e-05\n(0.00, Inf)\n-0.04\n0.969\n\n\nhomeless (homeless) × polydrug (multiple drugs used)\n1.27e+06\n5.12e+08\n(0.00, Inf)\n0.03\n0.972\n\n\n\n\n\n\n\n\n\nShow R codelibrary(ggfortify)\nautoplot(glm1)\n\n\nTable 4.3: Diagnostics for Poisson model\n\n\n\n\n\n\n\n\n\n\n\n–\n\n\nTable 4.4: Negative binomial model for needle-sharing data\n\nShow R codelibrary(MASS) #need this for glm.nb()\nglm1.nb = glm.nb(\n data = needles,\n shared_syr ~ age + sex + homeless*polydrug\n)\nsummary(glm1.nb)\n#> \n#> Call:\n#> glm.nb(formula = shared_syr ~ age + sex + homeless * polydrug, \n#> data = needles, init.theta = 0.08436295825, link = log)\n#> \n#> Coefficients:\n#> Estimate Std. Error z value\n#> (Intercept) 2.0557e+00 1.7133e+00 1.1999\n#> age -2.7630e-02 3.8153e-02 -0.7242\n#> sexM -1.0648e+00 8.0684e-01 -1.3197\n#> sexTrans -2.5336e+01 1.1484e+05 -0.0002\n#> homelesshomeless 1.6546e+00 7.2207e-01 2.2916\n#> polydrugmultiple drugs used -2.4562e+01 3.6055e+04 -0.0007\n#> homelesshomeless:polydrugmultiple drugs used 2.3178e+01 3.6055e+04 0.0006\n#> Pr(>|z|) \n#> (Intercept) 0.23019 \n#> age 0.46895 \n#> sexM 0.18694 \n#> sexTrans 0.99982 \n#> homelesshomeless 0.02193 *\n#> polydrugmultiple drugs used 0.99946 \n#> homelesshomeless:polydrugmultiple drugs used 0.99949 \n#> ---\n#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n#> \n#> (Dispersion parameter for Negative Binomial(0.0844) family taken to be 1)\n#> \n#> Null deviance: 69.8027 on 120 degrees of freedom\n#> Residual deviance: 57.7820 on 114 degrees of freedom\n#> (7 observations deleted due to missingness)\n#> AIC: 317.506\n#> \n#> Number of Fisher Scoring iterations: 1\n#> \n#> \n#> Theta: 0.0844 \n#> Std. Err.: 0.0197 \n#> \n#> 2 x log-likelihood: -301.5060\n\n\n\n\n\nShow R codetibble(name = names(coef(glm1)), poisson = coef(glm1), nb = coef(glm1.nb))\n\n\nTable 4.5: Poisson versus Negative Binomial Regression coefficient estimates\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\nDobson, Annette J, and Adrian G Barnett. 2018. An Introduction to Generalized Linear Models. 4th ed. CRC press. https://doi.org/10.1201/9781315182780.\n\n\nVittinghoff, Eric, David V Glidden, Stephen C Shiboski, and Charles E McCulloch. 2012. Regression Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Measures Models. 2nd ed. Springer. https://doi.org/10.1007/978-1-4614-1353-0.", + "section": "\n4.9 model", + "text": "4.9 model\n\nShow R codeglm1 = glm(\n data = needles,\n family = stats::poisson,\n shared_syr ~ age + sex + homeless*polydrug\n)\n\nlibrary(parameters)\nglm1 |> parameters(exponentiate = TRUE) |> \n print_md()\n\n\nTable 4.2: Poisson model for needle-sharing data\n\n\n\n\n\n\n\n\n\n\n\n\nParameter\nIRR\nSE\n95% CI\nz\np\n\n\n\n(Intercept)\n4.52\n1.15\n(2.74, 7.45)\n5.90\n< .001\n\n\nage\n0.97\n5.58e-03\n(0.96, 0.98)\n-5.41\n< .001\n\n\nsex (F)\n1.98\n0.23\n(1.58, 2.49)\n5.88\n< .001\n\n\nhomeless (homeless)\n3.58\n0.45\n(2.79, 4.59)\n10.06\n< .001\n\n\npolydrug (multiple drugs used)\n1.45e-07\n5.82e-05\n(0.00, Inf)\n-0.04\n0.969\n\n\nhomeless (homeless) × polydrug (multiple drugs used)\n1.27e+06\n5.12e+08\n(0.00, Inf)\n0.03\n0.972\n\n\n\n\n\n\n\n\n\nShow R codelibrary(ggfortify)\nautoplot(glm1)\n\n\nTable 4.3: Diagnostics for Poisson model\n\n\n\n\n\n\n\n\n\n\n\n–\n\n\nTable 4.4: Negative binomial model for needle-sharing data\n\nShow R codelibrary(MASS) #need this for glm.nb()\nglm1.nb = glm.nb(\n data = needles,\n shared_syr ~ age + sex + homeless*polydrug\n)\nsummary(glm1.nb)\n#> \n#> Call:\n#> glm.nb(formula = shared_syr ~ age + sex + homeless * polydrug, \n#> data = needles, init.theta = 0.08436295825, link = log)\n#> \n#> Coefficients:\n#> Estimate Std. Error z value\n#> (Intercept) 9.91e-01 1.71e+00 0.58\n#> age -2.76e-02 3.82e-02 -0.72\n#> sexF 1.06e+00 8.07e-01 1.32\n#> homelesshomeless 1.65e+00 7.22e-01 2.29\n#> polydrugmultiple drugs used -2.46e+01 3.61e+04 0.00\n#> homelesshomeless:polydrugmultiple drugs used 2.32e+01 3.61e+04 0.00\n#> Pr(>|z|) \n#> (Intercept) 0.563 \n#> age 0.469 \n#> sexF 0.187 \n#> homelesshomeless 0.022 *\n#> polydrugmultiple drugs used 0.999 \n#> homelesshomeless:polydrugmultiple drugs used 0.999 \n#> ---\n#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n#> \n#> (Dispersion parameter for Negative Binomial(0.0844) family taken to be 1)\n#> \n#> Null deviance: 69.193 on 119 degrees of freedom\n#> Residual deviance: 57.782 on 114 degrees of freedom\n#> (7 observations deleted due to missingness)\n#> AIC: 315.5\n#> \n#> Number of Fisher Scoring iterations: 1\n#> \n#> \n#> Theta: 0.0844 \n#> Std. Err.: 0.0197 \n#> \n#> 2 x log-likelihood: -301.5060\n\n\n\n\n\nShow R codetibble(name = names(coef(glm1)), poisson = coef(glm1), nb = coef(glm1.nb))\n\n\nTable 4.5: Poisson versus Negative Binomial Regression coefficient estimates\n\n\n\n \n\n\n\n\n\n\nzero-inflation\n\nShow R codelibrary(glmmTMB)\nzinf_fit1 = glmmTMB(\n family = \"poisson\",\n data = needles,\n formula = shared_syr ~ age + sex + homeless*polydrug,\n ziformula = ~ age + sex + homeless + polydrug # fit won't converge with interaction\n)\n\nzinf_fit1 |> \n parameters(exponentiate = TRUE) |> \n print_md()\n\n\nTable 4.6: Zero-inflated poisson model\n\n\n\n# Fixed Effects\n\n\n\n\n\n\n\n\n\nParameter\nIRR\nSE\n95% CI\nz\np\n\n\n\n(Intercept)\n3.16\n0.82\n(1.90, 5.25)\n4.44\n< .001\n\n\nage\n1.01\n5.88e-03\n(1.00, 1.02)\n1.74\n0.081\n\n\nsex [F]\n3.43\n0.44\n(2.67, 4.40)\n9.68\n< .001\n\n\nhomeless [homeless]\n3.44\n0.47\n(2.63, 4.50)\n9.03\n< .001\n\n\npolydrug [multiple drugs used]\n1.85e-09\n1.21e-05\n(0.00, Inf)\n-3.08e-03\n0.998\n\n\nhomeless [homeless] × polydrug [multiple drugs used]\n1.38e+08\n9.04e+11\n(0.00, Inf)\n2.87e-03\n0.998\n\n\n\n\n# Zero-Inflation\n\n\n\n\n\n\n\n\n\nParameter\nOdds Ratio\nSE\n95% CI\nz\np\n\n\n\n(Intercept)\n0.49\n0.54\n(0.06, 4.25)\n-0.65\n0.514\n\n\nage\n1.05\n0.03\n(1.00, 1.10)\n1.95\n0.051\n\n\nsex [F]\n1.44\n0.84\n(0.46, 4.50)\n0.62\n0.533\n\n\nhomeless [homeless]\n0.68\n0.34\n(0.26, 1.80)\n-0.78\n0.436\n\n\npolydrug [multiple drugs used]\n1.15\n0.91\n(0.24, 5.43)\n0.18\n0.858\n\n\n\n\n\n\n\n\nzero-inflated negative binomial model\n\nShow R codelibrary(glmmTMB)\nzinf_fit1 = glmmTMB(\n family = nbinom2,\n data = needles,\n formula = shared_syr ~ age + sex + homeless*polydrug,\n ziformula = ~ age + sex + homeless + polydrug # fit won't converge with interaction\n)\n\nzinf_fit1 |> \n parameters(exponentiate = TRUE) |> \n print_md()\n\n\nTable 4.7: Zero-inflated negative binomial model\n\n\n\n# Fixed Effects\n\n\n\n\n\n\n\n\n\nParameter\nIRR\nSE\n95% CI\nz\np\n\n\n\n(Intercept)\n1.06\n1.48\n(0.07, 16.52)\n0.04\n0.969\n\n\nage\n1.02\n0.03\n(0.96, 1.08)\n0.53\n0.599\n\n\nsex [F]\n6.86\n6.36\n(1.12, 42.16)\n2.08\n0.038\n\n\nhomeless [homeless]\n6.44\n4.59\n(1.60, 26.01)\n2.62\n0.009\n\n\npolydrug [multiple drugs used]\n8.25e-10\n7.07e-06\n(0.00, Inf)\n-2.44e-03\n0.998\n\n\nhomeless [homeless] × polydrug [multiple drugs used]\n2.36e+08\n2.02e+12\n(0.00, Inf)\n2.25e-03\n0.998\n\n\n\n\n# Zero-Inflation\n\n\n\n\n\n\n\n\n\nParameter\nOdds Ratio\nSE\n95% CI\nz\np\n\n\n\n(Intercept)\n0.10\n0.20\n(1.47e-03, 6.14)\n-1.11\n0.269\n\n\nage\n1.07\n0.04\n(0.99, 1.15)\n1.78\n0.075\n\n\nsex [F]\n2.72\n2.40\n(0.48, 15.33)\n1.13\n0.258\n\n\nhomeless [homeless]\n1.15\n0.86\n(0.27, 4.96)\n0.19\n0.853\n\n\npolydrug [multiple drugs used]\n0.75\n0.86\n(0.08, 7.12)\n-0.25\n0.799\n\n\n\n\n# Dispersion\n\nParameter\nCoefficient\n95% CI\n\n\n(Intercept)\n0.44\n(0.11, 1.71)\n\n\n\n\n\n\n\n\n\n\n\n\n\nDobson, Annette J, and Adrian G Barnett. 2018. An Introduction to Generalized Linear Models. 4th ed. CRC press. https://doi.org/10.1201/9781315182780.\n\n\nVittinghoff, Eric, David V Glidden, Stephen C Shiboski, and Charles E McCulloch. 2012. Regression Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Measures Models. 2nd ed. Springer. https://doi.org/10.1007/978-1-4614-1353-0.", "crumbs": [ "Generalized Linear Models", "4  Models for Count Outcomes" @@ -544,7 +555,7 @@ "href": "intro-to-survival-analysis.html#distribution-functions-for-time-to-event-variables", "title": "\n5  Introduction to Survival Analysis\n", "section": "\n5.3 Distribution functions for time-to-event variables", - "text": "5.3 Distribution functions for time-to-event variables\n\n5.3.1 The Probability Density Function (PDF)\nFor a time-to-event variable \\(T\\) with a continuous distribution, the probability density function is defined as usual:\n\\[f(t)\\stackrel{\\text{def}}{=}p(t) \\stackrel{\\text{def}}{=}p(T=t)\\]\nTypically, this density is assumed to be 0 for all \\(t<0\\); that is, \\(f(t) = 0, \\forall t<0\\). In other words, the range of \\(T\\) is typically \\([0,\\infty)\\).\n\nExample 5.1 (exponential distribution) Recall from Epi 202: the pdf of the exponential distribution family of models is:\n\\[p(T=t) = \\mathbb{1}_{t \\ge 0} \\cdot \\lambda \\text{e}^{-\\lambda t}\\]\nwhere \\(\\lambda > 0\\).\nHere are some examples of exponential pdfs:\n\n\n\n\n\n\n\n\n\n\n5.3.2 The Cumulative Distribution Function (CDF)\nThe cumulative distribution function is defined as:\n\\[\n\\begin{aligned}\nF(t) &\\stackrel{\\text{def}}{=}\\Pr(T \\le t)\\\\\n&=\\int_{u=0}^t f(u) du\n\\end{aligned}\n\\]\n\nExample 5.2 (exponential distribution) Recall from Epi 202: the cdf of the exponential distribution family of models is:\n\\[\nP(T\\le t) = \\mathbb{1}_{t \\ge 0} \\cdot (1- \\text{e}^{-\\lambda t})\n\\] where \\(\\lambda > 0\\).\nHere are some examples of exponential cdfs:\n\n\n\n\n\n\n\n\n\n\n5.3.3 The Survival Function\nFor survival data, a more important quantity is the survival function:\n\\[\n\\begin{aligned}\nS(t) &\\stackrel{\\text{def}}{=}\\Pr(T > t)\\\\\n&=\\int_{u=t}^\\infty p(u) du\\\\\n&=1-F(t)\\\\\n\\end{aligned}\n\\]\nThe survival function \\(S(t)\\) is the probability that the event time is later than \\(t\\). If the event in a clinical trial is death, then \\(S(t)\\) is the expected fraction of the original population at time 0 who have survived up to time \\(t\\) and are still alive at time \\(t\\); that is, if \\(X_t\\) represents survival status at time \\(t\\), with \\(X_t = 1\\) denoting alive at time \\(t\\) and \\(X_t = 0\\) denoting deceased at time \\(t\\), then:\n\\[S(t) = \\mathbb{E}\\left[X_t\\right]\\]\n\nExample 5.3 (exponential distribution) Since \\(S(t) = 1 - F(t)\\), the survival function of the exponential distribution family of models is:\n\\[\nP(T> t) = \\left\\{ {{\\text{e}^{-\\lambda t}, t\\ge0} \\atop {1, t \\le 0}}\\right.\n\\] where \\(\\lambda > 0\\).\nHere are some examples of exponential pdfs:\n\n\n\n\n\n\n\n\n\n\n5.3.4 The Hazard Function\nAnother important quantity is the hazard function:\n\nDefinition 5.1 (Hazard function) The hazard function for a random variable \\(T\\) at value \\(t\\) is the conditional density of \\(T\\) at \\(t\\), given \\(T\\ge t\\); that is:\n\\[h(t) \\stackrel{\\text{def}}{=}p(T=t|T\\ge t)\\]\nIf \\(T\\) represents the time at which an event occurs, then \\(h(t)\\) is the probability that the event occurs at time \\(t\\), given that it has not occurred prior to time \\(t\\).\n\nThe hazard function has an important relationship to the density and survival functions, which we can use to derive the hazard function for a given probability distribution.\n\nTheorem 5.1 \\[h(t)=\\frac{f(t)}{S(t)}\\]\n\nProof. \n\nLemma 5.1 (Joint probability of a variable with itself) \\[p(T=t, T\\ge t) = p(T=t)\\]\n\nProof. Recall from Epi 202: if \\(A\\) and \\(B\\) are statistical events and \\(A\\subseteq B\\), then \\(p(A, B) = p(A)\\). In particular, \\(\\{T=t\\} \\subseteq \\{T\\geq t\\}\\), so \\(p(T=t, T\\ge t) = p(T=t)\\).\n\n\nHence:\n\\[\n\\begin{aligned}\nh(t) &=p(T=t|T\\ge t)\\\\\n&=\\frac{p(T=t, T\\ge t)}{p(T \\ge t)}\\\\\n&=\\frac{p(T=t)}{p(T \\ge t)}\\\\\n&=\\frac{f(t)}{S(t)}\n\\end{aligned}\n\\]\n\n\n\nExample 5.4 (exponential distribution) The hazard function of the exponential distribution family of models is:\n\\[\n\\begin{aligned}\nP(T=t|T \\ge t)\n&= \\frac{f(t)}{S(t)}\\\\\n&= \\frac{\\mathbb{1}_{t \\ge 0}\\cdot \\lambda \\text{e}^{-\\lambda t}}{\\text{e}^{-\\lambda t}}\\\\\n&=\\mathbb{1}_{t \\ge 0}\\cdot \\lambda\n\\end{aligned}\n\\]\nFigure 5.1 shows some examples of exponential hazard functions:\n\n\n\n\nFigure 5.1: Examples of hazard functions for exponential distributions\n\n\n\n\n\n\n\n\nWe can also view the hazard function as the derivative of the negative of the logarithm of the survival function:\n\nTheorem 5.2 \\[h(t) = \\frac{\\partial}{\\partial t}\\left\\{-\\text{log}\\left\\{S(t)\\right\\}\\right\\}\\]\n\nProof. \\[\n\\begin{aligned}\nh(t)\n&= \\frac{f(t)}{S(t)}\\\\\n&= \\frac{-S'(t)}{S(t)}\\\\\n&= -\\frac{S'(t)}{S(t)}\\\\\n&=-\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\}\\\\\n&=\\frac{\\partial}{\\partial t}\\left\\{-\\text{log}\\left\\{S(t)\\right\\}\\right\\}\n\\end{aligned}\n\\]\n\n\n\n5.3.5 The Cumulative Hazard Function\nSince \\(h(t) = \\frac{\\partial}{\\partial t}\\left\\{-\\text{log}\\left\\{S(t)\\right\\}\\right\\}\\) (see Theorem 5.2), we also have:\n\nCorollary 5.1 \\[S(t) = \\text{exp}\\left\\{-\\int_{u=0}^t h(u)du\\right\\} \\tag{5.1}\\]\n\nThe integral in Equation 5.1 is important enough to have its own name: cumulative hazard.\n\nDefinition 5.2 (cumulative hazard) The cumulative hazard function \\(H(t)\\) is defined as:\n\\[H(t) \\stackrel{\\text{def}}{=}\\int_{u=0}^t h(u) du\\]\n\nAs we will see below, \\(H(t)\\) is tractable to estimate, and we can then derive an estimate of the hazard function using an approximate derivative of the estimated cumulative hazard.\n\nExample 5.5 The cumulative hazard function of the exponential distribution family of models is:\n\\[\nH(t) = \\mathbb{1}_{t \\ge 0}\\cdot \\lambda t\n\\]\nHere are some examples of exponential cumulative hazard functions:\n\n\n\n\n\n\n\n\n\n\n5.3.6 Some Key Mathematical Relationships among Survival Concepts\nDiagram:\n\\[\nh(t) \\xrightarrow[]{\\int_{u=0}^t h(u)du} H(t)\n\\xrightarrow[]{\\text{exp}\\left\\{-H(t)\\right\\}} S(t)\n\\xrightarrow[]{1-S(t)} F(t)\n\\]\n\\[\nh(t) \\xleftarrow[\\frac{\\partial}{\\partial t}H(t)]{} H(t)\n\\xleftarrow[-\\text{log}\\left\\{S(t)\\right\\}]{} S(t)\n\\xleftarrow[1-F(t)]{} F(t)\n\\]\nIdentities:\n\\[\n\\begin{aligned}\nS(t) &= 1 - F(t)\\\\\n&= \\text{exp}\\left\\{-H(t)\\right\\}\\\\\nS'(t) &= -f(t)\\\\\nH(t) &= -\\text{log}\\left\\{S(t)\\right\\}\\\\\nH'(t) &= h(t)\\\\\nh(t) &= \\frac{f(t)}{S(t)}\\\\\n&= -\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\} \\\\\nf(t) &= h(t)\\cdot S(t)\\\\\n\\end{aligned}\n\\]\nSome proofs (others left as exercises):\n\\[\n\\begin{aligned}\nS'(t) &= \\frac{\\partial}{\\partial t}(1-F(t))\\\\\n&= -F'(t)\\\\\n&= -f(t)\\\\\n\\end{aligned}\n\\]\n\\[\n\\begin{aligned}\n\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\}\n&= \\frac{S'(t)}{S(t)}\\\\\n&= -\\frac{f(t)}{S(t)}\\\\\n&= -h(t)\\\\\n\\end{aligned}\n\\]\n\\[\n\\begin{aligned}\nH(t)\n&\\stackrel{\\text{def}}{=}\\int_{u=0}^t h(u) du\\\\\n&= \\int_0^t -\\frac{\\partial}{\\partial u}\\text{log}\\left\\{S(u)\\right\\} du\\\\\n&= \\left[-\\text{log}\\left\\{S(u)\\right\\}\\right]_{u=0}^{u=t}\\\\\n&= \\left[\\text{log}\\left\\{S(u)\\right\\}\\right]_{u=t}^{u=0}\\\\\n&= \\text{log}\\left\\{S(0)\\right\\} - \\text{log}\\left\\{S(t)\\right\\}\\\\\n&= \\text{log}\\left\\{1\\right\\} - \\text{log}\\left\\{S(t)\\right\\}\\\\\n&= 0 - \\text{log}\\left\\{S(t)\\right\\}\\\\\n&=-\\text{log}\\left\\{S(t)\\right\\}\n\\end{aligned}\n\\]\nEquivalently:\n\\[S(t) = \\text{exp}\\left\\{-H(t)\\right\\}\\]\nExample: Time to death the US in 2004\nDaily hazard rates for US Females in 2004\nThe first day is the most dangerous:\n\n\n\nDaily Hazard Rates in 2004 for US Females\n\n\n\n\nDaily hazard rates for US Males and Females in 2004\nExercise: hypothesize why these curves differ where they do?\n\n\n\nDaily Hazard Rates in 2004 for US Males and Females 1-40\n\n\n\n\nSurvival curve for US females\nExercise: compare and contrast this curve with the corresponding hazard curve.\n\n\n\nSurvival Curve in 2004 for US Females\n\n\n\n\nLikelihood with censoring *\n\n\n\n\n\n\nNote\n\n\n\nThis subsection was not presented in class in 2023; it is not necessary to understand for the qualifying exam.\n\n\nIf an event time \\(T\\) is observed exactly as \\(T=t\\), then the likelihood of that observation is just its probability density function:\n\\[\n\\begin{aligned}\n\\mathcal L(t)\n&= p(T=t)\\\\\n&\\stackrel{\\text{def}}{=}f_T(t)\\\\\n&= h_T(t)S_T(t)\\\\\n\\ell(t)\n&\\stackrel{\\text{def}}{=}\\text{log}\\left\\{\\mathcal L(t)\\right\\}\\\\\n&= \\text{log}\\left\\{h_T(t)S_T(t)\\right\\}\\\\\n&= \\text{log}\\left\\{h_T(t)\\right\\} + \\text{log}\\left\\{S_T(t)\\right\\}\\\\\n&= \\text{log}\\left\\{h_T(t)\\right\\} - H_T(t)\\\\\n\\end{aligned}\n\\]\nIf instead the event time \\(T\\) is censored and only known to be after time \\(y\\), then the likelihood of that censored observation is instead the survival function evaluated at the censoring time:\n\\[\n\\begin{aligned}\n\\mathcal L(y)\n&=p_T(T>y)\\\\\n&\\stackrel{\\text{def}}{=}S_T(y)\\\\\n\\ell(y)\n&\\stackrel{\\text{def}}{=}\\text{log}\\left\\{\\mathcal L(y)\\right\\}\\\\\n&=\\text{log}\\left\\{S(y)\\right\\}\\\\\n&=-H(y)\\\\\n\\end{aligned}\n\\]\nWhat’s written above is incomplete. We also observed whether or not the observation was censored. Let \\(C\\) denote the time when censoring would occur (if the event did not occur first); let \\(f_C(y)\\) and \\(S_C(y)\\) be the corresponding density and survival functions for the censoring event.\nLet \\(Y\\) denote the time when observation ended (either by censoring or by the event of interest occurring), and let \\(D\\) be an indicator variable for the event occurring at \\(Y\\) (so \\(D=0\\) represents a censored observation and \\(D=1\\) represents an uncensored observation). In other words, let \\(Y \\stackrel{\\text{def}}{=}\\min(T,C)\\) and \\(D \\stackrel{\\text{def}}{=}\\mathbb 1{\\{T<=C\\}}\\).\nThen the complete likelihood of the observed data \\((Y,D)\\) is:\n\\[\n\\begin{aligned}\n\\mathcal L(y,d)\n&= p(Y=y, D=d)\\\\\n&= \\left[p(T=y,C> y)\\right]^d \\cdot\n\\left[p(T>y,C=y)\\right]^{1-d}\\\\\n\\end{aligned}\n\\]\nTypically, survival analyses assume that \\(C\\) and \\(T\\) are mutually independent; this assumption is called “non-informative” censoring.\nThen the joint likelihood \\(p(Y,D)\\) factors into the product \\(p(Y), p(D)\\), and the likelihood reduces to:\n\\[\n\\begin{aligned}\n\\mathcal L(y,d)\n&= \\left[p(T=y,C> y)\\right]^d\\cdot\n\\left[p(T>y,C=y)\\right]^{1-d}\\\\\n&= \\left[p(T=y)p(C> y)\\right]^d\\cdot\n\\left[p(T>y)p(C=y)\\right]^{1-d}\\\\\n&= \\left[f_T(y)S_C(y)\\right]^d\\cdot\n\\left[S(y)f_C(y)\\right]^{1-d}\\\\\n&= \\left[f_T(y)^d S_C(y)^d\\right]\\cdot\n\\left[S_T(y)^{1-d}f_C(y)^{1-d}\\right]\\\\\n&= \\left(f_T(y)^d \\cdot S_T(y)^{1-d}\\right)\\cdot\n\\left(f_C(y)^{1-d} \\cdot S_C(y)^{d}\\right)\n\\end{aligned}\n\\]\nThe corresponding log-likelihood is:\n\\[\n\\begin{aligned}\n\\ell(y,d)\n&= \\text{log}\\left\\{\\mathcal L(y,d) \\right\\}\\\\\n&= \\text{log}\\left\\{\n\\left(f_T(y)^d \\cdot S_T(y)^{1-d}\\right)\\cdot\n\\left(f_C(y)^{1-d} \\cdot S_C(y)^{d}\\right)\n\\right\\}\\\\\n&= \\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n+\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\\\\\n\\end{aligned}\n\\] Let\n\n\n\\(\\theta_T\\) represent the parameters of \\(p_T(t)\\),\n\n\\(\\theta_C\\) represent the parameters of \\(p_C(c)\\),\n\n\\(\\theta = (\\theta_T, \\theta_C)\\) be the combined vector of all parameters.\n\nThen corresponding score function is:\n\\[\n\\begin{aligned}\n\\ell'(y,d)\n&= \\frac{\\partial}{\\partial \\theta}\n\\left[\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n+\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\n\\right]\\\\\n&=\n\\left(\n\\frac{\\partial}{\\partial \\theta}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n\\right)\n+\n\\left(\n\\frac{\\partial}{\\partial \\theta}\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\n\\right)\\\\\n\\end{aligned}\n\\]\nAs long as \\(\\theta_C\\) and \\(\\theta_T\\) don’t share any parameters, then if censoring is non-informative, the partial derivative with respect to \\(\\theta_T\\) is:\n\\[\n\\begin{aligned}\n\\ell'_{\\theta_T}(y,d)\n&\\stackrel{\\text{def}}{=}\\frac{\\partial}{\\partial \\theta_T}\\ell(y,d)\\\\\n&=\n\\left(\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n\\right)\n+\n\\left(\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\n\\right)\\\\\n&=\n\\left(\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n\\right) + 0\\\\\n&=\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\\\\\n\\end{aligned}\n\\] Thus, the MLE for \\(\\theta_T\\) won’t depend on \\(\\theta_C\\), and we can ignore the distribution of \\(C\\) when estimating the parameters of \\(f_T(t)=p(T=t)\\).\nThen:\n\\[\n\\begin{aligned}\n\\mathcal L(y,d)\n&= f_T(y)^d \\cdot S_T(y)^{1-d}\\\\\n&= \\left(h_T(y)^d S_T(y)^d\\right) \\cdot S_T(y)^{1-d}\\\\\n&= h_T(y)^d \\cdot S_T(y)^d \\cdot S_T(y)^{1-d}\\\\\n&= h_T(y)^d \\cdot S_T(y)\\\\\n&= S_T(y) \\cdot h_T(y)^d \\\\\n\\end{aligned}\n\\]\nThat is, if the event occurred at time \\(y\\) (i.e., if \\(d=1\\)), then the likelihood of \\((Y,D) = (y,d)\\) is equal to the hazard function at \\(y\\) times the survival function at \\(y\\). Otherwise, the likelihood is equal to just the survival function at \\(y\\).\nThe corresponding log-likelihood is:\n\\[\n\\begin{aligned}\n\\ell(y,d)\n&=\\text{log}\\left\\{\\mathcal L(y,d)\\right\\}\\\\\n&= \\text{log}\\left\\{S_T(y) \\cdot h_T(y)^d\\right\\}\\\\\n&= \\text{log}\\left\\{S_T(y)\\right\\} + \\text{log}\\left\\{h_T(y)^d\\right\\}\\\\\n&= \\text{log}\\left\\{S_T(y)\\right\\} + d\\cdot \\text{log}\\left\\{h_T(y)\\right\\}\\\\\n&= -H_T(y) + d\\cdot \\text{log}\\left\\{h_T(y)\\right\\}\\\\\n\\end{aligned}\n\\]\nIn other words, the log-likelihood contribution from a single observation \\((Y,D) = (y,d)\\) is equal to the negative cumulative hazard at \\(y\\), plus the log of the hazard at \\(y\\) if the event occurred at time \\(y\\).\n\n\n\n\n\n\nNote\n\n\n\nEnd of extra section.", + "text": "5.3 Distribution functions for time-to-event variables\n\n5.3.1 The Probability Density Function (PDF)\nFor a time-to-event variable \\(T\\) with a continuous distribution, the probability density function is defined as usual:\n\\[f(t)\\stackrel{\\text{def}}{=}p(t) \\stackrel{\\text{def}}{=}p(T=t)\\]\nTypically, this density is assumed to be 0 for all \\(t<0\\); that is, \\(f(t) = 0, \\forall t<0\\). In other words, the range of \\(T\\) is typically \\([0,\\infty)\\).\n\n\nExample 5.1 (exponential distribution) Recall from Epi 202: the pdf of the exponential distribution family of models is:\n\\[p(T=t) = \\mathbb{1}_{t \\ge 0} \\cdot \\lambda \\text{e}^{-\\lambda t}\\]\nwhere \\(\\lambda > 0\\).\n\nHere are some examples of exponential pdfs:\n\n\n\n\n\n\n\n\n\n\n5.3.2 The Cumulative Distribution Function (CDF)\nThe cumulative distribution function is defined as:\n\\[\n\\begin{aligned}\nF(t) &\\stackrel{\\text{def}}{=}\\Pr(T \\le t)\\\\\n&=\\int_{u=-\\infty}^t f(u) du\n\\end{aligned}\n\\]\n\nExample 5.2 (exponential distribution) Recall from Epi 202: the cdf of the exponential distribution family of models is:\n\\[\nP(T\\le t) = \\mathbb{1}_{t \\ge 0} \\cdot (1- \\text{e}^{-\\lambda t})\n\\] where \\(\\lambda > 0\\).\n\nHere are some examples of exponential cdfs:\n\n\n\n\n\n\n\n\n\n5.3.3 The Survival Function\nFor survival data, a more important quantity is the survival function:\n\\[\n\\begin{aligned}\nS(t) &\\stackrel{\\text{def}}{=}\\Pr(T > t)\\\\\n&=\\int_{u=t}^\\infty p(u) du\\\\\n&=1-F(t)\\\\\n\\end{aligned}\n\\]\n\n\nThe survival function \\(S(t)\\) is the probability that the event time is later than \\(t\\). If the event in a clinical trial is death, then \\(S(t)\\) is the expected fraction of the original population at time 0 who have survived up to time \\(t\\) and are still alive at time \\(t\\); that is:\n\nIf \\(X_t\\) represents survival status at time \\(t\\), with \\(X_t = 1\\) denoting alive at time \\(t\\) and \\(X_t = 0\\) denoting deceased at time \\(t\\), then:\n\\[S(t) = \\mathbb{E}\\left[X_t\\right]\\]\n\n\nExample 5.3 (exponential distribution) Since \\(S(t) = 1 - F(t)\\), the survival function of the exponential distribution family of models is:\n\\[\nP(T> t) = \\left\\{ {{\\text{e}^{-\\lambda t}, t\\ge0} \\atop {1, t \\le 0}}\\right.\n\\] where \\(\\lambda > 0\\).\n\n\nHere are some examples of exponential pdfs:\n\n\n\n\n\n\n\n\n\n5.3.4 The Hazard Function\nAnother important quantity is the hazard function:\n\nDefinition 5.1 (Hazard function) The hazard function for a random variable \\(T\\) at value \\(t\\) is the conditional density of \\(T\\) at \\(t\\), given \\(T\\ge t\\); that is:\n\\[h(t) \\stackrel{\\text{def}}{=}p(T=t|T\\ge t)\\]\nIf \\(T\\) represents the time at which an event occurs, then \\(h(t)\\) is the probability that the event occurs at time \\(t\\), given that it has not occurred prior to time \\(t\\).\n\n\nThe hazard function has an important relationship to the density and survival functions, which we can use to derive the hazard function for a given probability distribution.\n\nTheorem 5.1 \\[h(t)=\\frac{f(t)}{S(t)}\\]\n\n\n\nProof. \\[\n\\begin{aligned}\nh(t) &=p(T=t|T\\ge t)\\\\\n&=\\frac{p(T=t, T\\ge t)}{p(T \\ge t)}\\\\\n&=\\frac{p(T=t)}{p(T \\ge t)}\\\\\n&=\\frac{f(t)}{S(t)}\n\\end{aligned}\n\\]\n\n\n\nExample 5.4 (exponential distribution) The hazard function of the exponential distribution family of models is:\n\\[\n\\begin{aligned}\nP(T=t|T \\ge t)\n&= \\frac{f(t)}{S(t)}\\\\\n&= \\frac{\\mathbb{1}_{t \\ge 0}\\cdot \\lambda \\text{e}^{-\\lambda t}}{\\text{e}^{-\\lambda t}}\\\\\n&=\\mathbb{1}_{t \\ge 0}\\cdot \\lambda\n\\end{aligned}\n\\] Figure 5.1 shows some examples of exponential hazard functions.\n\n\n\n\n\n\nFigure 5.1: Examples of hazard functions for exponential distributions\n\n\n\n\n\n\n\n\nWe can also view the hazard function as the derivative of the negative of the logarithm of the survival function:\n\nTheorem 5.2 \\[h(t) = \\frac{\\partial}{\\partial t}\\left\\{-\\text{log}\\left\\{S(t)\\right\\}\\right\\}\\]\n\n\n\nProof. \\[\n\\begin{aligned}\nh(t)\n&= \\frac{f(t)}{S(t)}\\\\\n&= \\frac{-S'(t)}{S(t)}\\\\\n&= -\\frac{S'(t)}{S(t)}\\\\\n&=-\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\}\\\\\n&=\\frac{\\partial}{\\partial t}\\left\\{-\\text{log}\\left\\{S(t)\\right\\}\\right\\}\n\\end{aligned}\n\\]\n\n\n5.3.5 The Cumulative Hazard Function\nSince \\(h(t) = \\frac{\\partial}{\\partial t}\\left\\{-\\text{log}\\left\\{S(t)\\right\\}\\right\\}\\) (see Theorem 5.2), we also have:\n\nCorollary 5.1 \\[S(t) = \\text{exp}\\left\\{-\\int_{u=0}^t h(u)du\\right\\} \\tag{5.1}\\]\n\n\n\nThe integral in Equation 5.1 is important enough to have its own name: cumulative hazard.\n\n\nDefinition 5.2 (cumulative hazard) The cumulative hazard function \\(H(t)\\) is defined as:\n\\[H(t) \\stackrel{\\text{def}}{=}\\int_{u=0}^t h(u) du\\]\n\nAs we will see below, \\(H(t)\\) is tractable to estimate, and we can then derive an estimate of the hazard function using an approximate derivative of the estimated cumulative hazard.\n\nExample 5.5 The cumulative hazard function of the exponential distribution family of models is:\n\\[\nH(t) = \\mathbb{1}_{t \\ge 0}\\cdot \\lambda t\n\\]\nFigure 5.2 shows some examples of exponential cumulative hazard functions.\n\n\n\n\n\n\nFigure 5.2: Examples of exponential cumulative hazard functions\n\n\n\n\n\n\n\n\n5.3.6 Some Key Mathematical Relationships among Survival Concepts\nDiagram:\n\\[\nh(t) \\xrightarrow[]{\\int_{u=0}^t h(u)du} H(t)\n\\xrightarrow[]{\\text{exp}\\left\\{-H(t)\\right\\}} S(t)\n\\xrightarrow[]{1-S(t)} F(t)\n\\]\n\\[\nh(t) \\xleftarrow[\\frac{\\partial}{\\partial t}H(t)]{} H(t)\n\\xleftarrow[-\\text{log}\\left\\{S(t)\\right\\}]{} S(t)\n\\xleftarrow[1-F(t)]{} F(t)\n\\]\n\nIdentities:\n\\[\n\\begin{aligned}\nS(t) &= 1 - F(t)\\\\\n&= \\text{exp}\\left\\{-H(t)\\right\\}\\\\\nS'(t) &= -f(t)\\\\\nH(t) &= -\\text{log}\\left\\{S(t)\\right\\}\\\\\nH'(t) &= h(t)\\\\\nh(t) &= \\frac{f(t)}{S(t)}\\\\\n&= -\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\} \\\\\nf(t) &= h(t)\\cdot S(t)\\\\\n\\end{aligned}\n\\]\n\nSome proofs (others left as exercises):\n\\[\n\\begin{aligned}\nS'(t) &= \\frac{\\partial}{\\partial t}(1-F(t))\\\\\n&= -F'(t)\\\\\n&= -f(t)\\\\\n\\end{aligned}\n\\]\n\n\\[\n\\begin{aligned}\n\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\}\n&= \\frac{S'(t)}{S(t)}\\\\\n&= -\\frac{f(t)}{S(t)}\\\\\n&= -h(t)\\\\\n\\end{aligned}\n\\]\n\n\\[\n\\begin{aligned}\nH(t)\n&\\stackrel{\\text{def}}{=}\\int_{u=0}^t h(u) du\\\\\n&= \\int_0^t -\\frac{\\partial}{\\partial u}\\text{log}\\left\\{S(u)\\right\\} du\\\\\n&= \\left[-\\text{log}\\left\\{S(u)\\right\\}\\right]_{u=0}^{u=t}\\\\\n&= \\left[\\text{log}\\left\\{S(u)\\right\\}\\right]_{u=t}^{u=0}\\\\\n&= \\text{log}\\left\\{S(0)\\right\\} - \\text{log}\\left\\{S(t)\\right\\}\\\\\n&= \\text{log}\\left\\{1\\right\\} - \\text{log}\\left\\{S(t)\\right\\}\\\\\n&= 0 - \\text{log}\\left\\{S(t)\\right\\}\\\\\n&=-\\text{log}\\left\\{S(t)\\right\\}\n\\end{aligned}\n\\]\n\nCorollary:\n\\[S(t) = \\text{exp}\\left\\{-H(t)\\right\\}\\]\n\nExample: Time to death the US in 2004\nThe first day is the most dangerous:\n\n\n\nDaily Hazard Rates in 2004 for US Females\n\n\n\n\n\nExercise: hypothesize why these curves differ where they do?\n\n\n\nDaily Hazard Rates in 2004 for US Males and Females 1-40\n\n\n\n\n\nExercise: compare and contrast this curve with the corresponding hazard curve.\n\n\n\nSurvival Curve in 2004 for US Females\n\n\n\n\n\n5.3.7 Likelihood with censoring\nIf an event time \\(T\\) is observed exactly as \\(T=t\\), then the likelihood of that observation is just its probability density function:\n\\[\n\\begin{aligned}\n\\mathcal L(t)\n&= p(T=t)\\\\\n&\\stackrel{\\text{def}}{=}f_T(t)\\\\\n&= h_T(t)S_T(t)\\\\\n\\ell(t)\n&\\stackrel{\\text{def}}{=}\\text{log}\\left\\{\\mathcal L(t)\\right\\}\\\\\n&= \\text{log}\\left\\{h_T(t)S_T(t)\\right\\}\\\\\n&= \\text{log}\\left\\{h_T(t)\\right\\} + \\text{log}\\left\\{S_T(t)\\right\\}\\\\\n&= \\text{log}\\left\\{h_T(t)\\right\\} - H_T(t)\\\\\n\\end{aligned}\n\\]\n\nIf instead the event time \\(T\\) is censored and only known to be after time \\(y\\), then the likelihood of that censored observation is instead the survival function evaluated at the censoring time:\n\\[\n\\begin{aligned}\n\\mathcal L(y)\n&=p_T(T>y)\\\\\n&\\stackrel{\\text{def}}{=}S_T(y)\\\\\n\\ell(y)\n&\\stackrel{\\text{def}}{=}\\text{log}\\left\\{\\mathcal L(y)\\right\\}\\\\\n&=\\text{log}\\left\\{S(y)\\right\\}\\\\\n&=-H(y)\\\\\n\\end{aligned}\n\\]\n\n\nWhat’s written above is incomplete. We also observed whether or not the observation was censored. Let \\(C\\) denote the time when censoring would occur (if the event did not occur first); let \\(f_C(y)\\) and \\(S_C(y)\\) be the corresponding density and survival functions for the censoring event.\nLet \\(Y\\) denote the time when observation ended (either by censoring or by the event of interest occurring), and let \\(D\\) be an indicator variable for the event occurring at \\(Y\\) (so \\(D=0\\) represents a censored observation and \\(D=1\\) represents an uncensored observation). In other words, let \\(Y \\stackrel{\\text{def}}{=}\\min(T,C)\\) and \\(D \\stackrel{\\text{def}}{=}\\mathbb 1{\\{T<=C\\}}\\).\nThen the complete likelihood of the observed data \\((Y,D)\\) is:\n\n\\[\n\\begin{aligned}\n\\mathcal L(y,d)\n&= p(Y=y, D=d)\\\\\n&= \\left[p(T=y,C> y)\\right]^d \\cdot\n\\left[p(T>y,C=y)\\right]^{1-d}\\\\\n\\end{aligned}\n\\]\n\n\nTypically, survival analyses assume that \\(C\\) and \\(T\\) are mutually independent; this assumption is called “non-informative” censoring.\nThen the joint likelihood \\(p(Y,D)\\) factors into the product \\(p(Y), p(D)\\), and the likelihood reduces to:\n\n\\[\n\\begin{aligned}\n\\mathcal L(y,d)\n&= \\left[p(T=y,C> y)\\right]^d\\cdot\n\\left[p(T>y,C=y)\\right]^{1-d}\\\\\n&= \\left[p(T=y)p(C> y)\\right]^d\\cdot\n\\left[p(T>y)p(C=y)\\right]^{1-d}\\\\\n&= \\left[f_T(y)S_C(y)\\right]^d\\cdot\n\\left[S(y)f_C(y)\\right]^{1-d}\\\\\n&= \\left[f_T(y)^d S_C(y)^d\\right]\\cdot\n\\left[S_T(y)^{1-d}f_C(y)^{1-d}\\right]\\\\\n&= \\left(f_T(y)^d \\cdot S_T(y)^{1-d}\\right)\\cdot\n\\left(f_C(y)^{1-d} \\cdot S_C(y)^{d}\\right)\n\\end{aligned}\n\\]\n\n\nThe corresponding log-likelihood is:\n\n\\[\n\\begin{aligned}\n\\ell(y,d)\n&= \\text{log}\\left\\{\\mathcal L(y,d) \\right\\}\\\\\n&= \\text{log}\\left\\{\n\\left(f_T(y)^d \\cdot S_T(y)^{1-d}\\right)\\cdot\n\\left(f_C(y)^{1-d} \\cdot S_C(y)^{d}\\right)\n\\right\\}\\\\\n&= \\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n+\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\\\\\n\\end{aligned}\n\\] Let\n\n\n\\(\\theta_T\\) represent the parameters of \\(p_T(t)\\),\n\n\\(\\theta_C\\) represent the parameters of \\(p_C(c)\\),\n\n\\(\\theta = (\\theta_T, \\theta_C)\\) be the combined vector of all parameters.\n\n\n\nThe corresponding score function is:\n\n\\[\n\\begin{aligned}\n\\ell'(y,d)\n&= \\frac{\\partial}{\\partial \\theta}\n\\left[\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n+\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\n\\right]\\\\\n&=\n\\left(\n\\frac{\\partial}{\\partial \\theta}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n\\right)\n+\n\\left(\n\\frac{\\partial}{\\partial \\theta}\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\n\\right)\\\\\n\\end{aligned}\n\\]\n\n\nAs long as \\(\\theta_C\\) and \\(\\theta_T\\) don’t share any parameters, then if censoring is non-informative, the partial derivative with respect to \\(\\theta_T\\) is:\n\n\\[\n\\begin{aligned}\n\\ell'_{\\theta_T}(y,d)\n&\\stackrel{\\text{def}}{=}\\frac{\\partial}{\\partial \\theta_T}\\ell(y,d)\\\\\n&=\n\\left(\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n\\right)\n+\n\\left(\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\n\\right)\\\\\n&=\n\\left(\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n\\right) + 0\\\\\n&=\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\\\\\n\\end{aligned}\n\\]\n\n\nThus, the MLE for \\(\\theta_T\\) won’t depend on \\(\\theta_C\\), and we can ignore the distribution of \\(C\\) when estimating the parameters of \\(f_T(t)=p(T=t)\\).\n\nThen:\n\\[\n\\begin{aligned}\n\\mathcal L(y,d)\n&= f_T(y)^d \\cdot S_T(y)^{1-d}\\\\\n&= \\left(h_T(y)^d S_T(y)^d\\right) \\cdot S_T(y)^{1-d}\\\\\n&= h_T(y)^d \\cdot S_T(y)^d \\cdot S_T(y)^{1-d}\\\\\n&= h_T(y)^d \\cdot S_T(y)\\\\\n&= S_T(y) \\cdot h_T(y)^d \\\\\n\\end{aligned}\n\\]\n\nThat is, if the event occurred at time \\(y\\) (i.e., if \\(d=1\\)), then the likelihood of \\((Y,D) = (y,d)\\) is equal to the hazard function at \\(y\\) times the survival function at \\(y\\). Otherwise, the likelihood is equal to just the survival function at \\(y\\).\n\n\n\nThe corresponding log-likelihood is:\n\n\\[\n\\begin{aligned}\n\\ell(y,d)\n&=\\text{log}\\left\\{\\mathcal L(y,d)\\right\\}\\\\\n&= \\text{log}\\left\\{S_T(y) \\cdot h_T(y)^d\\right\\}\\\\\n&= \\text{log}\\left\\{S_T(y)\\right\\} + \\text{log}\\left\\{h_T(y)^d\\right\\}\\\\\n&= \\text{log}\\left\\{S_T(y)\\right\\} + d\\cdot \\text{log}\\left\\{h_T(y)\\right\\}\\\\\n&= -H_T(y) + d\\cdot \\text{log}\\left\\{h_T(y)\\right\\}\\\\\n\\end{aligned}\n\\]\n\nIn other words, the log-likelihood contribution from a single observation \\((Y,D) = (y,d)\\) is equal to the negative cumulative hazard at \\(y\\), plus the log of the hazard at \\(y\\) if the event occurred at time \\(y\\).", "crumbs": [ "Time to Event Models", "5  Introduction to Survival Analysis" @@ -555,7 +566,7 @@ "href": "intro-to-survival-analysis.html#parametric-models-for-time-to-event-outcomes", "title": "\n5  Introduction to Survival Analysis\n", "section": "\n5.4 Parametric Models for Time-to-Event Outcomes", - "text": "5.4 Parametric Models for Time-to-Event Outcomes\n\n5.4.1 Exponential Distribution\n\nThe exponential distribution is the base distribution for survival analysis.\nThe distribution has a constant hazard \\(\\lambda\\)\n\nThe mean survival time is \\(\\lambda^{-1}\\)\n\n\nMathematical details of exponential distribution\n\\[\n\\begin{aligned}\nf(t) &= \\lambda \\text{e}^{-\\lambda t}\\\\\nE(t) &= \\lambda^{-1}\\\\\nVar(t) &= \\lambda^{-2}\\\\\nF(t) &= 1-\\text{e}^{-\\lambda x}\\\\\nS(t)&= \\text{e}^{-\\lambda x}\\\\\n\\ln(S(t))&=-\\lambda x\\\\\nh(t) &= -\\frac{f(t)}{S(t)} = -\\frac{\\lambda \\text{e}^{-\\lambda t}}{\\text{e}^{-\\lambda t}}=\\lambda\n\\end{aligned}\n\\]\nEstimation of \\(\\lambda\\)\n\n\nSuppose we have \\(m\\) exponential survival times of \\(t_1, t_2,\\ldots,t_m\\) and \\(k\\) right-censored values at \\(u_1,u_2,\\ldots,u_k\\).\nA survival time of \\(t_i=10\\) means that subject \\(i\\) died at time 10. A right-censored time \\(u_i=10\\) means that at time 10, subject \\(i\\) was still alive and that we have no further follow-up.\nFor the moment we will assume that the survival distribution is exponential and that all the subjects have the same parameter \\(\\lambda\\).\n\nWe have \\(m\\) exponential survival times of \\(t_1, t_2,\\ldots,t_m\\) and \\(k\\) right-censored values at \\(u_1,u_2,\\ldots,u_k\\). The log-likelihood of an observed survival time \\(t_i\\) is \\[\n\\text{log}\\left\\{\\lambda \\text{e}^{-\\lambda t_i}\\right\\} =\n\\text{log}\\left\\{\\lambda\\right\\}-\\lambda t_i\n\\] and the likelihood of a censored value is the probability of that outcome (survival greater than \\(u_j\\)) so the log-likelihood is\n\\[\n\\text{log}\\left\\{\\lambda \\text{e}^{u_j}\\right\\} =-\\lambda u_j.\n\\]\nLet \\(T=\\sum t_i\\) and \\(U=\\sum u_j\\). Then:\n\\[\n\\begin{aligned}\n\\ell(\\lambda) &= \\sum_{i=1}^m( \\ln \\lambda-\\lambda t_i) + \\sum_{j=1}^k (-\\lambda u_j)\\\\\n&= m \\ln \\lambda -(T+U)\\lambda\\\\\n\\ell'(\\lambda)\n&=m\\lambda^{-1} -(T+U)\\\\\n\\hat{\\lambda} &= \\frac{m}{T+U}\\\\\n\\ell''&=-m/\\lambda^2\\\\\n&< 0\\\\\n\\hat E[T] &= \\hat\\lambda^{-1}\\\\\n&= \\frac{T+U}{m}\n\\end{aligned}\n\\]\nFisher Information and Standard Error\n\\[\n\\begin{aligned}\nE[-\\ell'']\n& = m/\\lambda^2\\\\\n\\text{Var}\\left(\\hat\\lambda\\right)\n&\\approx \\left(E[-\\ell'']\\right)^{-1}\\\\\n&=\\lambda^2/m\\\\\n\\text{SE}\\left(\\hat\\lambda\\right)\n&= \\sqrt{\\text{Var}\\left(\\hat\\lambda\\right)}\\\\\n&\\approx \\lambda/\\sqrt{m}\n\\end{aligned}\n\\]\n\\(\\hat\\lambda\\) depends on the censoring times of the censored observations, but \\(\\text{Var}\\left(\\hat\\lambda\\right)\\) only depends on the number of uncensored observations, \\(m\\), and not on the number of censored observations (\\(k\\)).\nOther Parametric Survival Distributions\n\nAny density on \\([0,\\infty)\\) can be a survival distribution, but the most useful ones are all skew right.\nThe commonest generalization of the exponential is the Weibull.\nOther common choices are the gamma, log-normal, log-logistic, Gompertz, inverse Gaussian, and Pareto.\nMost of what we do going forward is non-parametric or semi-parametric, but sometimes these parametric distributions provide a useful approach.\n\n5.4.2 Weibull Distribution\n\\[\n\\begin{aligned}\np(t)&= \\alpha\\lambda x^{\\alpha-1}\\text{e}^{-\\lambda x^\\alpha}\\\\\nh(t)&=\\alpha\\lambda x^{\\alpha-1}\\\\\nS(t)&=\\text{e}^{-\\lambda x^\\alpha}\\\\\nE(T)&= \\Gamma(1+1/\\alpha)\\cdot \\lambda^{-1/\\alpha}\n\\end{aligned}\n\\]\nWhen \\(\\alpha=1\\) this is the exponential. When \\(\\alpha>1\\) the hazard is increasing and when \\(\\alpha < 1\\) the hazard is decreasing. This provides more flexibility than the exponential.\nWe will see more of this distribution later.", + "text": "5.4 Parametric Models for Time-to-Event Outcomes\n\n5.4.1 Exponential Distribution\n\nThe exponential distribution is the base distribution for survival analysis.\nThe distribution has a constant hazard \\(\\lambda\\)\n\nThe mean survival time is \\(\\lambda^{-1}\\)\n\n\n\nMathematical details of exponential distribution\n\\[\n\\begin{aligned}\nf(t) &= \\lambda \\text{e}^{-\\lambda t}\\\\\nE(t) &= \\lambda^{-1}\\\\\nVar(t) &= \\lambda^{-2}\\\\\nF(t) &= 1-\\text{e}^{-\\lambda x}\\\\\nS(t)&= \\text{e}^{-\\lambda x}\\\\\n\\ln(S(t))&=-\\lambda x\\\\\nh(t) &= -\\frac{f(t)}{S(t)} = -\\frac{\\lambda \\text{e}^{-\\lambda t}}{\\text{e}^{-\\lambda t}}=\\lambda\n\\end{aligned}\n\\]\nEstimating \\(\\lambda\\)\n\n\nSuppose we have \\(m\\) exponential survival times of \\(t_1, t_2,\\ldots,t_m\\) and \\(k\\) right-censored values at \\(u_1,u_2,\\ldots,u_k\\).\nA survival time of \\(t_i=10\\) means that subject \\(i\\) died at time 10. A right-censored time \\(u_i=10\\) means that at time 10, subject \\(i\\) was still alive and that we have no further follow-up.\nFor the moment we will assume that the survival distribution is exponential and that all the subjects have the same parameter \\(\\lambda\\).\n\nWe have \\(m\\) exponential survival times of \\(t_1, t_2,\\ldots,t_m\\) and \\(k\\) right-censored values at \\(u_1,u_2,\\ldots,u_k\\). The log-likelihood of an observed survival time \\(t_i\\) is \\[\n\\text{log}\\left\\{\\lambda \\text{e}^{-\\lambda t_i}\\right\\} =\n\\text{log}\\left\\{\\lambda\\right\\}-\\lambda t_i\n\\] and the likelihood of a censored value is the probability of that outcome (survival greater than \\(u_j\\)) so the log-likelihood is\n\\[\n\\begin{aligned}\n\\ell_j(\\lambda) &= \\text{log}\\left\\{\\lambda \\text{e}^{u_j}\\right\\}\n\\\\ &= -\\lambda u_j\n\\end{aligned}\n\\]\n\nLet \\(T=\\sum t_i\\) and \\(U=\\sum u_j\\). Then:\n\\[\n\\begin{aligned}\n\\ell(\\lambda) &= \\sum_{i=1}^m( \\ln \\lambda-\\lambda t_i) + \\sum_{j=1}^k (-\\lambda u_j)\\\\\n&= m \\ln \\lambda -(T+U)\\lambda\\\\\n\\ell'(\\lambda)\n&=m\\lambda^{-1} -(T+U)\\\\\n\\hat{\\lambda} &= \\frac{m}{T+U}\\\\\n\\ell''&=-m/\\lambda^2\\\\\n&< 0\\\\\n\\hat E[T] &= \\hat\\lambda^{-1}\\\\\n&= \\frac{T+U}{m}\n\\end{aligned}\n\\]\nFisher Information and Standard Error\n\\[\n\\begin{aligned}\nE[-\\ell'']\n& = m/\\lambda^2\\\\\n\\text{Var}\\left(\\hat\\lambda\\right)\n&\\approx \\left(E[-\\ell'']\\right)^{-1}\\\\\n&=\\lambda^2/m\\\\\n\\text{SE}\\left(\\hat\\lambda\\right)\n&= \\sqrt{\\text{Var}\\left(\\hat\\lambda\\right)}\\\\\n&\\approx \\lambda/\\sqrt{m}\n\\end{aligned}\n\\]\n\n\\(\\hat\\lambda\\) depends on the censoring times of the censored observations, but \\(\\text{Var}\\left(\\hat\\lambda\\right)\\) only depends on the number of uncensored observations, \\(m\\), and not on the number of censored observations (\\(k\\)).\n\n\n5.4.2 Other Parametric Survival Distributions\n\nAny density on \\([0,\\infty)\\) can be a survival distribution, but the most useful ones are all skew right.\nThe most frequently used generalization of the exponential is the Weibull.\nOther common choices are the gamma, log-normal, log-logistic, Gompertz, inverse Gaussian, and Pareto.\nMost of what we do going forward is non-parametric or semi-parametric, but sometimes these parametric distributions provide a useful approach.", "crumbs": [ "Time to Event Models", "5  Introduction to Survival Analysis" @@ -720,7 +731,7 @@ "href": "coxph-model-building.html#predictions-and-residuals", "title": "\n7  Building Cox Proportional Hazards models\n", "section": "\n7.3 Predictions and Residuals", - "text": "7.3 Predictions and Residuals\n\n7.3.1 Review: Predictions in Linear Regression\n\nIn linear regression, we have a linear predictor for each data point \\(i\\)\n\n\n\\[\n\\begin{aligned}\n\\eta_i &= \\beta_0+\\beta_1x_{1i}+\\cdots+\\beta_px_{pi}\\\\\n\\hat y_i &=\\hat\\eta_i = \\hat\\beta_0+\\hat\\beta_1x_{1i}+\\cdots+\\hat\\beta_px_{pi}\\\\\ny_i &\\sim N(\\eta_i,\\sigma^2)\n\\end{aligned}\n\\]\n\n\n\\(\\hat y_i\\) estimates the conditional mean of \\(y_i\\) given the covariate values \\(\\tilde{x}_i\\). This together with the prediction error says that we are predicting the distribution of values of \\(y\\).\n\n7.3.2 Review: Residuals in Linear Regression\n\nThe usual residual is \\(r_i=y_i-\\hat y_i\\), the difference between the actual value of \\(y\\) and a prediction of its mean.\nThe residuals are also the quantities the sum of whose squares is being minimized by the least squares/MLE estimation.\n\n7.3.3 Predictions and Residuals in survival models\n\nIn survival analysis, the equivalent of \\(y_i\\) is the event time \\(t_i\\), which is unknown for the censored observations.\nThe expected event time can be tricky to calculate:\n\n\\[\n\\hat{\\text{E}}[T|X=x] = \\int_{t=0}^{\\infty} \\hat S(t)dt\n\\]\n\n7.3.4 Wide prediction intervals\nThe nature of time-to-event data results in very wide prediction intervals:\n\nSuppose a cancer patient is predicted to have a mean lifetime of 5 years after diagnosis and suppose the distribution is exponential.\nIf we want a 95% interval for survival, the lower end is at the 0.025 percentage point of the exponential which is qexp(.025, rate = 1/5) = 0.12658904 years, or 1/40 of the mean lifetime.\nThe upper end is at the 0.975 point which is qexp(.975, rate = 1/5) = 18.44439727 years, or 3.7 times the mean lifetime.\nSaying that the survival time is somewhere between 6 weeks and 18 years does not seem very useful, but it may be the best we can do.\nFor survival analysis, something is like a residual if it is small when the model is accurate or if the accumulation of them is in some way minimized by the estimation algorithm, but there is no exact equivalence to linear regression residuals.\nAnd if there is, they are mostly quite large!\n\n7.3.5 Types of Residuals in Time-to-Event Models\n\nIt is often hard to make a decision from graph appearances, though the process can reveal much.\nSome diagnostic tests are based on residuals as with other regression methods:\n\nSchoenfeld residuals (via cox.zph) for proportionality.\n\nCox-Snell residuals for goodness of fit.\n\nmartingale residuals for non-linearity.\n\ndfbeta for influence.\n\n7.3.6 Schoenfeld residuals\n\nThere is a Schoenfeld residual for each subject \\(i\\) with an event (not censored) and for each predictor \\(x_{k}\\).\nAt the event time \\(t\\) for that subject, there is a risk set \\(R\\), and each subject \\(j\\) in the risk set has a risk coefficient \\(\\theta_j\\) and also a value \\(x_{jk}\\) of the predictor.\nThe Schoenfeld residual is the difference between \\(x_{ik}\\) and the risk-weighted average of all the \\(x_{jk}\\) over the risk set.\n\n\\[\nr^S_{ik} =\nx_{ik}-\\frac{\\sum_{k\\in R}x_{jk}\\theta_k}{\\sum_{k\\in R}\\theta_k}\n\\]\nThis residual measures how typical the individual subject is with respect to the covariate at the time of the event. Since subjects should fail more or less uniformly according to risk, the Schoenfeld residuals should be approximately level over time, not increasing or decreasing.\nWe can test this with the correlation with time on some scale, which could be the time itself, the log time, or the rank in the set of failure times.\nThe default is to use the KM curve as a transform, which is similar to the rank but deals better with censoring.\nThe cox.zph() function implements a score test proposed in Grambsch and Therneau (1994).\n\nShow R codehodg.zph = cox.zph(hodg.cox1)\nprint(hodg.zph)\n#> chisq df p\n#> gtype 0.5400 1 0.462\n#> dtype 1.8012 1 0.180\n#> score 3.8805 1 0.049\n#> wtime 0.0173 1 0.895\n#> gtype:dtype 4.0474 1 0.044\n#> GLOBAL 13.7573 5 0.017\n\n\ngtype\n\nShow R codeggcoxzph(hodg.zph, var = \"gtype\")\n\n\n\n\n\n\n\ndtype\n\nShow R codeggcoxzph(hodg.zph, var = \"dtype\")\n\n\n\n\n\n\n\nscore\n\nShow R codeggcoxzph(hodg.zph, var = \"score\")\n\n\n\n\n\n\n\nwtime\n\nShow R codeggcoxzph(hodg.zph, var = \"wtime\")\n\n\n\n\n\n\n\ngtype:dtype\n\nShow R codeggcoxzph(hodg.zph, var = \"gtype:dtype\")\n\n\n\n\n\n\n\nConclusions\n\nFrom the correlation test, the Karnofsky score and the interaction with graft type disease type induce modest but statistically significant non-proportionality.\nThe sample size here is relatively small (26 events in 43 subjects). If the sample size is large, very small amounts of non-proportionality can induce a significant result.\nAs time goes on, autologous grafts are over-represented at their own event times, but those from HOD patients become less represented.\nBoth the statistical tests and the plots are useful.", + "text": "7.3 Predictions and Residuals\n\n7.3.1 Review: Predictions in Linear Regression\n\nIn linear regression, we have a linear predictor for each data point \\(i\\)\n\n\n\\[\n\\begin{aligned}\n\\eta_i &= \\beta_0+\\beta_1x_{1i}+\\cdots+\\beta_px_{pi}\\\\\n\\hat y_i &=\\hat\\eta_i = \\hat\\beta_0+\\hat\\beta_1x_{1i}+\\cdots+\\hat\\beta_px_{pi}\\\\\ny_i &\\sim N(\\eta_i,\\sigma^2)\n\\end{aligned}\n\\]\n\n\n\\(\\hat y_i\\) estimates the conditional mean of \\(y_i\\) given the covariate values \\(\\tilde{x}_i\\). This together with the prediction error says that we are predicting the distribution of values of \\(y\\).\n\n7.3.2 Review: Residuals in Linear Regression\n\nThe usual residual is \\(r_i=y_i-\\hat y_i\\), the difference between the actual value of \\(y\\) and a prediction of its mean.\nThe residuals are also the quantities the sum of whose squares is being minimized by the least squares/MLE estimation.\n\n7.3.3 Predictions and Residuals in survival models\n\nIn survival analysis, the equivalent of \\(y_i\\) is the event time \\(t_i\\), which is unknown for the censored observations.\nThe expected event time can be tricky to calculate:\n\n\\[\n\\hat{\\text{E}}[T|X=x] = \\int_{t=0}^{\\infty} \\hat S(t)dt\n\\]\n\n7.3.4 Wide prediction intervals\nThe nature of time-to-event data results in very wide prediction intervals:\n\nSuppose a cancer patient is predicted to have a mean lifetime of 5 years after diagnosis and suppose the distribution is exponential.\nIf we want a 95% interval for survival, the lower end is at the 0.025 percentage point of the exponential which is qexp(.025, rate = 1/5) = 0.1266 years, or 1/40 of the mean lifetime.\nThe upper end is at the 0.975 point which is qexp(.975, rate = 1/5) = 18.4444 years, or 3.7 times the mean lifetime.\nSaying that the survival time is somewhere between 6 weeks and 18 years does not seem very useful, but it may be the best we can do.\nFor survival analysis, something is like a residual if it is small when the model is accurate or if the accumulation of them is in some way minimized by the estimation algorithm, but there is no exact equivalence to linear regression residuals.\nAnd if there is, they are mostly quite large!\n\n7.3.5 Types of Residuals in Time-to-Event Models\n\nIt is often hard to make a decision from graph appearances, though the process can reveal much.\nSome diagnostic tests are based on residuals as with other regression methods:\n\nSchoenfeld residuals (via cox.zph) for proportionality.\n\nCox-Snell residuals for goodness of fit.\n\nmartingale residuals for non-linearity.\n\ndfbeta for influence.\n\n7.3.6 Schoenfeld residuals\n\nThere is a Schoenfeld residual for each subject \\(i\\) with an event (not censored) and for each predictor \\(x_{k}\\).\nAt the event time \\(t\\) for that subject, there is a risk set \\(R\\), and each subject \\(j\\) in the risk set has a risk coefficient \\(\\theta_j\\) and also a value \\(x_{jk}\\) of the predictor.\nThe Schoenfeld residual is the difference between \\(x_{ik}\\) and the risk-weighted average of all the \\(x_{jk}\\) over the risk set.\n\n\\[\nr^S_{ik} =\nx_{ik}-\\frac{\\sum_{k\\in R}x_{jk}\\theta_k}{\\sum_{k\\in R}\\theta_k}\n\\]\nThis residual measures how typical the individual subject is with respect to the covariate at the time of the event. Since subjects should fail more or less uniformly according to risk, the Schoenfeld residuals should be approximately level over time, not increasing or decreasing.\nWe can test this with the correlation with time on some scale, which could be the time itself, the log time, or the rank in the set of failure times.\nThe default is to use the KM curve as a transform, which is similar to the rank but deals better with censoring.\nThe cox.zph() function implements a score test proposed in Grambsch and Therneau (1994).\n\nShow R codehodg.zph = cox.zph(hodg.cox1)\nprint(hodg.zph)\n#> chisq df p\n#> gtype 0.5400 1 0.462\n#> dtype 1.8012 1 0.180\n#> score 3.8805 1 0.049\n#> wtime 0.0173 1 0.895\n#> gtype:dtype 4.0474 1 0.044\n#> GLOBAL 13.7573 5 0.017\n\n\ngtype\n\nShow R codeggcoxzph(hodg.zph, var = \"gtype\")\n\n\n\n\n\n\n\ndtype\n\nShow R codeggcoxzph(hodg.zph, var = \"dtype\")\n\n\n\n\n\n\n\nscore\n\nShow R codeggcoxzph(hodg.zph, var = \"score\")\n\n\n\n\n\n\n\nwtime\n\nShow R codeggcoxzph(hodg.zph, var = \"wtime\")\n\n\n\n\n\n\n\ngtype:dtype\n\nShow R codeggcoxzph(hodg.zph, var = \"gtype:dtype\")\n\n\n\n\n\n\n\nConclusions\n\nFrom the correlation test, the Karnofsky score and the interaction with graft type disease type induce modest but statistically significant non-proportionality.\nThe sample size here is relatively small (26 events in 43 subjects). If the sample size is large, very small amounts of non-proportionality can induce a significant result.\nAs time goes on, autologous grafts are over-represented at their own event times, but those from HOD patients become less represented.\nBoth the statistical tests and the plots are useful.", "crumbs": [ "Time to Event Models", "7  Building Cox Proportional Hazards models" @@ -917,7 +928,7 @@ "href": "probability.html#random-variables", "title": "Appendix B — Probability", "section": "\nB.1 Random variables", - "text": "B.1 Random variables\n\nB.1.1 Binary variables\n\nDefinition B.1 (binary variable) A binary variable is a random variable which has only two possible values in its range.\n\n\nExercise B.1 (Examples of binary variables) What are some examples of binary variables in the health sciences?\n\n\n\nSolution. Examples of binary outcomes include:\n\nexposure (exposed vs unexposed)\ndisease (diseased vs healthy)\nrecovery (recovered vs unrecovered)\nrelapse (relapse vs remission)\nreturn to hospital (returned vs not)\nvital status (dead vs alive)\n\n\n\nB.1.2 Count variables\n\nDefinition B.2 (Count variable) A count variable is a random variable whose possible values are some subset of the non-negative integers; that is, a random variable \\(X\\) such that:\n\\[\\mathcal{R}{X} \\in \\mathbb{N}\\]\n\n\nExercise B.2 What are some examples of count variables?\n\n\n\nSolution. \n\nNumber of fish in a pond\nNumber of cyclones per season\nSeconds of tooth-brushing per session (if rounded)\nInfections per person-year\nVisits to ER per person-month\nCar accidents per 1000 miles driven\n\n\n\nExposure magnitude\n\nDefinition B.3 (Exposure magnitude)  \n\nFor many count outcomes, there is some sense of exposure magnitude, population size, or duration of observation (Table B.1).\n\n\n\n\nTable B.1: Examples of exposure units\n\n\n\n\n\n\n\noutcome\nexposure units\n\n\n\ninfectious disease incidence\nnumber of individuals exposed, time at risk\n\n\ncar accidents\nmiles driven\n\n\nworksite accidents\nperson-hours worked\n\n\npopulation size\nsize of habitat\n\n\n\n\n\n\n\nExposure units are similar to the number of trials in a binomial distribution, but in non-binomial count outcomes, there can be more than one event per unit of exposure.\nWe can use \\(t\\) to represent continuous-valued exposures/observation durations, and \\(n\\) to represent discrete-valued exposures.\n\n\n\nDefinition B.4 (Event rate)  \n\nWhen the concept of an exposure magnitude is meaningful, the mean of the outcome \\(Y\\) is typically modeled as an event rate (denoted \\(\\lambda\\)) times the exposure magnitude (\\(t\\)). That is:\n\n\\[\\mathbb{E}[Y|T=t] \\stackrel{\\text{def}}{=}\\mu = \\lambda \\cdot t\\]\n\\[\\lambda \\stackrel{\\text{def}}{=}\\frac{\\mu}{t} \\tag{B.1}\\]\n\n\n\nTheorem B.1 When the exposure magnitude is 0, there is no opportunity for events to occur:\n\\[\\mathbb{E}[Y|T=0] = 0\\]\n\n\n\nProof. \\[\\mathbb{E}[Y|T=0] = \\lambda \\cdot 0 = 0\\]\n\nProbability distributions for count outcomes\n\nPoisson distribution\nNegative binomial distribution", + "text": "B.1 Random variables\n\nB.1.1 Binary variables\n\nDefinition B.1 (binary variable) A binary variable is a random variable which has only two possible values in its range.\n\n\nExercise B.1 (Examples of binary variables) What are some examples of binary variables in the health sciences?\n\n\n\nSolution. Examples of binary outcomes include:\n\nexposure (exposed vs unexposed)\ndisease (diseased vs healthy)\nrecovery (recovered vs unrecovered)\nrelapse (relapse vs remission)\nreturn to hospital (returned vs not)\nvital status (dead vs alive)\n\n\n\nB.1.2 Count variables\n\nDefinition B.2 (Count variable) A count variable is a random variable whose possible values are some subset of the non-negative integers; that is, a random variable \\(X\\) such that:\n\\[\\mathcal{R}(X) \\in \\mathbb{N}\\]\n\n\n\nExercise B.2 What are some examples of count variables?\n\n\n\nSolution. \n\nNumber of fish in a pond\nNumber of cyclones per season\nSeconds of tooth-brushing per session (if rounded)\nInfections per person-year\nVisits to ER per person-month\nCar accidents per 1000 miles driven\n\n\n\nExposure magnitude\n\nDefinition B.3 (Exposure magnitude) For many count outcomes, there is some sense of exposure magnitude, population size, or duration of observation.\n\n\n\nExercise B.3 What are some examples of exposure magnitudes?\n\n\n\nSolution. \n\n\nTable B.1: Examples of exposure units\n\n\n\n\n\n\n\noutcome\nexposure units\n\n\n\ndisease incidence\nnumber of individuals exposed; time at risk\n\n\ncar accidents\nmiles driven\n\n\nworksite accidents\nperson-hours worked\n\n\npopulation size\nsize of habitat\n\n\n\n\n\n\n\n\nExposure units are similar to the number of trials in a binomial distribution, but in non-binomial count outcomes, there can be more than one event per unit of exposure.\nWe can use \\(t\\) to represent continuous-valued exposures/observation durations, and \\(n\\) to represent discrete-valued exposures.\n\n\n\nDefinition B.4 (Event rate)  \n\nFor a count outcome \\(Y\\) with exposure magnitude \\(t\\), the event rate (denoted \\(\\lambda\\)) is defined as the mean of \\(Y\\) divided by the the exposure magnitude. That is:\n\n\\[\\mu \\stackrel{\\text{def}}{=}\\mathbb{E}[Y|T=t]\\]\n\\[\\lambda \\stackrel{\\text{def}}{=}\\frac{\\mu}{t} \\tag{B.1}\\]\n\n\nEvent rate is somewhat analogous to odds in binary outcome models; it typically serves as an intermediate transformation between the mean of the outcome and the linear component of the model. However, in contrast with the odds function, the transformation \\(\\lambda = \\mu/t\\) is not considered part of the Poisson model’s link function, and it treats the exposure magnitude covariate differently from the other covariates.\n\n\n\nTheorem B.1 (Transformation function from event rate to mean) For a count variable with mean \\(\\mu\\), event rate \\(\\lambda\\), and exposure magnitude \\(t\\):\n\\[\\therefore\\mu = \\lambda \\cdot t \\tag{B.2}\\]\n\n\n\nSolution. Start from definition of event rate and use algebra to solve for \\(\\mu\\).\n\n\nEquation B.2 is analogous to the inverse-odds function for binary variables.\n\n\nTheorem B.2 When the exposure magnitude is 0, there is no opportunity for events to occur:\n\\[\\mathbb{E}[Y|T=0] = 0\\]\n\n\n\nProof. \\[\\mathbb{E}[Y|T=0] = \\lambda \\cdot 0 = 0\\]\n\nProbability distributions for count outcomes\n\nPoisson distribution\nNegative binomial distribution", "crumbs": [ "Appendices", "B  Probability" @@ -928,7 +939,7 @@ "href": "probability.html#key-probability-distributions", "title": "Appendix B — Probability", "section": "\nB.2 Key probability distributions", - "text": "B.2 Key probability distributions\n\nB.2.1 The Bernoulli distribution\n\nDefinition B.5 (Bernoulli distribution) The Bernoulli distribution family for a random variable \\(X\\) is defined as:\n\\[\n\\begin{aligned}\n\\Pr(X=x) &= \\mathbb{1}_{x\\in \\left\\{0,1\\right\\}}\\pi^x(1-\\pi)^{1-x}\\\\\n&= \\left\\{{\\pi, x=1}\\atop{1-\\pi, x=0}\\right.\n\\end{aligned}\n\\]\n\n\nB.2.2 The Poisson distribution\n\n\nFigure B.1: “Les Poissons”\n\n\n\n\nSiméon Denis Poisson\n\n\n\n\n\nLes Poissons\n\n\n\n\n\n\n\n\n\nDefinition B.6 (Poisson distribution) \\[\\mathcal{R}(Y) = \\left\\{0, 1, 2, ...\\right\\} = \\mathbb{N}\\]\n\\[\\text{P}(Y = y) = \\frac{\\mu^{y} e^{-\\mu}}{y!}, y \\in \\mathbb{N} \\tag{B.2}\\]\n\n(see Figure B.2)\n\n\\[\\text{P}(Y \\le y) = e^{-\\mu} \\sum_{j=0}^{\\left \\lfloor{y}\\right \\rfloor}\\frac{\\mu^j}{j!} \\tag{B.3}\\]\n\n(see Figure B.3)\n\n\n\n\nShow R codelibrary(dplyr)\npois_dists = tibble(\n mu = c(0.5, 1, 2, 5, 10, 20)) |> \n reframe(\n .by = mu,\n x = 0:30\n ) |> \n mutate(\n `P(X = x)` = dpois(x, lambda = mu),\n `P(X <= x)` = ppois(x, lambda = mu),\n mu = factor(mu)\n )\n\nlibrary(ggplot2)\nlibrary(latex2exp)\n\nplot0 = pois_dists |> \n ggplot(\n aes(\n x = x,\n y = `P(X = x)`,\n fill = mu,\n col = mu)) +\n theme(legend.position = \"bottom\") +\n labs(\n fill = latex2exp::TeX(\"$\\\\mu$\"),\n col = latex2exp::TeX(\"$\\\\mu$\"),\n y = latex2exp::TeX(\"$\\\\Pr_{\\\\mu}(X = x)$\"))\n\nplot1 = plot0 + \n geom_col(position = \"identity\", alpha = .5) +\n facet_wrap(~mu)\n # geom_point(alpha = 0.75) +\n # geom_line(alpha = 0.75)\nprint(plot1)\n\n\n\nFigure B.2: Poisson PMFs, by mean parameter \\(\\mu\\)\n\n\n\n\n\n\n\n\n\nShow R codelibrary(ggplot2)\n\nplot2 = \n plot0 + \n geom_step(alpha = 0.75) +\n aes(y = `P(X <= x)`) + \n labs(y = latex2exp::TeX(\"$\\\\Pr_{\\\\mu}(X \\\\leq x)$\"))\n\nprint(plot2)\n\n\n\nFigure B.3: Poisson CDFs\n\n\n\n\n\n\n\n\n\nExercise B.3 (Poisson distribution functions) Let \\(X \\sim \\text{Pois}(\\mu = 3.75)\\).\nCompute:\n\n\\(\\text{P}(X = 4 | \\mu = 3.75)\\)\n\\(\\text{P}(X \\le 7 | \\mu = 3.75)\\)\n\\(\\text{P}(X > 5 | \\mu = 3.75)\\)\n\n\n\n\nSolution. \n\n\\(\\text{P}(X=4) = 0.19378025\\)\n\\(\\text{P}(X\\le 7) = 0.96237866\\)\n\\(\\text{P}(X > 5) = 0.17711717\\)\n\n\n\n\nTheorem B.2 (Properties of the Poisson distribution) If \\(X \\sim \\text{Pois}(\\mu)\\), then:\n\n\\(\\mathbb{E}[Y] = \\mu\\)\n\\(\\text{Var}(Y) = \\mu\\)\n\n\n\nExercise B.4 Prove Theorem B.2.\n\n\n\nSolution. \\[\n\\begin{aligned}\n\\text{E}[X]\n&= \\sum_{x=0}^\\infty x \\cdot P(X=x)\\\\\n&= 0 \\cdot P(X=0) + \\sum_{x=1}^\\infty x \\cdot P(X=x)\\\\\n&= 0 + \\sum_{x=1}^\\infty x \\cdot P(X=x)\\\\\n&= \\sum_{x=1}^\\infty x \\cdot P(X=x)\\\\\n&= \\sum_{x=1}^\\infty x \\cdot \\frac{\\lambda^x e^{-\\lambda}}{x!}\\\\\n&= \\sum_{x=1}^\\infty x \\cdot \\frac{\\lambda^x e^{-\\lambda}}{x \\cdot (x-1)!} & [\\text{definition of factorial (\"!\") function}]\\\\\n&= \\sum_{x=1}^\\infty \\frac{\\lambda^x e^{-\\lambda}}{ (x-1)!}\\\\\n&= \\sum_{x=1}^\\infty \\frac{(\\lambda \\cdot \\lambda^{x-1}) e^{-\\lambda}}{ (x-1)!}\\\\\n&= \\lambda \\cdot \\sum_{x=1}^\\infty \\frac{( \\lambda^{x-1}) e^{-\\lambda}}{ (x-1)!}\\\\\n&= \\lambda \\cdot \\sum_{y=0}^\\infty \\frac{( \\lambda^{y}) e^{-\\lambda}}{ (y)!} &[\\text{substituting } y \\stackrel{\\text{def}}{=}x-1]\\\\\n&= \\lambda \\cdot 1 &[\\text{because PDFs sum to 1}]\\\\\n&= \\lambda\\\\\n\\end{aligned}\n\\]\n\n\nB.2.3 Accounting for exposure\nIf the exposures/observation durations, denoted \\(T=t\\) or \\(N=n\\), vary between observations, we model:\n\\[\\mu = \\lambda\\cdot t\\]\n\\(\\lambda\\) is interpreted as the “expected event rate per unit of exposure”; that is,\n\\[\\lambda = \\frac{\\mathbb{E}[Y|T=t]}{t}\\]\n\n\n\n\n\n\nImportant\n\n\n\nThe exposure magnitude, \\(T\\), is similar to a covariate in linear or logistic regression. However, there is an important difference: in count regression, there is no intercept corresponding to \\(\\mathbb{E}[Y|T=0]\\). In other words, this model assumes that if there is no exposure, there can’t be any events.\n\n\n\nTheorem B.3 If \\(\\mu = \\lambda\\cdot t\\), then:\n\\[\\text{log}\\left\\{\\mu \\right\\}= \\text{log}\\left\\{\\lambda\\right\\} + \\text{log}\\left\\{t\\right\\}\\]\n\n\nDefinition B.7 (Offset) When the linear component of a model involves a term without an unknown coefficient, that term is called an offset.\n\n\nB.2.4 The Negative-Binomial distribution\n{{}}", + "text": "B.2 Key probability distributions\n\nB.2.1 The Bernoulli distribution\n\nDefinition B.5 (Bernoulli distribution) The Bernoulli distribution family for a random variable \\(X\\) is defined as:\n\\[\n\\begin{aligned}\n\\Pr(X=x) &= \\mathbb{1}_{x\\in \\left\\{0,1\\right\\}}\\pi^x(1-\\pi)^{1-x}\\\\\n&= \\left\\{{\\pi, x=1}\\atop{1-\\pi, x=0}\\right.\n\\end{aligned}\n\\]\n\n\nB.2.2 The Poisson distribution\n\n\nFigure B.1: “Les Poissons”\n\n\n\n\nSiméon Denis Poisson\n\n\n\n\n\nLes Poissons\n\n\n\n\n\n\n\n\n\nDefinition B.6 (Poisson distribution) \\[\\mathcal{R}(Y) = \\left\\{0, 1, 2, ...\\right\\} = \\mathbb{N}\\]\n\\[\\text{P}(Y = y) = \\frac{\\mu^{y} e^{-\\mu}}{y!}, y \\in \\mathbb{N} \\tag{B.3}\\]\n\n(see Figure B.2)\n\n\\[\\text{P}(Y \\le y) = e^{-\\mu} \\sum_{j=0}^{\\left \\lfloor{y}\\right \\rfloor}\\frac{\\mu^j}{j!} \\tag{B.4}\\]\n\n(see Figure B.3)\n\n\n\n\nShow R codelibrary(dplyr)\npois_dists = tibble(\n mu = c(0.5, 1, 2, 5, 10, 20)) |> \n reframe(\n .by = mu,\n x = 0:30\n ) |> \n mutate(\n `P(X = x)` = dpois(x, lambda = mu),\n `P(X <= x)` = ppois(x, lambda = mu),\n mu = factor(mu)\n )\n\nlibrary(ggplot2)\nlibrary(latex2exp)\n\nplot0 = pois_dists |> \n ggplot(\n aes(\n x = x,\n y = `P(X = x)`,\n fill = mu,\n col = mu)) +\n theme(legend.position = \"bottom\") +\n labs(\n fill = latex2exp::TeX(\"$\\\\mu$\"),\n col = latex2exp::TeX(\"$\\\\mu$\"),\n y = latex2exp::TeX(\"$\\\\Pr_{\\\\mu}(X = x)$\"))\n\nplot1 = plot0 + \n geom_col(position = \"identity\", alpha = .5) +\n facet_wrap(~mu)\n # geom_point(alpha = 0.75) +\n # geom_line(alpha = 0.75)\nprint(plot1)\n\n\n\nFigure B.2: Poisson PMFs, by mean parameter \\(\\mu\\)\n\n\n\n\n\n\n\n\n\nShow R codelibrary(ggplot2)\n\nplot2 = \n plot0 + \n geom_step(alpha = 0.75) +\n aes(y = `P(X <= x)`) + \n labs(y = latex2exp::TeX(\"$\\\\Pr_{\\\\mu}(X \\\\leq x)$\"))\n\nprint(plot2)\n\n\n\nFigure B.3: Poisson CDFs\n\n\n\n\n\n\n\n\n\nExercise B.4 (Poisson distribution functions) Let \\(X \\sim \\text{Pois}(\\mu = 3.75)\\).\nCompute:\n\n\\(\\text{P}(X = 4 | \\mu = 3.75)\\)\n\\(\\text{P}(X \\le 7 | \\mu = 3.75)\\)\n\\(\\text{P}(X > 5 | \\mu = 3.75)\\)\n\n\n\n\nSolution. \n\n\\(\\text{P}(X=4) = 0.1938\\)\n\\(\\text{P}(X\\le 7) = 0.9624\\)\n\\(\\text{P}(X > 5) = 0.1771\\)\n\n\n\n\nTheorem B.3 (Properties of the Poisson distribution) If \\(X \\sim \\text{Pois}(\\mu)\\), then:\n\n\\(\\mathbb{E}[Y] = \\mu\\)\n\\(\\text{Var}(Y) = \\mu\\)\n\n\n\nExercise B.5 Prove Theorem B.3.\n\n\n\nSolution. \\[\n\\begin{aligned}\n\\text{E}[X]\n&= \\sum_{x=0}^\\infty x \\cdot P(X=x)\\\\\n&= 0 \\cdot P(X=0) + \\sum_{x=1}^\\infty x \\cdot P(X=x)\\\\\n&= 0 + \\sum_{x=1}^\\infty x \\cdot P(X=x)\\\\\n&= \\sum_{x=1}^\\infty x \\cdot P(X=x)\\\\\n&= \\sum_{x=1}^\\infty x \\cdot \\frac{\\lambda^x e^{-\\lambda}}{x!}\\\\\n&= \\sum_{x=1}^\\infty x \\cdot \\frac{\\lambda^x e^{-\\lambda}}{x \\cdot (x-1)!} & [\\text{definition of factorial (\"!\") function}]\\\\\n&= \\sum_{x=1}^\\infty \\frac{\\lambda^x e^{-\\lambda}}{ (x-1)!}\\\\\n&= \\sum_{x=1}^\\infty \\frac{(\\lambda \\cdot \\lambda^{x-1}) e^{-\\lambda}}{ (x-1)!}\\\\\n&= \\lambda \\cdot \\sum_{x=1}^\\infty \\frac{( \\lambda^{x-1}) e^{-\\lambda}}{ (x-1)!}\\\\\n&= \\lambda \\cdot \\sum_{y=0}^\\infty \\frac{( \\lambda^{y}) e^{-\\lambda}}{ (y)!} &[\\text{substituting } y \\stackrel{\\text{def}}{=}x-1]\\\\\n&= \\lambda \\cdot 1 &[\\text{because PDFs sum to 1}]\\\\\n&= \\lambda\\\\\n\\end{aligned}\n\\]\n\n\nAccounting for exposure\nIf the exposures/observation durations, denoted \\(T=t\\) or \\(N=n\\), vary between observations, we model:\n\\[\\mu = \\lambda\\cdot t\\]\n\\(\\lambda\\) is interpreted as the “expected event rate per unit of exposure”; that is,\n\\[\\lambda = \\frac{\\mathbb{E}[Y|T=t]}{t}\\]\n\n\n\n\n\n\nImportant\n\n\n\nThe exposure magnitude, \\(T\\), is similar to a covariate in linear or logistic regression. However, there is an important difference: in count regression, there is no intercept corresponding to \\(\\mathbb{E}[Y|T=0]\\). In other words, this model assumes that if there is no exposure, there can’t be any events.\n\n\n\nTheorem B.4 If \\(\\mu = \\lambda\\cdot t\\), then:\n\\[\\text{log}\\left\\{\\mu \\right\\}= \\text{log}\\left\\{\\lambda\\right\\} + \\text{log}\\left\\{t\\right\\}\\]\n\n\nDefinition B.7 (Offset) When the linear component of a model involves a term without an unknown coefficient, that term is called an offset.\n\n\nB.2.3 The Negative-Binomial distribution\n\nDefinition B.8 (Negative binomial distribution) \\[\n\\text{P}(Y=y)\n= \\frac{\\mu^y}{y!}\n\\cdot \\frac{\\Gamma(\\rho + y)}{\\Gamma(\\rho) \\cdot (\\rho + \\mu)^y}\n\\cdot \\left(1+\\frac{\\mu}{\\rho}\\right)^{-\\rho}\n\\]\nwhere \\(\\rho\\) is an overdispersion parameter and \\(\\Gamma(x) = (x-1)!\\) for integers \\(x\\).\n\n\nYou don’t need to memorize or understand this expression.\nAs \\(\\rho \\rightarrow \\infty\\), the second term converges to 1 and the third term converges to \\(\\text{exp}\\left\\{-\\mu\\right\\}\\), which brings us back to the Poisson distribution.\n\n\n\nTheorem B.5 If \\(Y \\sim \\text{NegBin}(\\mu, \\rho)\\), then:\n\n\\(\\mathbb{E}[Y] = \\mu\\)\n\\(\\text{Var}\\left(Y\\right) = \\mu + \\frac{\\mu^2}{\\rho} > \\mu\\)\n\n\n\nB.2.4 Weibull Distribution\n\\[\n\\begin{aligned}\np(t)&= \\alpha\\lambda x^{\\alpha-1}\\text{e}^{-\\lambda x^\\alpha}\\\\\nh(t)&=\\alpha\\lambda x^{\\alpha-1}\\\\\nS(t)&=\\text{e}^{-\\lambda x^\\alpha}\\\\\nE(T)&= \\Gamma(1+1/\\alpha)\\cdot \\lambda^{-1/\\alpha}\n\\end{aligned}\n\\]\nWhen \\(\\alpha=1\\) this is the exponential. When \\(\\alpha>1\\) the hazard is increasing and when \\(\\alpha < 1\\) the hazard is decreasing. This provides more flexibility than the exponential.\nWe will see more of this distribution later.", "crumbs": [ "Appendices", "B  Probability" @@ -939,7 +950,7 @@ "href": "probability.html#characteristics-of-probability-distributions", "title": "Appendix B — Probability", "section": "\nB.3 Characteristics of probability distributions", - "text": "B.3 Characteristics of probability distributions\n\nDefinition B.8 (Density function) The density function \\(f(t)\\) or \\(\\text{p}(T=t)\\) for a random variable \\(T\\) at value \\(t\\) can be defined as the derivative of the cumulative probability function \\(P(T\\le t)\\); that is:\n\\[f(t) \\stackrel{\\text{def}}{=}\\frac{\\partial}{\\partial t} \\Pr(T\\le t)\\]\n\n\nDefinition B.9 (Hazard function) The hazard function for a random variable \\(T\\) at value \\(t\\) is the conditional density of \\(T\\) at \\(t\\), given \\(T\\ge t\\); that is:\n\\[h(t) \\stackrel{\\text{def}}{=}p(T=t|T\\ge t)\\]\nIf \\(T\\) represents the time at which an event occurs, then \\(h(t)\\) is the probability that the event occurs at time \\(t\\), given that it has not occurred prior to time \\(t\\).\n\n\nDefinition B.10 (Expectation, expected value, population mean ) The expectation, expected value, or population mean of a continuous random variable \\(X\\), denoted \\(\\mathbb{E}\\left[X\\right]\\), \\(\\mu(X)\\), or \\(\\mu_X\\), is the weighted mean of \\(X\\)’s possible values, weighted by the probability density function of those values:\n\\[\\mathbb{E}\\left[X\\right] = \\int_{x\\in \\mathcal{R}(X)} x \\cdot \\text{p}(X=x)dx\\]\nThe expectation, expected value, or population mean of a discrete random variable \\(X\\), denoted \\(\\mathbb{E}\\left[X\\right]\\), \\(\\mu(X)\\), or \\(\\mu_X\\), is the mean of \\(X\\)’s possible values, weighted by the probability mass function of those values:\n\\[\\mathbb{E}\\left[X\\right] = \\sum_{x \\in \\mathcal{R}(X)} x \\cdot \\text{P}(X=x)\\]\n(c.f. https://en.wikipedia.org/wiki/Expected_value)\n\n\n\nTheorem B.4 (Expectation of the Bernoulli distribution) The expectation of a Bernoulli random variable with parameter \\(\\pi\\) is:\n\\[\\mathbb{E}\\left[X\\right] = \\pi\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n\\mathbb{E}\\left[X\\right]\n&= \\sum_{x\\in \\mathcal{R}(X)} x \\cdot\\text{P}(X=x)\n\\\\&= \\sum_{x\\in \\left\\{0,1\\right\\}} x \\cdot\\text{P}(X=x)\n\\\\&= \\left(0 \\cdot\\text{P}(X=0)\\right) + \\left(1 \\cdot\\text{P}(X=1)\\right)\n\\\\&= \\left(0 \\cdot(1-\\pi)\\right) + \\left(1 \\cdot\\pi\\right)\n\\\\&= 0 + \\pi\n\\\\&= \\pi\n\\end{aligned}\n\\]\n\n\n\nB.3.1 Variance and related characteristics\n\nDefinition B.11 (Variance) The variance of a random variable \\(X\\) is the expectation of the squared difference between \\(X\\) and \\(\\mathbb{E}\\left[X\\right]\\); that is:\n\\[\n\\text{Var}\\left(X\\right) \\stackrel{\\text{def}}{=}\\mathbb{E}\\left[(X-\\mathbb{E}\\left[X\\right])^2\\right]\n\\]\n\n\n\nTheorem B.5 (Simplified expression for variance) \\[\\text{Var}\\left(X\\right)=\\mathbb{E}\\left[X^2\\right] - \\left(\\mathbb{E}\\left[X\\right]\\right)^2\\]\n\n\nProof. By linearity of expectation, we have:\n\\[\n\\begin{aligned}\n\\text{Var}\\left(X\\right)\n&\\stackrel{\\text{def}}{=}\\mathbb{E}\\left[(X-\\mathbb{E}\\left[X\\right])^2\\right]\\\\\n&=\\mathbb{E}\\left[X^2 - 2X\\mathbb{E}\\left[X\\right] + \\left(\\mathbb{E}\\left[X\\right]\\right)^2\\right]\\\\\n&=\\mathbb{E}\\left[X^2\\right] - \\mathbb{E}\\left[2X\\mathbb{E}\\left[X\\right]\\right] + \\mathbb{E}\\left[\\left(\\mathbb{E}\\left[X\\right]\\right)^2\\right]\\\\\n&=\\mathbb{E}\\left[X^2\\right] - 2\\mathbb{E}\\left[X\\right]\\mathbb{E}\\left[X\\right] + \\left(\\mathbb{E}\\left[X\\right]\\right)^2\\\\\n&=\\mathbb{E}\\left[X^2\\right] - \\left(\\mathbb{E}\\left[X\\right]\\right)^2\\\\\n\\end{aligned}\n\\]\n\n\n\n\nDefinition B.12 (Precision) The precision of a random variable \\(X\\), often denoted \\(\\tau(X)\\), \\(\\tau_X\\), or shorthanded as \\(\\tau\\), is the inverse of that random variable’s variance; that is:\n\\[\\tau(X) \\stackrel{\\text{def}}{=}\\left(\\text{Var}\\left(X\\right)\\right)^{-1}\\]\n\n\nDefinition B.13 (Standard deviation) The standard deviation of a random variable \\(X\\) is the square-root of the variance of \\(X\\):\n\\[\\text{SD}\\left(X\\right) \\stackrel{\\text{def}}{=}\\sqrt{\\text{Var}\\left(X\\right)}\\]\n\n\n\nDefinition B.14 (Covariance) For any two one-dimensional random variables, \\(X,Y\\):\n\\[\\text{Cov}\\left(X,Y\\right) \\stackrel{\\text{def}}{=}\\mathbb{E}\\left[(X - \\mathbb{E}\\left[X\\right])(Y - \\mathbb{E}\\left[Y\\right])\\right]\\]\n\n\n\nTheorem B.6 \\[\\text{Cov}\\left(X,Y\\right)= \\mathbb{E}\\left[XY\\right] - \\mathbb{E}\\left[X\\right] \\mathbb{E}\\left[Y\\right]\\]\n\n\n\nProof. Left to the reader.\n\n\n\nLemma B.1 (The covariance of a variable with itself is its variance) For any random variable \\(X\\):\n\\[\\text{Cov}\\left(X,X\\right) = \\text{Var}\\left(X\\right)\\]\n\n\nProof. \\[\n\\begin{aligned}\n\\text{Cov}\\left(X,X\\right) &= E[XX] - E[X]E[X]\n\\\\ &= E[X^2]-(E[X])^2\n\\\\ &= \\text{Var}\\left(X\\right)\n\\end{aligned}\n\\]\n\n\n\nDefinition B.15 (Variance/covariance of a \\(p \\times 1\\) random vector) For a \\(p \\times 1\\) dimensional random vector \\(X\\),\n\\[\n\\begin{aligned}\n\\text{Var}(X)\n&\\stackrel{\\text{def}}{=}\\text{Cov}(X)\\\\\n&\\stackrel{\\text{def}}{=}E[ \\left( X - E\\lbrack X\\rbrack \\right)^{\\top}\\left( X - E\\lbrack X\\rbrack \\right) ]\\\\\n\\end{aligned}\n\\]\n\n\n\nTheorem B.7 (Alternate expression for variance of a random vector) \\[\n\\begin{aligned}\n\\text{Var}\\left(X\\right)\n&= E[ X^{\\top}X ] - {E\\lbrack X\\rbrack}^{\\top}E\\lbrack X\\rbrack\n\\end{aligned}\n\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n\\text{Var}\\left(X\\right)\n&= E[ \\left( X^{\\top} - E\\lbrack X\\rbrack^{\\top} \\right)\\left( X - E\\lbrack X\\rbrack \\right) ]\\\\\n&= E[ X^{\\top}X - E\\lbrack X\\rbrack^{\\top}X - X^{\\top}E\\lbrack X\\rbrack + E\\lbrack X\\rbrack^{\\top}E\\lbrack X\\rbrack ]\\\\\n&= E[ X^{\\top}X ] - E\\lbrack X\\rbrack^{\\top}E\\lbrack X\\rbrack - {E\\lbrack X\\rbrack}^{\\top}E\\lbrack X\\rbrack + E\\lbrack X\\rbrack^{\\top}E\\lbrack X\\rbrack\\\\\n&= E[ X^{\\top}X ] - 2{E\\lbrack X\\rbrack}^{\\top}E\\lbrack X\\rbrack + E\\lbrack X\\rbrack^{\\top}E\\lbrack X\\rbrack\\\\\n&= E[ X^{\\top}X ] - {E\\lbrack X\\rbrack}^{\\top}E\\lbrack X\\rbrack\n\\end{aligned}\n\\]\n\n\n\nTheorem B.8 (Variance of a linear combination) For any set of random variables \\(X_1, \\ldots, X_n\\) and corresponding constants \\(a_1, ... ,a_n\\):\n\\[\\text{Var}\\left(\\sum_{i=1}^na_i X_i\\right) = \\sum_{i=1}^n\\sum_{j=1}^n a_i a_j \\text{Cov}\\left(X_i,X_j\\right)\\]\n\n\n\nProof. Left to the reader…\n\n\n\nLemma B.2 For any two random variables \\(X\\) and \\(Y\\) and scalars \\(a\\) and \\(b\\):\n\\[\\text{Var}\\left(aX + bY\\right) = a^2 \\text{Var}\\left(X\\right) + b^2 \\text{Var}\\left(Y\\right) + 2(a \\cdot b) \\text{Cov}\\left(X,Y\\right)\\]\n\n\n\nProof. Apply Theorem B.8 with \\(n=2\\), \\(X_1 = X\\), and \\(X_2 = Y\\).\nOr, see https://statproofbook.github.io/P/var-lincomb.html\n\n\n\nDefinition B.16 (homoskedastic, heteroskedastic) A random variable \\(Y\\) is homoskedastic (with respect to covariates \\(X\\)) if the variance of \\(Y\\) does not vary with \\(X\\):\n\\[\\text{Var}(Y|X=x) = \\sigma^2, \\forall x\\]\nOtherwise it is heteroskedastic.\n\n\n\nDefinition B.17 (Statistical independence) A set of random variables \\(X_1, \\ldots, X_n\\) are statistically independent if their joint probability is equal to the product of their marginal probabilities:\n\\[\\Pr(X_1=x_1, \\ldots, X_n = x_n) = \\prod_{i=1}^n{\\Pr(X_i=x_i)}\\]\n\n\n\n\n\n\n\n\nTip\n\n\n\nThe symbol for independence, \\(⫫\\), is essentially just \\(\\prod\\) upside-down. So the symbol can remind you of its definition (Definition B.17).\n\n\n\n\n\nDefinition B.18 (Conditional independence) A set of random variables \\(Y_1, \\ldots, Y_n\\) are conditionally statistically independent given a set of covariates \\(X_1, \\ldots, X_n\\) if the joint probability of the \\(Y_i\\)s given the \\(X_i\\)s is equal to the product of their marginal probabilities:\n\\[\\Pr(Y_1=y_1, \\ldots, Y_n = y_n|X_1=x_1, \\ldots, X_n = x_n) = \\prod_{i=1}^n{\\Pr(Y_i=y_i|X_i=x_i)}\\]\n\n\n\nDefinition B.19 (Identically distributed) A set of random variables \\(X_1, \\ldots, X_n\\) are identically distributed if they have the same range \\(\\mathcal{R}(X)\\) and if their marginal distributions \\(\\text{P}(X_1=x_1), ..., \\text{P}(X_n=x_n)\\) are all equal to some shared distribution \\(\\text{P}(X=x)\\):\n\\[\n\\forall i\\in \\left\\{1:n\\right\\}, \\forall x \\in \\mathcal{R}(X): \\text{P}(X_i=x) = \\text{P}(X=x)\n\\]\n\n\n\nDefinition B.20 (Conditionally identically distributed) A set of random variables \\(Y_1, \\ldots, Y_n\\) are conditionally identically distributed given a set of covariates \\(X_1, \\ldots, X_n\\) if \\(Y_1, \\ldots, Y_n\\) have the same range \\(\\mathcal{R}(X)\\) and if the distributions \\(\\text{P}(Y_i=y_i|X_i =x_i)\\) are all equal to the same distribution \\(\\text{P}(Y=y|X=x)\\):\n\\[\n\\text{P}(Y_i=y|X_i=x) = \\text{P}(Y=y|X=x)\n\\]\n\n\n\nDefinition B.21 (Independent and identically distributed) A set of random variables \\(X_1, \\ldots, X_n\\) are independent and identically distributed (shorthand: “\\(X_i\\ \\text{iid}\\)”) if they are statistically independent and identically distributed.\n\n\n\nDefinition B.22 (Conditionally independent and identically distributed) A set of random variables \\(Y_1, \\ldots, Y_n\\) are conditionally independent and identically distributed (shorthand: “\\(Y_i | X_i\\ \\text{ciid}\\)” or just “\\(Y_i |X_i\\ \\text{iid}\\)”) given a set of covariates \\(X_1, \\ldots, X_n\\) if \\(Y_1, \\ldots, Y_n\\) are conditionally independent given \\(X_1, \\ldots, X_n\\) and \\(Y_1, \\ldots, Y_n\\) are identically distributed given \\(X_1, \\ldots, X_n\\).", + "text": "B.3 Characteristics of probability distributions\n\nDefinition B.9 (Density function) The density function \\(f(t)\\) or \\(\\text{p}(T=t)\\) for a random variable \\(T\\) at value \\(t\\) can be defined as the derivative of the cumulative probability function \\(P(T\\le t)\\); that is:\n\\[f(t) \\stackrel{\\text{def}}{=}\\frac{\\partial}{\\partial t} \\Pr(T\\le t)\\]\n\n\nDefinition B.10 (Hazard function) The hazard function for a random variable \\(T\\) at value \\(t\\) is the conditional density of \\(T\\) at \\(t\\), given \\(T\\ge t\\); that is:\n\\[h(t) \\stackrel{\\text{def}}{=}p(T=t|T\\ge t)\\]\nIf \\(T\\) represents the time at which an event occurs, then \\(h(t)\\) is the probability that the event occurs at time \\(t\\), given that it has not occurred prior to time \\(t\\).\n\n\nDefinition B.11 (Expectation, expected value, population mean ) The expectation, expected value, or population mean of a continuous random variable \\(X\\), denoted \\(\\mathbb{E}\\left[X\\right]\\), \\(\\mu(X)\\), or \\(\\mu_X\\), is the weighted mean of \\(X\\)’s possible values, weighted by the probability density function of those values:\n\\[\\mathbb{E}\\left[X\\right] = \\int_{x\\in \\mathcal{R}(X)} x \\cdot \\text{p}(X=x)dx\\]\nThe expectation, expected value, or population mean of a discrete random variable \\(X\\), denoted \\(\\mathbb{E}\\left[X\\right]\\), \\(\\mu(X)\\), or \\(\\mu_X\\), is the mean of \\(X\\)’s possible values, weighted by the probability mass function of those values:\n\\[\\mathbb{E}\\left[X\\right] = \\sum_{x \\in \\mathcal{R}(X)} x \\cdot \\text{P}(X=x)\\]\n(c.f. https://en.wikipedia.org/wiki/Expected_value)\n\n\n\nTheorem B.6 (Expectation of the Bernoulli distribution) The expectation of a Bernoulli random variable with parameter \\(\\pi\\) is:\n\\[\\mathbb{E}\\left[X\\right] = \\pi\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n\\mathbb{E}\\left[X\\right]\n&= \\sum_{x\\in \\mathcal{R}(X)} x \\cdot\\text{P}(X=x)\n\\\\&= \\sum_{x\\in \\left\\{0,1\\right\\}} x \\cdot\\text{P}(X=x)\n\\\\&= \\left(0 \\cdot\\text{P}(X=0)\\right) + \\left(1 \\cdot\\text{P}(X=1)\\right)\n\\\\&= \\left(0 \\cdot(1-\\pi)\\right) + \\left(1 \\cdot\\pi\\right)\n\\\\&= 0 + \\pi\n\\\\&= \\pi\n\\end{aligned}\n\\]\n\n\n\nB.3.1 Variance and related characteristics\n\nDefinition B.12 (Variance) The variance of a random variable \\(X\\) is the expectation of the squared difference between \\(X\\) and \\(\\mathbb{E}\\left[X\\right]\\); that is:\n\\[\n\\text{Var}\\left(X\\right) \\stackrel{\\text{def}}{=}\\mathbb{E}\\left[(X-\\mathbb{E}\\left[X\\right])^2\\right]\n\\]\n\n\n\nTheorem B.7 (Simplified expression for variance) \\[\\text{Var}\\left(X\\right)=\\mathbb{E}\\left[X^2\\right] - \\left(\\mathbb{E}\\left[X\\right]\\right)^2\\]\n\n\nProof. By linearity of expectation, we have:\n\\[\n\\begin{aligned}\n\\text{Var}\\left(X\\right)\n&\\stackrel{\\text{def}}{=}\\mathbb{E}\\left[(X-\\mathbb{E}\\left[X\\right])^2\\right]\\\\\n&=\\mathbb{E}\\left[X^2 - 2X\\mathbb{E}\\left[X\\right] + \\left(\\mathbb{E}\\left[X\\right]\\right)^2\\right]\\\\\n&=\\mathbb{E}\\left[X^2\\right] - \\mathbb{E}\\left[2X\\mathbb{E}\\left[X\\right]\\right] + \\mathbb{E}\\left[\\left(\\mathbb{E}\\left[X\\right]\\right)^2\\right]\\\\\n&=\\mathbb{E}\\left[X^2\\right] - 2\\mathbb{E}\\left[X\\right]\\mathbb{E}\\left[X\\right] + \\left(\\mathbb{E}\\left[X\\right]\\right)^2\\\\\n&=\\mathbb{E}\\left[X^2\\right] - \\left(\\mathbb{E}\\left[X\\right]\\right)^2\\\\\n\\end{aligned}\n\\]\n\n\n\n\nDefinition B.13 (Precision) The precision of a random variable \\(X\\), often denoted \\(\\tau(X)\\), \\(\\tau_X\\), or shorthanded as \\(\\tau\\), is the inverse of that random variable’s variance; that is:\n\\[\\tau(X) \\stackrel{\\text{def}}{=}\\left(\\text{Var}\\left(X\\right)\\right)^{-1}\\]\n\n\nDefinition B.14 (Standard deviation) The standard deviation of a random variable \\(X\\) is the square-root of the variance of \\(X\\):\n\\[\\text{SD}\\left(X\\right) \\stackrel{\\text{def}}{=}\\sqrt{\\text{Var}\\left(X\\right)}\\]\n\n\n\nDefinition B.15 (Covariance) For any two one-dimensional random variables, \\(X,Y\\):\n\\[\\text{Cov}\\left(X,Y\\right) \\stackrel{\\text{def}}{=}\\mathbb{E}\\left[(X - \\mathbb{E}\\left[X\\right])(Y - \\mathbb{E}\\left[Y\\right])\\right]\\]\n\n\n\nTheorem B.8 \\[\\text{Cov}\\left(X,Y\\right)= \\mathbb{E}\\left[XY\\right] - \\mathbb{E}\\left[X\\right] \\mathbb{E}\\left[Y\\right]\\]\n\n\n\nProof. Left to the reader.\n\n\n\nLemma B.1 (The covariance of a variable with itself is its variance) For any random variable \\(X\\):\n\\[\\text{Cov}\\left(X,X\\right) = \\text{Var}\\left(X\\right)\\]\n\n\nProof. \\[\n\\begin{aligned}\n\\text{Cov}\\left(X,X\\right) &= E[XX] - E[X]E[X]\n\\\\ &= E[X^2]-(E[X])^2\n\\\\ &= \\text{Var}\\left(X\\right)\n\\end{aligned}\n\\]\n\n\n\nDefinition B.16 (Variance/covariance of a \\(p \\times 1\\) random vector) For a \\(p \\times 1\\) dimensional random vector \\(X\\),\n\\[\n\\begin{aligned}\n\\text{Var}(X)\n&\\stackrel{\\text{def}}{=}\\text{Cov}(X)\\\\\n&\\stackrel{\\text{def}}{=}E[ \\left( X - E\\lbrack X\\rbrack \\right)^{\\top}\\left( X - E\\lbrack X\\rbrack \\right) ]\\\\\n\\end{aligned}\n\\]\n\n\n\nTheorem B.9 (Alternate expression for variance of a random vector) \\[\n\\begin{aligned}\n\\text{Var}\\left(X\\right)\n&= E[ X^{\\top}X ] - {E\\lbrack X\\rbrack}^{\\top}E\\lbrack X\\rbrack\n\\end{aligned}\n\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n\\text{Var}\\left(X\\right)\n&= E[ \\left( X^{\\top} - E\\lbrack X\\rbrack^{\\top} \\right)\\left( X - E\\lbrack X\\rbrack \\right) ]\\\\\n&= E[ X^{\\top}X - E\\lbrack X\\rbrack^{\\top}X - X^{\\top}E\\lbrack X\\rbrack + E\\lbrack X\\rbrack^{\\top}E\\lbrack X\\rbrack ]\\\\\n&= E[ X^{\\top}X ] - E\\lbrack X\\rbrack^{\\top}E\\lbrack X\\rbrack - {E\\lbrack X\\rbrack}^{\\top}E\\lbrack X\\rbrack + E\\lbrack X\\rbrack^{\\top}E\\lbrack X\\rbrack\\\\\n&= E[ X^{\\top}X ] - 2{E\\lbrack X\\rbrack}^{\\top}E\\lbrack X\\rbrack + E\\lbrack X\\rbrack^{\\top}E\\lbrack X\\rbrack\\\\\n&= E[ X^{\\top}X ] - {E\\lbrack X\\rbrack}^{\\top}E\\lbrack X\\rbrack\n\\end{aligned}\n\\]\n\n\n\nTheorem B.10 (Variance of a linear combination) For any set of random variables \\(X_1, \\ldots, X_n\\) and corresponding constants \\(a_1, ... ,a_n\\):\n\\[\\text{Var}\\left(\\sum_{i=1}^na_i X_i\\right) = \\sum_{i=1}^n\\sum_{j=1}^n a_i a_j \\text{Cov}\\left(X_i,X_j\\right)\\]\n\n\n\nProof. Left to the reader…\n\n\n\nLemma B.2 For any two random variables \\(X\\) and \\(Y\\) and scalars \\(a\\) and \\(b\\):\n\\[\\text{Var}\\left(aX + bY\\right) = a^2 \\text{Var}\\left(X\\right) + b^2 \\text{Var}\\left(Y\\right) + 2(a \\cdot b) \\text{Cov}\\left(X,Y\\right)\\]\n\n\n\nProof. Apply Theorem B.10 with \\(n=2\\), \\(X_1 = X\\), and \\(X_2 = Y\\).\nOr, see https://statproofbook.github.io/P/var-lincomb.html\n\n\n\nDefinition B.17 (homoskedastic, heteroskedastic) A random variable \\(Y\\) is homoskedastic (with respect to covariates \\(X\\)) if the variance of \\(Y\\) does not vary with \\(X\\):\n\\[\\text{Var}(Y|X=x) = \\sigma^2, \\forall x\\]\nOtherwise it is heteroskedastic.\n\n\n\nDefinition B.18 (Statistical independence) A set of random variables \\(X_1, \\ldots, X_n\\) are statistically independent if their joint probability is equal to the product of their marginal probabilities:\n\\[\\Pr(X_1=x_1, \\ldots, X_n = x_n) = \\prod_{i=1}^n{\\Pr(X_i=x_i)}\\]\n\n\n\n\n\n\n\n\nTip\n\n\n\nThe symbol for independence, \\(⫫\\), is essentially just \\(\\prod\\) upside-down. So the symbol can remind you of its definition (Definition B.18).\n\n\n\n\n\nDefinition B.19 (Conditional independence) A set of random variables \\(Y_1, \\ldots, Y_n\\) are conditionally statistically independent given a set of covariates \\(X_1, \\ldots, X_n\\) if the joint probability of the \\(Y_i\\)s given the \\(X_i\\)s is equal to the product of their marginal probabilities:\n\\[\\Pr(Y_1=y_1, \\ldots, Y_n = y_n|X_1=x_1, \\ldots, X_n = x_n) = \\prod_{i=1}^n{\\Pr(Y_i=y_i|X_i=x_i)}\\]\n\n\n\nDefinition B.20 (Identically distributed) A set of random variables \\(X_1, \\ldots, X_n\\) are identically distributed if they have the same range \\(\\mathcal{R}(X)\\) and if their marginal distributions \\(\\text{P}(X_1=x_1), ..., \\text{P}(X_n=x_n)\\) are all equal to some shared distribution \\(\\text{P}(X=x)\\):\n\\[\n\\forall i\\in \\left\\{1:n\\right\\}, \\forall x \\in \\mathcal{R}(X): \\text{P}(X_i=x) = \\text{P}(X=x)\n\\]\n\n\n\nDefinition B.21 (Conditionally identically distributed) A set of random variables \\(Y_1, \\ldots, Y_n\\) are conditionally identically distributed given a set of covariates \\(X_1, \\ldots, X_n\\) if \\(Y_1, \\ldots, Y_n\\) have the same range \\(\\mathcal{R}(X)\\) and if the distributions \\(\\text{P}(Y_i=y_i|X_i =x_i)\\) are all equal to the same distribution \\(\\text{P}(Y=y|X=x)\\):\n\\[\n\\text{P}(Y_i=y|X_i=x) = \\text{P}(Y=y|X=x)\n\\]\n\n\n\nDefinition B.22 (Independent and identically distributed) A set of random variables \\(X_1, \\ldots, X_n\\) are independent and identically distributed (shorthand: “\\(X_i\\ \\text{iid}\\)”) if they are statistically independent and identically distributed.\n\n\n\nDefinition B.23 (Conditionally independent and identically distributed) A set of random variables \\(Y_1, \\ldots, Y_n\\) are conditionally independent and identically distributed (shorthand: “\\(Y_i | X_i\\ \\text{ciid}\\)” or just “\\(Y_i |X_i\\ \\text{iid}\\)”) given a set of covariates \\(X_1, \\ldots, X_n\\) if \\(Y_1, \\ldots, Y_n\\) are conditionally independent given \\(X_1, \\ldots, X_n\\) and \\(Y_1, \\ldots, Y_n\\) are identically distributed given \\(X_1, \\ldots, X_n\\).", "crumbs": [ "Appendices", "B  Probability" @@ -1049,7 +1060,7 @@ "href": "intro-MLEs.html#example-maximum-likelihood-for-tropical-cyclones-in-australia", "title": "Appendix E — Introduction to Maximum Likelihood Inference", "section": "\nE.2 Example: Maximum likelihood for Tropical Cyclones in Australia", - "text": "E.2 Example: Maximum likelihood for Tropical Cyclones in Australia\n(Adapted from Dobson and Barnett (2018) §1.6.5)\n\nE.2.1 Data\nThe cyclones dataset in the dobson package (Table E.1) records the number of tropical cyclones in Northeastern Australia during 13 November-to-April cyclone seasons (more details in Dobson and Barnett (2018) §1.6.5 and help(cyclones, package = \"dobson\")). Figure E.1 graphs the number of cyclones (y-axis) by season (x-axis). Let’s use \\(Y_i\\) to represent these counts, where \\(i\\) is an indexing variable for the seasons and \\(Y_i\\) is the number of cyclones in season \\(i\\).\n\nE.2.2 Exploratory analysis\nSuppose we want to learn about how many cyclones to expect per season.\n\nShow R codelibrary(dobson)\nlibrary(dplyr)\ndata(cyclones)\nlibrary(pander)\npander(cyclones |> relocate(season, .before = everything()))\n\n\nTable E.1: Number of tropical cyclones during a season from November to April in Northeastern Australia\n\n\n\n\n\n\n\n\n\nseason\nyears\nnumber\n\n\n\n1\n1956/7\n6\n\n\n2\n1957/8\n5\n\n\n3\n1958/9\n4\n\n\n4\n1959/60\n6\n\n\n5\n1960/1\n6\n\n\n6\n1961/2\n3\n\n\n7\n1962/3\n12\n\n\n8\n1963/4\n7\n\n\n9\n1964/5\n4\n\n\n10\n1965/6\n2\n\n\n11\n1966/7\n6\n\n\n12\n1967/8\n7\n\n\n13\n1968/9\n4\n\n\n\n\n\n\n\n\n\nShow R codelibrary(ggplot2)\nlibrary(dplyr)\ncyclones |>\n mutate(years = years |> factor(levels = years)) |>\n ggplot(aes(x = years, y = number, group = 1)) +\n geom_point() +\n geom_line() +\n xlab(\"Season\") +\n ylab(\"Number of cyclones\") +\n expand_limits(y = 0) +\n theme(axis.text.x = element_text(vjust = .5, angle = 45))\n\n\n\nFigure E.1: Number of tropical cyclones per season in northeastern Australia, 1956-1969\n\n\n\n\n\n\n\nThere’s no obvious correlation between adjacent seasons, so let’s assume that each season is independent of the others.\nLet’s also assume that they are identically distributed; let’s denote this distribution as \\(P(Y=y)\\) (note that there’s no index \\(i\\) in this expression, since we are assuming the \\(Y_i\\)s are identically distributed). We can visualize the distribution using a bar plot (Figure E.2). Table E.2 provides summary statistics.\n\nShow R codecyclones |>\n ggplot() +\n geom_histogram(aes(x = number)) +\n expand_limits(x = 0) +\n xlab(\"Number of cyclones\") +\n ylab(\"Count (number of seasons)\")\n\n\n\nFigure E.2: Bar plot of cyclones per season\n\n\n\n\n\n\n\n\nShow R coden = nrow(cyclones)\nsumx = cyclones |> pull(number) |> sum()\nxbar = cyclones |> pull(number) |> mean()\n\ncyclones |> table1::table1(x = ~ number)\n\n\nTable E.2: Summary statistics for cyclones data\n\n\n\n\n\n\n\n\n\n\n\nOverall(N=13)\n\n\n\nnumber\n\n\n\nMean (SD)\n5.54 (2.47)\n\n\nMedian [Min, Max]\n6.00 [2.00, 12.0]\n\n\n\n\n\n\n\n\n\n\n\nE.2.3 Model\nWe want to estimate \\(P(Y=y)\\); that is, \\(P(Y=y)\\) is our estimand.\nWe could estimate \\(P(Y=y)\\) for each value of \\(y\\) in \\(0:\\infty\\) separately (“nonparametrically”) using the fraction of our data with \\(Y_i=y\\), but then we would be estimating an infinitely large set of parameters, and we would have low precision. We will probably do better with a parametric model.\n\nExercise E.1 What parametric probability distribution family might we use to model this empirical distribution?\n\n\nSolution. Let’s use the Poisson. The Poisson distribution is appropriate for this data , because the data are counts that could theoretically take any integer value (discrete) in the range \\(0:\\infty\\). Visually, the plot of our data closely resembles a Poisson or binomial distribution. Since cyclones do not have an “upper limit” on the number of events we could potentially observe in one season, the Poisson distribution is more appropriate than the binomial.\n\n\nExercise E.2 Write down the Poisson distribution’s probability mass function.\n\n\nSolution. \\[P(Y=y) = \\frac{\\lambda^{y} e^{-\\lambda}}{y!} \\tag{E.6}\\]\n\n\nE.2.4 Estimating the model parameters using maximum likelihood\nNow, we can estimate the parameter \\(\\lambda\\) for this distribution using maximum likelihood estimation.\nWhat is the likelihood?\n\nExercise E.3 Write down the likelihood (probability mass function or probability density function) of a single observation \\(x\\), according to your model.\n\n\nSolution. \\[\n\\begin{aligned}\n\\mathcal{L}(\\lambda; x)\n&= p(X=x|\\Lambda = \\lambda)\\\\\n&= \\frac{\\lambda^x e^{-\\lambda}}{x!}\\\\\n\\end{aligned}\n\\]\n\n\nExercise E.4 Write down the vector of parameters in your model.\n\n\nSolution. There is only one parameter, \\(\\lambda\\):\n\\[\\theta = (\\lambda)\\]\n\n\nExercise E.5 Write down the population mean and variance of a single observation from your chosen probability model, as a function of the parameters (extra credit - derive them).\n\n\nSolution. \n\nPopulation mean: \\(\\text{E}[X]=\\lambda\\)\n\nPopulation variance: \\(\\text{Var}(X)=\\lambda\\)\n\n\n\n\nExercise E.6 Write down the likelihood of the full dataset.\n\n\nSolution. \\[\n\\begin{aligned}\n\\mathcal{L}(\\lambda; \\tilde x)\n&= P(\\tilde X = \\tilde x) \\\\\n&= P(X_1 = x_1, X_2 = x_2, ..., X_{13} = x_{13}) \\\\\n&= \\prod_{i=1}^{13} P(X_i = x_i) \\\\\n&= \\prod_{i=1}^{13} \\frac{\\lambda^{x_i} e^{-\\lambda}}{x_i!}\n\\end{aligned}\n\\]\n\n\nExercise E.7 Graph the likelihood as a function of \\(\\lambda\\).\n\nSolution. \n\nShow R codelik = function(lambda, y = cyclones$number, n = length(y)) \n{\nlambda^sum(y) * exp(-n*lambda) / prod(factorial(y))\n}\n\nlibrary(ggplot2)\nlik_plot = \nggplot() +\ngeom_function(fun = lik, n = 1001) +\nxlim(min(cyclones$number), max(cyclones$number)) +\nylab(\"likelihood\") +\nxlab('lambda')\n\nprint(lik_plot)\n\n\n\nFigure E.3: Likelihood of Dobson cyclone data\n\n\n\n\n\n\n\n\n\n\nExercise E.8 Write down the log-likelihood of the full dataset.\n\nSolution. \\[\n\\begin{aligned}\n\\ell(\\lambda; \\tilde{x}) &= \\text{log}\\left\\{\\mathcal{L}(\\lambda;\\tilde{x})\\right\\}\\\\\n&= \\text{log}\\left\\{\\prod_{i = 1}^n\\frac{\\lambda^{x_i}\\text{e}^{-\\lambda}}{x_i!}\\right\\}\\\\\n&= \\sum_{i = 1}^n\\text{log}\\left\\{\\frac{\\lambda^{x_i}\\text{e}^{-\\lambda}}{x_i!}\\right\\}\\\\\n&= \\sum_{i = 1}^n{\\text{log}\\left\\{\\lambda^{x_i}\\right\\} +\\text{log}\\left\\{\\text{e}^{-\\lambda}\\right\\} - \\text{log}\\left\\{x_i!\\right\\}}\\\\\n&= \\sum_{i = 1}^n{x_i\\text{log}\\left\\{\\lambda\\right\\} -\\lambda - \\text{log}\\left\\{x_i!\\right\\}}\\\\\n&= \\sum_{i = 1}^nx_i\\text{log}\\left\\{\\lambda\\right\\} - \\sum_{i = 1}^n\\lambda - \\sum_{i = 1}^n\\text{log}\\left\\{x_i!\\right\\}\\\\\n&= \\sum_{i = 1}^nx_i\\text{log}\\left\\{\\lambda\\right\\} - n\\lambda - \\sum_{i = 1}^n\\text{log}\\left\\{x_i!\\right\\}\\\\\n\\end{aligned}\n\\]\n\n\n\nExercise E.9 Graph the log-likelihood as a function of \\(\\lambda\\).\n\nSolution. \n\nShow R code\nloglik = function(lambda, y = cyclones$number, n = length(y))\n{\nsum(y) * log(lambda) - n*lambda - sum(log(factorial(y)))\n}\n\nll_plot = ggplot() +\ngeom_function(fun = loglik, n = 1001) +\nxlim(min(cyclones$number), max(cyclones$number)) +\nylab(\"log-likelihood\") +\nxlab('lambda')\nll_plot\n\n\n\nFigure E.4: log-likelihood of Dobson cyclone data\n\n\n\n\n\n\n\n\n\nThe score function\n\nExercise E.10 Derive the score function for the dataset.\n\nSolution. The score function is the first derivative of the log-likelihood:\n\\[\n\\begin{aligned}\n\\ell'( \\lambda; \\tilde{x} ) &=\n\\frac{\\partial}{\\partial \\lambda}{\\sum_{i = 1}^nx_i\\text{log}\\left\\{\\lambda\\right\\} - n\\lambda - \\sum_{i = 1}^n\\text{log}\\left\\{x_i!\\right\\}}\\\\\n&= \\frac{\\partial}{\\partial \\lambda}\\sum_{i = 1}^nx_i\\text{log}\\left\\{\\lambda\\right\\} - \\frac{\\partial}{\\partial \\lambda}n\\lambda - \\frac{\\partial}{\\partial \\lambda}\\sum_{i = 1}^n\\text{log}\\left\\{x_i!\\right\\}\\\\\n&= \\sum_{i = 1}^nx_i\\frac{\\partial}{\\partial \\lambda}\\text{log}\\left\\{\\lambda\\right\\} - n\\frac{\\partial}{\\partial \\lambda}\\lambda - \\sum_{i = 1}^n\\frac{\\partial}{\\partial \\lambda}\\text{log}\\left\\{x_i!\\right\\}\\\\\n&= \\sum_{i = 1}^nx_i\\frac{1}{\\lambda} - n - 0\\\\\n&= \\frac{1}{\\lambda} \\sum_{i = 1}^nx_i- n\n\\\\&= \\left(\\frac{1}{\\lambda} n \\bar{x}\\right) - n\n\\\\&= \\left(\\frac{1}{\\lambda} 72\\right) - 13\n\\end{aligned}\n\\]\n\n\n\nExercise E.11 Graph the score function.\n\nSolution. \n\nShow R code\nscore = function(lambda, y = cyclones$number, n = length(y))\n{\n (sum(y) / lambda) - n\n}\n\nggplot() +\ngeom_function(fun = score, n = 1001) +\nxlim(min(cyclones$number), max(cyclones$number)) +\n\nylab(\"l'(lambda)\") +\nxlab('lambda') +\ngeom_hline(yintercept = 0, col = 'red')\n\n\n\nFigure E.5: score function of Dobson cyclone data\n\n\n\n\n\n\n\n\n\nThe Hessian matrix\n\nExercise E.12 Derive the Hessian matrix.\n\nSolution. The Hessian function for an iid sample is the 2nd derivative(s) of the log-likelihood:\n\\[\n\\begin{aligned}\n\\ell''( \\lambda; \\tilde{x} ) &= \\frac{\\partial}{\\partial \\lambda}\\left(\\frac{1}{\\lambda} \\sum_{i = 1}^nx_i- n\\right)\\\\\n&= \\frac{\\partial}{\\partial \\lambda}\\frac{1}{\\lambda} \\sum_{i = 1}^nx_i- \\frac{\\partial}{\\partial \\lambda}n\\\\\n&= -\\frac{1}{\\lambda^2} \\sum_{i = 1}^nx_i\\\\\n&= -\\frac{1}{\\lambda^2} n \\bar x\n\\\\&= -\\frac{1}{\\lambda^2} \\cdot 72\n\\end{aligned}\n\\]\n\n\n\nExercise E.13 Graph the Hessian.\n\nSolution. \n\nShow R code\nhessian = function(lambda, y = cyclones$number, n = length(y))\n{\n-sum(y)/(lambda^2)\n}\n\nggplot() +\ngeom_function(fun = hessian, n = 1001) +\nxlim(min(cyclones$number), max(cyclones$number)) +\n\nylab(\"l''(lambda)\") +\nxlab('lambda') +\ngeom_hline(yintercept = 0, col = 'red')\n\n\n\nFigure E.6: Hessian function of Dobson cyclone data\n\n\n\n\n\n\n\n\n\n\nExercise E.14 Write the score equation (estimating equation).\n\nSolution. \\[\\ell'( \\lambda; \\tilde{x} ) = 0\\]\n\n\n\nExercise E.15 Solve the estimating equation for \\(\\lambda\\):\n\nSolution. \\[\n\\begin{aligned}\n0 &= \\frac{1}{\\lambda}\\sum_{i = 1}^nx_i - n\\\\\nn &= \\frac{1}{\\lambda}\\sum_{i = 1}^nx_i\\\\\nn\\lambda &= \\sum_{i = 1}^nx_i\\\\\n\\lambda &=\n\\frac{1}{n}\\sum_{i = 1}^nx_i\\\\\n&=\\bar x\n\\end{aligned}\n\\]\n\n\nLet’s call this solution of the estimating equation \\(\\tilde \\lambda\\) for now:\n\\[\\tilde \\lambda \\stackrel{\\text{def}}{=}\\bar x\\]\n\nExercise E.16 Confirm that the Hessian \\(\\ell''(\\lambda; \\tilde{x})\\) is negative when evaluated at \\(\\tilde \\lambda\\).\n\nSolution. \\[\n\\begin{aligned}\n\\ell''( \\tilde\\lambda; \\tilde{x} ) &=\n-\\frac{1}{\\tilde\\lambda^2} n \\bar x\\\\\n&= -\\frac{1}{\\bar x^2} n\\bar x\\\\\n&= -\\frac{n}{\\bar x}\\\\\n&<0\\\\\n\\end{aligned}\n\\]\n\n\n\nExercise E.17 Find the MLE of \\(\\lambda\\).\n\nSolution. Since \\(\\ell''(\\tilde \\lambda; \\tilde{x})<0\\), \\(\\tilde \\lambda\\) is at least a local maximizer of the likelihood function \\(\\mathcal L(\\lambda)\\). Since there is only one solution to the estimating equation and the Hessian is negative definite everywhere, \\(\\tilde \\lambda\\) must also be the global maximizer of \\(\\mathcal L(\\lambda; \\tilde{x})\\):\n\nShow R codemle = mean(cyclones$number)\n\n\n\\[\\hat{\\lambda}_{\\text{ML}} = \\bar x = 5.53846154\\]\n\n\n\nExercise E.18 Graph the log-likelihood with the MLE superimposed.\n\nSolution. \n\nShow R codelibrary(dplyr)\n\nmle_data = tibble(x = mle, y = loglik(mle))\nll_plot + geom_point(data = mle_data, aes(x = x, y = y), col = 'red')\n\n\n\nFigure E.7: log-likelihood of Dobson cyclone data with MLE\n\n\n\n\n\n\n\n\n\nInformation matrices\n\nShow R codeobs_inf = function(...) -hessian(...)\nggplot() +\ngeom_function(fun = obs_inf, n = 1001) +\nxlim(min(cyclones$number), max(cyclones$number)) +\nylab(\"I(lambda)\") +\nxlab('lambda') +\ngeom_hline(yintercept = 0, col = 'red') \n\n\n\nFigure E.8: Observed information function of Dobson cyclone data\n\n\n\n\n\n\n\n\n\nExample E.1 (Finding the MLE using the Newton-Raphson algorithm)  \n\nWe found that the MLE was \\(\\hat{\\lambda} = \\bar{x}\\), by solving the score equation \\(\\ell'(\\lambda)=0\\) for \\(\\lambda\\).\nWhat if we hadn’t been able to solve the score equation?\nThen we could start with some initial guess \\({\\widehat{\\lambda}}^*\\), such as \\({\\widehat{\\lambda}}^*= 3\\), and use the Newton-Raphson algorithm.\n\n\nShow R code# specify initial guess:\ncur_lambda_est = 3\n\n\n\n\n\nIn Exercise E.10, we found that the score function was:\n\n\\[\n\\ell'( \\lambda; \\tilde{x} ) = \\left(\\frac{72}{\\lambda} \\right) - n\n\\]\n\nIn Exercise E.12, we found that the Hessian was:\n\n\\[\n\\ell''( \\lambda; \\tilde{x} ) = -\\frac{72}{\\lambda^2}\n\\]\n\n\nSo we can approximate the the score function using the first-order Taylor polynomial:\n\n\\[\n\\begin{aligned}\n\\ell'(\\lambda)\n&\\approx \\ell'^*(\\lambda)\n\\\\ &\\stackrel{\\text{def}}{=}\\ell'({\\widehat{\\lambda}}^*) + \\ell''({\\widehat{\\lambda}}^*)(\\lambda - {\\widehat{\\lambda}}^*)\n\\\\ &= \\left(\\frac{72}{{\\widehat{\\lambda}}^*} - n\\right) + \\left(-\\frac{72}{\\left({\\widehat{\\lambda}}^*\\right)^2}\\right) (\\lambda - {\\widehat{\\lambda}}^*)\n\\end{aligned}\n\\]\n\n\nFigure E.9 compares the true score function and the approximate score function at \\({\\widehat{\\lambda}}^*= 3\\).\n\n\nShow R codeapprox_score = function(lambda, lhat, ...)\n{\n score(lambda = lhat, ...) +\n hessian(lambda = lhat, ...) * (lambda - lhat)\n}\n\npoint_size = 5\n\nplot1 = ggplot() +\ngeom_function(\n fun = score, \n aes(col = \"true score function\"), \n n = 1001) +\ngeom_function(\n fun = approx_score, \n aes(col = \"approximate score function\"),\n n = 1001, \n args = list(lhat = cur_lambda_est)) +\ngeom_point(\n size = point_size,\n aes(x = cur_lambda_est, y = score(lambda = cur_lambda_est),\n col = \"current estimate\")\n) +\ngeom_point(\nsize = point_size,\naes(\nx = xbar,\ny = 0,\ncol = \"true MLE\"\n)\n) +\nxlim(min(cyclones$number), max(cyclones$number)) +\nylab(\"l'(lambda)\") +\nxlab('lambda') +\ngeom_hline(yintercept = 0)\n\nprint(plot1)\n\n\n\nFigure E.9: score function of Dobson cyclone data and approximate score function\n\n\n\n\n\n\n\n\n\nThis is equivalent to estimating the log-likelihood with a second-order Taylor polynomial:\n\n\\[\n\\ell^*(\\lambda) =\n\\ell({\\widehat{\\lambda}}^*) +\n(\\lambda - {\\widehat{\\lambda}}^*) \\ell'({\\widehat{\\lambda}}^*) +\n\\frac{1}{2}\\ell''({\\widehat{\\lambda}}^*)(\\lambda-{\\widehat{\\lambda}}^*)^2\n\\]\n\nShow R codeapprox_loglik = function(lambda, lhat, ...)\n{\nloglik(lambda = lhat, ...) +\nscore(lambda = lhat, ...) * (lambda - lhat) +\n 1/2 * hessian(lambda = lhat, ...) * (lambda - lhat)^2\n}\n\nplot_loglik = ggplot() +\ngeom_function(\n fun = loglik, \n aes(col = \"true log-likelihood\"), \n n = 1001) +\ngeom_function(\n fun = approx_loglik, \n aes(col = \"approximate log-likelihood\"),\n n = 1001, \n args = list(lhat = cur_lambda_est)) +\ngeom_point(\n size = point_size,\n aes(x = cur_lambda_est, y = loglik(lambda = cur_lambda_est),\n col = \"current estimate\")\n) +\ngeom_point(\nsize = point_size,\naes(\nx = xbar,\ny = loglik(xbar),\ncol = \"true MLE\"\n)\n) +\nxlim(min(cyclones$number) - 1, max(cyclones$number)) +\nylab(\"l'(lambda)\") +\nxlab('lambda')\n\nprint(plot_loglik)\n\n\n\nFigure E.10: log-likelihood of Dobson cyclone data and approximate log-likelihood function\n\n\n\n\n\n\n\n\n\nThe approximate score function, \\(\\ell'^*(\\lambda)\\), is a linear function of \\(\\lambda\\), so it is easy to solve the corresponding approximate score equation, \\(\\ell'^*(\\lambda) = 0\\), for \\(\\lambda\\):\n\n\\[\n\\begin{aligned}\n\\lambda\n&= {\\widehat{\\lambda}}^*- \\ell'({\\widehat{\\lambda}}^*) \\cdot\\left(\\ell''({\\widehat{\\lambda}}^*)\\right)^{-1}\n\\\\ &= 4.375\n\\end{aligned}\n\\]\n\nShow R codenew_lambda_est <- \n cur_lambda_est - \n score(cur_lambda_est) * hessian(cur_lambda_est)^-1\n\n\n\n\nShow R codeplot2 = plot1 + \ngeom_point(\nsize = point_size,\naes(\nx = new_lambda_est,\ny = 0,\ncol = \"new estimate\"\n)) +\ngeom_segment(\narrow = grid::arrow(),\nlinewidth = 2,\nalpha = .7,\naes(\nx = cur_lambda_est, \ny = approx_score(\nlhat = cur_lambda_est, \nlambda = cur_lambda_est),\nxend = new_lambda_est, \nyend = 0,\ncol = \"update\"\n)\n)\nprint(plot2)\n\n\n\nFigure E.11: score function of Dobson cyclone data and approximate score function\n\n\n\n\n\n\n\n\n\nSo we update \\({\\widehat{\\lambda}}^*\\leftarrow 4.375\\) and repeat our estimation process:\n\n\nShow R codeplot2 +\ngeom_function(\n fun = approx_score, \n aes(col = \"new approximate score function\"),\n n = 1001, \n args = list(lhat = new_lambda_est)) +\ngeom_point(\n size = point_size,\n aes(x = new_lambda_est, y = score(lambda = new_lambda_est),\n col = \"new estimate\")\n)\n\n\n\nFigure E.12: score function of Dobson cyclone data and approximate score function\n\n\n\n\n\n\n\n\n\nWe repeat this process until the likelihood converges:\n\n\nShow R codelibrary(tibble)\ncur_lambda_est = 3 # restarting\ndiff_loglik = Inf\ntolerance = 10^-4\nmax_iter = 100\nNR_info = tibble(\n iteration = 0,\n lambda = cur_lambda_est |> num(digits = 4),\n likelihood = lik(cur_lambda_est),\n `log(likelihood)` = loglik(cur_lambda_est) |> num(digits = 4),\n score = score(cur_lambda_est),\n hessian = hessian(cur_lambda_est)\n)\n\nfor (cur_iter in 1:max_iter)\n{\n\nnew_lambda_est <- \n cur_lambda_est - score(cur_lambda_est) * hessian(cur_lambda_est)^-1\n\ndiff_loglik = loglik(new_lambda_est) - loglik(cur_lambda_est)\n\nnew_NR_info = tibble(\n iteration = cur_iter,\n lambda = new_lambda_est,\n likelihood = lik(new_lambda_est),\n `log(likelihood)` = loglik(new_lambda_est),\n score = score(new_lambda_est),\n hessian = hessian(new_lambda_est),\n `diff(loglik)` = diff_loglik\n)\n\nNR_info = NR_info |> bind_rows(new_NR_info)\n\ncur_lambda_est = new_lambda_est\n\nif(abs(diff_loglik) < tolerance) break\n\n}\n\nNR_info\n\n\nTable E.3: Convergence of Newton-Raphson Algorithm for finding MLE of cyclone data\n\n\n\n \n\n\n\n\n\n\n\nCompare with Exercise E.17\n\n\n\nShow R codell_plot + \ngeom_segment(\ndata = NR_info,\narrow = grid::arrow(),\n# linewidth = 2,\nalpha = .7,\naes(\nx = lambda,\nxend = lead(lambda),\ny = `log(likelihood)`,\nyend = lead(`log(likelihood)`),\ncol = factor(iteration)\n)\n)\n\n\n\nFigure E.13: Newton-Raphson algorithm for finding MLE of model E.6 for cyclone data", + "text": "E.2 Example: Maximum likelihood for Tropical Cyclones in Australia\n(Adapted from Dobson and Barnett (2018) §1.6.5)\n\nE.2.1 Data\nThe cyclones dataset in the dobson package (Table E.1) records the number of tropical cyclones in Northeastern Australia during 13 November-to-April cyclone seasons (more details in Dobson and Barnett (2018) §1.6.5 and help(cyclones, package = \"dobson\")). Figure E.1 graphs the number of cyclones (y-axis) by season (x-axis). Let’s use \\(Y_i\\) to represent these counts, where \\(i\\) is an indexing variable for the seasons and \\(Y_i\\) is the number of cyclones in season \\(i\\).\n\nE.2.2 Exploratory analysis\nSuppose we want to learn about how many cyclones to expect per season.\n\nShow R codelibrary(dobson)\nlibrary(dplyr)\ndata(cyclones)\nlibrary(pander)\npander(cyclones |> relocate(season, .before = everything()))\n\n\nTable E.1: Number of tropical cyclones during a season from November to April in Northeastern Australia\n\n\n\n\n\n\n\n\n\nseason\nyears\nnumber\n\n\n\n1\n1956/7\n6\n\n\n2\n1957/8\n5\n\n\n3\n1958/9\n4\n\n\n4\n1959/60\n6\n\n\n5\n1960/1\n6\n\n\n6\n1961/2\n3\n\n\n7\n1962/3\n12\n\n\n8\n1963/4\n7\n\n\n9\n1964/5\n4\n\n\n10\n1965/6\n2\n\n\n11\n1966/7\n6\n\n\n12\n1967/8\n7\n\n\n13\n1968/9\n4\n\n\n\n\n\n\n\n\n\nShow R codelibrary(ggplot2)\nlibrary(dplyr)\ncyclones |>\n mutate(years = years |> factor(levels = years)) |>\n ggplot(aes(x = years, y = number, group = 1)) +\n geom_point() +\n geom_line() +\n xlab(\"Season\") +\n ylab(\"Number of cyclones\") +\n expand_limits(y = 0) +\n theme(axis.text.x = element_text(vjust = .5, angle = 45))\n\n\n\nFigure E.1: Number of tropical cyclones per season in northeastern Australia, 1956-1969\n\n\n\n\n\n\n\nThere’s no obvious correlation between adjacent seasons, so let’s assume that each season is independent of the others.\nLet’s also assume that they are identically distributed; let’s denote this distribution as \\(P(Y=y)\\) (note that there’s no index \\(i\\) in this expression, since we are assuming the \\(Y_i\\)s are identically distributed). We can visualize the distribution using a bar plot (Figure E.2). Table E.2 provides summary statistics.\n\nShow R codecyclones |>\n ggplot() +\n geom_histogram(aes(x = number)) +\n expand_limits(x = 0) +\n xlab(\"Number of cyclones\") +\n ylab(\"Count (number of seasons)\")\n\n\n\nFigure E.2: Bar plot of cyclones per season\n\n\n\n\n\n\n\n\nShow R coden = nrow(cyclones)\nsumx = cyclones |> pull(number) |> sum()\nxbar = cyclones |> pull(number) |> mean()\n\ncyclones |> table1::table1(x = ~ number)\n\n\nTable E.2: Summary statistics for cyclones data\n\n\n\n\n\n\n\n\n\n\n\nOverall(N=13)\n\n\n\nnumber\n\n\n\nMean (SD)\n5.54 (2.47)\n\n\nMedian [Min, Max]\n6.00 [2.00, 12.0]\n\n\n\n\n\n\n\n\n\n\n\nE.2.3 Model\nWe want to estimate \\(P(Y=y)\\); that is, \\(P(Y=y)\\) is our estimand.\nWe could estimate \\(P(Y=y)\\) for each value of \\(y\\) in \\(0:\\infty\\) separately (“nonparametrically”) using the fraction of our data with \\(Y_i=y\\), but then we would be estimating an infinitely large set of parameters, and we would have low precision. We will probably do better with a parametric model.\n\nExercise E.1 What parametric probability distribution family might we use to model this empirical distribution?\n\n\nSolution. Let’s use the Poisson. The Poisson distribution is appropriate for this data , because the data are counts that could theoretically take any integer value (discrete) in the range \\(0:\\infty\\). Visually, the plot of our data closely resembles a Poisson or binomial distribution. Since cyclones do not have an “upper limit” on the number of events we could potentially observe in one season, the Poisson distribution is more appropriate than the binomial.\n\n\nExercise E.2 Write down the Poisson distribution’s probability mass function.\n\n\nSolution. \\[P(Y=y) = \\frac{\\lambda^{y} e^{-\\lambda}}{y!} \\tag{E.6}\\]\n\n\nE.2.4 Estimating the model parameters using maximum likelihood\nNow, we can estimate the parameter \\(\\lambda\\) for this distribution using maximum likelihood estimation.\nWhat is the likelihood?\n\nExercise E.3 Write down the likelihood (probability mass function or probability density function) of a single observation \\(x\\), according to your model.\n\n\nSolution. \\[\n\\begin{aligned}\n\\mathcal{L}(\\lambda; x)\n&= p(X=x|\\Lambda = \\lambda)\\\\\n&= \\frac{\\lambda^x e^{-\\lambda}}{x!}\\\\\n\\end{aligned}\n\\]\n\n\nExercise E.4 Write down the vector of parameters in your model.\n\n\nSolution. There is only one parameter, \\(\\lambda\\):\n\\[\\theta = (\\lambda)\\]\n\n\nExercise E.5 Write down the population mean and variance of a single observation from your chosen probability model, as a function of the parameters (extra credit - derive them).\n\n\nSolution. \n\nPopulation mean: \\(\\text{E}[X]=\\lambda\\)\n\nPopulation variance: \\(\\text{Var}(X)=\\lambda\\)\n\n\n\n\nExercise E.6 Write down the likelihood of the full dataset.\n\n\nSolution. \\[\n\\begin{aligned}\n\\mathcal{L}(\\lambda; \\tilde{x})\n&= \\text{P}(\\tilde{X}= \\tilde{x}) \\\\\n&= \\text{P}(X_1 = x_1, X_2 = x_2, ..., X_{13} = x_{13}) \\\\\n&= \\prod_{i=1}^{13} \\text{P}(X_i = x_i) \\\\\n&= \\prod_{i=1}^{13} \\frac{\\lambda^{x_i} e^{-\\lambda}}{x_i!}\n\\end{aligned}\n\\]\n\n\nExercise E.7 Graph the likelihood as a function of \\(\\lambda\\).\n\nSolution. \n\nShow R codelik = function(lambda, y = cyclones$number, n = length(y)) \n{\nlambda^sum(y) * exp(-n*lambda) / prod(factorial(y))\n}\n\nlibrary(ggplot2)\nlik_plot = \nggplot() +\ngeom_function(fun = lik, n = 1001) +\nxlim(min(cyclones$number), max(cyclones$number)) +\nylab(\"likelihood\") +\nxlab('lambda')\n\nprint(lik_plot)\n\n\n\nFigure E.3: Likelihood of Dobson cyclone data\n\n\n\n\n\n\n\n\n\n\nExercise E.8 Write down the log-likelihood of the full dataset.\n\nSolution. \\[\n\\begin{aligned}\n\\ell(\\lambda; \\tilde{x}) &= \\text{log}\\left\\{\\mathcal{L}(\\lambda;\\tilde{x})\\right\\}\\\\\n&= \\text{log}\\left\\{\\prod_{i = 1}^n\\frac{\\lambda^{x_i}\\text{e}^{-\\lambda}}{x_i!}\\right\\}\\\\\n&= \\sum_{i = 1}^n\\text{log}\\left\\{\\frac{\\lambda^{x_i}\\text{e}^{-\\lambda}}{x_i!}\\right\\}\\\\\n&= \\sum_{i = 1}^n{\\text{log}\\left\\{\\lambda^{x_i}\\right\\} +\\text{log}\\left\\{\\text{e}^{-\\lambda}\\right\\} - \\text{log}\\left\\{x_i!\\right\\}}\\\\\n&= \\sum_{i = 1}^n{x_i\\text{log}\\left\\{\\lambda\\right\\} -\\lambda - \\text{log}\\left\\{x_i!\\right\\}}\\\\\n&= \\sum_{i = 1}^nx_i\\text{log}\\left\\{\\lambda\\right\\} - \\sum_{i = 1}^n\\lambda - \\sum_{i = 1}^n\\text{log}\\left\\{x_i!\\right\\}\\\\\n&= \\sum_{i = 1}^nx_i\\text{log}\\left\\{\\lambda\\right\\} - n\\lambda - \\sum_{i = 1}^n\\text{log}\\left\\{x_i!\\right\\}\\\\\n\\end{aligned}\n\\]\n\n\n\nExercise E.9 Graph the log-likelihood as a function of \\(\\lambda\\).\n\nSolution. \n\nShow R code\nloglik = function(lambda, y = cyclones$number, n = length(y))\n{\nsum(y) * log(lambda) - n*lambda - sum(log(factorial(y)))\n}\n\nll_plot = ggplot() +\ngeom_function(fun = loglik, n = 1001) +\nxlim(min(cyclones$number), max(cyclones$number)) +\nylab(\"log-likelihood\") +\nxlab('lambda')\nll_plot\n\n\n\nFigure E.4: log-likelihood of Dobson cyclone data\n\n\n\n\n\n\n\n\n\nThe score function\n\nExercise E.10 Derive the score function for the dataset.\n\nSolution. The score function is the first derivative of the log-likelihood:\n\\[\n\\begin{aligned}\n\\ell'( \\lambda; \\tilde{x} ) &=\n\\frac{\\partial}{\\partial \\lambda}{\\sum_{i = 1}^nx_i\\text{log}\\left\\{\\lambda\\right\\} - n\\lambda - \\sum_{i = 1}^n\\text{log}\\left\\{x_i!\\right\\}}\\\\\n&= \\frac{\\partial}{\\partial \\lambda}\\sum_{i = 1}^nx_i\\text{log}\\left\\{\\lambda\\right\\} - \\frac{\\partial}{\\partial \\lambda}n\\lambda - \\frac{\\partial}{\\partial \\lambda}\\sum_{i = 1}^n\\text{log}\\left\\{x_i!\\right\\}\\\\\n&= \\sum_{i = 1}^nx_i\\frac{\\partial}{\\partial \\lambda}\\text{log}\\left\\{\\lambda\\right\\} - n\\frac{\\partial}{\\partial \\lambda}\\lambda - \\sum_{i = 1}^n\\frac{\\partial}{\\partial \\lambda}\\text{log}\\left\\{x_i!\\right\\}\\\\\n&= \\sum_{i = 1}^nx_i\\frac{1}{\\lambda} - n - 0\\\\\n&= \\frac{1}{\\lambda} \\sum_{i = 1}^nx_i- n\n\\\\&= \\left(\\frac{1}{\\lambda} n \\bar{x}\\right) - n\n\\\\&= \\left(\\frac{1}{\\lambda} 72\\right) - 13\n\\end{aligned}\n\\]\n\n\n\nExercise E.11 Graph the score function.\n\nSolution. \n\nShow R code\nscore = function(lambda, y = cyclones$number, n = length(y))\n{\n (sum(y) / lambda) - n\n}\n\nggplot() +\ngeom_function(fun = score, n = 1001) +\nxlim(min(cyclones$number), max(cyclones$number)) +\n\nylab(\"l'(lambda)\") +\nxlab('lambda') +\ngeom_hline(yintercept = 0, col = 'red')\n\n\n\nFigure E.5: score function of Dobson cyclone data\n\n\n\n\n\n\n\n\n\nThe Hessian matrix\n\nExercise E.12 Derive the Hessian matrix.\n\nSolution. The Hessian function for an iid sample is the 2nd derivative(s) of the log-likelihood:\n\\[\n\\begin{aligned}\n\\ell''( \\lambda; \\tilde{x} ) &= \\frac{\\partial}{\\partial \\lambda}\\left(\\frac{1}{\\lambda} \\sum_{i = 1}^nx_i- n\\right)\\\\\n&= \\frac{\\partial}{\\partial \\lambda}\\frac{1}{\\lambda} \\sum_{i = 1}^nx_i- \\frac{\\partial}{\\partial \\lambda}n\\\\\n&= -\\frac{1}{\\lambda^2} \\sum_{i = 1}^nx_i\\\\\n&= -\\frac{1}{\\lambda^2} n \\bar x\n\\\\&= -\\frac{1}{\\lambda^2} \\cdot 72\n\\end{aligned}\n\\]\n\n\n\nExercise E.13 Graph the Hessian.\n\nSolution. \n\nShow R code\nhessian = function(lambda, y = cyclones$number, n = length(y))\n{\n-sum(y)/(lambda^2)\n}\n\nggplot() +\ngeom_function(fun = hessian, n = 1001) +\nxlim(min(cyclones$number), max(cyclones$number)) +\n\nylab(\"l''(lambda)\") +\nxlab('lambda') +\ngeom_hline(yintercept = 0, col = 'red')\n\n\n\nFigure E.6: Hessian function of Dobson cyclone data\n\n\n\n\n\n\n\n\n\n\nExercise E.14 Write the score equation (estimating equation).\n\nSolution. \\[\\ell'( \\lambda; \\tilde{x} ) = 0\\]\n\n\n\nExercise E.15 Solve the estimating equation for \\(\\lambda\\):\n\nSolution. \\[\n\\begin{aligned}\n0 &= \\frac{1}{\\lambda}\\sum_{i = 1}^nx_i - n\\\\\nn &= \\frac{1}{\\lambda}\\sum_{i = 1}^nx_i\\\\\nn\\lambda &= \\sum_{i = 1}^nx_i\\\\\n\\lambda &=\n\\frac{1}{n}\\sum_{i = 1}^nx_i\\\\\n&=\\bar x\n\\end{aligned}\n\\]\n\n\nLet’s call this solution of the estimating equation \\(\\tilde \\lambda\\) for now:\n\\[\\tilde \\lambda \\stackrel{\\text{def}}{=}\\bar x\\]\n\nExercise E.16 Confirm that the Hessian \\(\\ell''(\\lambda; \\tilde{x})\\) is negative when evaluated at \\(\\tilde \\lambda\\).\n\nSolution. \\[\n\\begin{aligned}\n\\ell''( \\tilde\\lambda; \\tilde{x} ) &=\n-\\frac{1}{\\tilde\\lambda^2} n \\bar x\\\\\n&= -\\frac{1}{\\bar x^2} n\\bar x\\\\\n&= -\\frac{n}{\\bar x}\\\\\n&<0\\\\\n\\end{aligned}\n\\]\n\n\n\nExercise E.17 Find the MLE of \\(\\lambda\\).\n\nSolution. Since \\(\\ell''(\\tilde \\lambda; \\tilde{x})<0\\), \\(\\tilde \\lambda\\) is at least a local maximizer of the likelihood function \\(\\mathcal L(\\lambda)\\). Since there is only one solution to the estimating equation and the Hessian is negative definite everywhere, \\(\\tilde \\lambda\\) must also be the global maximizer of \\(\\mathcal L(\\lambda; \\tilde{x})\\):\n\nShow R codemle = mean(cyclones$number)\n\n\n\\[\\hat{\\lambda}_{\\text{ML}} = \\bar x = 5.5385\\]\n\n\n\nExercise E.18 Graph the log-likelihood with the MLE superimposed.\n\nSolution. \n\nShow R codelibrary(dplyr)\n\nmle_data = tibble(x = mle, y = loglik(mle))\nll_plot + geom_point(data = mle_data, aes(x = x, y = y), col = 'red')\n\n\n\nFigure E.7: log-likelihood of Dobson cyclone data with MLE\n\n\n\n\n\n\n\n\n\nInformation matrices\n\nShow R codeobs_inf = function(...) -hessian(...)\nggplot() +\ngeom_function(fun = obs_inf, n = 1001) +\nxlim(min(cyclones$number), max(cyclones$number)) +\nylab(\"I(lambda)\") +\nxlab('lambda') +\ngeom_hline(yintercept = 0, col = 'red') \n\n\n\nFigure E.8: Observed information function of Dobson cyclone data\n\n\n\n\n\n\n\n\n\nExample E.1 (Finding the MLE using the Newton-Raphson algorithm)  \n\nWe found that the MLE was \\(\\hat{\\lambda} = \\bar{x}\\), by solving the score equation \\(\\ell'(\\lambda)=0\\) for \\(\\lambda\\).\nWhat if we hadn’t been able to solve the score equation?\nThen we could start with some initial guess \\({\\widehat{\\lambda}}^*\\), such as \\({\\widehat{\\lambda}}^*= 3\\), and use the Newton-Raphson algorithm.\n\n\nShow R code# specify initial guess:\ncur_lambda_est = 3\n\n\n\n\n\nIn Exercise E.10, we found that the score function was:\n\n\\[\n\\ell'( \\lambda; \\tilde{x} ) = \\left(\\frac{72}{\\lambda} \\right) - n\n\\]\n\nIn Exercise E.12, we found that the Hessian was:\n\n\\[\n\\ell''( \\lambda; \\tilde{x} ) = -\\frac{72}{\\lambda^2}\n\\]\n\n\nSo we can approximate the the score function using the first-order Taylor polynomial:\n\n\\[\n\\begin{aligned}\n\\ell'(\\lambda)\n&\\approx \\ell'^*(\\lambda)\n\\\\ &\\stackrel{\\text{def}}{=}\\ell'({\\widehat{\\lambda}}^*) + \\ell''({\\widehat{\\lambda}}^*)(\\lambda - {\\widehat{\\lambda}}^*)\n\\\\ &= \\left(\\frac{72}{{\\widehat{\\lambda}}^*} - n\\right) + \\left(-\\frac{72}{\\left({\\widehat{\\lambda}}^*\\right)^2}\\right) (\\lambda - {\\widehat{\\lambda}}^*)\n\\end{aligned}\n\\]\n\n\nFigure E.9 compares the true score function and the approximate score function at \\({\\widehat{\\lambda}}^*= 3\\).\n\n\nShow R codeapprox_score = function(lambda, lhat, ...)\n{\n score(lambda = lhat, ...) +\n hessian(lambda = lhat, ...) * (lambda - lhat)\n}\n\npoint_size = 5\n\nplot1 = ggplot() +\ngeom_function(\n fun = score, \n aes(col = \"true score function\"), \n n = 1001) +\ngeom_function(\n fun = approx_score, \n aes(col = \"approximate score function\"),\n n = 1001, \n args = list(lhat = cur_lambda_est)) +\ngeom_point(\n size = point_size,\n aes(x = cur_lambda_est, y = score(lambda = cur_lambda_est),\n col = \"current estimate\")\n) +\ngeom_point(\nsize = point_size,\naes(\nx = xbar,\ny = 0,\ncol = \"true MLE\"\n)\n) +\nxlim(min(cyclones$number), max(cyclones$number)) +\nylab(\"l'(lambda)\") +\nxlab('lambda') +\ngeom_hline(yintercept = 0)\n\nprint(plot1)\n\n\n\nFigure E.9: score function of Dobson cyclone data and approximate score function\n\n\n\n\n\n\n\n\n\nThis is equivalent to estimating the log-likelihood with a second-order Taylor polynomial:\n\n\\[\n\\ell^*(\\lambda) =\n\\ell({\\widehat{\\lambda}}^*) +\n(\\lambda - {\\widehat{\\lambda}}^*) \\ell'({\\widehat{\\lambda}}^*) +\n\\frac{1}{2}\\ell''({\\widehat{\\lambda}}^*)(\\lambda-{\\widehat{\\lambda}}^*)^2\n\\]\n\nShow R codeapprox_loglik = function(lambda, lhat, ...)\n{\nloglik(lambda = lhat, ...) +\nscore(lambda = lhat, ...) * (lambda - lhat) +\n 1/2 * hessian(lambda = lhat, ...) * (lambda - lhat)^2\n}\n\nplot_loglik = ggplot() +\ngeom_function(\n fun = loglik, \n aes(col = \"true log-likelihood\"), \n n = 1001) +\ngeom_function(\n fun = approx_loglik, \n aes(col = \"approximate log-likelihood\"),\n n = 1001, \n args = list(lhat = cur_lambda_est)) +\ngeom_point(\n size = point_size,\n aes(x = cur_lambda_est, y = loglik(lambda = cur_lambda_est),\n col = \"current estimate\")\n) +\ngeom_point(\nsize = point_size,\naes(\nx = xbar,\ny = loglik(xbar),\ncol = \"true MLE\"\n)\n) +\nxlim(min(cyclones$number) - 1, max(cyclones$number)) +\nylab(\"l'(lambda)\") +\nxlab('lambda')\n\nprint(plot_loglik)\n\n\n\nFigure E.10: log-likelihood of Dobson cyclone data and approximate log-likelihood function\n\n\n\n\n\n\n\n\n\nThe approximate score function, \\(\\ell'^*(\\lambda)\\), is a linear function of \\(\\lambda\\), so it is easy to solve the corresponding approximate score equation, \\(\\ell'^*(\\lambda) = 0\\), for \\(\\lambda\\):\n\n\\[\n\\begin{aligned}\n\\lambda\n&= {\\widehat{\\lambda}}^*- \\ell'({\\widehat{\\lambda}}^*) \\cdot\\left(\\ell''({\\widehat{\\lambda}}^*)\\right)^{-1}\n\\\\ &= 4.375\n\\end{aligned}\n\\]\n\nShow R codenew_lambda_est <- \n cur_lambda_est - \n score(cur_lambda_est) * hessian(cur_lambda_est)^-1\n\n\n\n\nShow R codeplot2 = plot1 + \ngeom_point(\nsize = point_size,\naes(\nx = new_lambda_est,\ny = 0,\ncol = \"new estimate\"\n)) +\ngeom_segment(\narrow = grid::arrow(),\nlinewidth = 2,\nalpha = .7,\naes(\nx = cur_lambda_est, \ny = approx_score(\nlhat = cur_lambda_est, \nlambda = cur_lambda_est),\nxend = new_lambda_est, \nyend = 0,\ncol = \"update\"\n)\n)\nprint(plot2)\n\n\n\nFigure E.11: score function of Dobson cyclone data and approximate score function\n\n\n\n\n\n\n\n\n\nSo we update \\({\\widehat{\\lambda}}^*\\leftarrow 4.375\\) and repeat our estimation process:\n\n\nShow R codeplot2 +\ngeom_function(\n fun = approx_score, \n aes(col = \"new approximate score function\"),\n n = 1001, \n args = list(lhat = new_lambda_est)) +\ngeom_point(\n size = point_size,\n aes(x = new_lambda_est, y = score(lambda = new_lambda_est),\n col = \"new estimate\")\n)\n\n\n\nFigure E.12: score function of Dobson cyclone data and approximate score function\n\n\n\n\n\n\n\n\n\nWe repeat this process until the likelihood converges:\n\n\nShow R codelibrary(tibble)\ncur_lambda_est = 3 # restarting\ndiff_loglik = Inf\ntolerance = 10^-4\nmax_iter = 100\nNR_info = tibble(\n iteration = 0,\n lambda = cur_lambda_est |> num(digits = 4),\n likelihood = lik(cur_lambda_est),\n `log(likelihood)` = loglik(cur_lambda_est) |> num(digits = 4),\n score = score(cur_lambda_est),\n hessian = hessian(cur_lambda_est)\n)\n\nfor (cur_iter in 1:max_iter)\n{\n\nnew_lambda_est <- \n cur_lambda_est - score(cur_lambda_est) * hessian(cur_lambda_est)^-1\n\ndiff_loglik = loglik(new_lambda_est) - loglik(cur_lambda_est)\n\nnew_NR_info = tibble(\n iteration = cur_iter,\n lambda = new_lambda_est,\n likelihood = lik(new_lambda_est),\n `log(likelihood)` = loglik(new_lambda_est),\n score = score(new_lambda_est),\n hessian = hessian(new_lambda_est),\n `diff(loglik)` = diff_loglik\n)\n\nNR_info = NR_info |> bind_rows(new_NR_info)\n\ncur_lambda_est = new_lambda_est\n\nif(abs(diff_loglik) < tolerance) break\n\n}\n\nNR_info\n\n\nTable E.3: Convergence of Newton-Raphson Algorithm for finding MLE of cyclone data\n\n\n\n \n\n\n\n\n\n\n\nCompare with Exercise E.17\n\n\n\nShow R codell_plot + \ngeom_segment(\ndata = NR_info,\narrow = grid::arrow(),\n# linewidth = 2,\nalpha = .7,\naes(\nx = lambda,\nxend = lead(lambda),\ny = `log(likelihood)`,\nyend = lead(`log(likelihood)`),\ncol = factor(iteration)\n)\n)\n\n\n\nFigure E.13: Newton-Raphson algorithm for finding MLE of model E.6 for cyclone data", "crumbs": [ "Appendices", "E  Introduction to Maximum Likelihood Inference" diff --git a/site_libs/crosstalk-1.2.0/css/crosstalk.min.css b/site_libs/crosstalk-1.2.0/css/crosstalk.min.css deleted file mode 100644 index 6b4538284..000000000 --- a/site_libs/crosstalk-1.2.0/css/crosstalk.min.css +++ /dev/null @@ -1 +0,0 @@ -.container-fluid.crosstalk-bscols{margin-left:-30px;margin-right:-30px;white-space:normal}body>.container-fluid.crosstalk-bscols{margin-left:auto;margin-right:auto}.crosstalk-input-checkboxgroup .crosstalk-options-group .crosstalk-options-column{display:inline-block;padding-right:12px;vertical-align:top}@media only screen and (max-width: 480px){.crosstalk-input-checkboxgroup .crosstalk-options-group .crosstalk-options-column{display:block;padding-right:inherit}}.crosstalk-input{margin-bottom:15px}.crosstalk-input .control-label{margin-bottom:0;vertical-align:middle}.crosstalk-input input[type="checkbox"]{margin:4px 0 0;margin-top:1px;line-height:normal}.crosstalk-input .checkbox{position:relative;display:block;margin-top:10px;margin-bottom:10px}.crosstalk-input .checkbox>label{padding-left:20px;margin-bottom:0;font-weight:400;cursor:pointer}.crosstalk-input .checkbox input[type="checkbox"],.crosstalk-input .checkbox-inline input[type="checkbox"]{position:absolute;margin-top:2px;margin-left:-20px}.crosstalk-input .checkbox+.checkbox{margin-top:-5px}.crosstalk-input .checkbox-inline{position:relative;display:inline-block;padding-left:20px;margin-bottom:0;font-weight:400;vertical-align:middle;cursor:pointer}.crosstalk-input .checkbox-inline+.checkbox-inline{margin-top:0;margin-left:10px} diff --git a/site_libs/crosstalk-1.2.0/js/crosstalk.js b/site_libs/crosstalk-1.2.0/js/crosstalk.js deleted file mode 100644 index fd9eb53d2..000000000 --- a/site_libs/crosstalk-1.2.0/js/crosstalk.js +++ /dev/null @@ -1,1474 +0,0 @@ -(function(){function e(t,n,r){function s(o,u){if(!n[o]){if(!t[o]){var a=typeof require=="function"&&require;if(!u&&a)return a(o,!0);if(i)return i(o,!0);var f=new Error("Cannot find module '"+o+"'");throw f.code="MODULE_NOT_FOUND",f}var l=n[o]={exports:{}};t[o][0].call(l.exports,function(e){var n=t[o][1][e];return s(n?n:e)},l,l.exports,e,t,n,r)}return n[o].exports}var i=typeof require=="function"&&require;for(var o=0;o b) { - 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"symbol" : typeof obj; }; - -exports.default = group; - -var _var2 = require("./var"); - -var _var3 = _interopRequireDefault(_var2); - -function _interopRequireDefault(obj) { return obj && obj.__esModule ? obj : { default: obj }; } - -function _classCallCheck(instance, Constructor) { if (!(instance instanceof Constructor)) { throw new TypeError("Cannot call a class as a function"); } } - -// Use a global so that multiple copies of crosstalk.js can be loaded and still -// have groups behave as singletons across all copies. -global.__crosstalk_groups = global.__crosstalk_groups || {}; -var groups = global.__crosstalk_groups; - -function group(groupName) { - if (groupName && typeof groupName === "string") { - if (!groups.hasOwnProperty(groupName)) { - groups[groupName] = new Group(groupName); - } - return groups[groupName]; - } else if ((typeof groupName === "undefined" ? "undefined" : _typeof(groupName)) === "object" && groupName._vars && groupName.var) { - // Appears to already be a group object - return groupName; - } else if (Array.isArray(groupName) && groupName.length == 1 && typeof groupName[0] === "string") { - return group(groupName[0]); - } else { - throw new Error("Invalid groupName argument"); - } -} - -var Group = function () { - function Group(name) { - _classCallCheck(this, Group); - - this.name = name; - this._vars = {}; - } - - _createClass(Group, [{ - key: "var", - value: function _var(name) { - if (!name || typeof name !== "string") { - throw new Error("Invalid var name"); - } - - if (!this._vars.hasOwnProperty(name)) this._vars[name] = new _var3.default(this, name); - return this._vars[name]; - } - }, { - key: "has", - value: function has(name) { - if (!name || typeof name !== "string") { - throw new Error("Invalid var name"); - } - - return this._vars.hasOwnProperty(name); - } - }]); - - return Group; -}(); - -}).call(this,typeof global !== "undefined" ? global : typeof self !== "undefined" ? self : typeof window !== "undefined" ? window : {}) - -},{"./var":12}],5:[function(require,module,exports){ -(function (global){ -"use strict"; - -Object.defineProperty(exports, "__esModule", { - value: true -}); - -var _group = require("./group"); - -var _group2 = _interopRequireDefault(_group); - -var _selection = require("./selection"); - -var _filter = require("./filter"); - -var _input = require("./input"); - -require("./input_selectize"); - -require("./input_checkboxgroup"); - -require("./input_slider"); - -function _interopRequireDefault(obj) { return obj && obj.__esModule ? obj : { default: obj }; } - -var defaultGroup = (0, _group2.default)("default"); - -function var_(name) { - return defaultGroup.var(name); -} - -function has(name) { - return defaultGroup.has(name); -} - -if (global.Shiny) { - global.Shiny.addCustomMessageHandler("update-client-value", function (message) { - if (typeof message.group === "string") { - (0, _group2.default)(message.group).var(message.name).set(message.value); - } else { - var_(message.name).set(message.value); - } - }); -} - -var crosstalk = { - group: _group2.default, - var: var_, - has: has, - SelectionHandle: _selection.SelectionHandle, - FilterHandle: _filter.FilterHandle, - bind: _input.bind -}; - -/** - * @namespace crosstalk - */ -exports.default = crosstalk; - -global.crosstalk = crosstalk; - -}).call(this,typeof global !== "undefined" ? global : typeof self !== "undefined" ? self : typeof window !== "undefined" ? window : {}) - -},{"./filter":2,"./group":4,"./input":6,"./input_checkboxgroup":7,"./input_selectize":8,"./input_slider":9,"./selection":10}],6:[function(require,module,exports){ -(function (global){ -"use strict"; - -Object.defineProperty(exports, "__esModule", { - value: true -}); -exports.register = register; -exports.bind = bind; -var $ = global.jQuery; - -var bindings = {}; - -function register(reg) { - bindings[reg.className] = reg; - if (global.document && global.document.readyState !== "complete") { - $(function () { - bind(); - }); - } else if (global.document) { - setTimeout(bind, 100); - } -} - -function bind() { - Object.keys(bindings).forEach(function (className) { - var binding = bindings[className]; - $("." + binding.className).not(".crosstalk-input-bound").each(function (i, el) { - bindInstance(binding, el); - }); - }); -} - -// Escape jQuery identifier -function $escape(val) { - return val.replace(/([!"#$%&'()*+,./:;<=>?@[\\\]^`{|}~])/g, "\\$1"); -} - -function bindEl(el) { - var $el = $(el); - Object.keys(bindings).forEach(function (className) { - if ($el.hasClass(className) && !$el.hasClass("crosstalk-input-bound")) { - var binding = bindings[className]; - bindInstance(binding, el); - } - }); -} - -function bindInstance(binding, el) { - var jsonEl = $(el).find("script[type='application/json'][data-for='" + $escape(el.id) + "']"); - var data = JSON.parse(jsonEl[0].innerText); - - var instance = binding.factory(el, data); - $(el).data("crosstalk-instance", instance); - $(el).addClass("crosstalk-input-bound"); -} - -if (global.Shiny) { - var inputBinding = new global.Shiny.InputBinding(); - var _$ = global.jQuery; - _$.extend(inputBinding, { - find: function find(scope) { - return _$(scope).find(".crosstalk-input"); - }, - initialize: function initialize(el) { - if (!_$(el).hasClass("crosstalk-input-bound")) { - bindEl(el); - } - }, - getId: function getId(el) { - return el.id; - }, - getValue: function getValue(el) {}, - setValue: function setValue(el, value) {}, - receiveMessage: function receiveMessage(el, data) {}, - subscribe: function subscribe(el, callback) { - _$(el).data("crosstalk-instance").resume(); - }, - unsubscribe: function unsubscribe(el) { - _$(el).data("crosstalk-instance").suspend(); - } - }); - global.Shiny.inputBindings.register(inputBinding, "crosstalk.inputBinding"); -} - -}).call(this,typeof global !== "undefined" ? global : typeof self !== "undefined" ? self : typeof window !== "undefined" ? window : {}) - -},{}],7:[function(require,module,exports){ -(function (global){ -"use strict"; - -var _input = require("./input"); - -var input = _interopRequireWildcard(_input); - -var _filter = require("./filter"); - -function _interopRequireWildcard(obj) { if (obj && obj.__esModule) { return obj; } else { var newObj = {}; if (obj != null) { for (var key in obj) { if (Object.prototype.hasOwnProperty.call(obj, key)) newObj[key] = obj[key]; } } newObj.default = obj; return newObj; } } - -var $ = global.jQuery; - -input.register({ - className: "crosstalk-input-checkboxgroup", - - factory: function factory(el, data) { - /* - * map: {"groupA": ["keyA", "keyB", ...], ...} - * group: "ct-groupname" - */ - var ctHandle = new _filter.FilterHandle(data.group); - - var lastKnownKeys = void 0; - var $el = $(el); - $el.on("change", "input[type='checkbox']", function () { - var checked = $el.find("input[type='checkbox']:checked"); - if (checked.length === 0) { - lastKnownKeys = null; - ctHandle.clear(); - } else { - var keys = {}; - checked.each(function () { - data.map[this.value].forEach(function (key) { - keys[key] = true; - }); - }); - var keyArray = Object.keys(keys); - keyArray.sort(); - lastKnownKeys = keyArray; - ctHandle.set(keyArray); - } - }); - - return { - suspend: function suspend() { - ctHandle.clear(); - }, - resume: function resume() { - if (lastKnownKeys) ctHandle.set(lastKnownKeys); - } - }; - } -}); - -}).call(this,typeof global !== "undefined" ? global : typeof self !== "undefined" ? self : typeof window !== "undefined" ? window : {}) - -},{"./filter":2,"./input":6}],8:[function(require,module,exports){ -(function (global){ -"use strict"; - -var _input = require("./input"); - -var input = _interopRequireWildcard(_input); - -var _util = require("./util"); - -var util = _interopRequireWildcard(_util); - -var _filter = require("./filter"); - -function _interopRequireWildcard(obj) { if (obj && obj.__esModule) { return obj; } else { var newObj = {}; if (obj != null) { for (var key in obj) { if (Object.prototype.hasOwnProperty.call(obj, key)) newObj[key] = obj[key]; } } newObj.default = obj; return newObj; } } - -var $ = global.jQuery; - -input.register({ - className: "crosstalk-input-select", - - factory: function factory(el, data) { - /* - * items: {value: [...], label: [...]} - * map: {"groupA": ["keyA", "keyB", ...], ...} - * group: "ct-groupname" - */ - - var first = [{ value: "", label: "(All)" }]; - var items = util.dataframeToD3(data.items); - var opts = { - options: first.concat(items), - valueField: "value", - labelField: "label", - searchField: "label" - }; - - var select = $(el).find("select")[0]; - - var selectize = $(select).selectize(opts)[0].selectize; - - var ctHandle = new _filter.FilterHandle(data.group); - - var lastKnownKeys = void 0; - selectize.on("change", function () { - if (selectize.items.length === 0) { - lastKnownKeys = null; - ctHandle.clear(); - } else { - var keys = {}; - selectize.items.forEach(function (group) { - data.map[group].forEach(function (key) { - keys[key] = true; - }); - }); - var keyArray = Object.keys(keys); - keyArray.sort(); - lastKnownKeys = keyArray; - ctHandle.set(keyArray); - } - }); - - return { - suspend: function suspend() { - ctHandle.clear(); - }, - resume: function resume() { - if (lastKnownKeys) ctHandle.set(lastKnownKeys); - } - }; - } -}); - -}).call(this,typeof global !== "undefined" ? global : typeof self !== "undefined" ? self : typeof window !== "undefined" ? window : {}) - -},{"./filter":2,"./input":6,"./util":11}],9:[function(require,module,exports){ -(function (global){ -"use strict"; - -var _slicedToArray = function () { function sliceIterator(arr, i) { var _arr = []; var _n = true; var _d = false; var _e = undefined; try { for (var _i = arr[Symbol.iterator](), _s; !(_n = (_s = _i.next()).done); _n = true) { _arr.push(_s.value); if (i && _arr.length === i) break; } } catch (err) { _d = true; _e = err; } finally { try { if (!_n && _i["return"]) _i["return"](); } finally { if (_d) throw _e; } } return _arr; } return function (arr, i) { if (Array.isArray(arr)) { return arr; } else if (Symbol.iterator in Object(arr)) { return sliceIterator(arr, i); } else { throw new TypeError("Invalid attempt to destructure non-iterable instance"); } }; }(); - -var _input = require("./input"); - -var input = _interopRequireWildcard(_input); - -var _filter = require("./filter"); - -function _interopRequireWildcard(obj) { if (obj && obj.__esModule) { return obj; } else { var newObj = {}; if (obj != null) { for (var key in obj) { if (Object.prototype.hasOwnProperty.call(obj, key)) newObj[key] = obj[key]; } } newObj.default = obj; return newObj; } } - -var $ = global.jQuery; -var strftime = global.strftime; - -input.register({ - className: "crosstalk-input-slider", - - factory: function factory(el, data) { - /* - * map: {"groupA": ["keyA", "keyB", ...], ...} - * group: "ct-groupname" - */ - var ctHandle = new _filter.FilterHandle(data.group); - - var opts = {}; - var $el = $(el).find("input"); - var dataType = $el.data("data-type"); - var timeFormat = $el.data("time-format"); - var round = $el.data("round"); - var timeFormatter = void 0; - - // Set up formatting functions - if (dataType === "date") { - timeFormatter = strftime.utc(); - opts.prettify = function (num) { - return timeFormatter(timeFormat, new Date(num)); - }; - } else if (dataType === "datetime") { - var timezone = $el.data("timezone"); - if (timezone) timeFormatter = strftime.timezone(timezone);else timeFormatter = strftime; - - opts.prettify = function (num) { - return timeFormatter(timeFormat, new Date(num)); - }; - } else if (dataType === "number") { - if (typeof round !== "undefined") opts.prettify = function (num) { - var factor = Math.pow(10, round); - return Math.round(num * factor) / factor; - }; - } - - $el.ionRangeSlider(opts); - - function getValue() { - var result = $el.data("ionRangeSlider").result; - - // Function for converting numeric value from slider to appropriate type. - var convert = void 0; - var dataType = $el.data("data-type"); - if (dataType === "date") { - convert = function convert(val) { - return formatDateUTC(new Date(+val)); - }; - } else if (dataType === "datetime") { - convert = function convert(val) { - // Convert ms to s - return +val / 1000; - }; - } else { - convert = function convert(val) { - return +val; - }; - } - - if ($el.data("ionRangeSlider").options.type === "double") { - return [convert(result.from), convert(result.to)]; - } else { - return convert(result.from); - } - } - - var lastKnownKeys = null; - - $el.on("change.crosstalkSliderInput", function (event) { - if (!$el.data("updating") && !$el.data("animating")) { - var _getValue = getValue(), - _getValue2 = _slicedToArray(_getValue, 2), - from = _getValue2[0], - to = _getValue2[1]; - - var keys = []; - for (var i = 0; i < data.values.length; i++) { - var val = data.values[i]; - if (val >= from && val <= to) { - keys.push(data.keys[i]); - } - } - keys.sort(); - ctHandle.set(keys); - lastKnownKeys = keys; - } - }); - - // let $el = $(el); - // $el.on("change", "input[type="checkbox"]", function() { - // let checked = $el.find("input[type="checkbox"]:checked"); - // if (checked.length === 0) { - // ctHandle.clear(); - // } else { - // let keys = {}; - // checked.each(function() { - // data.map[this.value].forEach(function(key) { - // keys[key] = true; - // }); - // }); - // let keyArray = Object.keys(keys); - // keyArray.sort(); - // ctHandle.set(keyArray); - // } - // }); - - return { - suspend: function suspend() { - ctHandle.clear(); - }, - resume: function resume() { - if (lastKnownKeys) ctHandle.set(lastKnownKeys); - } - }; - } -}); - -// Convert a number to a string with leading zeros -function padZeros(n, digits) { - var str = n.toString(); - while (str.length < digits) { - str = "0" + str; - }return str; -} - -// Given a Date object, return a string in yyyy-mm-dd format, using the -// UTC date. This may be a day off from the date in the local time zone. -function formatDateUTC(date) { - if (date instanceof Date) { - return date.getUTCFullYear() + "-" + padZeros(date.getUTCMonth() + 1, 2) + "-" + padZeros(date.getUTCDate(), 2); - } else { - return null; - } -} - -}).call(this,typeof global !== "undefined" ? global : typeof self !== "undefined" ? self : typeof window !== "undefined" ? window : {}) - -},{"./filter":2,"./input":6}],10:[function(require,module,exports){ -"use strict"; - -Object.defineProperty(exports, "__esModule", { - value: true -}); -exports.SelectionHandle = undefined; - -var _createClass = function () { function defineProperties(target, props) { for (var i = 0; i < props.length; i++) { var descriptor = props[i]; descriptor.enumerable = descriptor.enumerable || false; descriptor.configurable = true; if ("value" in descriptor) descriptor.writable = true; Object.defineProperty(target, descriptor.key, descriptor); } } return function (Constructor, protoProps, staticProps) { if (protoProps) defineProperties(Constructor.prototype, protoProps); if (staticProps) defineProperties(Constructor, staticProps); return Constructor; }; }(); - -var _events = require("./events"); - -var _events2 = _interopRequireDefault(_events); - -var _group = require("./group"); - -var _group2 = _interopRequireDefault(_group); - -var _util = require("./util"); - -var util = _interopRequireWildcard(_util); - -function _interopRequireWildcard(obj) { if (obj && obj.__esModule) { return obj; } else { var newObj = {}; if (obj != null) { for (var key in obj) { if (Object.prototype.hasOwnProperty.call(obj, key)) newObj[key] = obj[key]; } } newObj.default = obj; return newObj; } } - -function _interopRequireDefault(obj) { return obj && obj.__esModule ? obj : { default: obj }; } - -function _classCallCheck(instance, Constructor) { if (!(instance instanceof Constructor)) { throw new TypeError("Cannot call a class as a function"); } } - -/** - * Use this class to read and write (and listen for changes to) the selection - * for a Crosstalk group. This is intended to be used for linked brushing. - * - * If two (or more) `SelectionHandle` instances in the same webpage share the - * same group name, they will share the same state. Setting the selection using - * one `SelectionHandle` instance will result in the `value` property instantly - * changing across the others, and `"change"` event listeners on all instances - * (including the one that initiated the sending) will fire. - * - * @param {string} [group] - The name of the Crosstalk group, or if none, - * null or undefined (or any other falsy value). This can be changed later - * via the [SelectionHandle#setGroup](#setGroup) method. - * @param {Object} [extraInfo] - An object whose properties will be copied to - * the event object whenever an event is emitted. - */ -var SelectionHandle = exports.SelectionHandle = function () { - function SelectionHandle() { - var group = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : null; - var extraInfo = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : null; - - _classCallCheck(this, SelectionHandle); - - this._eventRelay = new _events2.default(); - this._emitter = new util.SubscriptionTracker(this._eventRelay); - - // Name of the group we're currently tracking, if any. Can change over time. - this._group = null; - // The Var we're currently tracking, if any. Can change over time. - this._var = null; - // The event handler subscription we currently have on var.on("change"). - this._varOnChangeSub = null; - - this._extraInfo = util.extend({ sender: this }, extraInfo); - - this.setGroup(group); - } - - /** - * Changes the Crosstalk group membership of this SelectionHandle. The group - * being switched away from (if any) will not have its selection value - * modified as a result of calling `setGroup`, even if this handle was the - * most recent handle to set the selection of the group. - * - * The group being switched to (if any) will also not have its selection value - * modified as a result of calling `setGroup`. If you want to set the - * selection value of the new group, call `set` explicitly. - * - * @param {string} group - The name of the Crosstalk group, or null (or - * undefined) to clear the group. - */ - - - _createClass(SelectionHandle, [{ - key: "setGroup", - value: function setGroup(group) { - var _this = this; - - // If group is unchanged, do nothing - if (this._group === group) return; - // Treat null, undefined, and other falsy values the same - if (!this._group && !group) return; - - if (this._var) { - this._var.off("change", this._varOnChangeSub); - this._var = null; - this._varOnChangeSub = null; - } - - this._group = group; - - if (group) { - this._var = (0, _group2.default)(group).var("selection"); - var sub = this._var.on("change", function (e) { - _this._eventRelay.trigger("change", e, _this); - }); - this._varOnChangeSub = sub; - } - } - - /** - * Retrieves the current selection for the group represented by this - * `SelectionHandle`. - * - * - If no selection is active, then this value will be falsy. - * - If a selection is active, but no data points are selected, then this - * value will be an empty array. - * - If a selection is active, and data points are selected, then the keys - * of the selected data points will be present in the array. - */ - - }, { - key: "_mergeExtraInfo", - - - /** - * Combines the given `extraInfo` (if any) with the handle's default - * `_extraInfo` (if any). - * @private - */ - value: function _mergeExtraInfo(extraInfo) { - // Important incidental effect: shallow clone is returned - return util.extend({}, this._extraInfo ? this._extraInfo : null, extraInfo ? extraInfo : null); - } - - /** - * Overwrites the current selection for the group, and raises the `"change"` - * event among all of the group's '`SelectionHandle` instances (including - * this one). - * - * @fires SelectionHandle#change - * @param {string[]} selectedKeys - Falsy, empty array, or array of keys (see - * {@link SelectionHandle#value}). - * @param {Object} [extraInfo] - Extra properties to be included on the event - * object that's passed to listeners (in addition to any options that were - * passed into the `SelectionHandle` constructor). - */ - - }, { - key: "set", - value: function set(selectedKeys, extraInfo) { - if (this._var) this._var.set(selectedKeys, this._mergeExtraInfo(extraInfo)); - } - - /** - * Overwrites the current selection for the group, and raises the `"change"` - * event among all of the group's '`SelectionHandle` instances (including - * this one). - * - * @fires SelectionHandle#change - * @param {Object} [extraInfo] - Extra properties to be included on the event - * object that's passed to listeners (in addition to any that were passed - * into the `SelectionHandle` constructor). - */ - - }, { - key: "clear", - value: function clear(extraInfo) { - if (this._var) this.set(void 0, this._mergeExtraInfo(extraInfo)); - } - - /** - * Subscribes to events on this `SelectionHandle`. - * - * @param {string} eventType - Indicates the type of events to listen to. - * Currently, only `"change"` is supported. - * @param {SelectionHandle~listener} listener - The callback function that - * will be invoked when the event occurs. - * @return {string} - A token to pass to {@link SelectionHandle#off} to cancel - * this subscription. - */ - - }, { - key: "on", - value: function on(eventType, listener) { - return this._emitter.on(eventType, listener); - } - - /** - * Cancels event subscriptions created by {@link SelectionHandle#on}. - * - * @param {string} eventType - The type of event to unsubscribe. - * @param {string|SelectionHandle~listener} listener - Either the callback - * function previously passed into {@link SelectionHandle#on}, or the - * string that was returned from {@link SelectionHandle#on}. - */ - - }, { - key: "off", - value: function off(eventType, listener) { - return this._emitter.off(eventType, listener); - } - - /** - * Shuts down the `SelectionHandle` object. - * - * Removes all event listeners that were added through this handle. - */ - - }, { - key: "close", - value: function close() { - this._emitter.removeAllListeners(); - this.setGroup(null); - } - }, { - key: "value", - get: function get() { - return this._var ? this._var.get() : null; - } - }]); - - return SelectionHandle; -}(); - -/** - * @callback SelectionHandle~listener - * @param {Object} event - An object containing details of the event. For - * `"change"` events, this includes the properties `value` (the new - * value of the selection, or `undefined` if no selection is active), - * `oldValue` (the previous value of the selection), and `sender` (the - * `SelectionHandle` instance that made the change). - */ - -/** - * @event SelectionHandle#change - * @type {object} - * @property {object} value - The new value of the selection, or `undefined` - * if no selection is active. - * @property {object} oldValue - The previous value of the selection. - * @property {SelectionHandle} sender - The `SelectionHandle` instance that - * changed the value. - */ - -},{"./events":1,"./group":4,"./util":11}],11:[function(require,module,exports){ -"use strict"; - -Object.defineProperty(exports, "__esModule", { - value: true -}); - -var _createClass = function () { function defineProperties(target, props) { for (var i = 0; i < props.length; i++) { var descriptor = props[i]; descriptor.enumerable = descriptor.enumerable || false; descriptor.configurable = true; if ("value" in descriptor) descriptor.writable = true; Object.defineProperty(target, descriptor.key, descriptor); } } return function (Constructor, protoProps, staticProps) { if (protoProps) defineProperties(Constructor.prototype, protoProps); if (staticProps) defineProperties(Constructor, staticProps); return Constructor; }; }(); - -var _typeof = typeof Symbol === "function" && typeof Symbol.iterator === "symbol" ? function (obj) { return typeof obj; } : function (obj) { return obj && typeof Symbol === "function" && obj.constructor === Symbol && obj !== Symbol.prototype ? "symbol" : typeof obj; }; - -exports.extend = extend; -exports.checkSorted = checkSorted; -exports.diffSortedLists = diffSortedLists; -exports.dataframeToD3 = dataframeToD3; - -function _classCallCheck(instance, Constructor) { if (!(instance instanceof Constructor)) { throw new TypeError("Cannot call a class as a function"); } } - -function extend(target) { - for (var _len = arguments.length, sources = Array(_len > 1 ? _len - 1 : 0), _key = 1; _key < _len; _key++) { - sources[_key - 1] = arguments[_key]; - } - - for (var i = 0; i < sources.length; i++) { - var src = sources[i]; - if (typeof src === "undefined" || src === null) continue; - - for (var key in src) { - if (src.hasOwnProperty(key)) { - target[key] = src[key]; - } - } - } - return target; -} - -function checkSorted(list) { - for (var i = 1; i < list.length; i++) { - if (list[i] <= list[i - 1]) { - throw new Error("List is not sorted or contains duplicate"); - } - } -} - -function diffSortedLists(a, b) { - var i_a = 0; - var i_b = 0; - - if (!a) a = []; - if (!b) b = []; - - var a_only = []; - var b_only = []; - - checkSorted(a); - checkSorted(b); - - while (i_a < a.length && i_b < b.length) { - if (a[i_a] === b[i_b]) { - i_a++; - i_b++; - } else if (a[i_a] < b[i_b]) { - a_only.push(a[i_a++]); - } else { - b_only.push(b[i_b++]); - } - } - - if (i_a < a.length) a_only = a_only.concat(a.slice(i_a)); - if (i_b < b.length) b_only = b_only.concat(b.slice(i_b)); - return { - removed: a_only, - added: b_only - }; -} - -// Convert from wide: { colA: [1,2,3], colB: [4,5,6], ... } -// to long: [ {colA: 1, colB: 4}, {colA: 2, colB: 5}, ... ] -function dataframeToD3(df) { - var names = []; - var length = void 0; - for (var name in df) { - if (df.hasOwnProperty(name)) names.push(name); - if (_typeof(df[name]) !== "object" || typeof df[name].length === "undefined") { - throw new Error("All fields must be arrays"); - } else if (typeof length !== "undefined" && length !== df[name].length) { - throw new Error("All fields must be arrays of the same length"); - } - length = df[name].length; - } - var results = []; - var item = void 0; - for (var row = 0; row < length; row++) { - item = {}; - for (var col = 0; col < names.length; col++) { - item[names[col]] = df[names[col]][row]; - } - results.push(item); - } - return results; -} - -/** - * Keeps track of all event listener additions/removals and lets all active - * listeners be removed with a single operation. - * - * @private - */ - -var SubscriptionTracker = exports.SubscriptionTracker = function () { - function SubscriptionTracker(emitter) { - _classCallCheck(this, SubscriptionTracker); - - this._emitter = emitter; - this._subs = {}; - } - - _createClass(SubscriptionTracker, [{ - key: "on", - value: function on(eventType, listener) { - var sub = this._emitter.on(eventType, listener); - this._subs[sub] = eventType; - return sub; - } - }, { - key: "off", - value: function off(eventType, listener) { - var sub = this._emitter.off(eventType, listener); - if (sub) { - delete this._subs[sub]; - } - return sub; - } - }, { - key: "removeAllListeners", - value: function removeAllListeners() { - var _this = this; - - var current_subs = this._subs; - this._subs = {}; - Object.keys(current_subs).forEach(function (sub) { - _this._emitter.off(current_subs[sub], sub); - }); - } - }]); - - return SubscriptionTracker; -}(); - -},{}],12:[function(require,module,exports){ -(function (global){ -"use strict"; - -Object.defineProperty(exports, "__esModule", { - value: true -}); - -var _typeof = typeof Symbol === "function" && typeof Symbol.iterator === "symbol" ? function (obj) { return typeof obj; } : function (obj) { return obj && typeof Symbol === "function" && obj.constructor === Symbol && obj !== Symbol.prototype ? "symbol" : typeof obj; }; - -var _createClass = function () { function defineProperties(target, props) { for (var i = 0; i < props.length; i++) { var descriptor = props[i]; descriptor.enumerable = descriptor.enumerable || false; descriptor.configurable = true; if ("value" in descriptor) descriptor.writable = true; Object.defineProperty(target, descriptor.key, descriptor); } } return function (Constructor, protoProps, staticProps) { if (protoProps) defineProperties(Constructor.prototype, protoProps); if (staticProps) defineProperties(Constructor, staticProps); return Constructor; }; }(); - -var _events = require("./events"); - -var _events2 = _interopRequireDefault(_events); - -function _interopRequireDefault(obj) { return obj && obj.__esModule ? obj : { default: obj }; } - -function _classCallCheck(instance, Constructor) { if (!(instance instanceof Constructor)) { throw new TypeError("Cannot call a class as a function"); } } - -var Var = function () { - function Var(group, name, /*optional*/value) { - _classCallCheck(this, Var); - - this._group = group; - this._name = name; - this._value = value; - this._events = new _events2.default(); - } - - _createClass(Var, [{ - key: "get", - value: function get() { - return this._value; - } - }, { - key: "set", - value: function set(value, /*optional*/event) { - if (this._value === value) { - // Do nothing; the value hasn't changed - return; - } - var oldValue = this._value; - this._value = value; - // Alert JavaScript listeners that the value has changed - var evt = {}; - if (event && (typeof event === "undefined" ? "undefined" : _typeof(event)) === "object") { - for (var k in event) { - if (event.hasOwnProperty(k)) evt[k] = event[k]; - } - } - evt.oldValue = oldValue; - evt.value = value; - this._events.trigger("change", evt, this); - - // TODO: Make this extensible, to let arbitrary back-ends know that - // something has changed - if (global.Shiny && global.Shiny.onInputChange) { - global.Shiny.onInputChange(".clientValue-" + (this._group.name !== null ? this._group.name + "-" : "") + this._name, typeof value === "undefined" ? null : value); - } - } - }, { - key: "on", - value: function on(eventType, listener) { - return this._events.on(eventType, listener); - } - }, { - key: "off", - value: function off(eventType, listener) { - return this._events.off(eventType, listener); - } - }]); - - return Var; -}(); - -exports.default = Var; - -}).call(this,typeof global !== "undefined" ? global : typeof self !== "undefined" ? self : typeof window !== "undefined" ? window : {}) - -},{"./events":1}]},{},[5]) -//# sourceMappingURL=crosstalk.js.map diff --git a/site_libs/crosstalk-1.2.0/js/crosstalk.js.map b/site_libs/crosstalk-1.2.0/js/crosstalk.js.map deleted file mode 100644 index cff94f089..000000000 --- a/site_libs/crosstalk-1.2.0/js/crosstalk.js.map +++ /dev/null @@ -1,37 +0,0 @@ -{ - "version": 3, - "sources": [ - "node_modules/browser-pack/_prelude.js", - "javascript/src/events.js", - "javascript/src/filter.js", - "javascript/src/filterset.js", - "javascript/src/group.js", - "javascript/src/index.js", - "javascript/src/input.js", - "javascript/src/input_checkboxgroup.js", - "javascript/src/input_selectize.js", - "javascript/src/input_slider.js", - "javascript/src/selection.js", - "javascript/src/util.js", - "javascript/src/var.js" - ], - "names": [], - "mappings": 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- "file": "generated.js", - "sourceRoot": "", - "sourcesContent": [ - "(function(){function e(t,n,r){function s(o,u){if(!n[o]){if(!t[o]){var a=typeof require==\"function\"&&require;if(!u&&a)return a(o,!0);if(i)return i(o,!0);var f=new Error(\"Cannot find module '\"+o+\"'\");throw f.code=\"MODULE_NOT_FOUND\",f}var l=n[o]={exports:{}};t[o][0].call(l.exports,function(e){var n=t[o][1][e];return s(n?n:e)},l,l.exports,e,t,n,r)}return n[o].exports}var i=typeof require==\"function\"&&require;for(var o=0;o {\n this._eventRelay.trigger(\"change\", e, this);\n });\n this._varOnChangeSub = sub;\n }\n }\n\n /**\n * Combine the given `extraInfo` (if any) with the handle's default\n * `_extraInfo` (if any).\n * @private\n */\n _mergeExtraInfo(extraInfo) {\n return util.extend({},\n this._extraInfo ? this._extraInfo : null,\n extraInfo ? extraInfo : null);\n }\n\n /**\n * Close the handle. This clears this handle's contribution to the filter set,\n * and unsubscribes all event listeners.\n */\n close() {\n this._emitter.removeAllListeners();\n this.clear();\n this.setGroup(null);\n }\n\n /**\n * Clear this handle's contribution to the filter set.\n *\n * @param {Object} [extraInfo] - Extra properties to be included on the event\n * object that's passed to listeners (in addition to any options that were\n * passed into the `FilterHandle` constructor).\n * \n * @fires FilterHandle#change\n */\n clear(extraInfo) {\n if (!this._filterSet)\n return;\n this._filterSet.clear(this._id);\n this._onChange(extraInfo);\n }\n\n /**\n * Set this handle's contribution to the filter set. This array should consist\n * of the keys of the rows that _should_ be displayed; any keys that are not\n * present in the array will be considered _filtered out_. Note that multiple\n * `FilterHandle` instances in the group may each contribute an array of keys,\n * and only those keys that appear in _all_ of the arrays make it through the\n * filter.\n *\n * @param {string[]} keys - Empty array, or array of keys. To clear the\n * filter, don't pass an empty array; instead, use the\n * {@link FilterHandle#clear} method.\n * @param {Object} [extraInfo] - Extra properties to be included on the event\n * object that's passed to listeners (in addition to any options that were\n * passed into the `FilterHandle` constructor).\n * \n * @fires FilterHandle#change\n */\n set(keys, extraInfo) {\n if (!this._filterSet)\n return;\n this._filterSet.update(this._id, keys);\n this._onChange(extraInfo);\n }\n\n /**\n * @return {string[]|null} - Either: 1) an array of keys that made it through\n * all of the `FilterHandle` instances, or, 2) `null`, which means no filter\n * is being applied (all data should be displayed).\n */\n get filteredKeys() {\n return this._filterSet ? this._filterSet.value : null;\n }\n\n /**\n * Subscribe to events on this `FilterHandle`.\n *\n * @param {string} eventType - Indicates the type of events to listen to.\n * Currently, only `\"change\"` is supported.\n * @param {FilterHandle~listener} listener - The callback function that\n * will be invoked when the event occurs.\n * @return {string} - A token to pass to {@link FilterHandle#off} to cancel\n * this subscription.\n */\n on(eventType, listener) {\n return this._emitter.on(eventType, listener);\n }\n\n /**\n * Cancel event subscriptions created by {@link FilterHandle#on}.\n *\n * @param {string} eventType - The type of event to unsubscribe.\n * @param {string|FilterHandle~listener} listener - Either the callback\n * function previously passed into {@link FilterHandle#on}, or the\n * string that was returned from {@link FilterHandle#on}.\n */\n off(eventType, listener) {\n return this._emitter.off(eventType, listener);\n }\n\n _onChange(extraInfo) {\n if (!this._filterSet)\n return;\n this._filterVar.set(this._filterSet.value, this._mergeExtraInfo(extraInfo));\n }\n\n /**\n * @callback FilterHandle~listener\n * @param {Object} event - An object containing details of the event. For\n * `\"change\"` events, this includes the properties `value` (the new\n * value of the filter set, or `null` if no filter set is active),\n * `oldValue` (the previous value of the filter set), and `sender` (the\n * `FilterHandle` instance that made the change).\n */\n\n}\n\n/**\n * @event FilterHandle#change\n * @type {object}\n * @property {object} value - The new value of the filter set, or `null`\n * if no filter set is active.\n * @property {object} oldValue - The previous value of the filter set.\n * @property {FilterHandle} sender - The `FilterHandle` instance that\n * changed the value.\n */\n", - "import { diffSortedLists } from \"./util\";\n\nfunction naturalComparator(a, b) {\n if (a === b) {\n return 0;\n } else if (a < b) {\n return -1;\n } else if (a > b) {\n return 1;\n }\n}\n\n/**\n * @private\n */\nexport default class FilterSet {\n constructor() {\n this.reset();\n }\n\n reset() {\n // Key: handle ID, Value: array of selected keys, or null\n this._handles = {};\n // Key: key string, Value: count of handles that include it\n this._keys = {};\n this._value = null;\n this._activeHandles = 0;\n }\n\n get value() {\n return this._value;\n }\n\n update(handleId, keys) {\n if (keys !== null) {\n keys = keys.slice(0); // clone before sorting\n keys.sort(naturalComparator);\n }\n\n let {added, removed} = diffSortedLists(this._handles[handleId], keys);\n this._handles[handleId] = keys;\n\n for (let i = 0; i < added.length; i++) {\n this._keys[added[i]] = (this._keys[added[i]] || 0) + 1;\n }\n for (let i = 0; i < removed.length; i++) {\n this._keys[removed[i]]--;\n }\n\n this._updateValue(keys);\n }\n\n /**\n * @param {string[]} keys Sorted array of strings that indicate\n * a superset of possible keys.\n * @private\n */\n _updateValue(keys = this._allKeys) {\n let handleCount = Object.keys(this._handles).length;\n if (handleCount === 0) {\n this._value = null;\n } else {\n this._value = [];\n for (let i = 0; i < keys.length; i++) {\n let count = this._keys[keys[i]];\n if (count === handleCount) {\n this._value.push(keys[i]);\n }\n }\n }\n }\n\n clear(handleId) {\n if (typeof(this._handles[handleId]) === \"undefined\") {\n return;\n }\n\n let keys = this._handles[handleId];\n if (!keys) {\n keys = [];\n }\n\n for (let i = 0; i < keys.length; i++) {\n this._keys[keys[i]]--;\n }\n delete this._handles[handleId];\n\n this._updateValue();\n }\n\n get _allKeys() {\n let allKeys = Object.keys(this._keys);\n allKeys.sort(naturalComparator);\n return allKeys;\n }\n}\n", - "import Var from \"./var\";\n\n// Use a global so that multiple copies of crosstalk.js can be loaded and still\n// have groups behave as singletons across all copies.\nglobal.__crosstalk_groups = global.__crosstalk_groups || {};\nlet groups = global.__crosstalk_groups;\n\nexport default function group(groupName) {\n if (groupName && typeof(groupName) === \"string\") {\n if (!groups.hasOwnProperty(groupName)) {\n groups[groupName] = new Group(groupName);\n }\n return groups[groupName];\n } else if (typeof(groupName) === \"object\" && groupName._vars && groupName.var) {\n // Appears to already be a group object\n return groupName;\n } else if (Array.isArray(groupName) &&\n groupName.length == 1 &&\n typeof(groupName[0]) === \"string\") {\n return group(groupName[0]);\n } else {\n throw new Error(\"Invalid groupName argument\");\n }\n}\n\nclass Group {\n constructor(name) {\n this.name = name;\n this._vars = {};\n }\n\n var(name) {\n if (!name || typeof(name) !== \"string\") {\n throw new Error(\"Invalid var name\");\n }\n\n if (!this._vars.hasOwnProperty(name))\n this._vars[name] = new Var(this, name);\n return this._vars[name];\n }\n\n has(name) {\n if (!name || typeof(name) !== \"string\") {\n throw new Error(\"Invalid var name\");\n }\n\n return this._vars.hasOwnProperty(name);\n }\n}\n", - "import group from \"./group\";\nimport { SelectionHandle } from \"./selection\";\nimport { FilterHandle } from \"./filter\";\nimport { bind } from \"./input\";\nimport \"./input_selectize\";\nimport \"./input_checkboxgroup\";\nimport \"./input_slider\";\n\nconst defaultGroup = group(\"default\");\n\nfunction var_(name) {\n return defaultGroup.var(name);\n}\n\nfunction has(name) {\n return defaultGroup.has(name);\n}\n\nif (global.Shiny) {\n global.Shiny.addCustomMessageHandler(\"update-client-value\", function(message) {\n if (typeof(message.group) === \"string\") {\n group(message.group).var(message.name).set(message.value);\n } else {\n var_(message.name).set(message.value);\n }\n });\n}\n\nconst crosstalk = {\n group: group,\n var: var_,\n has: has,\n SelectionHandle: SelectionHandle,\n FilterHandle: FilterHandle,\n bind: bind\n};\n\n/**\n * @namespace crosstalk\n */\nexport default crosstalk;\nglobal.crosstalk = crosstalk;\n", - "let $ = global.jQuery;\n\nlet bindings = {};\n\nexport function register(reg) {\n bindings[reg.className] = reg;\n if (global.document && global.document.readyState !== \"complete\") {\n $(() => {\n bind();\n });\n } else if (global.document) {\n setTimeout(bind, 100);\n }\n}\n\nexport function bind() {\n Object.keys(bindings).forEach(function(className) {\n let binding = bindings[className];\n $(\".\" + binding.className).not(\".crosstalk-input-bound\").each(function(i, el) {\n bindInstance(binding, el);\n });\n });\n}\n\n// Escape jQuery identifier\nfunction $escape(val) {\n return val.replace(/([!\"#$%&'()*+,./:;<=>?@[\\\\\\]^`{|}~])/g, \"\\\\$1\");\n}\n\nfunction bindEl(el) {\n let $el = $(el);\n Object.keys(bindings).forEach(function(className) {\n if ($el.hasClass(className) && !$el.hasClass(\"crosstalk-input-bound\")) {\n let binding = bindings[className];\n bindInstance(binding, el);\n }\n });\n}\n\nfunction bindInstance(binding, el) {\n let jsonEl = $(el).find(\"script[type='application/json'][data-for='\" + $escape(el.id) + \"']\");\n let data = JSON.parse(jsonEl[0].innerText);\n\n let instance = binding.factory(el, data);\n $(el).data(\"crosstalk-instance\", instance);\n $(el).addClass(\"crosstalk-input-bound\");\n}\n\nif (global.Shiny) {\n let inputBinding = new global.Shiny.InputBinding();\n let $ = global.jQuery;\n $.extend(inputBinding, {\n find: function(scope) {\n return $(scope).find(\".crosstalk-input\");\n },\n initialize: function(el) {\n if (!$(el).hasClass(\"crosstalk-input-bound\")) {\n bindEl(el);\n }\n },\n getId: function(el) {\n return el.id;\n },\n getValue: function(el) {\n\n },\n setValue: function(el, value) {\n\n },\n receiveMessage: function(el, data) {\n\n },\n subscribe: function(el, callback) {\n $(el).data(\"crosstalk-instance\").resume();\n },\n unsubscribe: function(el) {\n $(el).data(\"crosstalk-instance\").suspend();\n }\n });\n global.Shiny.inputBindings.register(inputBinding, \"crosstalk.inputBinding\");\n}\n", - "import * as input from \"./input\";\nimport { FilterHandle } from \"./filter\";\n\nlet $ = global.jQuery;\n\ninput.register({\n className: \"crosstalk-input-checkboxgroup\",\n\n factory: function(el, data) {\n /*\n * map: {\"groupA\": [\"keyA\", \"keyB\", ...], ...}\n * group: \"ct-groupname\"\n */\n let ctHandle = new FilterHandle(data.group);\n\n let lastKnownKeys;\n let $el = $(el);\n $el.on(\"change\", \"input[type='checkbox']\", function() {\n let checked = $el.find(\"input[type='checkbox']:checked\");\n if (checked.length === 0) {\n lastKnownKeys = null;\n ctHandle.clear();\n } else {\n let keys = {};\n checked.each(function() {\n data.map[this.value].forEach(function(key) {\n keys[key] = true;\n });\n });\n let keyArray = Object.keys(keys);\n keyArray.sort();\n lastKnownKeys = keyArray;\n ctHandle.set(keyArray);\n }\n });\n\n return {\n suspend: function() {\n ctHandle.clear();\n },\n resume: function() {\n if (lastKnownKeys)\n ctHandle.set(lastKnownKeys);\n }\n };\n }\n});\n", - "import * as input from \"./input\";\nimport * as util from \"./util\";\nimport { FilterHandle } from \"./filter\";\n\nlet $ = global.jQuery;\n\ninput.register({\n className: \"crosstalk-input-select\",\n\n factory: function(el, data) {\n /*\n * items: {value: [...], label: [...]}\n * map: {\"groupA\": [\"keyA\", \"keyB\", ...], ...}\n * group: \"ct-groupname\"\n */\n\n let first = [{value: \"\", label: \"(All)\"}];\n let items = util.dataframeToD3(data.items);\n let opts = {\n options: first.concat(items),\n valueField: \"value\",\n labelField: \"label\",\n searchField: \"label\"\n };\n\n let select = $(el).find(\"select\")[0];\n\n let selectize = $(select).selectize(opts)[0].selectize;\n\n let ctHandle = new FilterHandle(data.group);\n\n let lastKnownKeys;\n selectize.on(\"change\", function() {\n if (selectize.items.length === 0) {\n lastKnownKeys = null;\n ctHandle.clear();\n } else {\n let keys = {};\n selectize.items.forEach(function(group) {\n data.map[group].forEach(function(key) {\n keys[key] = true;\n });\n });\n let keyArray = Object.keys(keys);\n keyArray.sort();\n lastKnownKeys = keyArray;\n ctHandle.set(keyArray);\n }\n });\n\n return {\n suspend: function() {\n ctHandle.clear();\n },\n resume: function() {\n if (lastKnownKeys)\n ctHandle.set(lastKnownKeys);\n }\n };\n }\n});\n", - "import * as input from \"./input\";\nimport { FilterHandle } from \"./filter\";\n\nlet $ = global.jQuery;\nlet strftime = global.strftime;\n\ninput.register({\n className: \"crosstalk-input-slider\",\n\n factory: function(el, data) {\n /*\n * map: {\"groupA\": [\"keyA\", \"keyB\", ...], ...}\n * group: \"ct-groupname\"\n */\n let ctHandle = new FilterHandle(data.group);\n\n let opts = {};\n let $el = $(el).find(\"input\");\n let dataType = $el.data(\"data-type\");\n let timeFormat = $el.data(\"time-format\");\n let round = $el.data(\"round\");\n let timeFormatter;\n\n // Set up formatting functions\n if (dataType === \"date\") {\n timeFormatter = strftime.utc();\n opts.prettify = function(num) {\n return timeFormatter(timeFormat, new Date(num));\n };\n\n } else if (dataType === \"datetime\") {\n let timezone = $el.data(\"timezone\");\n if (timezone)\n timeFormatter = strftime.timezone(timezone);\n else\n timeFormatter = strftime;\n\n opts.prettify = function(num) {\n return timeFormatter(timeFormat, new Date(num));\n };\n } else if (dataType === \"number\") {\n if (typeof round !== \"undefined\")\n opts.prettify = function(num) {\n let factor = Math.pow(10, round);\n return Math.round(num * factor) / factor;\n };\n }\n\n $el.ionRangeSlider(opts);\n\n function getValue() {\n let result = $el.data(\"ionRangeSlider\").result;\n\n // Function for converting numeric value from slider to appropriate type.\n let convert;\n let dataType = $el.data(\"data-type\");\n if (dataType === \"date\") {\n convert = function(val) {\n return formatDateUTC(new Date(+val));\n };\n } else if (dataType === \"datetime\") {\n convert = function(val) {\n // Convert ms to s\n return +val / 1000;\n };\n } else {\n convert = function(val) { return +val; };\n }\n\n if ($el.data(\"ionRangeSlider\").options.type === \"double\") {\n return [convert(result.from), convert(result.to)];\n } else {\n return convert(result.from);\n }\n }\n\n let lastKnownKeys = null;\n\n $el.on(\"change.crosstalkSliderInput\", function(event) {\n if (!$el.data(\"updating\") && !$el.data(\"animating\")) {\n let [from, to] = getValue();\n let keys = [];\n for (let i = 0; i < data.values.length; i++) {\n let val = data.values[i];\n if (val >= from && val <= to) {\n keys.push(data.keys[i]);\n }\n }\n keys.sort();\n ctHandle.set(keys);\n lastKnownKeys = keys;\n }\n });\n\n\n // let $el = $(el);\n // $el.on(\"change\", \"input[type=\"checkbox\"]\", function() {\n // let checked = $el.find(\"input[type=\"checkbox\"]:checked\");\n // if (checked.length === 0) {\n // ctHandle.clear();\n // } else {\n // let keys = {};\n // checked.each(function() {\n // data.map[this.value].forEach(function(key) {\n // keys[key] = true;\n // });\n // });\n // let keyArray = Object.keys(keys);\n // keyArray.sort();\n // ctHandle.set(keyArray);\n // }\n // });\n\n return {\n suspend: function() {\n ctHandle.clear();\n },\n resume: function() {\n if (lastKnownKeys)\n ctHandle.set(lastKnownKeys);\n }\n };\n }\n});\n\n\n// Convert a number to a string with leading zeros\nfunction padZeros(n, digits) {\n let str = n.toString();\n while (str.length < digits)\n str = \"0\" + str;\n return str;\n}\n\n// Given a Date object, return a string in yyyy-mm-dd format, using the\n// UTC date. This may be a day off from the date in the local time zone.\nfunction formatDateUTC(date) {\n if (date instanceof Date) {\n return date.getUTCFullYear() + \"-\" +\n padZeros(date.getUTCMonth()+1, 2) + \"-\" +\n padZeros(date.getUTCDate(), 2);\n\n } else {\n return null;\n }\n}\n", - "import Events from \"./events\";\nimport grp from \"./group\";\nimport * as util from \"./util\";\n\n/**\n * Use this class to read and write (and listen for changes to) the selection\n * for a Crosstalk group. This is intended to be used for linked brushing.\n *\n * If two (or more) `SelectionHandle` instances in the same webpage share the\n * same group name, they will share the same state. Setting the selection using\n * one `SelectionHandle` instance will result in the `value` property instantly\n * changing across the others, and `\"change\"` event listeners on all instances\n * (including the one that initiated the sending) will fire.\n *\n * @param {string} [group] - The name of the Crosstalk group, or if none,\n * null or undefined (or any other falsy value). This can be changed later\n * via the [SelectionHandle#setGroup](#setGroup) method.\n * @param {Object} [extraInfo] - An object whose properties will be copied to\n * the event object whenever an event is emitted.\n */\nexport class SelectionHandle {\n\n constructor(group = null, extraInfo = null) {\n this._eventRelay = new Events();\n this._emitter = new util.SubscriptionTracker(this._eventRelay);\n\n // Name of the group we're currently tracking, if any. Can change over time.\n this._group = null;\n // The Var we're currently tracking, if any. Can change over time.\n this._var = null;\n // The event handler subscription we currently have on var.on(\"change\").\n this._varOnChangeSub = null;\n\n this._extraInfo = util.extend({ sender: this }, extraInfo);\n\n this.setGroup(group);\n }\n\n /**\n * Changes the Crosstalk group membership of this SelectionHandle. The group\n * being switched away from (if any) will not have its selection value\n * modified as a result of calling `setGroup`, even if this handle was the\n * most recent handle to set the selection of the group.\n *\n * The group being switched to (if any) will also not have its selection value\n * modified as a result of calling `setGroup`. If you want to set the\n * selection value of the new group, call `set` explicitly.\n *\n * @param {string} group - The name of the Crosstalk group, or null (or\n * undefined) to clear the group.\n */\n setGroup(group) {\n // If group is unchanged, do nothing\n if (this._group === group)\n return;\n // Treat null, undefined, and other falsy values the same\n if (!this._group && !group)\n return;\n\n if (this._var) {\n this._var.off(\"change\", this._varOnChangeSub);\n this._var = null;\n this._varOnChangeSub = null;\n }\n\n this._group = group;\n\n if (group) {\n this._var = grp(group).var(\"selection\");\n let sub = this._var.on(\"change\", (e) => {\n this._eventRelay.trigger(\"change\", e, this);\n });\n this._varOnChangeSub = sub;\n }\n }\n\n /**\n * Retrieves the current selection for the group represented by this\n * `SelectionHandle`.\n *\n * - If no selection is active, then this value will be falsy.\n * - If a selection is active, but no data points are selected, then this\n * value will be an empty array.\n * - If a selection is active, and data points are selected, then the keys\n * of the selected data points will be present in the array.\n */\n get value() {\n return this._var ? this._var.get() : null;\n }\n\n /**\n * Combines the given `extraInfo` (if any) with the handle's default\n * `_extraInfo` (if any).\n * @private\n */\n _mergeExtraInfo(extraInfo) {\n // Important incidental effect: shallow clone is returned\n return util.extend({},\n this._extraInfo ? this._extraInfo : null,\n extraInfo ? extraInfo : null);\n }\n\n /**\n * Overwrites the current selection for the group, and raises the `\"change\"`\n * event among all of the group's '`SelectionHandle` instances (including\n * this one).\n *\n * @fires SelectionHandle#change\n * @param {string[]} selectedKeys - Falsy, empty array, or array of keys (see\n * {@link SelectionHandle#value}).\n * @param {Object} [extraInfo] - Extra properties to be included on the event\n * object that's passed to listeners (in addition to any options that were\n * passed into the `SelectionHandle` constructor).\n */\n set(selectedKeys, extraInfo) {\n if (this._var)\n this._var.set(selectedKeys, this._mergeExtraInfo(extraInfo));\n }\n\n /**\n * Overwrites the current selection for the group, and raises the `\"change\"`\n * event among all of the group's '`SelectionHandle` instances (including\n * this one).\n *\n * @fires SelectionHandle#change\n * @param {Object} [extraInfo] - Extra properties to be included on the event\n * object that's passed to listeners (in addition to any that were passed\n * into the `SelectionHandle` constructor).\n */\n clear(extraInfo) {\n if (this._var)\n this.set(void 0, this._mergeExtraInfo(extraInfo));\n }\n\n /**\n * Subscribes to events on this `SelectionHandle`.\n *\n * @param {string} eventType - Indicates the type of events to listen to.\n * Currently, only `\"change\"` is supported.\n * @param {SelectionHandle~listener} listener - The callback function that\n * will be invoked when the event occurs.\n * @return {string} - A token to pass to {@link SelectionHandle#off} to cancel\n * this subscription.\n */\n on(eventType, listener) {\n return this._emitter.on(eventType, listener);\n }\n\n /**\n * Cancels event subscriptions created by {@link SelectionHandle#on}.\n *\n * @param {string} eventType - The type of event to unsubscribe.\n * @param {string|SelectionHandle~listener} listener - Either the callback\n * function previously passed into {@link SelectionHandle#on}, or the\n * string that was returned from {@link SelectionHandle#on}.\n */\n off(eventType, listener) {\n return this._emitter.off(eventType, listener);\n }\n\n /**\n * Shuts down the `SelectionHandle` object.\n *\n * Removes all event listeners that were added through this handle.\n */\n close() {\n this._emitter.removeAllListeners();\n this.setGroup(null);\n }\n}\n\n/**\n * @callback SelectionHandle~listener\n * @param {Object} event - An object containing details of the event. For\n * `\"change\"` events, this includes the properties `value` (the new\n * value of the selection, or `undefined` if no selection is active),\n * `oldValue` (the previous value of the selection), and `sender` (the\n * `SelectionHandle` instance that made the change).\n */\n\n/**\n * @event SelectionHandle#change\n * @type {object}\n * @property {object} value - The new value of the selection, or `undefined`\n * if no selection is active.\n * @property {object} oldValue - The previous value of the selection.\n * @property {SelectionHandle} sender - The `SelectionHandle` instance that\n * changed the value.\n */\n", - "export function extend(target, ...sources) {\n for (let i = 0; i < sources.length; i++) {\n let src = sources[i];\n if (typeof(src) === \"undefined\" || src === null)\n continue;\n\n for (let key in src) {\n if (src.hasOwnProperty(key)) {\n target[key] = src[key];\n }\n }\n }\n return target;\n}\n\nexport function checkSorted(list) {\n for (let i = 1; i < list.length; i++) {\n if (list[i] <= list[i-1]) {\n throw new Error(\"List is not sorted or contains duplicate\");\n }\n }\n}\n\nexport function diffSortedLists(a, b) {\n let i_a = 0;\n let i_b = 0;\n\n if (!a) a = [];\n if (!b) b = [];\n\n let a_only = [];\n let b_only = [];\n\n checkSorted(a);\n checkSorted(b);\n\n while (i_a < a.length && i_b < b.length) {\n if (a[i_a] === b[i_b]) {\n i_a++;\n i_b++;\n } else if (a[i_a] < b[i_b]) {\n a_only.push(a[i_a++]);\n } else {\n b_only.push(b[i_b++]);\n }\n }\n\n if (i_a < a.length)\n a_only = a_only.concat(a.slice(i_a));\n if (i_b < b.length)\n b_only = b_only.concat(b.slice(i_b));\n return {\n removed: a_only,\n added: b_only\n };\n}\n\n// Convert from wide: { colA: [1,2,3], colB: [4,5,6], ... }\n// to long: [ {colA: 1, colB: 4}, {colA: 2, colB: 5}, ... ]\nexport function dataframeToD3(df) {\n let names = [];\n let length;\n for (let name in df) {\n if (df.hasOwnProperty(name))\n names.push(name);\n if (typeof(df[name]) !== \"object\" || typeof(df[name].length) === \"undefined\") {\n throw new Error(\"All fields must be arrays\");\n } else if (typeof(length) !== \"undefined\" && length !== df[name].length) {\n throw new Error(\"All fields must be arrays of the same length\");\n }\n length = df[name].length;\n }\n let results = [];\n let item;\n for (let row = 0; row < length; row++) {\n item = {};\n for (let col = 0; col < names.length; col++) {\n item[names[col]] = df[names[col]][row];\n }\n results.push(item);\n }\n return results;\n}\n\n/**\n * Keeps track of all event listener additions/removals and lets all active\n * listeners be removed with a single operation.\n *\n * @private\n */\nexport class SubscriptionTracker {\n constructor(emitter) {\n this._emitter = emitter;\n this._subs = {};\n }\n\n on(eventType, listener) {\n let sub = this._emitter.on(eventType, listener);\n this._subs[sub] = eventType;\n return sub;\n }\n\n off(eventType, listener) {\n let sub = this._emitter.off(eventType, listener);\n if (sub) {\n delete this._subs[sub];\n }\n return sub;\n }\n\n removeAllListeners() {\n let current_subs = this._subs;\n this._subs = {};\n Object.keys(current_subs).forEach((sub) => {\n this._emitter.off(current_subs[sub], sub);\n });\n }\n}\n", - "import Events from \"./events\";\n\nexport default class Var {\n constructor(group, name, /*optional*/ value) {\n this._group = group;\n this._name = name;\n this._value = value;\n this._events = new Events();\n }\n\n get() {\n return this._value;\n }\n\n set(value, /*optional*/ event) {\n if (this._value === value) {\n // Do nothing; the value hasn't changed\n return;\n }\n let oldValue = this._value;\n this._value = value;\n // Alert JavaScript listeners that the value has changed\n let evt = {};\n if (event && typeof(event) === \"object\") {\n for (let k in event) {\n if (event.hasOwnProperty(k))\n evt[k] = event[k];\n }\n }\n evt.oldValue = oldValue;\n evt.value = value;\n this._events.trigger(\"change\", evt, this);\n\n // TODO: Make this extensible, to let arbitrary back-ends know that\n // something has changed\n if (global.Shiny && global.Shiny.onInputChange) {\n global.Shiny.onInputChange(\n \".clientValue-\" +\n (this._group.name !== null ? this._group.name + \"-\" : \"\") +\n this._name,\n typeof(value) === \"undefined\" ? null : value\n );\n }\n }\n\n on(eventType, listener) {\n return this._events.on(eventType, listener);\n }\n\n off(eventType, listener) {\n return this._events.off(eventType, listener);\n }\n}\n" - ] -} \ No newline at end of file diff --git a/site_libs/crosstalk-1.2.0/js/crosstalk.min.js b/site_libs/crosstalk-1.2.0/js/crosstalk.min.js deleted file mode 100644 index b7ec0ac9f..000000000 --- a/site_libs/crosstalk-1.2.0/js/crosstalk.min.js +++ /dev/null @@ -1,2 +0,0 @@ -!function o(u,a,l){function s(n,e){if(!a[n]){if(!u[n]){var t="function"==typeof require&&require;if(!e&&t)return t(n,!0);if(f)return f(n,!0);var r=new Error("Cannot find module '"+n+"'");throw r.code="MODULE_NOT_FOUND",r}var i=a[n]={exports:{}};u[n][0].call(i.exports,function(e){var t=u[n][1][e];return s(t||e)},i,i.exports,o,u,a,l)}return a[n].exports}for(var f="function"==typeof require&&require,e=0;e?@[\\\]^`{|}~])/g,"\\$1")+"']"),r=JSON.parse(n[0].innerText),i=e.factory(t,r);o(t).data("crosstalk-instance",i),o(t).addClass("crosstalk-input-bound")}if(t.Shiny){var e=new t.Shiny.InputBinding,u=t.jQuery;u.extend(e,{find:function(e){return u(e).find(".crosstalk-input")},initialize:function(e){var t,n;u(e).hasClass("crosstalk-input-bound")||(n=o(t=e),Object.keys(r).forEach(function(e){n.hasClass(e)&&!n.hasClass("crosstalk-input-bound")&&i(r[e],t)}))},getId:function(e){return e.id},getValue:function(e){},setValue:function(e,t){},receiveMessage:function(e,t){},subscribe:function(e,t){u(e).data("crosstalk-instance").resume()},unsubscribe:function(e){u(e).data("crosstalk-instance").suspend()}}),t.Shiny.inputBindings.register(e,"crosstalk.inputBinding")}}).call(this,"undefined"!=typeof global?global:"undefined"!=typeof self?self:"undefined"!=typeof window?window:{})},{}],7:[function(r,e,t){(function(e){"use strict";var t=function(e){{if(e&&e.__esModule)return e;var t={};if(null!=e)for(var n in e)Object.prototype.hasOwnProperty.call(e,n)&&(t[n]=e[n]);return t.default=e,t}}(r("./input")),n=r("./filter");var a=e.jQuery;t.register({className:"crosstalk-input-checkboxgroup",factory:function(e,r){var i=new n.FilterHandle(r.group),o=void 0,u=a(e);return u.on("change","input[type='checkbox']",function(){var e=u.find("input[type='checkbox']:checked");if(0===e.length)o=null,i.clear();else{var t={};e.each(function(){r.map[this.value].forEach(function(e){t[e]=!0})});var n=Object.keys(t);n.sort(),o=n,i.set(n)}}),{suspend:function(){i.clear()},resume:function(){o&&i.set(o)}}}})}).call(this,"undefined"!=typeof global?global:"undefined"!=typeof self?self:"undefined"!=typeof window?window:{})},{"./filter":2,"./input":6}],8:[function(r,e,t){(function(e){"use strict";var t=n(r("./input")),l=n(r("./util")),s=r("./filter");function n(e){if(e&&e.__esModule)return e;var t={};if(null!=e)for(var n in e)Object.prototype.hasOwnProperty.call(e,n)&&(t[n]=e[n]);return t.default=e,t}var f=e.jQuery;t.register({className:"crosstalk-input-select",factory:function(e,n){var t=l.dataframeToD3(n.items),r={options:[{value:"",label:"(All)"}].concat(t),valueField:"value",labelField:"label",searchField:"label"},i=f(e).find("select")[0],o=f(i).selectize(r)[0].selectize,u=new s.FilterHandle(n.group),a=void 0;return o.on("change",function(){if(0===o.items.length)a=null,u.clear();else{var t={};o.items.forEach(function(e){n.map[e].forEach(function(e){t[e]=!0})});var e=Object.keys(t);e.sort(),a=e,u.set(e)}}),{suspend:function(){u.clear()},resume:function(){a&&u.set(a)}}}})}).call(this,"undefined"!=typeof global?global:"undefined"!=typeof self?self:"undefined"!=typeof window?window:{})},{"./filter":2,"./input":6,"./util":11}],9:[function(n,e,t){(function(e){"use strict";var d=function(e,t){if(Array.isArray(e))return e;if(Symbol.iterator in Object(e))return function(e,t){var n=[],r=!0,i=!1,o=void 0;try{for(var u,a=e[Symbol.iterator]();!(r=(u=a.next()).done)&&(n.push(u.value),!t||n.length!==t);r=!0);}catch(e){i=!0,o=e}finally{try{!r&&a.return&&a.return()}finally{if(i)throw o}}return n}(e,t);throw new TypeError("Invalid attempt to destructure non-iterable instance")},t=function(e){{if(e&&e.__esModule)return e;var t={};if(null!=e)for(var n in e)Object.prototype.hasOwnProperty.call(e,n)&&(t[n]=e[n]);return t.default=e,t}}(n("./input")),a=n("./filter");var v=e.jQuery,p=e.strftime;function y(e,t){for(var n=e.toString();n.length {\n this._eventRelay.trigger(\"change\", e, this);\n });\n this._varOnChangeSub = sub;\n }\n }\n\n /**\n * Combine the given `extraInfo` (if any) with the handle's default\n * `_extraInfo` (if any).\n * @private\n */\n _mergeExtraInfo(extraInfo) {\n return util.extend({},\n this._extraInfo ? this._extraInfo : null,\n extraInfo ? extraInfo : null);\n }\n\n /**\n * Close the handle. This clears this handle's contribution to the filter set,\n * and unsubscribes all event listeners.\n */\n close() {\n this._emitter.removeAllListeners();\n this.clear();\n this.setGroup(null);\n }\n\n /**\n * Clear this handle's contribution to the filter set.\n *\n * @param {Object} [extraInfo] - Extra properties to be included on the event\n * object that's passed to listeners (in addition to any options that were\n * passed into the `FilterHandle` constructor).\n * \n * @fires FilterHandle#change\n */\n clear(extraInfo) {\n if (!this._filterSet)\n return;\n this._filterSet.clear(this._id);\n this._onChange(extraInfo);\n }\n\n /**\n * Set this handle's contribution to the filter set. This array should consist\n * of the keys of the rows that _should_ be displayed; any keys that are not\n * present in the array will be considered _filtered out_. Note that multiple\n * `FilterHandle` instances in the group may each contribute an array of keys,\n * and only those keys that appear in _all_ of the arrays make it through the\n * filter.\n *\n * @param {string[]} keys - Empty array, or array of keys. To clear the\n * filter, don't pass an empty array; instead, use the\n * {@link FilterHandle#clear} method.\n * @param {Object} [extraInfo] - Extra properties to be included on the event\n * object that's passed to listeners (in addition to any options that were\n * passed into the `FilterHandle` constructor).\n * \n * @fires FilterHandle#change\n */\n set(keys, extraInfo) {\n if (!this._filterSet)\n return;\n this._filterSet.update(this._id, keys);\n this._onChange(extraInfo);\n }\n\n /**\n * @return {string[]|null} - Either: 1) an array of keys that made it through\n * all of the `FilterHandle` instances, or, 2) `null`, which means no filter\n * is being applied (all data should be displayed).\n */\n get filteredKeys() {\n return this._filterSet ? this._filterSet.value : null;\n }\n\n /**\n * Subscribe to events on this `FilterHandle`.\n *\n * @param {string} eventType - Indicates the type of events to listen to.\n * Currently, only `\"change\"` is supported.\n * @param {FilterHandle~listener} listener - The callback function that\n * will be invoked when the event occurs.\n * @return {string} - A token to pass to {@link FilterHandle#off} to cancel\n * this subscription.\n */\n on(eventType, listener) {\n return this._emitter.on(eventType, listener);\n }\n\n /**\n * Cancel event subscriptions created by {@link FilterHandle#on}.\n *\n * @param {string} eventType - The type of event to unsubscribe.\n * @param {string|FilterHandle~listener} listener - Either the callback\n * function previously passed into {@link FilterHandle#on}, or the\n * string that was returned from {@link FilterHandle#on}.\n */\n off(eventType, listener) {\n return this._emitter.off(eventType, listener);\n }\n\n _onChange(extraInfo) {\n if (!this._filterSet)\n return;\n this._filterVar.set(this._filterSet.value, this._mergeExtraInfo(extraInfo));\n }\n\n /**\n * @callback FilterHandle~listener\n * @param {Object} event - An object containing details of the event. For\n * `\"change\"` events, this includes the properties `value` (the new\n * value of the filter set, or `null` if no filter set is active),\n * `oldValue` (the previous value of the filter set), and `sender` (the\n * `FilterHandle` instance that made the change).\n */\n\n}\n\n/**\n * @event FilterHandle#change\n * @type {object}\n * @property {object} value - The new value of the filter set, or `null`\n * if no filter set is active.\n * @property {object} oldValue - The previous value of the filter set.\n * @property {FilterHandle} sender - The `FilterHandle` instance that\n * changed the value.\n */\n","import { diffSortedLists } from \"./util\";\n\nfunction naturalComparator(a, b) {\n if (a === b) {\n return 0;\n } else if (a < b) {\n return -1;\n } else if (a > b) {\n return 1;\n }\n}\n\n/**\n * @private\n */\nexport default class FilterSet {\n constructor() {\n this.reset();\n }\n\n reset() {\n // Key: handle ID, Value: array of selected keys, or null\n this._handles = {};\n // Key: key string, Value: count of handles that include it\n this._keys = {};\n this._value = null;\n this._activeHandles = 0;\n }\n\n get value() {\n return this._value;\n }\n\n update(handleId, keys) {\n if (keys !== null) {\n keys = keys.slice(0); // clone before sorting\n keys.sort(naturalComparator);\n }\n\n let {added, removed} = diffSortedLists(this._handles[handleId], keys);\n this._handles[handleId] = keys;\n\n for (let i = 0; i < added.length; i++) {\n this._keys[added[i]] = (this._keys[added[i]] || 0) + 1;\n }\n for (let i = 0; i < removed.length; i++) {\n this._keys[removed[i]]--;\n }\n\n this._updateValue(keys);\n }\n\n /**\n * @param {string[]} keys Sorted array of strings that indicate\n * a superset of possible keys.\n * @private\n */\n _updateValue(keys = this._allKeys) {\n let handleCount = Object.keys(this._handles).length;\n if (handleCount === 0) {\n this._value = null;\n } else {\n this._value = [];\n for (let i = 0; i < keys.length; i++) {\n let count = this._keys[keys[i]];\n if (count === handleCount) {\n this._value.push(keys[i]);\n }\n }\n }\n }\n\n clear(handleId) {\n if (typeof(this._handles[handleId]) === \"undefined\") {\n return;\n }\n\n let keys = this._handles[handleId];\n if (!keys) {\n keys = [];\n }\n\n for (let i = 0; i < keys.length; i++) {\n this._keys[keys[i]]--;\n }\n delete this._handles[handleId];\n\n this._updateValue();\n }\n\n get _allKeys() {\n let allKeys = Object.keys(this._keys);\n allKeys.sort(naturalComparator);\n return allKeys;\n }\n}\n","import Var from \"./var\";\n\n// Use a global so that multiple copies of crosstalk.js can be loaded and still\n// have groups behave as singletons across all copies.\nglobal.__crosstalk_groups = global.__crosstalk_groups || {};\nlet groups = global.__crosstalk_groups;\n\nexport default function group(groupName) {\n if (groupName && typeof(groupName) === \"string\") {\n if (!groups.hasOwnProperty(groupName)) {\n groups[groupName] = new Group(groupName);\n }\n return groups[groupName];\n } else if (typeof(groupName) === \"object\" && groupName._vars && groupName.var) {\n // Appears to already be a group object\n return groupName;\n } else if (Array.isArray(groupName) &&\n groupName.length == 1 &&\n typeof(groupName[0]) === \"string\") {\n return group(groupName[0]);\n } else {\n throw new Error(\"Invalid groupName argument\");\n }\n}\n\nclass Group {\n constructor(name) {\n this.name = name;\n this._vars = {};\n }\n\n var(name) {\n if (!name || typeof(name) !== \"string\") {\n throw new Error(\"Invalid var name\");\n }\n\n if (!this._vars.hasOwnProperty(name))\n this._vars[name] = new Var(this, name);\n return this._vars[name];\n }\n\n has(name) {\n if (!name || typeof(name) !== \"string\") {\n throw new Error(\"Invalid var name\");\n }\n\n return this._vars.hasOwnProperty(name);\n }\n}\n","import group from \"./group\";\nimport { SelectionHandle } from \"./selection\";\nimport { FilterHandle } from \"./filter\";\nimport { bind } from \"./input\";\nimport \"./input_selectize\";\nimport \"./input_checkboxgroup\";\nimport \"./input_slider\";\n\nconst defaultGroup = group(\"default\");\n\nfunction var_(name) {\n return defaultGroup.var(name);\n}\n\nfunction has(name) {\n return defaultGroup.has(name);\n}\n\nif (global.Shiny) {\n global.Shiny.addCustomMessageHandler(\"update-client-value\", function(message) {\n if (typeof(message.group) === \"string\") {\n group(message.group).var(message.name).set(message.value);\n } else {\n var_(message.name).set(message.value);\n }\n });\n}\n\nconst crosstalk = {\n group: group,\n var: var_,\n has: has,\n SelectionHandle: SelectionHandle,\n FilterHandle: FilterHandle,\n bind: bind\n};\n\n/**\n * @namespace crosstalk\n */\nexport default crosstalk;\nglobal.crosstalk = crosstalk;\n","let $ = global.jQuery;\n\nlet bindings = {};\n\nexport function register(reg) {\n bindings[reg.className] = reg;\n if (global.document && global.document.readyState !== \"complete\") {\n $(() => {\n bind();\n });\n } else if (global.document) {\n setTimeout(bind, 100);\n }\n}\n\nexport function bind() {\n Object.keys(bindings).forEach(function(className) {\n let binding = bindings[className];\n $(\".\" + binding.className).not(\".crosstalk-input-bound\").each(function(i, el) {\n bindInstance(binding, el);\n });\n });\n}\n\n// Escape jQuery identifier\nfunction $escape(val) {\n return val.replace(/([!\"#$%&'()*+,./:;<=>?@[\\\\\\]^`{|}~])/g, \"\\\\$1\");\n}\n\nfunction bindEl(el) {\n let $el = $(el);\n Object.keys(bindings).forEach(function(className) {\n if ($el.hasClass(className) && !$el.hasClass(\"crosstalk-input-bound\")) {\n let binding = bindings[className];\n bindInstance(binding, el);\n }\n });\n}\n\nfunction bindInstance(binding, el) {\n let jsonEl = $(el).find(\"script[type='application/json'][data-for='\" + $escape(el.id) + \"']\");\n let data = JSON.parse(jsonEl[0].innerText);\n\n let instance = binding.factory(el, data);\n $(el).data(\"crosstalk-instance\", instance);\n $(el).addClass(\"crosstalk-input-bound\");\n}\n\nif (global.Shiny) {\n let inputBinding = new global.Shiny.InputBinding();\n let $ = global.jQuery;\n $.extend(inputBinding, {\n find: function(scope) {\n return $(scope).find(\".crosstalk-input\");\n },\n initialize: function(el) {\n if (!$(el).hasClass(\"crosstalk-input-bound\")) {\n bindEl(el);\n }\n },\n getId: function(el) {\n return el.id;\n },\n getValue: function(el) {\n\n },\n setValue: function(el, value) {\n\n },\n receiveMessage: function(el, data) {\n\n },\n subscribe: function(el, callback) {\n $(el).data(\"crosstalk-instance\").resume();\n },\n unsubscribe: function(el) {\n $(el).data(\"crosstalk-instance\").suspend();\n }\n });\n global.Shiny.inputBindings.register(inputBinding, \"crosstalk.inputBinding\");\n}\n","import * as input from \"./input\";\nimport { FilterHandle } from \"./filter\";\n\nlet $ = global.jQuery;\n\ninput.register({\n className: \"crosstalk-input-checkboxgroup\",\n\n factory: function(el, data) {\n /*\n * map: {\"groupA\": [\"keyA\", \"keyB\", ...], ...}\n * group: \"ct-groupname\"\n */\n let ctHandle = new FilterHandle(data.group);\n\n let lastKnownKeys;\n let $el = $(el);\n $el.on(\"change\", \"input[type='checkbox']\", function() {\n let checked = $el.find(\"input[type='checkbox']:checked\");\n if (checked.length === 0) {\n lastKnownKeys = null;\n ctHandle.clear();\n } else {\n let keys = {};\n checked.each(function() {\n data.map[this.value].forEach(function(key) {\n keys[key] = true;\n });\n });\n let keyArray = Object.keys(keys);\n keyArray.sort();\n lastKnownKeys = keyArray;\n ctHandle.set(keyArray);\n }\n });\n\n return {\n suspend: function() {\n ctHandle.clear();\n },\n resume: function() {\n if (lastKnownKeys)\n ctHandle.set(lastKnownKeys);\n }\n };\n }\n});\n","import * as input from \"./input\";\nimport * as util from \"./util\";\nimport { FilterHandle } from \"./filter\";\n\nlet $ = global.jQuery;\n\ninput.register({\n className: \"crosstalk-input-select\",\n\n factory: function(el, data) {\n /*\n * items: {value: [...], label: [...]}\n * map: {\"groupA\": [\"keyA\", \"keyB\", ...], ...}\n * group: \"ct-groupname\"\n */\n\n let first = [{value: \"\", label: \"(All)\"}];\n let items = util.dataframeToD3(data.items);\n let opts = {\n options: first.concat(items),\n valueField: \"value\",\n labelField: \"label\",\n searchField: \"label\"\n };\n\n let select = $(el).find(\"select\")[0];\n\n let selectize = $(select).selectize(opts)[0].selectize;\n\n let ctHandle = new FilterHandle(data.group);\n\n let lastKnownKeys;\n selectize.on(\"change\", function() {\n if (selectize.items.length === 0) {\n lastKnownKeys = null;\n ctHandle.clear();\n } else {\n let keys = {};\n selectize.items.forEach(function(group) {\n data.map[group].forEach(function(key) {\n keys[key] = true;\n });\n });\n let keyArray = Object.keys(keys);\n keyArray.sort();\n lastKnownKeys = keyArray;\n ctHandle.set(keyArray);\n }\n });\n\n return {\n suspend: function() {\n ctHandle.clear();\n },\n resume: function() {\n if (lastKnownKeys)\n ctHandle.set(lastKnownKeys);\n }\n };\n }\n});\n","import * as input from \"./input\";\nimport { FilterHandle } from \"./filter\";\n\nlet $ = global.jQuery;\nlet strftime = global.strftime;\n\ninput.register({\n className: \"crosstalk-input-slider\",\n\n factory: function(el, data) {\n /*\n * map: {\"groupA\": [\"keyA\", \"keyB\", ...], ...}\n * group: \"ct-groupname\"\n */\n let ctHandle = new FilterHandle(data.group);\n\n let opts = {};\n let $el = $(el).find(\"input\");\n let dataType = $el.data(\"data-type\");\n let timeFormat = $el.data(\"time-format\");\n let round = $el.data(\"round\");\n let timeFormatter;\n\n // Set up formatting functions\n if (dataType === \"date\") {\n timeFormatter = strftime.utc();\n opts.prettify = function(num) {\n return timeFormatter(timeFormat, new Date(num));\n };\n\n } else if (dataType === \"datetime\") {\n let timezone = $el.data(\"timezone\");\n if (timezone)\n timeFormatter = strftime.timezone(timezone);\n else\n timeFormatter = strftime;\n\n opts.prettify = function(num) {\n return timeFormatter(timeFormat, new Date(num));\n };\n } else if (dataType === \"number\") {\n if (typeof round !== \"undefined\")\n opts.prettify = function(num) {\n let factor = Math.pow(10, round);\n return Math.round(num * factor) / factor;\n };\n }\n\n $el.ionRangeSlider(opts);\n\n function getValue() {\n let result = $el.data(\"ionRangeSlider\").result;\n\n // Function for converting numeric value from slider to appropriate type.\n let convert;\n let dataType = $el.data(\"data-type\");\n if (dataType === \"date\") {\n convert = function(val) {\n return formatDateUTC(new Date(+val));\n };\n } else if (dataType === \"datetime\") {\n convert = function(val) {\n // Convert ms to s\n return +val / 1000;\n };\n } else {\n convert = function(val) { return +val; };\n }\n\n if ($el.data(\"ionRangeSlider\").options.type === \"double\") {\n return [convert(result.from), convert(result.to)];\n } else {\n return convert(result.from);\n }\n }\n\n let lastKnownKeys = null;\n\n $el.on(\"change.crosstalkSliderInput\", function(event) {\n if (!$el.data(\"updating\") && !$el.data(\"animating\")) {\n let [from, to] = getValue();\n let keys = [];\n for (let i = 0; i < data.values.length; i++) {\n let val = data.values[i];\n if (val >= from && val <= to) {\n keys.push(data.keys[i]);\n }\n }\n keys.sort();\n ctHandle.set(keys);\n lastKnownKeys = keys;\n }\n });\n\n\n // let $el = $(el);\n // $el.on(\"change\", \"input[type=\"checkbox\"]\", function() {\n // let checked = $el.find(\"input[type=\"checkbox\"]:checked\");\n // if (checked.length === 0) {\n // ctHandle.clear();\n // } else {\n // let keys = {};\n // checked.each(function() {\n // data.map[this.value].forEach(function(key) {\n // keys[key] = true;\n // });\n // });\n // let keyArray = Object.keys(keys);\n // keyArray.sort();\n // ctHandle.set(keyArray);\n // }\n // });\n\n return {\n suspend: function() {\n ctHandle.clear();\n },\n resume: function() {\n if (lastKnownKeys)\n ctHandle.set(lastKnownKeys);\n }\n };\n }\n});\n\n\n// Convert a number to a string with leading zeros\nfunction padZeros(n, digits) {\n let str = n.toString();\n while (str.length < digits)\n str = \"0\" + str;\n return str;\n}\n\n// Given a Date object, return a string in yyyy-mm-dd format, using the\n// UTC date. This may be a day off from the date in the local time zone.\nfunction formatDateUTC(date) {\n if (date instanceof Date) {\n return date.getUTCFullYear() + \"-\" +\n padZeros(date.getUTCMonth()+1, 2) + \"-\" +\n padZeros(date.getUTCDate(), 2);\n\n } else {\n return null;\n }\n}\n","import Events from \"./events\";\nimport grp from \"./group\";\nimport * as util from \"./util\";\n\n/**\n * Use this class to read and write (and listen for changes to) the selection\n * for a Crosstalk group. This is intended to be used for linked brushing.\n *\n * If two (or more) `SelectionHandle` instances in the same webpage share the\n * same group name, they will share the same state. Setting the selection using\n * one `SelectionHandle` instance will result in the `value` property instantly\n * changing across the others, and `\"change\"` event listeners on all instances\n * (including the one that initiated the sending) will fire.\n *\n * @param {string} [group] - The name of the Crosstalk group, or if none,\n * null or undefined (or any other falsy value). This can be changed later\n * via the [SelectionHandle#setGroup](#setGroup) method.\n * @param {Object} [extraInfo] - An object whose properties will be copied to\n * the event object whenever an event is emitted.\n */\nexport class SelectionHandle {\n\n constructor(group = null, extraInfo = null) {\n this._eventRelay = new Events();\n this._emitter = new util.SubscriptionTracker(this._eventRelay);\n\n // Name of the group we're currently tracking, if any. Can change over time.\n this._group = null;\n // The Var we're currently tracking, if any. Can change over time.\n this._var = null;\n // The event handler subscription we currently have on var.on(\"change\").\n this._varOnChangeSub = null;\n\n this._extraInfo = util.extend({ sender: this }, extraInfo);\n\n this.setGroup(group);\n }\n\n /**\n * Changes the Crosstalk group membership of this SelectionHandle. The group\n * being switched away from (if any) will not have its selection value\n * modified as a result of calling `setGroup`, even if this handle was the\n * most recent handle to set the selection of the group.\n *\n * The group being switched to (if any) will also not have its selection value\n * modified as a result of calling `setGroup`. If you want to set the\n * selection value of the new group, call `set` explicitly.\n *\n * @param {string} group - The name of the Crosstalk group, or null (or\n * undefined) to clear the group.\n */\n setGroup(group) {\n // If group is unchanged, do nothing\n if (this._group === group)\n return;\n // Treat null, undefined, and other falsy values the same\n if (!this._group && !group)\n return;\n\n if (this._var) {\n this._var.off(\"change\", this._varOnChangeSub);\n this._var = null;\n this._varOnChangeSub = null;\n }\n\n this._group = group;\n\n if (group) {\n this._var = grp(group).var(\"selection\");\n let sub = this._var.on(\"change\", (e) => {\n this._eventRelay.trigger(\"change\", e, this);\n });\n this._varOnChangeSub = sub;\n }\n }\n\n /**\n * Retrieves the current selection for the group represented by this\n * `SelectionHandle`.\n *\n * - If no selection is active, then this value will be falsy.\n * - If a selection is active, but no data points are selected, then this\n * value will be an empty array.\n * - If a selection is active, and data points are selected, then the keys\n * of the selected data points will be present in the array.\n */\n get value() {\n return this._var ? this._var.get() : null;\n }\n\n /**\n * Combines the given `extraInfo` (if any) with the handle's default\n * `_extraInfo` (if any).\n * @private\n */\n _mergeExtraInfo(extraInfo) {\n // Important incidental effect: shallow clone is returned\n return util.extend({},\n this._extraInfo ? this._extraInfo : null,\n extraInfo ? extraInfo : null);\n }\n\n /**\n * Overwrites the current selection for the group, and raises the `\"change\"`\n * event among all of the group's '`SelectionHandle` instances (including\n * this one).\n *\n * @fires SelectionHandle#change\n * @param {string[]} selectedKeys - Falsy, empty array, or array of keys (see\n * {@link SelectionHandle#value}).\n * @param {Object} [extraInfo] - Extra properties to be included on the event\n * object that's passed to listeners (in addition to any options that were\n * passed into the `SelectionHandle` constructor).\n */\n set(selectedKeys, extraInfo) {\n if (this._var)\n this._var.set(selectedKeys, this._mergeExtraInfo(extraInfo));\n }\n\n /**\n * Overwrites the current selection for the group, and raises the `\"change\"`\n * event among all of the group's '`SelectionHandle` instances (including\n * this one).\n *\n * @fires SelectionHandle#change\n * @param {Object} [extraInfo] - Extra properties to be included on the event\n * object that's passed to listeners (in addition to any that were passed\n * into the `SelectionHandle` constructor).\n */\n clear(extraInfo) {\n if (this._var)\n this.set(void 0, this._mergeExtraInfo(extraInfo));\n }\n\n /**\n * Subscribes to events on this `SelectionHandle`.\n *\n * @param {string} eventType - Indicates the type of events to listen to.\n * Currently, only `\"change\"` is supported.\n * @param {SelectionHandle~listener} listener - The callback function that\n * will be invoked when the event occurs.\n * @return {string} - A token to pass to {@link SelectionHandle#off} to cancel\n * this subscription.\n */\n on(eventType, listener) {\n return this._emitter.on(eventType, listener);\n }\n\n /**\n * Cancels event subscriptions created by {@link SelectionHandle#on}.\n *\n * @param {string} eventType - The type of event to unsubscribe.\n * @param {string|SelectionHandle~listener} listener - Either the callback\n * function previously passed into {@link SelectionHandle#on}, or the\n * string that was returned from {@link SelectionHandle#on}.\n */\n off(eventType, listener) {\n return this._emitter.off(eventType, listener);\n }\n\n /**\n * Shuts down the `SelectionHandle` object.\n *\n * Removes all event listeners that were added through this handle.\n */\n close() {\n this._emitter.removeAllListeners();\n this.setGroup(null);\n }\n}\n\n/**\n * @callback SelectionHandle~listener\n * @param {Object} event - An object containing details of the event. For\n * `\"change\"` events, this includes the properties `value` (the new\n * value of the selection, or `undefined` if no selection is active),\n * `oldValue` (the previous value of the selection), and `sender` (the\n * `SelectionHandle` instance that made the change).\n */\n\n/**\n * @event SelectionHandle#change\n * @type {object}\n * @property {object} value - The new value of the selection, or `undefined`\n * if no selection is active.\n * @property {object} oldValue - The previous value of the selection.\n * @property {SelectionHandle} sender - The `SelectionHandle` instance that\n * changed the value.\n */\n","export function extend(target, ...sources) {\n for (let i = 0; i < sources.length; i++) {\n let src = sources[i];\n if (typeof(src) === \"undefined\" || src === null)\n continue;\n\n for (let key in src) {\n if (src.hasOwnProperty(key)) {\n target[key] = src[key];\n }\n }\n }\n return target;\n}\n\nexport function checkSorted(list) {\n for (let i = 1; i < list.length; i++) {\n if (list[i] <= list[i-1]) {\n throw new Error(\"List is not sorted or contains duplicate\");\n }\n }\n}\n\nexport function diffSortedLists(a, b) {\n let i_a = 0;\n let i_b = 0;\n\n if (!a) a = [];\n if (!b) b = [];\n\n let a_only = [];\n let b_only = [];\n\n checkSorted(a);\n checkSorted(b);\n\n while (i_a < a.length && i_b < b.length) {\n if (a[i_a] === b[i_b]) {\n i_a++;\n i_b++;\n } else if (a[i_a] < b[i_b]) {\n a_only.push(a[i_a++]);\n } else {\n b_only.push(b[i_b++]);\n }\n }\n\n if (i_a < a.length)\n a_only = a_only.concat(a.slice(i_a));\n if (i_b < b.length)\n b_only = b_only.concat(b.slice(i_b));\n return {\n removed: a_only,\n added: b_only\n };\n}\n\n// Convert from wide: { colA: [1,2,3], colB: [4,5,6], ... }\n// to long: [ {colA: 1, colB: 4}, {colA: 2, colB: 5}, ... ]\nexport function dataframeToD3(df) {\n let names = [];\n let length;\n for (let name in df) {\n if (df.hasOwnProperty(name))\n names.push(name);\n if (typeof(df[name]) !== \"object\" || typeof(df[name].length) === \"undefined\") {\n throw new Error(\"All fields must be arrays\");\n } else if (typeof(length) !== \"undefined\" && length !== df[name].length) {\n throw new Error(\"All fields must be arrays of the same length\");\n }\n length = df[name].length;\n }\n let results = [];\n let item;\n for (let row = 0; row < length; row++) {\n item = {};\n for (let col = 0; col < names.length; col++) {\n item[names[col]] = df[names[col]][row];\n }\n results.push(item);\n }\n return results;\n}\n\n/**\n * Keeps track of all event listener additions/removals and lets all active\n * listeners be removed with a single operation.\n *\n * @private\n */\nexport class SubscriptionTracker {\n constructor(emitter) {\n this._emitter = emitter;\n this._subs = {};\n }\n\n on(eventType, listener) {\n let sub = this._emitter.on(eventType, listener);\n this._subs[sub] = eventType;\n return sub;\n }\n\n off(eventType, listener) {\n let sub = this._emitter.off(eventType, listener);\n if (sub) {\n delete this._subs[sub];\n }\n return sub;\n }\n\n removeAllListeners() {\n let current_subs = this._subs;\n this._subs = {};\n Object.keys(current_subs).forEach((sub) => {\n this._emitter.off(current_subs[sub], sub);\n });\n }\n}\n","import Events from \"./events\";\n\nexport default class Var {\n constructor(group, name, /*optional*/ value) {\n this._group = group;\n this._name = name;\n this._value = value;\n this._events = new Events();\n }\n\n get() {\n return this._value;\n }\n\n set(value, /*optional*/ event) {\n if (this._value === value) {\n // Do nothing; the value hasn't changed\n return;\n }\n let oldValue = this._value;\n this._value = value;\n // Alert JavaScript listeners that the value has changed\n let evt = {};\n if (event && typeof(event) === \"object\") {\n for (let k in event) {\n if (event.hasOwnProperty(k))\n evt[k] = event[k];\n }\n }\n evt.oldValue = oldValue;\n evt.value = value;\n this._events.trigger(\"change\", evt, this);\n\n // TODO: Make this extensible, to let arbitrary back-ends know that\n // something has changed\n if (global.Shiny && global.Shiny.onInputChange) {\n global.Shiny.onInputChange(\n \".clientValue-\" +\n (this._group.name !== null ? this._group.name + \"-\" : \"\") +\n this._name,\n typeof(value) === \"undefined\" ? null : value\n );\n }\n }\n\n on(eventType, listener) {\n return this._events.on(eventType, listener);\n }\n\n off(eventType, listener) {\n return this._events.off(eventType, listener);\n }\n}\n"]} \ No newline at end of file diff --git a/site_libs/crosstalk-1.2.0/scss/crosstalk.scss b/site_libs/crosstalk-1.2.0/scss/crosstalk.scss deleted file mode 100644 index 35665616f..000000000 --- a/site_libs/crosstalk-1.2.0/scss/crosstalk.scss +++ /dev/null @@ -1,75 +0,0 @@ -/* Adjust margins outwards, so column contents line up with the edges of the - parent of container-fluid. */ -.container-fluid.crosstalk-bscols { - margin-left: -30px; - margin-right: -30px; - white-space: normal; -} - -/* But don't adjust the margins outwards if we're directly under the body, - i.e. we were the top-level of something at the console. */ -body > .container-fluid.crosstalk-bscols { - margin-left: auto; - margin-right: auto; -} - -.crosstalk-input-checkboxgroup .crosstalk-options-group .crosstalk-options-column { - display: inline-block; - padding-right: 12px; - vertical-align: top; -} - -@media only screen and (max-width:480px) { - .crosstalk-input-checkboxgroup .crosstalk-options-group .crosstalk-options-column { - display: block; - padding-right: inherit; - } -} - -/* Relevant BS3 styles to make filter_checkbox() look reasonable without Bootstrap */ -.crosstalk-input { - margin-bottom: 15px; /* a la .form-group */ - .control-label { - margin-bottom: 0; - vertical-align: middle; - } - input[type="checkbox"] { - margin: 4px 0 0; - margin-top: 1px; - line-height: normal; - } - .checkbox { - position: relative; - display: block; - margin-top: 10px; - margin-bottom: 10px; - } - .checkbox > label{ - padding-left: 20px; - margin-bottom: 0; - font-weight: 400; - cursor: pointer; - } - .checkbox input[type="checkbox"], - .checkbox-inline input[type="checkbox"] { - position: absolute; - margin-top: 2px; - margin-left: -20px; - } - .checkbox + .checkbox { - margin-top: -5px; - } - .checkbox-inline { - position: relative; - display: inline-block; - padding-left: 20px; - margin-bottom: 0; - font-weight: 400; - vertical-align: middle; - cursor: pointer; - } - .checkbox-inline + .checkbox-inline { - margin-top: 0; - margin-left: 10px; - } -} diff --git a/site_libs/htmlwidgets-1.6.2/htmlwidgets.js b/site_libs/htmlwidgets-1.6.2/htmlwidgets.js deleted file mode 100644 index 1067d029f..000000000 --- a/site_libs/htmlwidgets-1.6.2/htmlwidgets.js +++ /dev/null @@ -1,901 +0,0 @@ -(function() { - // If window.HTMLWidgets is already defined, then use it; otherwise create a - // new object. This allows preceding code to set options that affect the - // initialization process (though none currently exist). - window.HTMLWidgets = window.HTMLWidgets || {}; - - // See if we're running in a viewer pane. If not, we're in a web browser. - var viewerMode = window.HTMLWidgets.viewerMode = - /\bviewer_pane=1\b/.test(window.location); - - // See if we're running in Shiny mode. If not, it's a static document. - // Note that static widgets can appear in both Shiny and static modes, but - // obviously, Shiny widgets can only appear in Shiny apps/documents. - var shinyMode = window.HTMLWidgets.shinyMode = - typeof(window.Shiny) !== "undefined" && !!window.Shiny.outputBindings; - - // We can't count on jQuery being available, so we implement our own - // version if necessary. - function querySelectorAll(scope, selector) { - if (typeof(jQuery) !== "undefined" && scope instanceof jQuery) { - return scope.find(selector); - } - if (scope.querySelectorAll) { - return scope.querySelectorAll(selector); - } - } - - function asArray(value) { - if (value === null) - return []; - if ($.isArray(value)) - return value; - return [value]; - } - - // Implement jQuery's extend - function extend(target /*, ... */) { - if (arguments.length == 1) { - return target; - } - for (var i = 1; i < arguments.length; i++) { - var source = arguments[i]; - for (var prop in source) { - if (source.hasOwnProperty(prop)) { - target[prop] = source[prop]; - } - } - } - return target; - } - - // IE8 doesn't support Array.forEach. - function forEach(values, callback, thisArg) { - if (values.forEach) { - values.forEach(callback, thisArg); - } else { - for (var i = 0; i < values.length; i++) { - callback.call(thisArg, values[i], i, values); - } - } - } - - // Replaces the specified method with the return value of funcSource. - // - // Note that funcSource should not BE the new method, it should be a function - // that RETURNS the new method. funcSource receives a single argument that is - // the overridden method, it can be called from the new method. The overridden - // method can be called like a regular function, it has the target permanently - // bound to it so "this" will work correctly. - function overrideMethod(target, methodName, funcSource) { - var superFunc = target[methodName] || function() {}; - var superFuncBound = function() { - return superFunc.apply(target, arguments); - }; - target[methodName] = funcSource(superFuncBound); - } - - // Add a method to delegator that, when invoked, calls - // delegatee.methodName. If there is no such method on - // the delegatee, but there was one on delegator before - // delegateMethod was called, then the original version - // is invoked instead. - // For example: - // - // var a = { - // method1: function() { console.log('a1'); } - // method2: function() { console.log('a2'); } - // }; - // var b = { - // method1: function() { console.log('b1'); } - // }; - // delegateMethod(a, b, "method1"); - // delegateMethod(a, b, "method2"); - // a.method1(); - // a.method2(); - // - // The output would be "b1", "a2". - function delegateMethod(delegator, delegatee, methodName) { - var inherited = delegator[methodName]; - delegator[methodName] = function() { - var target = delegatee; - var method = delegatee[methodName]; - - // The method doesn't exist on the delegatee. Instead, - // call the method on the delegator, if it exists. - if (!method) { - target = delegator; - method = inherited; - } - - if (method) { - return method.apply(target, arguments); - } - }; - } - - // Implement a vague facsimilie of jQuery's data method - function elementData(el, name, value) { - if (arguments.length == 2) { - return el["htmlwidget_data_" + name]; - } else if (arguments.length == 3) { - el["htmlwidget_data_" + name] = value; - return el; - } else { - throw new Error("Wrong number of arguments for elementData: " + - arguments.length); - } - } - - // http://stackoverflow.com/questions/3446170/escape-string-for-use-in-javascript-regex - function escapeRegExp(str) { - return str.replace(/[\-\[\]\/\{\}\(\)\*\+\?\.\\\^\$\|]/g, "\\$&"); - } - - function hasClass(el, className) { - var re = new RegExp("\\b" + escapeRegExp(className) + "\\b"); - return re.test(el.className); - } - - // elements - array (or array-like object) of HTML elements - // className - class name to test for - // include - if true, only return elements with given className; - // if false, only return elements *without* given className - function filterByClass(elements, className, include) { - var results = []; - for (var i = 0; i < elements.length; i++) { - if (hasClass(elements[i], className) == include) - results.push(elements[i]); - } - return results; - } - - function on(obj, eventName, func) { - if (obj.addEventListener) { - obj.addEventListener(eventName, func, false); - } else if (obj.attachEvent) { - obj.attachEvent(eventName, func); - } - } - - function off(obj, eventName, func) { - if (obj.removeEventListener) - obj.removeEventListener(eventName, func, false); - else if (obj.detachEvent) { - obj.detachEvent(eventName, func); - } - } - - // Translate array of values to top/right/bottom/left, as usual with - // the "padding" CSS property - // https://developer.mozilla.org/en-US/docs/Web/CSS/padding - function unpackPadding(value) { - if (typeof(value) === "number") - value = [value]; - if (value.length === 1) { - return {top: value[0], right: value[0], bottom: value[0], left: value[0]}; - } - if (value.length === 2) { - return {top: value[0], right: value[1], bottom: value[0], left: value[1]}; - } - if (value.length === 3) { - return {top: value[0], right: value[1], bottom: value[2], left: value[1]}; - } - if (value.length === 4) { - return {top: value[0], right: value[1], bottom: value[2], left: value[3]}; - } - } - - // Convert an unpacked padding object to a CSS value - function paddingToCss(paddingObj) { - return paddingObj.top + "px " + paddingObj.right + "px " + paddingObj.bottom + "px " + paddingObj.left + "px"; - } - - // Makes a number suitable for CSS - function px(x) { - if (typeof(x) === "number") - return x + "px"; - else - return x; - } - - // Retrieves runtime widget sizing information for an element. - // The return value is either null, or an object with fill, padding, - // defaultWidth, defaultHeight fields. - function sizingPolicy(el) { - var sizingEl = document.querySelector("script[data-for='" + el.id + "'][type='application/htmlwidget-sizing']"); - if (!sizingEl) - return null; - var sp = JSON.parse(sizingEl.textContent || sizingEl.text || "{}"); - if (viewerMode) { - return sp.viewer; - } else { - return sp.browser; - } - } - - // @param tasks Array of strings (or falsy value, in which case no-op). - // Each element must be a valid JavaScript expression that yields a - // function. Or, can be an array of objects with "code" and "data" - // properties; in this case, the "code" property should be a string - // of JS that's an expr that yields a function, and "data" should be - // an object that will be added as an additional argument when that - // function is called. - // @param target The object that will be "this" for each function - // execution. - // @param args Array of arguments to be passed to the functions. (The - // same arguments will be passed to all functions.) - function evalAndRun(tasks, target, args) { - if (tasks) { - forEach(tasks, function(task) { - var theseArgs = args; - if (typeof(task) === "object") { - theseArgs = theseArgs.concat([task.data]); - task = task.code; - } - var taskFunc = tryEval(task); - if (typeof(taskFunc) !== "function") { - throw new Error("Task must be a function! Source:\n" + task); - } - taskFunc.apply(target, theseArgs); - }); - } - } - - // Attempt eval() both with and without enclosing in parentheses. - // Note that enclosing coerces a function declaration into - // an expression that eval() can parse - // (otherwise, a SyntaxError is thrown) - function tryEval(code) { - var result = null; - try { - result = eval("(" + code + ")"); - } catch(error) { - if (!(error instanceof SyntaxError)) { - throw error; - } - try { - result = eval(code); - } catch(e) { - if (e instanceof SyntaxError) { - throw error; - } else { - throw e; - } - } - } - return result; - } - - function initSizing(el) { - var sizing = sizingPolicy(el); - if (!sizing) - return; - - var cel = document.getElementById("htmlwidget_container"); - if (!cel) - return; - - if (typeof(sizing.padding) !== "undefined") { - document.body.style.margin = "0"; - document.body.style.padding = paddingToCss(unpackPadding(sizing.padding)); - } - - if (sizing.fill) { - document.body.style.overflow = "hidden"; - document.body.style.width = "100%"; - document.body.style.height = "100%"; - document.documentElement.style.width = "100%"; - document.documentElement.style.height = "100%"; - cel.style.position = "absolute"; - var pad = unpackPadding(sizing.padding); - cel.style.top = pad.top + "px"; - cel.style.right = pad.right + "px"; - cel.style.bottom = pad.bottom + "px"; - cel.style.left = pad.left + "px"; - el.style.width = "100%"; - el.style.height = "100%"; - - return { - getWidth: function() { return cel.getBoundingClientRect().width; }, - getHeight: function() { return cel.getBoundingClientRect().height; } - }; - - } else { - el.style.width = px(sizing.width); - el.style.height = px(sizing.height); - - return { - getWidth: function() { return cel.getBoundingClientRect().width; }, - getHeight: function() { return cel.getBoundingClientRect().height; } - }; - } - } - - // Default implementations for methods - var defaults = { - find: function(scope) { - return querySelectorAll(scope, "." + this.name); - }, - renderError: function(el, err) { - var $el = $(el); - - this.clearError(el); - - // Add all these error classes, as Shiny does - var errClass = "shiny-output-error"; - if (err.type !== null) { - // use the classes of the error condition as CSS class names - errClass = errClass + " " + $.map(asArray(err.type), function(type) { - return errClass + "-" + type; - }).join(" "); - } - errClass = errClass + " htmlwidgets-error"; - - // Is el inline or block? If inline or inline-block, just display:none it - // and add an inline error. - var display = $el.css("display"); - $el.data("restore-display-mode", display); - - if (display === "inline" || display === "inline-block") { - $el.hide(); - if (err.message !== "") { - var errorSpan = $("").addClass(errClass); - errorSpan.text(err.message); - $el.after(errorSpan); - } - } else if (display === "block") { - // If block, add an error just after the el, set visibility:none on the - // el, and position the error to be on top of the el. - // Mark it with a unique ID and CSS class so we can remove it later. - $el.css("visibility", "hidden"); - if (err.message !== "") { - var errorDiv = $("