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Published
-

Last modified: 2024-05-12: 21:41:28 (PM)

+

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

@@ -900,12 +900,17 @@

Exercise 2.1 According to this model, what’s the mean birthweight for a female born at 36 weeks?

-
+
+
+Table 2.4: Estimated coefficients for model 2.1 +
Show R code
coef(bw_lm1)
 #> (Intercept)     sexmale         age 
-#>     -1773.3       163.0       120.9
+#> -1773.32184 163.03930 120.89433
+ +
@@ -924,7 +929,7 @@ \begin{aligned} E[Y|A = 0, A = 36] &= \beta_0 + \beta_M \cdot 0+ \beta_A \cdot 36 \\ -&= 2578.8739 +&= 2578.873934 \end{aligned} \]

@@ -955,7 +960,7 @@ \begin{aligned} E[Y|M = 1, A = 36] &= \beta_0 + \beta_M \cdot 1+ \beta_A \cdot 36 \\ -&= 2741.9132 +&= 2741.91323693 \end{aligned} \]

@@ -975,9 +980,9 @@ \begin{aligned} & E[Y|M = 1, A = 36] - E[Y|M = 0, A = 36]\\ &= -2741.9132 - 2578.8739\\ +2741.91323693 - 2578.873934\\ &= -163.0393 +163.03930293 \end{aligned} \]

Shortcut:

@@ -987,7 +992,7 @@ &= (\beta_0 + \beta_M \cdot 1+ \beta_A \cdot 36) - (\beta_0 + \beta_M \cdot 0+ \beta_A \cdot 36) \\ &= \beta_M \\ -&= 163.0393 +&= 163.03930293 \end{aligned} \]

@@ -1019,7 +1024,7 @@ select = "{estimate}")
-Table 2.4: Birthweight model with interaction term +Table 2.5: Birthweight model with interaction term
@@ -1126,7 +1131,7 @@

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

@@ -1148,7 +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\\ -&= 2762.7069 +&= 2762.70689655 \end{aligned} \]

@@ -1164,7 +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\\ -&= 209.9736 +&= 209.97356322 \end{aligned} \]

@@ -1200,7 +1205,7 @@ select = "{estimate}")
-Table 2.5: Birthweight model with interaction term +Table 2.6: Birthweight model with interaction term
@@ -1253,7 +1258,7 @@ select = "{estimate}")
-Table 2.6: Birthweight model - stratified betas +Table 2.7: Birthweight model - stratified betas
@@ -1447,7 +1452,7 @@
Show R code
bw_lm2 |> parameters() |> print_md()
-Table 2.7: Estimated model for birthweight data with interaction term +Table 2.8: Estimated model for birthweight data with interaction term
@@ -1597,7 +1602,7 @@

Note

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

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

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

-

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

+

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

2.4.2 Wald tests and CIs

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

@@ -1721,17 +1726,17 @@

Note

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

@@ -2024,12 +2029,12 @@

Note

n_unique.bw = nrow(bw.X.unique)
-

For example, in the birthweight data, there are \(q = 12\) unique patterns (Table 2.8).

+

For example, in the birthweight data, there are \(q = 12\) unique patterns (Table 2.9).

Show R code
bw.X.unique
-Table 2.8: Unique covariate combinations in the birthweight data, with replicate counts +Table 2.9: Unique covariate combinations in the birthweight data, with replicate counts
@@ -2112,7 +2117,7 @@

Note

-

Example 2.1 (Replicates in the birthweight data) In the birthweight dataset, there are 2 replicates of the combination “female, age 36” (Table 2.8).

+

Example 2.1 (Replicates in the birthweight data) In the birthweight dataset, there are 2 replicates of the combination “female, age 36” (Table 2.9).

@@ -2162,7 +2167,7 @@

Note

print_md()
-Table 2.9: Saturated model for the birthweight data +Table 2.10: Saturated model for the birthweight data
@@ -2324,12 +2329,12 @@

Note

-

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

+

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

  • -\(\ell_{\text{max}}\) is the log-likelihood of the full model: -151.4016
    • +\(\ell_{\text{max}}\) is the log-likelihood of the full model: -151.40160056
  • -\(\ell\) is the log-likelihood of our comparison model (two slopes, two intercepts): -156.5793
    • +\(\ell\) is the log-likelihood of our comparison model (two slopes, two intercepts): -156.57928767

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.)

@@ -2408,12 +2413,12 @@

Note

-

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

+

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

In R, this test is:

@@ -3329,7 +3334,7 @@

Note

This should be true for every value of \(x\).

-

If we didn’t correctly guess the functional form of linear component of the mean, \[\text{E}[Y|X=x] = \beta_0 + \beta_1 X_1 + ... + \beta_p X_p\]

+

If we didn’t correctly guess the functional form of the linear component of the mean, \[\text{E}[Y|X=x] = \beta_0 + \beta_1 X_1 + ... + \beta_p X_p\]

Then the the residuals might have nonzero mean.

Regardless of whether we guessed the mean function correctly, ther the variance of the residuals might differ between values of \(x\).

@@ -3604,7 +3609,7 @@

Note

Show R code
anova(lm_PLOS_linear, lm_PLOS_quad)
-Table 2.10: linear vs quadratic +Table 2.11: linear vs quadratic
@@ -3644,7 +3649,7 @@

Note

Show R code
anova(lm_PLOS_quad, lm_PLOS_cub)
-Table 2.11: quadratic vs cubic +Table 2.12: quadratic vs cubic
@@ -3921,7 +3926,7 @@

Note

- + @@ -3929,7 +3934,7 @@

Note

- + @@ -4066,10 +4071,10 @@

Note

Show R code
coef(cvfit, s = "lambda.1se")
 #> 4 x 1 sparse Matrix of class "dgCMatrix"
 #>                  s1
-#> (Intercept) 34.2044
+#> (Intercept) 34.4241
 #> age          .     
-#> weight      -0.0926
-#> protein      0.8582
+#> weight -0.0662 +#> protein 0.6607

@@ -4088,13 +4093,13 @@

Note

2.10.2

-

For example, Table 2.12 shows the (in)famous iris data (Anderson (1935)), and Table 2.13 provides summary statistics. The data include three species: “setosa”, “versicolor”, and “virginica”.

+

For example, Table 2.13 shows the (in)famous iris data (Anderson (1935)), and Table 2.14 provides summary statistics. The data include three species: “setosa”, “versicolor”, and “virginica”.

Show R code
head(iris)
-Table 2.12: The iris data +Table 2.13: The iris data
full6.7216.946 0.4805 0.3831 5.956
reduced6.5766.627 0.4454 0.3802 5.971
@@ -4164,7 +4169,7 @@

Note

)
-Table 2.13: Summary statistics for the iris data +Table 2.14: Summary statistics for the iris data
@@ -4285,7 +4290,7 @@

Note

distinct(Species, X)
-Table 2.14: iris data with numeric coding of species +Table 2.15: iris data with numeric coding of species
@@ -4319,7 +4324,7 @@

Note

iris_lm1 |> parameters() |> print_md()
-Table 2.15: Model of iris data with numeric coding of Species +Table 2.16: Model of iris data with numeric coding of Species
@@ -4389,7 +4394,7 @@

Note

iris_lm2 |> parameters() |> print_md()
-Table 2.16: Model of iris data with Species as a categorical variable +Table 2.17: Model of iris data with Species as a categorical variable
@@ -4610,7 +4615,7 @@

Note

Show R code
hers |> head()
-Table 2.17: HERS dataset +Table 2.18: HERS dataset
@@ -5841,7 +5846,7 @@

Note

```{r}#| tbl-cap: "Estimated coefficients for [model @eq-lm-parallel]" -#| label: coef-tab-model1 +#| label: tbl-coef-model1coef(bw_lm1)``` @@ -7360,493 +7365,494 @@

Note

--- -If we didn't correctly guess the functional form of linear component of the mean, $$\text{E}[Y|X=x] = \beta_0 + \beta_1 X_1 + ... + \beta_p X_p$$ - -Then the the residuals might have nonzero mean. - -Regardless of whether we guessed the mean function correctly, ther the variance of the residuals might differ between values of $x$. - ---- - -#### Residuals versus fitted values - -::: notes -To look for these issues, we can plot the residuals $e_i$ against the fitted values $\hat y_i$ (@fig-bw_lm2-resid-vs-fitted). -::: -```{r} -#| label: fig-bw_lm2-resid-vs-fitted -#| fig-cap: "`birthweight` model (@eq-BW-lm-interact): residuals versus fitted values" -autoplot(bw_lm2, which = 1, ncol = 1) |> print() -``` - -::: notes -If the model is correct, the blue line should stay flat and close to 0, and the cloud of dots should have the same vertical spread regardless of the fitted value. - -If not, we probably need to change the functional form of linear component of the mean, $$\text{E}[Y|X=x] = \beta_0 + \beta_1 X_1 + ... + \beta_p X_p$$ - -::: - ---- - -{{< include exm-plot-fits.qmd >}} ---- - -#### Scale-location plot - -::: notes -We can also plot the square roots of the absolute values of the standardized residuals against the fitted values (@fig-bw-scale-loc). -::: - -```{r} -#| label: fig-bw-scale-loc -#| fig-cap: "Scale-location plot of `birthweight` data" -autoplot(bw_lm2, which = 3, ncol = 1) |> print() -``` - -::: notes -Here, the blue line doesn't need to be near 0, -but it should be flat. -If not, the residual variance $\sigma^2$ might not be constant, -and we might need to transform our outcome $Y$ -(or use a model that allows non-constant variance). -::: - ---- - -#### Residuals versus leverage - -::: notes - -We can also plot our standardized residuals against "leverage", which roughly speaking is a measure of how unusual each $x_i$ value is. Very unusual $x_i$ values can have extreme effects on the model fit, so we might want to remove those observations as outliers, particularly if they have large residuals. - -::: - -```{r} -#| labels: fig-bw_lm2_resid-vs-leverage -#| fig-cap: "`birthweight` model with interactions (@eq-BW-lm-interact): residuals versus leverage" -autoplot(bw_lm2, which = 5, ncol = 1) |> print() -``` - -::: notes -The blue line should be relatively flat and close to 0 here. -::: - ---- - -### Diagnostics constructed by hand - -```{r} -bw = - bw |> - mutate( - predlm2 = predict(bw_lm2), - residlm2 = weight - predlm2, - std_resid = residlm2 / sigma(bw_lm2), - # std_resid_builtin = rstandard(bw_lm2), # uses leverage - sqrt_abs_std_resid = std_resid |> abs() |> sqrt() - - ) - -``` - -##### Residuals vs fitted - -```{r} -resid_vs_fit = bw |> - ggplot( - aes(x = predlm2, y = residlm2, col = sex, shape = sex) - ) + - geom_point() + - theme_classic() + - geom_hline(yintercept = 0) - -``` - -::: {.content-visible when-format="html"} -```{r} -print(resid_vs_fit) -``` -::: - -::: {.content-visible when-format="pdf"} -```{r} -print(resid_vs_fit) -``` -::: - -##### Standardized residuals vs fitted - -```{r} -bw |> - ggplot( - aes(x = predlm2, y = std_resid, col = sex, shape = sex) - ) + - geom_point() + - theme_classic() + - geom_hline(yintercept = 0) - -``` - -##### Standardized residuals vs gestational age - -```{r} -bw |> - ggplot( - aes(x = age, y = std_resid, col = sex, shape = sex) - ) + - geom_point() + - theme_classic() + - geom_hline(yintercept = 0) - -``` - -##### `sqrt(abs(rstandard()))` vs fitted - -Compare with `autoplot(bw_lm2, 3)` - -```{r} -bw |> - ggplot( - aes(x = predlm2, y = sqrt_abs_std_resid, col = sex, shape = sex) - ) + - geom_point() + - theme_classic() + - geom_hline(yintercept = 0) - +If we didn't correctly guess the functional form of the linear component of the mean, +$$\text{E}[Y|X=x] = \beta_0 + \beta_1 X_1 + ... + \beta_p X_p$$ + +Then the the residuals might have nonzero mean. + +Regardless of whether we guessed the mean function correctly, ther the variance of the residuals might differ between values of $x$. + +--- + +#### Residuals versus fitted values + +::: notes +To look for these issues, we can plot the residuals $e_i$ against the fitted values $\hat y_i$ (@fig-bw_lm2-resid-vs-fitted). +::: +```{r} +#| label: fig-bw_lm2-resid-vs-fitted +#| fig-cap: "`birthweight` model (@eq-BW-lm-interact): residuals versus fitted values" +autoplot(bw_lm2, which = 1, ncol = 1) |> print() +``` + +::: notes +If the model is correct, the blue line should stay flat and close to 0, and the cloud of dots should have the same vertical spread regardless of the fitted value. + +If not, we probably need to change the functional form of linear component of the mean, $$\text{E}[Y|X=x] = \beta_0 + \beta_1 X_1 + ... + \beta_p X_p$$ + +::: + +--- + +{{< include exm-plot-fits.qmd >}} +--- + +#### Scale-location plot + +::: notes +We can also plot the square roots of the absolute values of the standardized residuals against the fitted values (@fig-bw-scale-loc). +::: + +```{r} +#| label: fig-bw-scale-loc +#| fig-cap: "Scale-location plot of `birthweight` data" +autoplot(bw_lm2, which = 3, ncol = 1) |> print() +``` + +::: notes +Here, the blue line doesn't need to be near 0, +but it should be flat. +If not, the residual variance $\sigma^2$ might not be constant, +and we might need to transform our outcome $Y$ +(or use a model that allows non-constant variance). +::: + +--- + +#### Residuals versus leverage + +::: notes + +We can also plot our standardized residuals against "leverage", which roughly speaking is a measure of how unusual each $x_i$ value is. Very unusual $x_i$ values can have extreme effects on the model fit, so we might want to remove those observations as outliers, particularly if they have large residuals. + +::: + +```{r} +#| labels: fig-bw_lm2_resid-vs-leverage +#| fig-cap: "`birthweight` model with interactions (@eq-BW-lm-interact): residuals versus leverage" +autoplot(bw_lm2, which = 5, ncol = 1) |> print() +``` + +::: notes +The blue line should be relatively flat and close to 0 here. +::: + +--- + +### Diagnostics constructed by hand + +```{r} +bw = + bw |> + mutate( + predlm2 = predict(bw_lm2), + residlm2 = weight - predlm2, + std_resid = residlm2 / sigma(bw_lm2), + # std_resid_builtin = rstandard(bw_lm2), # uses leverage + sqrt_abs_std_resid = std_resid |> abs() |> sqrt() + + ) + +``` + +##### Residuals vs fitted + +```{r} +resid_vs_fit = bw |> + ggplot( + aes(x = predlm2, y = residlm2, col = sex, shape = sex) + ) + + geom_point() + + theme_classic() + + geom_hline(yintercept = 0) + +``` + +::: {.content-visible when-format="html"} +```{r} +print(resid_vs_fit) +``` +::: + +::: {.content-visible when-format="pdf"} +```{r} +print(resid_vs_fit) +``` +::: + +##### Standardized residuals vs fitted + +```{r} +bw |> + ggplot( + aes(x = predlm2, y = std_resid, col = sex, shape = sex) + ) + + geom_point() + + theme_classic() + + geom_hline(yintercept = 0) + +``` + +##### Standardized residuals vs gestational age + +```{r} +bw |> + ggplot( + aes(x = age, y = std_resid, col = sex, shape = sex) + ) + + geom_point() + + theme_classic() + + geom_hline(yintercept = 0) + +``` + +##### `sqrt(abs(rstandard()))` vs fitted + +Compare with `autoplot(bw_lm2, 3)` + +```{r} +bw |> + ggplot( + aes(x = predlm2, y = sqrt_abs_std_resid, col = sex, shape = sex) + ) + + geom_point() + + theme_classic() + + geom_hline(yintercept = 0) -``` - -## Model selection - -(adapted from @dobson4e §6.3.3; for more information on prediction, see @james2013introduction and @rms2e). - -::: notes -If we have a lot of covariates in our dataset, we might want to choose a small subset to use in our model. - -There are a few possible metrics to consider for choosing a "best" model. -::: - -### Mean squared error - -We 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. - -Unfortunately, $$\frac{1}{n}\sum_{i=1}^n (y_i-\hat y_i)^2$$ 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. - ---- - -#### Cross-validation - -```{r} -data("carbohydrate", package = "dobson") -library(cvTools) -full_model <- lm(carbohydrate ~ ., data = carbohydrate) -cv_full = - full_model |> cvFit( - data = carbohydrate, K = 5, R = 10, - y = carbohydrate$carbohydrate) - -reduced_model = update(full_model, - formula = ~ . - age) - -cv_reduced = - reduced_model |> cvFit( - data = carbohydrate, K = 5, R = 10, - y = carbohydrate$carbohydrate) -``` - ---- - -```{r} -results_reduced = - tibble( - model = "wgt+protein", - errs = cv_reduced$reps[]) -results_full = - tibble(model = "wgt+age+protein", - errs = cv_full$reps[]) - -cv_results = - bind_rows(results_reduced, results_full) - -cv_results |> - ggplot(aes(y = model, x = errs)) + - geom_boxplot() - -``` - ---- - -##### comparing metrics - -```{r} -compare_results = tribble( - ~ model, ~ cvRMSE, ~ r.squared, ~adj.r.squared, ~ trainRMSE, ~loglik, - "full", cv_full$cv, summary(full_model)$r.squared, summary(full_model)$adj.r.squared, sigma(full_model), logLik(full_model) |> as.numeric(), - "reduced", cv_reduced$cv, summary(reduced_model)$r.squared, summary(reduced_model)$adj.r.squared, sigma(reduced_model), logLik(reduced_model) |> as.numeric()) - -compare_results -``` - ---- - -```{r} -anova(full_model, reduced_model) -``` - ---- - -#### stepwise regression - -```{r} -library(olsrr) -olsrr:::ols_step_both_aic(full_model) -``` - ---- - -#### Lasso - -$$\arg min_{\theta} \llik(\th) + \lambda \sum_{j=1}^p|\beta_j|$$ - -```{r} -library(glmnet) -y = carbohydrate$carbohydrate -x = carbohydrate |> - select(age, weight, protein) |> - as.matrix() -fit = glmnet(x,y) -``` - ---- - -```{r} -#| fig-cap: "Lasso selection" -#| label: fig-carbs-lasso -autoplot(fit, xvar = 'lambda') - -``` - ---- - -```{r} -cvfit = cv.glmnet(x,y) -plot(cvfit) -``` - ---- - -```{r} -coef(cvfit, s = "lambda.1se") -``` - + +``` + +## Model selection + +(adapted from @dobson4e §6.3.3; for more information on prediction, see @james2013introduction and @rms2e). + +::: notes +If we have a lot of covariates in our dataset, we might want to choose a small subset to use in our model. + +There are a few possible metrics to consider for choosing a "best" model. +::: + +### Mean squared error + +We 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. + +Unfortunately, $$\frac{1}{n}\sum_{i=1}^n (y_i-\hat y_i)^2$$ 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. + +--- + +#### Cross-validation + +```{r} +data("carbohydrate", package = "dobson") +library(cvTools) +full_model <- lm(carbohydrate ~ ., data = carbohydrate) +cv_full = + full_model |> cvFit( + data = carbohydrate, K = 5, R = 10, + y = carbohydrate$carbohydrate) + +reduced_model = update(full_model, + formula = ~ . - age) + +cv_reduced = + reduced_model |> cvFit( + data = carbohydrate, K = 5, R = 10, + y = carbohydrate$carbohydrate) +``` + +--- + +```{r} +results_reduced = + tibble( + model = "wgt+protein", + errs = cv_reduced$reps[]) +results_full = + tibble(model = "wgt+age+protein", + errs = cv_full$reps[]) + +cv_results = + bind_rows(results_reduced, results_full) + +cv_results |> + ggplot(aes(y = model, x = errs)) + + geom_boxplot() + +``` + +--- + +##### comparing metrics + +```{r} +compare_results = tribble( + ~ model, ~ cvRMSE, ~ r.squared, ~adj.r.squared, ~ trainRMSE, ~loglik, + "full", cv_full$cv, summary(full_model)$r.squared, summary(full_model)$adj.r.squared, sigma(full_model), logLik(full_model) |> as.numeric(), + "reduced", cv_reduced$cv, summary(reduced_model)$r.squared, summary(reduced_model)$adj.r.squared, sigma(reduced_model), logLik(reduced_model) |> as.numeric()) + +compare_results +``` + +--- + +```{r} +anova(full_model, reduced_model) +``` + +--- + +#### stepwise regression + +```{r} +library(olsrr) +olsrr:::ols_step_both_aic(full_model) +``` + +--- + +#### Lasso + +$$\arg min_{\theta} \llik(\th) + \lambda \sum_{j=1}^p|\beta_j|$$ + +```{r} +library(glmnet) +y = carbohydrate$carbohydrate +x = carbohydrate |> + select(age, weight, protein) |> + as.matrix() +fit = glmnet(x,y) +``` + +--- + +```{r} +#| fig-cap: "Lasso selection" +#| label: fig-carbs-lasso +autoplot(fit, xvar = 'lambda') + +``` + +--- + +```{r} +cvfit = cv.glmnet(x,y) +plot(cvfit) +``` + +--- + +```{r} +coef(cvfit, s = "lambda.1se") +``` -## Categorical covariates with more than two levels - -### Example: `birthweight` - -In the birthweight example, the variable `sex` had only two observed values: - -```{r} -unique(bw$sex) -``` - -If there are more than two observed values, we can't just use a single variable with 0s and 1s. - -### - -:::{.notes} -For example, @tbl-iris-data shows the [(in)famous](https://www.meganstodel.com/posts/no-to-iris/) `iris` data (@anderson1935irises), -and @tbl-iris-summary provides summary statistics. -The data include three species: "setosa", "versicolor", and "virginica". -::: - -```{r} -#| tbl-cap: "The `iris` data" -#| label: tbl-iris-data -head(iris) -``` - -```{r} -#| tbl-cap: "Summary statistics for the `iris` data" -#| label: tbl-iris-summary - -library(table1) -table1( - x = ~ . | Species, - data = iris, - overall = FALSE -) - -``` - ---- - -If we want to model `Sepal.Length` by species, we could create a variable $X$ that represents "setosa" as $X=1$, "virginica" as $X=2$, and "versicolor" as $X=3$. - -```{r} -#| label: tbl-numeric-coding -#| tbl-cap: "`iris` data with numeric coding of species" -data(iris) # this step is not always necessary, but ensures you're starting -# from the original version of a dataset stored in a loaded package - -iris = - iris |> - tibble() |> - mutate( - X = case_when( - Species == "setosa" ~ 1, - Species == "virginica" ~ 2, - Species == "versicolor" ~ 3 - ) - ) - -iris |> - distinct(Species, X) - -``` - -Then we could fit a model like: - -```{r} -#| label: tbl-iris-numeric-species -#| tbl-cap: "Model of `iris` data with numeric coding of `Species`" -iris_lm1 = lm(Sepal.Length ~ X, data = iris) -iris_lm1 |> parameters() |> print_md() - -``` - -### Let's see how that model looks: - -```{r} -#| label: fig-iris-numeric-species-model -#| fig-cap: "Model of `iris` data with numeric coding of `Species`" -iris_plot1 = iris |> - ggplot( - aes( - x = X, - y = Sepal.Length) - ) + - geom_point(alpha = .1) + - geom_abline( - intercept = coef(iris_lm1)[1], - slope = coef(iris_lm1)[2]) + - theme_bw(base_size = 18) -print(iris_plot1) - -``` - -We have forced the model to use a straight line for the three estimated means. Maybe not a good idea? - -### Let's see what R does with categorical variables by default: - -```{r} -#| label: tbl-iris-model-factor1 -#| tbl-cap: "Model of `iris` data with `Species` as a categorical variable" -iris_lm2 = lm(Sepal.Length ~ Species, data = iris) -iris_lm2 |> parameters() |> print_md() - -``` - -### Re-parametrize with no intercept - -If you don't want the default and offset option, you can use "-1" like we've seen previously: - -```{r} -#| label: tbl-iris-no-intcpt -iris.lm2b = lm(Sepal.Length ~ Species - 1, data = iris) -iris.lm2b |> parameters() |> print_md() - -``` - -### Let's see what these new models look like: - -```{r} -#| label: fig-iris-no-intcpt -iris_plot2 = - iris |> - mutate( - predlm2 = predict(iris_lm2)) |> - arrange(X) |> - ggplot(aes(x = X, y = Sepal.Length)) + - geom_point(alpha = .1) + - geom_line(aes(y = predlm2), col = "red") + - geom_abline( - intercept = coef(iris_lm1)[1], - slope = coef(iris_lm1)[2]) + - theme_bw(base_size = 18) - -print(iris_plot2) - -``` - -### Let's see how R did that: - -```{r} -#| label: tbl-iris-model-matrix-factor -formula(iris_lm2) -model.matrix(iris_lm2) |> as_tibble() |> unique() - -``` - -This is called a "corner point parametrization". - -```{r} -#| label: tbl-iris-group-point-parameterization -formula(iris.lm2b) -model.matrix(iris.lm2b) |> as_tibble() |> unique() - -``` - -This can be called a "group point parametrization". - -There are more options; see @dobson4e §6.4.1 and the -[`codingMatrices` package](https://CRAN.R-project.org/package=codingMatrices) -[vignette](https://cran.r-project.org/web/packages/codingMatrices/vignettes/codingMatrices.pdf) -(@venablescodingMatrices). - -## Ordinal covariates - -(c.f. @dobson4e §2.4.4) - ---- - -::: notes -We can create ordinal variables in R using the `ordered()` function^[or equivalently, `factor(ordered = TRUE)`]. -::: - -:::{#exm-ordinal-variable} - -```{r} -#| eval: false -url = paste0( - "https://regression.ucsf.edu/sites/g/files/tkssra6706/", - "f/wysiwyg/home/data/hersdata.dta") -library(haven) -hers = read_dta(url) -``` - -```{r} -#| include: false -library(haven) -hers = read_stata("data/hersdata.dta") -``` - -```{r} -#| tbl-cap: "HERS dataset" -#| label: tbl-HERS -hers |> head() -``` - + +## Categorical covariates with more than two levels + +### Example: `birthweight` + +In the birthweight example, the variable `sex` had only two observed values: + +```{r} +unique(bw$sex) +``` + +If there are more than two observed values, we can't just use a single variable with 0s and 1s. + +### + +:::{.notes} +For example, @tbl-iris-data shows the [(in)famous](https://www.meganstodel.com/posts/no-to-iris/) `iris` data (@anderson1935irises), +and @tbl-iris-summary provides summary statistics. +The data include three species: "setosa", "versicolor", and "virginica". +::: + +```{r} +#| tbl-cap: "The `iris` data" +#| label: tbl-iris-data +head(iris) +``` + +```{r} +#| tbl-cap: "Summary statistics for the `iris` data" +#| label: tbl-iris-summary + +library(table1) +table1( + x = ~ . | Species, + data = iris, + overall = FALSE +) + +``` + +--- + +If we want to model `Sepal.Length` by species, we could create a variable $X$ that represents "setosa" as $X=1$, "virginica" as $X=2$, and "versicolor" as $X=3$. + +```{r} +#| label: tbl-numeric-coding +#| tbl-cap: "`iris` data with numeric coding of species" +data(iris) # this step is not always necessary, but ensures you're starting +# from the original version of a dataset stored in a loaded package + +iris = + iris |> + tibble() |> + mutate( + X = case_when( + Species == "setosa" ~ 1, + Species == "virginica" ~ 2, + Species == "versicolor" ~ 3 + ) + ) + +iris |> + distinct(Species, X) + +``` + +Then we could fit a model like: + +```{r} +#| label: tbl-iris-numeric-species +#| tbl-cap: "Model of `iris` data with numeric coding of `Species`" +iris_lm1 = lm(Sepal.Length ~ X, data = iris) +iris_lm1 |> parameters() |> print_md() + +``` + +### Let's see how that model looks: + +```{r} +#| label: fig-iris-numeric-species-model +#| fig-cap: "Model of `iris` data with numeric coding of `Species`" +iris_plot1 = iris |> + ggplot( + aes( + x = X, + y = Sepal.Length) + ) + + geom_point(alpha = .1) + + geom_abline( + intercept = coef(iris_lm1)[1], + slope = coef(iris_lm1)[2]) + + theme_bw(base_size = 18) +print(iris_plot1) + +``` + +We have forced the model to use a straight line for the three estimated means. Maybe not a good idea? + +### Let's see what R does with categorical variables by default: + +```{r} +#| label: tbl-iris-model-factor1 +#| tbl-cap: "Model of `iris` data with `Species` as a categorical variable" +iris_lm2 = lm(Sepal.Length ~ Species, data = iris) +iris_lm2 |> parameters() |> print_md() + +``` + +### Re-parametrize with no intercept + +If you don't want the default and offset option, you can use "-1" like we've seen previously: + +```{r} +#| label: tbl-iris-no-intcpt +iris.lm2b = lm(Sepal.Length ~ Species - 1, data = iris) +iris.lm2b |> parameters() |> print_md() + +``` + +### Let's see what these new models look like: + +```{r} +#| label: fig-iris-no-intcpt +iris_plot2 = + iris |> + mutate( + predlm2 = predict(iris_lm2)) |> + arrange(X) |> + ggplot(aes(x = X, y = Sepal.Length)) + + geom_point(alpha = .1) + + geom_line(aes(y = predlm2), col = "red") + + geom_abline( + intercept = coef(iris_lm1)[1], + slope = coef(iris_lm1)[2]) + + theme_bw(base_size = 18) + +print(iris_plot2) + +``` + +### Let's see how R did that: + +```{r} +#| label: tbl-iris-model-matrix-factor +formula(iris_lm2) +model.matrix(iris_lm2) |> as_tibble() |> unique() + +``` + +This is called a "corner point parametrization". + +```{r} +#| label: tbl-iris-group-point-parameterization +formula(iris.lm2b) +model.matrix(iris.lm2b) |> as_tibble() |> unique() + +``` + +This can be called a "group point parametrization". + +There are more options; see @dobson4e §6.4.1 and the +[`codingMatrices` package](https://CRAN.R-project.org/package=codingMatrices) +[vignette](https://cran.r-project.org/web/packages/codingMatrices/vignettes/codingMatrices.pdf) +(@venablescodingMatrices). + +## Ordinal covariates + +(c.f. @dobson4e §2.4.4) + +--- + +::: notes +We can create ordinal variables in R using the `ordered()` function^[or equivalently, `factor(ordered = TRUE)`]. +::: + +:::{#exm-ordinal-variable} + +```{r} +#| eval: false +url = paste0( + 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  • F.2.2 library(printr) currently breaks df-print: paged
    • +
  • F.3 LaTeX
    • @@ -359,7 +360,7 @@
    Published
    -

    Last modified: 2024-05-07: 11:55:30 (AM)

    +

    Last modified: 2024-05-16: 13:17:29 (PM)

    @@ -466,11 +467,14 @@ F.2.2 library(printr) currently breaks df-print: paged

    See https://github.com/yihui/printr/issues/41

    +

    +F.3 LaTeX

    +

    Double superscript issues: https://www.overleaf.com/learn/latex/Errors/Double_superscript

    -
    @@ -3159,398 +3171,410 @@ ${Cov}\left\{ \ell'(x,\theta) \right\}$, is also very important; we call it the "expected information matrix", "Fisher information matrix", or just "information matrix", and we represent it -using the symbol $\mathfrak{I}$ (`\frakturI` in Unicode, `\mathfrak{I}` in LaTeX). +using the symbol $\einff{I}$ (`\frakturI` in Unicode, `\mathfrak{I}` in LaTeX). -$$\mathfrak{I \eqdef I(}\theta) \eqdef {Cov}\left( \ell'|\theta \right) = E[ \ell'{\ell'}^{\top} ] - E[ \ell' ]\ E[ \ell' ]^{\top} -$$ - -The elements of $\mathfrak{I}$ are: - -$$\left\{ \mathfrak{I}_{ij} \eqdef {Cov}\left( {\ell'}_{i},{\ell'}_{j} \right) = E[ \ell_{i}'\ell_{j}' ] - E[ {\ell'}_{i} ] E[ {\ell'}_{j} ] \right\}$$ - -Here, +$$ +\ba +\einf +\eqdef \einf(\theta) +\\ &\eqdef \Covf{\ell'|\theta} +\\ &= \Expp[ \ell'{\ell'}\' ] - \Expp[ \ell' ] \ \Expp[ \ell' ]\' +\ea +$$ -$$ -\ba -\E{\ell'} -&\eqdef \int_{x \in \rangef{x}} -{ -\ell'(x,\th) \p(X = x | \th) dx -} -\\ &= \int_{x \in \rangef{X}} -{ -\paren -{ -\deriv{\th} -\log{\p(X = x | \th)} -} -\p(X = x | \theta) dx -} -\\ &= -\int_{x \in \rangef{X}} -{ -\frac -{\deriv{\theta} \p(X = x | \th)} -{\p(X = x | \theta)} -\p(X = x | \theta) dx -} -\\ &= -\int_{x \in \rangef{X}} -{ -\deriv{\theta} \p(X = x | \th) dx -} -\ea -$$ - - - -And similarly - -$$ -\Exp{\ell' \ell'^{\top}} -\eqdef -\int_{x \in R(x)} -{\ell'(x,\theta)\ell'(x,\theta)^{\top}\ -\pf{X = x | \th}\ dx} +The elements of $\mathfrak{I}$ are: + +$$ +\ba +\mathfrak{I}_{ij} +&\eqdef \Covf{{\ell'}_{i},{\ell'}_{j}} +\\ &= \Expp[ \ell_{i}'\ell_{j}' ] - \Expp[ {\ell'}_{i} ] \Expp[ {\ell'}_{j} ] +\ea +$$ + +Here, + +$$ +\ba +\E{\ell'} +&\eqdef \int_{x \in \rangef{x}} +{ +\ell'(x,\th) \p(X = x | \th) dx +} +\\ &= \int_{x \in \rangef{X}} +{ +\paren +{ +\deriv{\th} +\log{\p(X = x | \th)} +} +\p(X = x | \theta) dx +} +\\ &= +\int_{x \in \rangef{X}} +{ +\frac +{\deriv{\theta} \p(X = x | \th)} +{\p(X = x | \theta)} +\p(X = x | \theta) dx +} +\\ &= +\int_{x \in \rangef{X}} +{ +\deriv{\theta} \p(X = x | \th) dx +} +\ea $$ -Note that $\Exp{\ell'}$ and -$\Exp{\ell'{\ell'}^{\top}}$ -are functions of $\theta$ but not of $x$; -the expectation operator removed $x$. - -Also note that for most of the distributions you are familiar with -(including Gaussian, binomial, Poisson, exponential): - -$$\Exp{\ell'} = 0$$ - -So + + +And similarly + +$$ +\Exp{\ell' \ell'^{\top}} +\eqdef +\int_{x \in R(x)} +{\ell'(x,\theta)\ell'(x,\theta)^{\top}\ +\pf{X = x | \th}\ dx} +$$ -$$\einff{\theta} = \Exp{\ell'{\ell'}^{\top} }$$ - -Moreover, for those distributions (called the "exponential family"), we -have: +Note that $\Exp{\ell'}$ and +$\Exp{\ell'{\ell'}^{\top}}$ +are functions of $\theta$ but not of $x$; +the expectation operator removed $x$. -$$ -\mathfrak{I} = -\Exp{\ell''} -= \Exp{- \ell''} -$$ +Also note that for most of the distributions you are familiar with +(including Gaussian, binomial, Poisson, exponential): + +$$\Exp{\ell'} = 0$$ -(see @dobson4e, §3.17), where +So -$$\ell'' \eqdef \deriv{\theta}\ell^{'(x,\theta)^{\top}} = \deriv{\theta}\deriv{\theta^{\top}}\ell(x,\theta)$$ +$$\einff{\theta} = \Exp{\ell'{\ell'}^{\top} }$$ -is the $p \times p$ matrix whose elements are: - -$$\ell_{ij}'' \eqdef \deriv{\theta_{i}}\deriv{\theta_{j}}\log{ p\left( X = x \mid \theta \right)}$$ - -$\ell''$ is called the "Hessian"^[named after mathematician [Otto Hesse](https://en.wikipedia.org/wiki/Otto_Hesse)] of the log-likelihood -function. - -Sometimes, we use $I(\theta;x) \eqdef - \ell''$ (note the -standard-font "I" here). $I(\theta;x)$ is the observed information, precision, or concentration -matrix (Negative Hessian). - -:::{.callout-important} -#### Key point - The asymptotics of MLEs gives us -${\widehat{\theta}}_{ML} \sim N\left( \theta,\mathfrak{I}^{- 1}(\theta) \right)$, -approximately, for large sample sizes. -::: - -We can estimate $\einf^{- 1}(\theta)$ by working out -$\Ef{-\ell''}$ or -$\Ef{\ell'{\ell'}^{\top}}$ -and plugging in $\hthml$, but sometimes we instead use -$\oinf(\hthml,\vx)$ for convenience; there are -some cases where it’s provably better according to some criteria (@efron1978assessing). - -### Iterative maximization {#sec-newton-raphson} - -(c.f., @dobson4e, Chapter 4) - -::: notes -Later, -when we are trying to find MLEs for likelihoods which we can’t easily differentiate, -we will "hill-climb" using the Newton-Raphson algorithm: -::: - -$$ -\begin{aligned} -\esttmp{\theta} -&\leftarrow \esttmp{\theta} + \inv{\oinff{\vec y;\esttmp{\theta}}} -\scoref{\vec y;\esttmp{\theta}} -\\ &= \esttmp{\theta} - \inv{\hessf{\vec y;\esttmp{\theta}}} -\scoref{\vec y;\esttmp{\theta}} -\end{aligned} -$$ - ---- +Moreover, for those distributions (called the "exponential family"), we +have: + +$$ +\mathfrak{I} = -\Exp{\ell''} += \Exp{- \ell''} +$$ + +(see @dobson4e, §3.17), where + +$$\ell'' \eqdef \deriv{\theta}\ell^{'(x,\theta)^{\top}} = \deriv{\theta}\deriv{\theta^{\top}}\ell(x,\theta)$$ + +is the $p \times p$ matrix whose elements are: + +$$\ell_{ij}'' \eqdef \deriv{\theta_{i}}\deriv{\theta_{j}}\log{ p\left( X = x \mid \theta \right)}$$ + +$\ell''$ is called the "Hessian"^[named after mathematician [Otto Hesse](https://en.wikipedia.org/wiki/Otto_Hesse)] of the log-likelihood +function. + +Sometimes, we use $I(\theta;x) \eqdef - \ell''$ (note the +standard-font "I" here). $I(\theta;x)$ is the observed information, precision, or concentration +matrix (Negative Hessian). + +:::{.callout-important} +#### Key point + The asymptotics of MLEs gives us +${\widehat{\theta}}_{ML} \sim N\left( \theta,\mathfrak{I}^{- 1}(\theta) \right)$, +approximately, for large sample sizes. +::: + +We can estimate $\einf^{- 1}(\theta)$ by working out +$\Ef{-\ell''}$ or +$\Ef{\ell'{\ell'}^{\top}}$ +and plugging in $\hthml$, but sometimes we instead use +$\oinf(\hthml,\vx)$ for convenience; there are +some cases where it’s provably better according to some criteria (@efron1978assessing). + +### Iterative maximization {#sec-newton-raphson} + +(c.f., @dobson4e, Chapter 4) + +::: notes +Later, +when we are trying to find MLEs for likelihoods which we can’t easily differentiate, +we will "hill-climb" using the Newton-Raphson algorithm: +::: -::: notes -The reasoning for this algorithm is that we can approximate the the score function using the first-order [Taylor polynomial](https://en.wikipedia.org/wiki/Taylor%27s_theorem): -::: - -$$ -\ba -\score(\th) -&\approx \score^*(\th) -\\ &\eqdef \score(\esttmp{\th}) + \hessian(\esttmp{\th})(\th - \esttmp{\th}) -\ea -$$ +$$ +\begin{aligned} +\esttmp{\theta} +&\leftarrow \esttmp{\theta} + \inv{\oinff{\vec y;\esttmp{\theta}}} +\scoref{\vec y;\esttmp{\theta}} +\\ &= \esttmp{\theta} - \inv{\hessf{\vec y;\esttmp{\theta}}} +\scoref{\vec y;\esttmp{\theta}} +\end{aligned} +$$ + +--- ---- - -::: notes -The approximate score function, $\score^*(\th)$, is a linear function of $\th$, so it is easy to solve the corresponding approximate score equation, $\score^*(\th) = 0$, for $\th$: - -::: - -$$ -\ba -\th -&= \esttmp{\th} - \score(\esttmp{\th}) \cd \inv{\hessian(\esttmp{\th})} -\ea -$$ +::: notes +The reasoning for this algorithm is that we can approximate the the score function using the first-order [Taylor polynomial](https://en.wikipedia.org/wiki/Taylor%27s_theorem): +::: + +$$ +\ba +\score(\th) +&\approx \score^*(\th) +\\ &\eqdef \score(\esttmp{\th}) + \hessian(\esttmp{\th})(\th - \esttmp{\th}) +\ea +$$ + +--- ---- - -For computational simplicity, we will sometimes use -$\mathfrak{I}^{- 1}(\theta)$ in place of -$I\left( \widehat{\theta},y \right)$; -doing so is called "Fisher scoring" or the "method of scoring". -Note that this is the opposite of the substitution that we are making for estimating the variance of the MLE; -this time we should technically use the observed information but we use the expected information instead. - - ---- +::: notes +The approximate score function, $\score^*(\th)$, is a linear function of $\th$, so it is easy to solve the corresponding approximate score equation, $\score^*(\th) = 0$, for $\th$: + +::: + +$$ +\ba +\th +&= \esttmp{\th} - \score(\esttmp{\th}) \cd \inv{\hessian(\esttmp{\th})} +\ea +$$ -There’s also an "empirical information matrix" (see @mclachlan2007em): +--- -$$I_{e}(\theta,y) \eqdef \sum_{i = 1}^{n}{\ell_{i}'\ {\ell_{i}'}^{\top}} - \frac{1}{n}\ell'{\ell'}^{\top}$$ - -where $\ell_{i}$ is the log-likelihood of the ith observation. -Note that $\ell' = \sum_{i = 1}^{n}\ell_{i}'$. - -$\frac{1}{n}I_{e}(\theta,y)$ is the sample equivalent of +For computational simplicity, we will sometimes use +$\mathfrak{I}^{- 1}(\theta)$ in place of +$I\left( \widehat{\theta},y \right)$; +doing so is called "Fisher scoring" or the "method of scoring". +Note that this is the opposite of the substitution that we are making for estimating the variance of the MLE; +this time we should technically use the observed information but we use the expected information instead. -$$\mathfrak{I \eqdef I(}\theta) \eqdef {Cov}\left( \ell'|\theta \right) = E[ \ell'{\ell'}^{\top} ] - E[ \ell' ]\ E[ \ell' ]^{\top}$$ - -$$\left\{ \mathfrak{I}_{jk} \eqdef {Cov}\left( {\ell'}_{j},{\ell'}_{k} \right) = E[ \ell_{j}'\ell_{k}' ] - E[ {\ell'}_{j} ] E[ {\ell'}_{k} ] \right\}$$ - -$I_{e}(\theta,y)$ is sometimes computationally easier to compute for -Newton-Raphson-type maximization algorithms. + +--- + +There’s also an "empirical information matrix" (see @mclachlan2007em): + +$$I_{e}(\theta,y) \eqdef \sum_{i = 1}^{n}{\ell_{i}'\ {\ell_{i}'}^{\top}} - \frac{1}{n}\ell'{\ell'}^{\top}$$ -c.f. <https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization> - -### Quantifying (un)certainty of MLEs - -#### Confidence intervals for MLEs - -An asymptotic approximation of a 95% confidence interval for $\theta_k$ is - -$$ -\hthml \pm z_{0.975} \times \HSE{\hth_k} -$$ +where $\ell_{i}$ is the log-likelihood of the ith observation. +Note that $\ell' = \sum_{i = 1}^{n}\ell_{i}'$. + +$\frac{1}{n}I_{e}(\theta,y)$ is the sample equivalent of + +$$\mathfrak{I \eqdef I(}\theta) \eqdef {Cov}\left( \ell'|\theta \right) = E[ \ell'{\ell'}^{\top} ] - E[ \ell' ]\ E[ \ell' ]^{\top}$$ + +$$\left\{ \mathfrak{I}_{jk} \eqdef {Cov}\left( {\ell'}_{j},{\ell'}_{k} \right) = E[ \ell_{j}'\ell_{k}' ] - E[ {\ell'}_{j} ] E[ {\ell'}_{k} ] \right\}$$ + +$I_{e}(\theta,y)$ is sometimes computationally easier to compute for +Newton-Raphson-type maximization algorithms. -where $z_\beta$ the $\beta$ quantile of the standard Gaussian distribution. +c.f. <https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization> -#### p-values and hypothesis tests for MLEs +### Quantifying (un)certainty of MLEs -(to add) +#### Confidence intervals for MLEs -#### Likelihood ratio tests for MLEs +An asymptotic approximation of a 95% confidence interval for $\theta_k$ is -log(likelihood ratio) tests [c.f. @dobson4e §5.7]: - -$$-2\ell_{0} \sim \chi^{2}(p - q)$$ +$$ +\hthml \pm z_{0.975} \times \HSE{\hth_k} +$$ -See also <https://online.stat.psu.edu/stat504/book/export/html/657> +where $z_\beta$ the $\beta$ quantile of the standard Gaussian distribution. -#### Prediction intervals for MLEs +#### p-values and hypothesis tests for MLEs -$$\overline{X} \in [ \widehat{\mu} \pm z_{1 - \alpha\text{/}2}\frac{\sigma}{m} ]$$ +(to add) -Where $m$ is the sample size of the new data to be predicted (typically -1, except for binary outcomes, where it needs to be bigger for -prediction intervals to make sense) +#### Likelihood ratio tests for MLEs + +log(likelihood ratio) tests [c.f. @dobson4e §5.7]: -[^1]: I might sometimes switch the order of $x,$ $\theta$; this is - unintentional and not meaningful. - -## Example: Maximum likelihood for Tropical Cyclones in Australia - -{{< include dobson-cyclone-example.qmd >}} - -## Maximum likelihood inference for univariate Gaussian models - -Suppose $X_{1}, ..., X_{n} \siid N(\mu, \sigma^{2})$. -Let $X = (X_{1},\ldots,X_{n})^{\top}$ be these random -variables in vector format. Let $x_{i}$ and $x$ denote the corresponding -observed data. Then $\theta = (\mu,\sigma^{2})$ is -the vector of true parameters, and $\Theta = (\Mu, \Sigma^2)$ is the vector of parameters as a random -vector. - -Then the log-likelihood -is: +$$-2\ell_{0} \sim \chi^{2}(p - q)$$ + +See also <https://online.stat.psu.edu/stat504/book/export/html/657> + +#### Prediction intervals for MLEs + +$$\overline{X} \in [ \widehat{\mu} \pm z_{1 - \alpha\text{/}2}\frac{\sigma}{m} ]$$ + +Where $m$ is the sample size of the new data to be predicted (typically +1, except for binary outcomes, where it needs to be bigger for +prediction intervals to make sense) + +[^1]: I might sometimes switch the order of $x,$ $\theta$; this is + unintentional and not meaningful. + +## Example: Maximum likelihood for Tropical Cyclones in Australia + +{{< include dobson-cyclone-example.qmd >}} -$$ -\begin{aligned} -\ell -&\propto - \frac{n}{2}\log{\sigma^{2}} - \frac{1}{2}\sum_{i = 1}^{n}\frac{( x_{i} - \mu)^{2}}{\sigma^{2}}\\ -&= - \frac{n}{2}\log{\sigma^{2}} - \frac{1}{2\sigma^{2}}\sum_{i = 1}^{n}{x_{i}^{2} - 2x_{i}\mu + \mu^{2}} -\end{aligned} -$$ - +## Maximum likelihood inference for univariate Gaussian models + +Suppose $X_{1}, ..., X_{n} \siid N(\mu, \sigma^{2})$. +Let $X = (X_{1},\ldots,X_{n})^{\top}$ be these random +variables in vector format. Let $x_{i}$ and $x$ denote the corresponding +observed data. Then $\theta = (\mu,\sigma^{2})$ is +the vector of true parameters, and $\Theta = (\Mu, \Sigma^2)$ is the vector of parameters as a random +vector. -### The score function - -$$\ell'(x,\theta) \eqdef \deriv{\theta}\ell(x,\theta) = \left( \begin{array}{r} -\deriv{\mu}\ell(\theta;x) \\ -\deriv{\sigma^{2}}\ell(\theta;x) -\end{array} \right) = \left( \begin{array}{r} -\ell_{\mu}'(\theta;x) \\ -\ell_{\sigma^{2}}'(\theta;x) -\end{array} \right)$$. - -$\ell'(x,\theta)$ is the function we set equal to 0 and solve -to find the MLE: - -$${\widehat{\theta}}_{ML} = \left\{ \theta:\ell'(x,\theta) = 0 \right\}$$ - -### MLE of $\mu$ - -$$ -\ba -\frac{d\ell}{d\mu} -&= - \frac{1}{2}\sum_{i = 1}^{n} -\frac{- 2(x_{i} - \mu)}{\sigma^{2}} -\\ &= \frac{1}{\sigma^{2}} -\sb{ - \paren{ - \sum_{i = 1}^{n}x_{i} - } - - n\mu -} -\ea -$$ - -If $\frac{d\ell}{d\mu} = 0$, then -$\mu = \overline{x} \eqdef \frac{1}{n}\sum_{i = 1}^{n}x_{i}$. - -$$\frac{d^{2}\ell}{(d\mu)^{2}} = \frac{- n}{\sigma^{2}} < 0$$ - -So ${\widehat{\mu}}_{ML} = \overline{x}$. - -### MLE of $\sigma^{2}$ - -:::{.callout-tip} -#### Reparametrizing the Gaussian distribution +Then the log-likelihood +is: + +$$ +\begin{aligned} +\ell +&\propto - \frac{n}{2}\log{\sigma^{2}} - \frac{1}{2}\sum_{i = 1}^{n}\frac{( x_{i} - \mu)^{2}}{\sigma^{2}}\\ +&= - \frac{n}{2}\log{\sigma^{2}} - \frac{1}{2\sigma^{2}}\sum_{i = 1}^{n}{x_{i}^{2} - 2x_{i}\mu + \mu^{2}} +\end{aligned} +$$ + + +### The score function + +$$\ell'(x,\theta) \eqdef \deriv{\theta}\ell(x,\theta) = \left( \begin{array}{r} +\deriv{\mu}\ell(\theta;x) \\ +\deriv{\sigma^{2}}\ell(\theta;x) +\end{array} \right) = \left( \begin{array}{r} +\ell_{\mu}'(\theta;x) \\ +\ell_{\sigma^{2}}'(\theta;x) +\end{array} \right)$$. + +$\ell'(x,\theta)$ is the function we set equal to 0 and solve +to find the MLE: + +$${\widehat{\theta}}_{ML} = \left\{ \theta:\ell'(x,\theta) = 0 \right\}$$ + +### MLE of $\mu$ + +$$ +\ba +\frac{d\ell}{d\mu} +&= - \frac{1}{2}\sum_{i = 1}^{n} +\frac{- 2(x_{i} - \mu)}{\sigma^{2}} +\\ &= \frac{1}{\sigma^{2}} +\sb{ + \paren{ + \sum_{i = 1}^{n}x_{i} + } + - n\mu +} +\ea +$$ -When solving for ${\widehat{\sigma}}_{ML}$, you can treat -$\sigma^{2}$ as an atomic variable (don’t differentiate with respect to -$\sigma$ or things get messy). In fact, you can replace $\sigma^{2}$ -with $1/\tau$ and differentiate with respect to $\tau$ instead, and the -process might be even easier. -::: +If $\frac{d\ell}{d\mu} = 0$, then +$\mu = \overline{x} \eqdef \frac{1}{n}\sum_{i = 1}^{n}x_{i}$. + +$$\frac{d^{2}\ell}{(d\mu)^{2}} = \frac{- n}{\sigma^{2}} < 0$$ + +So ${\widehat{\mu}}_{ML} = \overline{x}$. -$$\frac{d\ell}{d\sigma^{2}} = \deriv{\sigma^{2}}\left( - \frac{n}{2}\log{\sigma^{2}} - \frac{1}{2}\sum_{i = 1}^{n}\frac{\left( x_{i} - \mu \right)^{2}}{\sigma^{2}} \right)\ $$ +### MLE of $\sigma^{2}$ -$$= - \frac{n}{2}\left( \sigma^{2} \right)^{- 1} + \frac{1}{2}\left( \sigma^{2} \right)^{- 2}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2}$$ - -If $\frac{d\ell}{d\sigma^{2}} = 0$, then: - -$$\frac{n}{2}\left( \sigma^{2} \right)^{- 1} = \frac{1}{2}\left( \sigma^{2} \right)^{- 2}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2}$$ - -$$\sigma^{2} = \frac{1}{n}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2}$$ - -We plug in ${\widehat{\mu}}_{ML} = \overline{x}$ to maximize globally (a -technique called profiling): - -$$\sigma_{ML}^{2} = \frac{1}{n}\sum_{i = 1}^{n}\left( x_{i} - \overline{x} \right)^{2}$$ - -Now: - -$$ -\begin{aligned} -\frac{d^{2}\ell}{\left( d\sigma^{2} \right)^{2}} -&= \deriv{\sigma^{2}}\left\{ - \frac{n}{2}\left( \sigma^{2} \right)^{- 1} + \frac{1}{2}\left( \sigma^{2} \right)^{- 2}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2} \right\}\\ -&= \left\{ - \frac{n}{2}\deriv{\sigma^{2}}\left( \sigma^{2} \right)^{- 1} + \frac{1}{2}\deriv{\sigma^{2}}\left( \sigma^{2} \right)^{- 2}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2} \right\}\\ -&= \left\{ \frac{n}{2}\left( \sigma^{2} \right)^{- 2} - \left( \sigma^{2} \right)^{- 3}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2} \right\}\\ -&= \left( \sigma^{2} \right)^{- 2}\left\{ \frac{n}{2} - \left( \sigma^{2} \right)^{- 1}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2} \right\} -\end{aligned} -$$ +:::{.callout-tip} +#### Reparametrizing the Gaussian distribution + +When solving for ${\widehat{\sigma}}_{ML}$, you can treat +$\sigma^{2}$ as an atomic variable (don’t differentiate with respect to +$\sigma$ or things get messy). In fact, you can replace $\sigma^{2}$ +with $1/\tau$ and differentiate with respect to $\tau$ instead, and the +process might be even easier. +::: + +$$\frac{d\ell}{d\sigma^{2}} = \deriv{\sigma^{2}}\left( - \frac{n}{2}\log{\sigma^{2}} - \frac{1}{2}\sum_{i = 1}^{n}\frac{\left( x_{i} - \mu \right)^{2}}{\sigma^{2}} \right)\ $$ + +$$= - \frac{n}{2}\left( \sigma^{2} \right)^{- 1} + \frac{1}{2}\left( \sigma^{2} \right)^{- 2}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2}$$ + +If $\frac{d\ell}{d\sigma^{2}} = 0$, then: + +$$\frac{n}{2}\left( \sigma^{2} \right)^{- 1} = \frac{1}{2}\left( \sigma^{2} \right)^{- 2}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2}$$ + +$$\sigma^{2} = \frac{1}{n}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2}$$ + +We plug in ${\widehat{\mu}}_{ML} = \overline{x}$ to maximize globally (a +technique called profiling): + +$$\sigma_{ML}^{2} = \frac{1}{n}\sum_{i = 1}^{n}\left( x_{i} - \overline{x} \right)^{2}$$ -Evaluated at -$\mu = \overline{x},\sigma^{2} = \frac{1}{n}\sum_{i = 1}^{n}\left( x_{i} - \overline{x} \right)^{2}$, -we have: - -$$ -\begin{aligned} -\frac{d^{2}\ell}{\left( d\sigma^{2} \right)^{2}} -&= \left( {\widehat{\sigma}}^{2} \right)^{- 2}\left\{ \frac{n}{2} - \left( {\widehat{\sigma}}^{2} \right)^{- 1}\sum_{i = 1}^{n}\left( x_{i} - \overline{x} \right)^{2} \right\}\\ -&= \left( {\widehat{\sigma}}^{2} \right)^{- 2}\left\{ \frac{n}{2} - \left( {\widehat{\sigma}}^{2} \right)^{- 1}n{\widehat{\sigma}}^{2} \right\}\\ -&= \left( {\widehat{\sigma}}^{2} \right)^{- 2}\left\{ \frac{n}{2} - n \right\}\\ -&= \left( {\widehat{\sigma}}^{2} \right)^{- 2}n\left\{ \frac{1}{2} - 1 \right\}\\ -&= \left( {\widehat{\sigma}}^{2} \right)^{- 2}n\left( - \frac{1}{2} \right) < 0 -\end{aligned} -$$ - -Finally, we have: - -$$ -\begin{aligned} -\frac{d^{2}\ell}{d\mu\ d\sigma^{2}} -&= \deriv{\mu}\left\{ - \frac{n}{2}\left( \sigma^{2} \right)^{- 1} + \frac{1}{2}\left( \sigma^{2} \right)^{- 2}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2} \right\}\\ -&= \frac{1}{2}\left( \sigma^{2} \right)^{- 2}\deriv{\mu}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2}\\ -&= \frac{1}{2}\left( \sigma^{2} \right)^{- 2}\sum_{i = 1}^{n}{- 2(x_{i} - \mu)}\\ -&= - \left( \sigma^{2} \right)^{- 2}\sum_{i = 1}^{n}{(x_{i} - \mu)} +Now: + +$$ +\begin{aligned} +\frac{d^{2}\ell}{\left( d\sigma^{2} \right)^{2}} +&= \deriv{\sigma^{2}}\left\{ - \frac{n}{2}\left( \sigma^{2} \right)^{- 1} + \frac{1}{2}\left( \sigma^{2} \right)^{- 2}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2} \right\}\\ +&= \left\{ - \frac{n}{2}\deriv{\sigma^{2}}\left( \sigma^{2} \right)^{- 1} + \frac{1}{2}\deriv{\sigma^{2}}\left( \sigma^{2} \right)^{- 2}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2} \right\}\\ +&= \left\{ \frac{n}{2}\left( \sigma^{2} \right)^{- 2} - \left( \sigma^{2} \right)^{- 3}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2} \right\}\\ +&= \left( \sigma^{2} \right)^{- 2}\left\{ \frac{n}{2} - \left( \sigma^{2} \right)^{- 1}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2} \right\} +\end{aligned} +$$ + +Evaluated at +$\mu = \overline{x},\sigma^{2} = \frac{1}{n}\sum_{i = 1}^{n}\left( x_{i} - \overline{x} \right)^{2}$, +we have: + +$$ +\begin{aligned} +\frac{d^{2}\ell}{\left( d\sigma^{2} \right)^{2}} +&= \left( {\widehat{\sigma}}^{2} \right)^{- 2}\left\{ \frac{n}{2} - \left( {\widehat{\sigma}}^{2} \right)^{- 1}\sum_{i = 1}^{n}\left( x_{i} - \overline{x} \right)^{2} \right\}\\ +&= \left( {\widehat{\sigma}}^{2} \right)^{- 2}\left\{ \frac{n}{2} - \left( {\widehat{\sigma}}^{2} \right)^{- 1}n{\widehat{\sigma}}^{2} \right\}\\ +&= \left( {\widehat{\sigma}}^{2} \right)^{- 2}\left\{ \frac{n}{2} - n \right\}\\ +&= \left( {\widehat{\sigma}}^{2} \right)^{- 2}n\left\{ \frac{1}{2} - 1 \right\}\\ +&= \left( {\widehat{\sigma}}^{2} \right)^{- 2}n\left( - \frac{1}{2} \right) < 0 \end{aligned} $$ -Evaluated at -$\mu = \widehat{\mu} = \overline{x},\sigma^{2} = {\widehat{\sigma}}^{2} = \frac{1}{n}\sum_{i = 1}^{n}\left( x_{i} - \overline{x} \right)^{2}$, -we have: - -$$\frac{d^{2}\ell}{d\mu\ d\sigma^{2}} = - \left( {\widehat{\sigma}}^{2} \right)^{- 2}\left( n\overline{x} - n\overline{x} \right) = 0$$ - -### Covariance matrix - -$$I = \begin{bmatrix} -\frac{n}{\sigma^{2}} & 0 \\ -0 & \left( {\widehat{\sigma}}^{2} \right)^{- 2}n\left( - \frac{1}{2} \right) -\end{bmatrix} = \begin{bmatrix} -a & 0 \\ -0 & d -\end{bmatrix}$$ +Finally, we have: + +$$ +\begin{aligned} +\frac{d^{2}\ell}{d\mu\ d\sigma^{2}} +&= \deriv{\mu}\left\{ - \frac{n}{2}\left( \sigma^{2} \right)^{- 1} + \frac{1}{2}\left( \sigma^{2} \right)^{- 2}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2} \right\}\\ +&= \frac{1}{2}\left( \sigma^{2} \right)^{- 2}\deriv{\mu}\sum_{i = 1}^{n}\left( x_{i} - \mu \right)^{2}\\ +&= \frac{1}{2}\left( \sigma^{2} \right)^{- 2}\sum_{i = 1}^{n}{- 2(x_{i} - \mu)}\\ +&= - \left( \sigma^{2} \right)^{- 2}\sum_{i = 1}^{n}{(x_{i} - \mu)} +\end{aligned} +$$ + +Evaluated at +$\mu = \widehat{\mu} = \overline{x},\sigma^{2} = {\widehat{\sigma}}^{2} = \frac{1}{n}\sum_{i = 1}^{n}\left( x_{i} - \overline{x} \right)^{2}$, +we have: -So: +$$\frac{d^{2}\ell}{d\mu\ d\sigma^{2}} = - \left( {\widehat{\sigma}}^{2} \right)^{- 2}\left( n\overline{x} - n\overline{x} \right) = 0$$ -$$I^{- 1} = \frac{1}{ad}\begin{bmatrix} -d & 0 \\ -0 & a -\end{bmatrix} = \begin{bmatrix} -\frac{1}{a} & 0 \\ -0 & \frac{1}{d} -\end{bmatrix}$$ - -$$I^{- 1} = \begin{bmatrix} -\frac{{\widehat{\sigma}}^{2}}{n} & 0 \\ -0 & \frac{{2\left( {\widehat{\sigma}}^{2} \right)}^{2}}{n} -\end{bmatrix}$$ - -See @CaseBerg01 p322, example 7.2.12. - -To prove it’s a maximum, we need: - -- $\ell' = 0$ - -- At least one diagonal element of $\ell''$ is negative. - -- Determinant of $\ell''$ is positive. - - -{{< include HERS-example.qmd >}} +### Covariance matrix + +$$I = \begin{bmatrix} +\frac{n}{\sigma^{2}} & 0 \\ +0 & \left( {\widehat{\sigma}}^{2} \right)^{- 2}n\left( - \frac{1}{2} \right) +\end{bmatrix} = \begin{bmatrix} +a & 0 \\ +0 & d +\end{bmatrix}$$ + +So: + +$$I^{- 1} = \frac{1}{ad}\begin{bmatrix} +d & 0 \\ +0 & a +\end{bmatrix} = \begin{bmatrix} +\frac{1}{a} & 0 \\ +0 & \frac{1}{d} +\end{bmatrix}$$ + +$$I^{- 1} = \begin{bmatrix} +\frac{{\widehat{\sigma}}^{2}}{n} & 0 \\ +0 & \frac{{2\left( {\widehat{\sigma}}^{2} \right)}^{2}}{n} +\end{bmatrix}$$ + +See @CaseBerg01 p322, example 7.2.12. + +To prove it’s a maximum, we need: + +- $\ell' = 0$ + +- At least one diagonal element of $\ell''$ is negative. + +- Determinant of $\ell''$ is positive. + + +{{< include HERS-example.qmd >}}
    diff --git a/intro-to-GLMs.html b/intro-to-GLMs.html index 7c9fec244..d17883848 100644 --- a/intro-to-GLMs.html +++ b/intro-to-GLMs.html @@ -370,7 +370,7 @@
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    Table of contents

  • H.1.1 Methods versus functions
  • H.1.2 Debugging R and C code
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  • H.2 The tidyverse
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  • H.3 Piping +
  • H.2 data.frames and tibbles
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  • H.4 Quarto
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  • H.5 Packages
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  • H.6 Git
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  • H.7 Spatial data science
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  • H.8 Shiny apps
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  • H.3 The tidyverse
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  • H.4 Piping +
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  • H.5 Quarto
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  • H.6 Packages
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  • H.7 Git
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  • H.8 Spatial data science
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  • H.9 Shiny apps
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    Table of contents

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    See https://www.maths.ed.ac.uk/~swood34/RCdebug/RCdebug.html

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    H.2 The tidyverse

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    H.2 data.frames and tibbles

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    H.2.1 Displaying tibbles

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    See vignette("digits", package = "tibble")

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    H.3 The tidyverse

    The tidyverse is an opinionated collection of R packages designed for data science. All packages share an underlying design philosophy, grammar, and data structures.

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    Kuhn and Silge (2022)

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    H.3 Piping

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    H.4 Piping

    See Wickham, Çetinkaya-Rundel, and Grolemund (2023) for details.

    There are currently (2024) two commonly-used pipe operators in R:

    • %>%: the “magrittr pipe”, from the magrittr package (Bache and Wickham (2022); re-exported by dplyr and others) .

    • |>: the “native pipe”, from base R (≥4.1.0)

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    H.3.1 Which pipe should I use?

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    H.4.1 Which pipe should I use?

    Wickham, Çetinkaya-Rundel, and Grolemund (2023) recommends the native pipe:

    For simple cases, |> and %>% behave identically. So why do we recommend the base pipe? Firstly, because it’s part of base R, it’s always available for you to use, even when you’re not using the tidyverse. Secondly, |> is quite a bit simpler than %>%: in the time between the invention of %>% in 2014 and the inclusion of |> in R 4.1.0 in 2021, we gained a better understanding of the pipe. This allowed the base implementation to jettison infrequently used and less important features.

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    H.3.2 Why doesn’t ggplot2 use piping?

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    H.4.2 Why doesn’t ggplot2 use piping?

    Here’s tidyverse creator Hadley Wickham’s answer (from 2018):

    I think it’s worth unpacking this question into a few smaller pieces:

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    H.4 Quarto

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    H.5 Quarto

    Quarto is a system for writing documents with embedded R code and/or results:

    • Read this ASAP: https://r4ds.hadley.nz/communicate
    • Then use this for reference: https://quarto.org/docs/reference/
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    H.5 Packages

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    +

    H.6 Packages

    This book espouses our philosophy of package development: anything that can be automated, should be automated. Do as little as possible by hand. Do as much as possible with functions. The goal is to spend your time thinking about what you want your package to do rather than thinking about the minutiae of package structure.

    @@ -486,8 +497,8 @@

    Wickham and Bryan (2023) as a reference

    -
    -

    H.6 Git

    +
    +

    H.7 Git

    94% of respondents to a 2022 Stack Overflow survey reported using git for version control link

    More details

      @@ -495,14 +506,14 @@

      https://usethis.r-lib.org/articles/pr-functions.html

    -
    -

    H.7 Spatial data science

    +
    +

    H.8 Spatial data science

    • Pebesma and Bivand (2023)
    -
    -

    H.8 Shiny apps

    +
    +

    H.9 Shiny apps

    • Read this first: Wickham (2021)
    • Use this as a reference: Fay et al. (2021)
      • @@ -1197,82 +1208,88 @@

      See <https://www.maths.ed.ac.uk/~swood34/RCdebug/RCdebug.html> -## The `tidyverse` +## `data.frame`s and `tibble`s -> The tidyverse is an opinionated collection of R packages designed for data science. All packages share an underlying design philosophy, grammar, and data structures. +### Displaying `tibble`s -- https://www.tidyverse.org/ +See `vignette("digits", package = "tibble")` -These packages are being actively developed by [Hadley Wickham](https://hadley.nz/) and his colleagues at [posit](https://posit.co/)^[[the company formerly known as RStudio]( https://posit.co/blog/rstudio-is-becoming-posit/)]. +## The `tidyverse` -Details: +> The tidyverse is an opinionated collection of R packages designed for data science. All packages share an underlying design philosophy, grammar, and data structures. -- @welcometidyverse -- @r4ds -- @kuhn2022tidy +- https://www.tidyverse.org/ + +These packages are being actively developed by [Hadley Wickham](https://hadley.nz/) and his colleagues at [posit](https://posit.co/)^[[the company formerly known as RStudio]( https://posit.co/blog/rstudio-is-becoming-posit/)]. -## Piping +Details: -See [@r4ds](https://r4ds.hadley.nz/data-transform.html#sec-the-pipe) for details. - -There are currently (2024) two commonly-used pipe operators in R: +- @welcometidyverse +- @r4ds +- @kuhn2022tidy -- `%>%`: the "`magrittr` pipe", from the [`magrittr`](https://cran.r-project.org/web/packages/magrittr/index.html) package (@magrittr; [re-exported](https://r-pkgs.org/dependencies-in-practice.html#re-exporting) by [`dplyr`](https://cran.r-project.org/web/packages/dplyr/index.html) and others) . +## Piping -- `|>`: the "native pipe", from base R (≥4.1.0) +See [@r4ds](https://r4ds.hadley.nz/data-transform.html#sec-the-pipe) for details. -### Which pipe should I use? +There are currently (2024) two commonly-used pipe operators in R: -@r4ds [recommends the native pipe](https://r4ds.hadley.nz/data-transform.html#sec-the-pipe:~:text=So%20why%20do%20we%20recommend%20the%20base%20pipe%3F): +- `%>%`: the "`magrittr` pipe", from the [`magrittr`](https://cran.r-project.org/web/packages/magrittr/index.html) package (@magrittr; [re-exported](https://r-pkgs.org/dependencies-in-practice.html#re-exporting) by [`dplyr`](https://cran.r-project.org/web/packages/dplyr/index.html) and others) . -> For simple cases, |> and %>% behave identically. So why do we recommend the base pipe? Firstly, because it’s part of base R, it’s always available for you to use, even when you’re not using the tidyverse. Secondly, |> is quite a bit simpler than %>%: in the time between the invention of %>% in 2014 and the inclusion of |> in R 4.1.0 in 2021, we gained a better understanding of the pipe. This allowed the base implementation to jettison infrequently used and less important features. +- `|>`: the "native pipe", from base R (≥4.1.0) -### Why doesn't `ggplot2` use piping? +### Which pipe should I use? -Here's `tidyverse` creator Hadley Wickham's answer (from 2018): +@r4ds [recommends the native pipe](https://r4ds.hadley.nz/data-transform.html#sec-the-pipe:~:text=So%20why%20do%20we%20recommend%20the%20base%20pipe%3F): -> I think it's worth unpacking this question into a few smaller pieces: -> -> - Should ggplot2 use the pipe? IMO, yes. -> - Could ggplot2 support both the pipe and plus? No -> - Would it be worth it to create a ggplot3 that uses the pipe? No. +> For simple cases, |> and %>% behave identically. So why do we recommend the base pipe? Firstly, because it’s part of base R, it’s always available for you to use, even when you’re not using the tidyverse. Secondly, |> is quite a bit simpler than %>%: in the time between the invention of %>% in 2014 and the inclusion of |> in R 4.1.0 in 2021, we gained a better understanding of the pipe. This allowed the base implementation to jettison infrequently used and less important features. + +### Why doesn't `ggplot2` use piping? + +Here's `tidyverse` creator Hadley Wickham's answer (from 2018): -https://forum.posit.co/t/why-cant-ggplot2-use/4372/7 - -## Quarto - -Quarto is a system for writing documents with embedded R code and/or results: +> I think it's worth unpacking this question into a few smaller pieces: +> +> - Should ggplot2 use the pipe? IMO, yes. +> - Could ggplot2 support both the pipe and plus? No +> - Would it be worth it to create a ggplot3 that uses the pipe? No. -- Read this ASAP: https://r4ds.hadley.nz/communicate -- Then use this for reference: https://quarto.org/docs/reference/ - -## Packages - -> This book espouses our philosophy of package development: anything that can be automated, should be automated. Do as little as possible by hand. Do as much as possible with functions. The goal is to spend your time thinking about what you want your package to do rather than thinking about the minutiae of package structure. - -- https://r-pkgs.org/introduction.html#:~:text=This%20book%20espouses,of%20package%20structure. +https://forum.posit.co/t/why-cant-ggplot2-use/4372/7 + +## Quarto + +Quarto is a system for writing documents with embedded R code and/or results: + +- Read this ASAP: https://r4ds.hadley.nz/communicate +- Then use this for reference: https://quarto.org/docs/reference/ -- Read this ASAP: <https://r-pkgs.org/whole-game.html> -- Use the rest of @rpkgs as a reference - -## Git - -94% of respondents to a 2022 Stack Overflow survey reported using git for version control [link](https://survey.stackoverflow.co/2022/#section-version-control-version-control-systems) - -[More details](https://r-pkgs.org/software-development-practices.html#sec-sw-dev-practices-git-github) +## Packages + +> This book espouses our philosophy of package development: anything that can be automated, should be automated. Do as little as possible by hand. Do as much as possible with functions. The goal is to spend your time thinking about what you want your package to do rather than thinking about the minutiae of package structure. + +- https://r-pkgs.org/introduction.html#:~:text=This%20book%20espouses,of%20package%20structure. + +- Read this ASAP: <https://r-pkgs.org/whole-game.html> +- Use the rest of @rpkgs as a reference -- <https://happygitwithr.com/> +## Git -- <https://usethis.r-lib.org/articles/pr-functions.html> +94% of respondents to a 2022 Stack Overflow survey reported using git for version control [link](https://survey.stackoverflow.co/2022/#section-version-control-version-control-systems) -## Spatial data science +[More details](https://r-pkgs.org/software-development-practices.html#sec-sw-dev-practices-git-github) -- @spatialds +- <https://happygitwithr.com/> -## Shiny apps +- <https://usethis.r-lib.org/articles/pr-functions.html> -- Read this first: @masteringshiny -- Use this as a reference: @golembook +## Spatial data science + +- @spatialds + +## Shiny apps + +- Read this first: @masteringshiny +- Use this as a reference: @golembook
      diff --git a/logistic-regression.html b/logistic-regression.html index d374132f5..b1ef6f095 100644 --- a/logistic-regression.html +++ b/logistic-regression.html @@ -465,8 +465,12 @@

      Table of contents

    • 3.10.5 Deviance residuals
    • 3.10.6 Diagnostic plots

    -
  • 3.11 Other link functions for Bernoulli outcomes
    • +
  • 3.11 Other link functions for Bernoulli outcomes +
  • 3.12 Quasibinomial
    • +
  • 3.13 Further reading
    • @@ -487,7 +491,7 @@

    Table of contents

    Published
    -

    Last modified: 2024-05-12: 21:36:37 (PM)

    +

    Last modified: 2024-05-16: 14:01:28 (PM)

    @@ -775,11 +779,11 @@

    -

    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.

    +

    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.

    @@ -827,7 +831,7 @@

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

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

    @@ -836,7 +840,7 @@

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

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

    @@ -965,7 +969,7 @@

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

    Let’s recalculate this result using our shortcut.

    @@ -976,7 +980,7 @@

    Same answer as in Example 3.5!

    @@ -1019,7 +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 \(6.7776\times 10^{-6}\).

    +

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

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

    @@ -1217,7 +1221,7 @@

    @@ -1388,7 +1392,7 @@
    Case 1: rare events
    pi2/pi1     #> [1] 2 odds(pi2)/odds(pi1) -#> [1] 2.02 +#> [1] 2.0204082
    @@ -1438,9 +1442,9 @@

    Odds R

    Solution.

      -

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

      • -

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

      • -

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

      • +

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

      • +

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

      • +

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

    This is the same estimate we calculated in Example 3.7.

    @@ -1767,21 +1771,22 @@

    .by = everything(), outcome = c( rep(1, times = died), - rep(0, times = survived))) - -lm1 = beetles_long |> lm(formula = outcome ~ dose) -f.linear = function(x) predict(lm1, newdata = data.frame(dose = x)) - -range1 = range(beetles$dose) + c(-.2, .2) - -plot2 = - plot1 + - geom_function( - fun = f.linear, - aes(col = "Straight line")) + - labs(colour="Model", size = "") - -plot2 |> print()

    + rep(0, times = survived))) |> + as_tibble() + +lm1 = beetles_long |> lm(formula = outcome ~ dose) +f.linear = function(x) predict(lm1, newdata = data.frame(dose = x)) + +range1 = range(beetles$dose) + c(-.2, .2) + +plot2 = + plot1 + + geom_function( + fun = f.linear, + aes(col = "Straight line")) + + labs(colour="Model", size = "") + +plot2 |> print()
    @@ -1938,16 +1943,22 @@

    Show R code 
    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$n * fitted.values(beetles_glm_grouped)
    +#> 1 2 3 4 5 6 +#> 0.058601026 0.164027869 0.362119006 0.605314906 0.795171772 0.903235819 +#> 7 8 +#> 0.955196107 0.979049344 +predict(beetles_glm_grouped, type = "response") +#> 1 2 3 4 5 6 +#> 0.058601026 0.164027869 0.362119006 0.605314906 0.795171772 0.903235819 +#> 7 8 +#> 0.955196107 0.979049344 +predict(beetles_glm_grouped, type = "link") +#> 1 2 3 4 5 6 +#> -2.77661484 -1.62855893 -0.56617883 0.42766061 1.35638644 2.23370678 +#> 7 8 +#> 3.05962163 3.84441209 + +fit_y = beetles$n * fitted.values(beetles_glm_grouped)

    @@ -2089,9 +2100,9 @@ 

    
 logLik(beetles_glm_grouped)
-#> 'log Lik.' -18.72 (df=2)
+#> 'log Lik.' -18.715135 (df=2)
 logLik(beetles_glm_ungrouped)
-#> 'log Lik.' -186.2 (df=2)
    +#> 'log Lik.' -186.2354 (df=2)
    @@ -2761,19 +2772,22 @@

    wcgs |> summarize( .by = c(age, dibpat), - n = n(), - `p(chd)` = mean(chd69 == "Yes") |> - labelled(label = "CHD Event by 1969"), - `odds(chd)` = `p(chd)`/(1-`p(chd)`), - `logit(chd)` = log(`odds(chd)`) - ) - -chd_grouped_data

    + n = sum(chd69 %in% c("Yes", "No")), + x = sum(chd69 == "Yes")) |> + mutate( + `n - x` = n - x, + `p(chd)` = (x / n) |> + labelled(label = "CHD Event by 1969"), + `odds(chd)` = `p(chd)` / (1 - `p(chd)`), + `logit(chd)` = log(`odds(chd)`) + ) + +chd_grouped_data
    @@ -2864,22 +2878,23 @@

    Log-odds (logit) scal
    Show R code -
    chd_plot_logit = 
-  chd_grouped_data |> 
-  ggplot(
-    aes(
-      x = age, 
-      y = `logit(chd)`, 
-      col = dibpat)
-  ) +
-  geom_point(aes(size = n), alpha = .7) + 
-  scale_size(range = c(1,4)) +
-  geom_line() +
-  theme_bw() +
-  ylab("log{odds(CHD Event by 1969)}") +
-  ggeasy::easy_labs()
-
-print(chd_plot_logit)
    +
    trans_logit = trans_new(
+  name = "logit", 
+  transform = logit, 
+  inverse = expit)
+
+chd_plot_logit = 
+  chd_plot_probs + 
+  scale_y_continuous(
+    trans = trans_logit, # this line changes the vertical spacing
+    name = chd_plot_probs$labels$y,
+    breaks = c(seq(.01, .1, by = .01), .15, .2),
+    minor_breaks = NULL,
+    sec.axis = sec_axis(
+      ~ logit(.),
+      name = "log(odds(CHD Event by 1969))"))
+
+print(chd_plot_logit)
    @@ -3102,18 +3117,18 @@

    odds scale

    Show R code -
    curve_type_A = function(x) 
-{
-  chd_glm_strat |> predict(
-    type = "link",
-    newdata = tibble(age = x, dibpat = "Type A")) |> exp()
-}
-curve_type_B = function(x) 
-{
-  chd_glm_strat |> predict(
-    type = "link",
-    newdata = tibble(age = x, dibpat = "Type B")) |> exp()
-}
+
    # curve_type_A = function(x) 
+# {
+#   chd_glm_strat |> predict(
+#     type = "link",
+#     newdata = tibble(age = x, dibpat = "Type A")) |> exp()
+# }
+# curve_type_B = function(x) 
+# {
+#   chd_glm_strat |> predict(
+#     type = "link",
+#     newdata = tibble(age = x, dibpat = "Type B")) |> exp()
+# }
 
 chd_plot_odds_2 =
   chd_plot_odds +
@@ -3147,18 +3162,18 @@ 
    log-odds (logit) sc
 
    
 
    
 Show R code
-
    curve_type_A = function(x) 
-{
-  chd_glm_strat |> predict(
-    type = "link",
-    newdata = tibble(age = x, dibpat = "Type A"))
-}
-curve_type_B = function(x) 
-{
-  chd_glm_strat |> predict(
-    type = "link",
-    newdata = tibble(age = x, dibpat = "Type B"))
-}
+
    # curve_type_A = function(x) 
+# {
+#   chd_glm_strat |> predict(
+#     type = "link",
+#     newdata = tibble(age = x, dibpat = "Type A"))
+# }
+# curve_type_B = function(x) 
+# {
+#   chd_glm_strat |> predict(
+#     type = "link",
+#     newdata = tibble(age = x, dibpat = "Type B"))
+# }
 
 chd_plot_logit_2 =
   chd_plot_logit +
@@ -3273,12 +3288,38 @@ 
    
-
    Compare with Table 3.8:
    
+
    Compare with Table 3.8.
    
 
    
 
    
 
    
 
    Exercise 3.15 If I give you model 1, how would you get the coefficients of model 2?
    
 
    
+
    
+
    
+
    Theorem 3.15 For the logistic regression model:
    
+
      
+
    • \(Y_i|\tilde{X}_i \ \sim_{⫫}\ \text{Ber}(\pi(\tilde{X}_i))\)
      • 
+
    • \(\pi(\tilde{x}) = \text{expit}\left\{\tilde{x}'\tilde{\beta}\right\}\)
      • 
+
    
+
    Consider two covariate patterns, \(\tilde{x}\) and \(\tilde{x^*}\).
    
+
    The odds ratio comparing these covariate patterns is:
    
+
    \[
+\omega(\tilde{x},\tilde{x^*}) = \text{exp}\left\{(\tilde{x}-\tilde{x^*})^{\top} \tilde{\beta}\right\}
+\]
    
+
    
+
    
+
    
+
    Proof. \[
+\begin{aligned}
+\omega(\tilde{x},\tilde{x^*})
+&= \frac {\omega(Y=1 | \tilde{X}= \tilde{x})} {\omega(Y=1 | \tilde{X}= \tilde{x^*})}
+\\ &= \frac{\text{exp}\left\{\tilde{x}^{\top}\tilde{\beta}\right\}}{\text{exp}\left\{{\tilde{x^*}}^{\top} \tilde{\beta}\right\}}
+\\ &= \text{exp}\left\{\tilde{x}^{\top}\tilde{\beta}- {\tilde{x^*}}^{\top} \tilde{\beta}\right\}
+\\ &= \text{exp}\left\{(\tilde{x}^{\top} - {\tilde{x^*}}^{\top}) \tilde{\beta}\right\}
+\\ &= \text{exp}\left\{{(\tilde{x}- \tilde{x^*})}^{\top} \tilde{\beta}\right\}
+\end{aligned}
+\]
    
+

    @@ -3468,13 +3509,14 @@

    < e = sum(pred_probs_glm1) ) -HL_table |> pander() +library(pander) +HL_table |> pander()
    -+ @@ -3501,7 +3543,7 @@

    <

    - + @@ -3667,8 +3709,8 @@

    < 
    ggplotly(HL_plot)
    -
    - +
    +
    @@ -3819,8 +3861,8 @@ 

    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 \(\pi = 0.5\)).

    @@ -3956,8 +3998,8 @@

    Re 
    wcgs_resid_plot1 |> ggplotly()
    -
    - +
    +
    @@ -4181,10 +4223,156 @@

    in practice, I often get errors about not finding good starting values for the estimation procedure. This is likely because the model is producing fitted probabilities greater than 1.

    When this happens, you can try to fit Poisson regression models instead (we will see those soon!). But then the outcome distribution isn’t quite right, and you won’t get warnings about fitted probabilities greater than 1. In my opinion, the Poisson model for binary outcomes is confusing and not very appealing.

    +

    [0,0.1]2,3052305 129 135.2
    ++++++ + + + + + + + + + + + + + + + + +
    (Intercept)dibpatType BagedibpatType B:age
    -0.21990.13740.0071-0.004194
    +
    + +
    +Show R code +
    library(ggfortify)
+wcgs_glm_logit_link |> autoplot(which = c(1), ncol = 1) + facet_wrap(~dibpat)
+wcgs_glm_identity_link |> autoplot(which = c(1), ncol = 1) + facet_wrap(~dibpat)
    +
    +
    +
    +
    +Figure 3.15: Residuals vs Fitted plot for wcgs models +
    +
    +
    +
    +
    +
    +
    +(a) Logistic link +
    +
    + +
    +
    +
    +
    +
    +
    +
    +
    +(b) Identity link +
    +
    + +
    +
    +
    +
    +
    +
    +
    +
    +
    +Show R code +
    beetles_lm = 
+  beetles_long |> 
+  lm(formula = died ~ dose)
+
+beetles = 
+  beetles |> mutate(
+    resid_logit = beetles_glm_grouped |> resid(type = "response"))
+beetles_glm_grouped |> autoplot(which = c(1), ncol = 1)
+beetles_lm |> autoplot(which = c(1), ncol = 1)
    +
    +
    +
    +
    +Figure 3.16: Residuals vs Fitted plot for BeetleMortality models +
    +
    +
    +
    +
    +
    +
    +(a) Logistic link +
    +
    + +
    +
    +
    +
    +
    +
    +
    +
    +(b) Identity link +
    +
    + +
    +
    +
    +
    +
    +
    +
    +
    +

    3.12 Quasibinomial

    See Hua Zhou’s lecture notes

    +
    +
    +

    3.13 Further reading

    +
      +
    • Hosmer Jr, Lemeshow, and Sturdivant (2013) is a classic textbook on logistic regression
      • +
    @@ -4193,6 +4381,9 @@

    Dobson, Annette J, and Adrian G Barnett. 2018. An Introduction to Generalized Linear Models. 4th ed. CRC press. https://doi.org/10.1201/9781315182780. +
    +Hosmer Jr, David W, Stanley Lemeshow, and Rodney X Sturdivant. 2013. Applied Logistic Regression. John Wiley & Sons. +
    Rosenman, Ray H, Richard J Brand, C David Jenkins, Meyer Friedman, Reuben Straus, and Moses Wurm. 1975. “Coronary Heart Disease in the Western Collaborative Group Study: Final Follow-up Experience of 8 1/2 Years.” JAMA 233 (8): 872–77. https://doi.org/10.1001/jama.1975.03260080034016.
    @@ -4821,2864 +5012,2964 @@

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    @@ -468,6 +468,13 @@

    \[\text{exp}\left\{a+b\right\} = \text{exp}\left\{a\right\} \cdot\text{exp}\left\{b\right\}\]

    +
    +

    Corollary A.2 (exponential of a difference)  

    +
    +

    The exponential of a difference is equal to the quotient of the exponentials of the addends:

    +
    +

    \[\text{exp}\left\{a-b\right\} = \frac{\text{exp}\left\{a\right\}}{\text{exp}\left\{b\right\}}\]

    +

    Theorem A.7 (derivative of logarithm) \[\text{log}'\left\{x\right\} = \frac{1}{x} = x^{-1}\]

    @@ -476,7 +483,7 @@

    -

    Corollary A.2 (Chain rule for logarithms) \[ +

    Corollary A.3 (Chain rule for logarithms) \[ \frac{\partial}{\partial x}\text{log}\left\{f(x)\right\} = \frac{f'(x)}{f(x)} \]

    @@ -1321,199 +1328,210 @@

    ::: -:::{#thm-deriv-log} - -#### derivative of logarithm -$$\dlogf{x} = \frac{1}{x} = x^{-1}$$ -::: - -:::{#thm-deriv-exp} -#### derivative of exponential -$$\dexpf{x} = \expf{x}$$ +:::{#cor-exp-sum} +#### exponential of a difference + +::: notes +The exponential of a difference is equal to the quotient of the exponentials of the addends: +::: + +$$\exp{a-b} = \frac{\exp{a}}{\exp{b}}$$ + ::: ---- +:::{#thm-deriv-log} -:::{#cor-deriv-log-chain} - -#### Chain rule for logarithms -$$ -\deriv{x}\log{f(x)} = \frac{f'(x)}{f(x)} -$$ - +#### derivative of logarithm +$$\dlogf{x} = \frac{1}{x} = x^{-1}$$ +::: + +:::{#thm-deriv-exp} +#### derivative of exponential +$$\dexpf{x} = \expf{x}$$ ::: -::: proof -Apply @thm-chain-rule and @thm-deriv-log. -::: +--- + +:::{#cor-deriv-log-chain} ---- - -## Vector Calculus {#sec-vector-calculus} - -(adapted from @fieller2018basics, §7.2) - -Let $\vx$ and $\vb$ be vectors of length $p$, -or in other words, matrices of length $p \times 1$: - -$$ -\vx = \begin{bmatrix} -x_{1} \\ -x_{2} \\ -\vdots \\ -x_{p} -\end{bmatrix} \\ -$$ - -$$ -\vb = \begin{bmatrix} -\beta_{1} \\ -\beta_{2} \\ -\vdots \\ -\beta_{p} -\end{bmatrix} -$$ - -:::{#def-transpose} -#### Transpose - -The transpose of a row vector is the column vector with the same -sequence of entries: - -$$ -\vx' \equiv \vx^\top \equiv [x_1, x_2, ..., x_p] -$$ - -::: - -:::{#exm-dot-product} -#### Dot product as matrix multiplication - -$$ -\vx\'\vb -= [x_1, x_2, ..., x_p] -\begin{bmatrix} -\beta_{1} \\ -\beta_{2} \\ -\vdots \\ -\beta_{p} -\end{bmatrix} = -x_1\beta_1+x_2\beta_2 +...+x_p \beta_p -$$ -::: - -:::{#thm-transpose-sum} -#### Transpose of a sum -$$(\vx+\vy)\' = \vx\' + \vy\'$$ -::: - ---- - -::: {#def-vector-derivative} -#### Vector derivative -If $f(\vb)$ is a function that takes a vector $\vb$ as input and outputs a scalar, -such as $f(\vb) = x'\vb$, -then: - -$$ -\deriv{ \vb} f(\vb) = -\begin{bmatrix} -\deriv{\beta_1}f(\vb) \\ -\deriv{\beta_2}f(\vb) \\ -\vdots \\ -\deriv{\beta_p}f(\vb) -\end{bmatrix} -$$ -::: +#### Chain rule for logarithms +$$ +\deriv{x}\log{f(x)} = \frac{f'(x)}{f(x)} +$$ + +::: + +::: proof +Apply @thm-chain-rule and @thm-deriv-log. +::: + +--- + +## Vector Calculus {#sec-vector-calculus} + +(adapted from @fieller2018basics, §7.2) + +Let $\vx$ and $\vb$ be vectors of length $p$, +or in other words, matrices of length $p \times 1$: + +$$ +\vx = \begin{bmatrix} +x_{1} \\ +x_{2} \\ +\vdots \\ +x_{p} +\end{bmatrix} \\ +$$ + +$$ +\vb = \begin{bmatrix} +\beta_{1} \\ +\beta_{2} \\ +\vdots \\ +\beta_{p} +\end{bmatrix} +$$ + +:::{#def-transpose} +#### Transpose + +The transpose of a row vector is the column vector with the same +sequence of entries: + +$$ +\vx' \equiv \vx^\top \equiv [x_1, x_2, ..., x_p] +$$ + +::: + +:::{#exm-dot-product} +#### Dot product as matrix multiplication + +$$ +\vx\'\vb += [x_1, x_2, ..., x_p] +\begin{bmatrix} +\beta_{1} \\ +\beta_{2} \\ +\vdots \\ +\beta_{p} +\end{bmatrix} = +x_1\beta_1+x_2\beta_2 +...+x_p \beta_p +$$ +::: + +:::{#thm-transpose-sum} +#### Transpose of a sum +$$(\vx+\vy)\' = \vx\' + \vy\'$$ +::: + +--- + +::: {#def-vector-derivative} +#### Vector derivative +If $f(\vb)$ is a function that takes a vector $\vb$ as input and outputs a scalar, +such as $f(\vb) = x'\vb$, +then: ---- - -:::{#thm-deriv-lincom} - -#### Derivative of a linear combination - -$$ -\deriv{\vb} \vx\' \vb = x +$$ +\deriv{ \vb} f(\vb) = +\begin{bmatrix} +\deriv{\beta_1}f(\vb) \\ +\deriv{\beta_2}f(\vb) \\ +\vdots \\ +\deriv{\beta_p}f(\vb) +\end{bmatrix} $$ - -:::: notes -This looks a lot like non-vector calculus, except that you have to transpose the coefficient. -:::: -::: +::: + +--- + +:::{#thm-deriv-lincom} ---- +#### Derivative of a linear combination -::: proof - -If $f(\beta) = x\'\beta$, then: +$$ +\deriv{\vb} \vx\' \vb = x +$$ -$$ -\deriv{ \beta} x\'\beta= -\begin{bmatrix} -\deriv{\beta_1}(x_1\beta_1+x_2\beta_2 +...+x_p \beta_p ) \\ -\deriv{\beta_2}(x_1\beta_1+x_2\beta_2 +...+x_p \beta_p ) \\ -\vdots \\ -\deriv{\beta_p}(x_1\beta_1+x_2\beta_2 +...+x_p \beta_p ) -\end{bmatrix} -= -\begin{bmatrix} -x_{1} \\ -x_{2} \\ -\vdots \\ -x_{p} -\end{bmatrix} -= \vx -$$ - -::: - ---- - -:::{#thm-deriv-normsq} - -#### Derivative of a vector quadratic - -$$ -\deriv{\vb} \vb'\vb = 2\vb -$$ - -::: - -::: notes -This is like taking the derivative of $x^2$. -::: - ---- - -:::{#thm-quadratic-form} - -If $S$ is a $p\times p$ matrix, then: - -$$ -\deriv{\beta} \beta'S\beta = 2S\beta -$$ - -::: - -::: notes -Again, this is like taking the derivative of $cx^2$ with respect to $x$ in non-vector calculus. -::: - -## Additional resources - -### Calculus - -- @khuri2003advanced - -### Linear Algebra and Vector Calculus - -- @fieller2018basics -- @banerjee2014linear -- @searle2017matrix - -### Numerical Analysis - -- [Hua Zhou](https://hua-zhou.github.io/)'s [lecture notes for "UCLA Biostat 216 - Mathematical Methods for Biostatistics" (2023 Fall)](https://ucla-biostat-216.github.io/2023fall/schedule/schedule.html) +:::: notes +This looks a lot like non-vector calculus, except that you have to transpose the coefficient. +:::: +::: + +--- + +::: proof + +If $f(\beta) = x\'\beta$, then: + +$$ +\deriv{ \beta} x\'\beta= +\begin{bmatrix} +\deriv{\beta_1}(x_1\beta_1+x_2\beta_2 +...+x_p \beta_p ) \\ +\deriv{\beta_2}(x_1\beta_1+x_2\beta_2 +...+x_p \beta_p ) \\ +\vdots \\ +\deriv{\beta_p}(x_1\beta_1+x_2\beta_2 +...+x_p \beta_p ) +\end{bmatrix} += +\begin{bmatrix} +x_{1} \\ +x_{2} \\ +\vdots \\ +x_{p} +\end{bmatrix} += \vx +$$ + +::: + +--- + +:::{#thm-deriv-normsq} + +#### Derivative of a vector quadratic + +$$ +\deriv{\vb} \vb'\vb = 2\vb +$$ + +::: + +::: notes +This is like taking the derivative of $x^2$. +::: + +--- + +:::{#thm-quadratic-form} + +If $S$ is a $p\times p$ matrix, then: + +$$ +\deriv{\beta} \beta'S\beta = 2S\beta +$$ + +::: + +::: notes +Again, this is like taking the derivative of $cx^2$ with respect to $x$ in non-vector calculus. +::: + +## Additional resources + +### Calculus + +- @khuri2003advanced + +### Linear Algebra and Vector Calculus + +- @fieller2018basics +- @banerjee2014linear +- @searle2017matrix + +### Numerical Analysis + +- [Hua Zhou](https://hua-zhou.github.io/)'s [lecture notes for "UCLA Biostat 216 - Mathematical Methods for Biostatistics" (2023 Fall)](https://ucla-biostat-216.github.io/2023fall/schedule/schedule.html)
    diff --git a/probability.html b/probability.html index 5d8b5f029..e43e28a71 100644 --- a/probability.html +++ b/probability.html @@ -346,7 +346,9 @@ B.2 Key probability distributions
  • @@ -370,7 +372,7 @@
    Published
    -

    Last modified: 2024-05-11: 12:47:15 (PM)

    +

    Last modified: 2024-05-16: 13:36:09 (PM)

    @@ -462,12 +464,46 @@
  • vital status (dead vs alive)
    • +
    +
    +

    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:

    +

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

    +
    +
    +

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

    +
    +
    +
    +

    Solution.

    + +
    +
    +
    +

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

    +
      +
    • person*year
      • +
    • time at risk
      • +
    • length of toothbrushing session
      • +
    • miles driven
      • +
    • hours driven
      • +
    • size of pond
      • +
    +

    This concept is similar to the number of trials in a binomial distribution, except that in non-binomial count outcomes, there can be more than one event per trial.

    +

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

    +

    B.2 Key probability distributions

    B.2.1 The Bernoulli distribution

    -

    Definition B.2 (Bernoulli distribution) The Bernoulli distribution family for a random variable \(X\) is defined as:

    +

    Definition B.4 (Bernoulli distribution) The Bernoulli distribution family for a random variable \(X\) is defined as:

    \[ \begin{aligned} \Pr(X=x) &= \mathbb{1}_{x\in \left\{0,1\right\}}\pi^x(1-\pi)^{1-x}\\ @@ -475,24 +511,223 @@ \end{aligned} \]

    -

    +

    B.2.2 The Poisson distribution

    +
    +
    +Figure B.1: “Les Poissons” +
    +
    +
    +
    +
    Siméon Denis Poisson

    +
    +
    +
    +
    +
    +
    Les Poissons

    +
    +
    +
    +
    +
    +
    +
    +
    -

    Definition B.3 (Poisson distribution) \[P(Y=y) = \frac{\lambda^{y} e^{-\lambda}}{y!}\]

    +

    Definition B.5 (Poisson distribution) \[\mathcal{R}(Y) = \left\{0, 1, 2, ...\right\} = \mathbb{N}\]

    +

    \[\text{P}(Y = y) = \frac{\mu^{y} e^{-\mu}}{y!}, y \in \mathbb{N} \tag{B.1}\]

    +
    +

    (see Figure B.2)

    +
    +

    \[\text{P}(Y \le y) = e^{-\mu} \sum_{j=0}^{\left \lfloor{y}\right \rfloor}\frac{\mu^j}{j!} \tag{B.2}\]

    +
    +

    (see Figure B.3)

    +
    +
    +
    +
    +
    Show R code
    library(dplyr)
+pois_dists = tibble(
+  mu = c(0.5, 1, 2, 5, 10, 20)) |> 
+  reframe(
+    .by = mu,
+    x = 0:30
+  ) |> 
+  mutate(
+    `P(X = x)` = dpois(x, lambda = mu),
+    `P(X <= x)` = ppois(x, lambda = mu),
+    mu = factor(mu)
+  )
+
+library(ggplot2)
+library(latex2exp)
+
+plot0 = pois_dists |> 
+  ggplot(
+    aes(
+      x = x,
+      y = `P(X = x)`,
+      fill = mu,
+      col = mu)) +
+  theme(legend.position = "bottom") +
+  labs(
+    fill = latex2exp::TeX("$\\mu$"),
+    col = latex2exp::TeX("$\\mu$"),
+    y = latex2exp::TeX("$\\Pr_{\\mu}(X = x)$"))
+
+plot1 = plot0 + 
+  geom_col(position = "identity", alpha  = .5) +
+  facet_wrap(~mu)
+  # geom_point(alpha = 0.75) +
+  # geom_line(alpha = 0.75)
+print(plot1)
    +
    +
    +
    +Figure B.2: Poisson PMFs, by mean parameter \(\mu\) +
    + +
    +
    +
    +
    +
    +
    +
    +
    Show R code
    library(ggplot2)
+
+plot2 = 
+  plot0 + 
+  geom_step(alpha = 0.75) +
+  aes(y = `P(X <= x)`) + 
+  labs(y = latex2exp::TeX("$\\Pr_{\\mu}(X \\leq x)$"))
+
+print(plot2)
    +
    +
    +
    +Figure B.3: Poisson CDFs +
    + +
    +
    +
    +
    +
    +
    +
    +

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

    +

    Compute:

    +
      +
    • \(\text{P}(X = 4 | \mu = 3.75)\)
      • +
    • \(\text{P}(X \le 7 | \mu = 3.75)\)
      • +
    • \(\text{P}(X > 5 | \mu = 3.75)\)
      • +
    +
    +
    +
    +

    Solution.

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

    Theorem B.1 (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.1.

    +
    +
    +
    +

    Solution. \[ +\begin{aligned} +\text{E}[X] +&= \sum_{x=0}^\infty x \cdot P(X=x)\\ +&= 0 \cdot P(X=0) + \sum_{x=1}^\infty x \cdot P(X=x)\\ +&= 0 + \sum_{x=1}^\infty x \cdot P(X=x)\\ +&= \sum_{x=1}^\infty x \cdot P(X=x)\\ +&= \sum_{x=1}^\infty x \cdot \frac{\lambda^x e^{-\lambda}}{x!}\\ +&= \sum_{x=1}^\infty x \cdot \frac{\lambda^x e^{-\lambda}}{x \cdot (x-1)!} & [\text{definition of factorial ("!") function}]\\ +&= \sum_{x=1}^\infty \frac{\lambda^x e^{-\lambda}}{ (x-1)!}\\ +&= \sum_{x=1}^\infty \frac{(\lambda \cdot \lambda^{x-1}) e^{-\lambda}}{ (x-1)!}\\ +&= \lambda \cdot \sum_{x=1}^\infty \frac{( \lambda^{x-1}) e^{-\lambda}}{ (x-1)!}\\ +&= \lambda \cdot \sum_{y=0}^\infty \frac{( \lambda^{y}) e^{-\lambda}}{ (y)!} &[\text{substituting } y \stackrel{\text{def}}{=}x-1]\\ +&= \lambda \cdot 1 &[\text{because PDFs sum to 1}]\\ +&= \lambda\\ +\end{aligned} +\]

    +
    +

    +B.2.3 Accounting for exposure

    +

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

    +

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

    +

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

    +

    \[\lambda = \frac{\mathbb E[Y|T=t]}{t}\]

    +
    +
    +
    + +
    +
    +Important +
    +
    +
    +

    The 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.

    +
    +
    +
    +

    Theorem B.2 If \(\mu = \lambda\cdot t\), then:

    +

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

    +
    +
    +

    Definition B.6 (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

    +
    +

    Definition B.7 (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.3 If \(Y \sim \text{NegBin}(\mu, \rho)\), then:

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

    B.3 Characteristics of probability distributions

    -

    Definition B.4 (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.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:

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

    -

    Definition B.5 (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.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:

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

    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.6 (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.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:

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

    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:

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

    @@ -500,7 +735,7 @@
    -

    Theorem B.1 (Expectation of the Bernoulli distribution) The expectation of a Bernoulli random variable with parameter \(\pi\) is:

    +

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

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

    @@ -521,14 +756,14 @@

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Springer. https://doi.org/10.1007/978-3-319-19425-7. +

    +Hosmer Jr, David W, Stanley Lemeshow, and Rodney X Sturdivant. 2013. +Applied Logistic Regression. John Wiley & Sons. +
    James, Gareth, Daniela Witten, Trevor Hastie, Robert Tibshirani, et al. 2013. An Introduction to Statistical Learning. Vol. 112. @@ -545,6 +549,10 @@ Seber, George AF, and Alan J Lee. 2012. Linear Regression Analysis. 2nd ed. John Wiley & Sons. https://www.wiley.com/en-us/Linear+Regression+Analysis%2C+2nd+Edition-p-9781118274422.
    +
    +Selvin, Steve. 2001. Epidemiologic Analysis: A Case-Oriented +Approach. Oxford University Press. +
    Van Buuren, Stef. 2018. Flexible Imputation of Missing Data. CRC press. https://stefvanbuuren.name/fimd/. diff --git a/search.json b/search.json index e8fc22c8e..50f87ada9 100644 --- a/search.json +++ b/search.json @@ -24,7 +24,7 @@ "href": "index.html#other-resources", "title": "Regression Models for Epidemiology", "section": "Other resources", - "text": "Other resources\nThese notes represent my still-developing perspective on regression models in epidemiology. Many other statisticians and epidemiologists have published their own perspectives, and I encourage you to explore your many options and find ones that resonate with you. I have attempted to cite my sources throughout these notes.\nHere are some additional resources:\n\nDunn, Smyth, et al. (2018) is a recent textbook on GLMs. It doesn’t cover time-to-event models, and it doesn’t use the modern tidyverse packages (ggplot2, dplyr, etc.), but otherwise it seems great.\nMoore (2016) is a recent textbook on survival analysis. It also doesn’t use the tidyverse, but otherwise seems great.\nHarrell (2015) is another popular textbook. It uses ggplot2 but not dplyr, and covers GLMs and survival analysis. An abbreviated but continuously updated version with audio clips is available at https://hbiostat.org/rmsc/.\nFox (2015) is another standard text.\nMcCullagh and Nelder (1989) is a classic, theoretical textbook on GLMs\nDalgaard (2008) covers GLMs and survival analysis at an applied level, using base R\nVittinghoff et al. (2012) covers GLMs, survival analysis, and causal inference, using Stata\nFaraway (2016) has GLMs but not survival analysis\nhttps://ucla-biostat-200c-2020spring.github.io/schedule/schedule.html provides course notes for “Biostat 200C - Methods in Biostatistics C” at UCLA, which is at the Biostatistics MS level.\nhttps://online.stat.psu.edu/stat504/book/ provides course notes for “STAT 504 - Analysis of Discrete Data” at Penn State University. It includes logistic regression and Poisson regression, as well as 2-way tables and other related topics, and includes SAS code.\nNahhas (n.d.) is currently in-development", + "text": "Other resources\nThese notes represent my still-developing perspective on regression models in epidemiology. Many other statisticians and epidemiologists have published their own perspectives, and I encourage you to explore your many options and find ones that resonate with you. I have attempted to cite my sources throughout these notes.\nHere are some additional resources:\n\nDunn, Smyth, et al. (2018) is a recent textbook on GLMs. It doesn’t cover time-to-event models, and it doesn’t use the modern tidyverse packages (ggplot2, dplyr, etc.), but otherwise it seems great.\nMoore (2016) is a recent textbook on survival analysis. It also doesn’t use the tidyverse, but otherwise seems great.\nHarrell (2015) is another popular textbook. It uses ggplot2 but not dplyr, and covers GLMs and survival analysis. An abbreviated but continuously updated version with audio clips is available at https://hbiostat.org/rmsc/.\nFox (2015) is another standard text.\nMcCullagh and Nelder (1989) is a classic, theoretical textbook on GLMs\nDalgaard (2008) covers GLMs and survival analysis at an applied level, using base R\nVittinghoff et al. (2012) covers GLMs, survival analysis, and causal inference, using Stata\nFaraway (2016) has GLMs but not survival analysis\nSelvin (2001) provides worked-out examples of applications for a wide range of statistical analysis techniques.\nhttps://ucla-biostat-200c-2020spring.github.io/schedule/schedule.html provides course notes for “Biostat 200C - Methods in Biostatistics C” at UCLA, which is at the Biostatistics MS level.\nhttps://online.stat.psu.edu/stat504/book/ provides course notes for “STAT 504 - Analysis of Discrete Data” at Penn State University. It includes logistic regression and Poisson regression, as well as 2-way tables and other related topics, and includes SAS code.\nNahhas (n.d.) is currently in-development", "crumbs": [ "Preface" ] @@ -34,7 +34,7 @@ "href": "index.html#license", "title": "Regression Models for Epidemiology", "section": "License", - "text": "License\nThis book is licensed to you under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.\nThe code samples in this book are licensed under Creative Commons CC0 1.0 Universal (CC0 1.0), i.e. public domain.\n\n\n\n\n\n\nDalgaard, Peter. 2008. Introductory Statistics with r. New York, NY: Springer New York. https://link.springer.com/book/10.1007/978-0-387-79054-1.\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\nDunn, Peter K, Gordon K Smyth, et al. 2018. Generalized Linear Models with Examples in r. Vol. 53. Springer. https://link.springer.com/book/10.1007/978-1-4419-0118-7.\n\n\nFaraway, Julian J. 2016. Extending the Linear Model with r: Generalized Linear, Mixed Effects and Nonparametric Regression Models. 2nd ed. Chapman; Hall/CRC. https://doi.org/10.1201/9781315382722.\n\n\nFox, John. 2015. Applied Regression Analysis and Generalized Linear Models. Sage publications.\n\n\nHarrell, Frank E. 2015. Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis. 2nd ed. Springer. https://doi.org/10.1007/978-3-319-19425-7.\n\n\nMcCullagh, Peter, and J. A. Nelder. 1989. Generalized Linear Models. 2nd ed. Routledge. https://www.utstat.toronto.edu/~brunner/oldclass/2201s11/readings/glmbook.pdf.\n\n\nMoore, Dirk F. 2016. Applied Survival Analysis Using r. Vol. 473. Springer.\n\n\nNahhas, Ramzi W. n.d. Introduction to Regression Methods for Public Health Using r. CRC Press. https://www.bookdown.org/rwnahhas/RMPH/.\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.", + "text": "License\nThis book is licensed to you under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.\nThe code samples in this book are licensed under Creative Commons CC0 1.0 Universal (CC0 1.0), i.e. public domain.\n\n\n\n\n\n\nDalgaard, Peter. 2008. Introductory Statistics with r. New York, NY: Springer New York. https://link.springer.com/book/10.1007/978-0-387-79054-1.\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\nDunn, Peter K, Gordon K Smyth, et al. 2018. Generalized Linear Models with Examples in r. Vol. 53. Springer. https://link.springer.com/book/10.1007/978-1-4419-0118-7.\n\n\nFaraway, Julian J. 2016. Extending the Linear Model with r: Generalized Linear, Mixed Effects and Nonparametric Regression Models. 2nd ed. Chapman; Hall/CRC. https://doi.org/10.1201/9781315382722.\n\n\nFox, John. 2015. Applied Regression Analysis and Generalized Linear Models. Sage publications.\n\n\nHarrell, Frank E. 2015. Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis. 2nd ed. Springer. https://doi.org/10.1007/978-3-319-19425-7.\n\n\nMcCullagh, Peter, and J. A. Nelder. 1989. Generalized Linear Models. 2nd ed. Routledge. https://www.utstat.toronto.edu/~brunner/oldclass/2201s11/readings/glmbook.pdf.\n\n\nMoore, Dirk F. 2016. Applied Survival Analysis Using r. Vol. 473. Springer.\n\n\nNahhas, Ramzi W. n.d. Introduction to Regression Methods for Public Health Using r. CRC Press. https://www.bookdown.org/rwnahhas/RMPH/.\n\n\nSelvin, Steve. 2001. Epidemiologic Analysis: A Case-Oriented Approach. Oxford University Press.\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": [ "Preface" ] @@ -84,7 +84,7 @@ "href": "glms.html", "title": "Generalized Linear Models", "section": "", - "text": "This section is primarily adapted starting from the textbook “An Introduction to Generalized Linear Models” (4th edition, 2018) by Annette J. Dobson and Adrian G. Barnett:\nhttps://doi.org/10.1201/9781315182780\nThe type of predictive model one uses depends on several issues; one is the type of response.\n\nMeasured values such as quantity of a protein, age, weight usually can be handled in an ordinary linear regression model, possibly after a log transformation.\nPatient survival, which may be censored, calls for a different method (survival analysis, Cox regression).\nIf the response is binary, then can we use logistic regression models\nIf the response is a count, we can use Poisson regression\nIf the count has a higher variance than is consistent with the Poisson, we can use a negative binomial or over-dispersed Poisson\nOther forms of response can generate other types of generalized linear models\n\nWe need a linear predictor of the same form as in linear regression βx. In theory, such a linear predictor can generate any type of number as a prediction, positive, negative, or zero\nWe choose a suitable distribution for the type of data we are predicting (normal for any number, gamma for positive numbers, binomial for binary responses, Poisson for counts)\nWe create a link function which maps the mean of the distribution onto the set of all possible linear prediction results, which is the whole real line (-∞, ∞). The inverse of the link function takes the linear predictor to the actual prediction.\n\nOrdinary linear regression has identity link (no transformation by the link function) and uses the normal distribution\nIf one is predicting an inherently positive quantity, one may want to use the log link since ex is always positive.\nAn alternative to using a generalized linear model with a log link, is to transform the data using the log. This is a device that works well with measurement data and may be usable in other cases, but it cannot be used for 0/1 data or for count data that may be 0.\n\n\nR glm() Families\n\n\n\n\n\n\nFamily\nLinks\n\n\n\n\ngaussian\nidentity, log, inverse\n\n\nbinomial\nlogit, probit, cauchit, log, cloglog\n\n\ngamma\ninverse, identity, log\n\n\ninverse.gaussian\n1/mu^2, inverse, identity, log\n\n\nPoisson\nlog, identity, sqrt\n\n\nquasi\nidentity, logit, probit, cloglog, inverse, log, 1/mu^2 and sqrt\n\n\nquasibinomial\nlogit, probit, identity, cloglog, inverse, log, 1/mu^2 and sqrt\n\n\nquasipoisson\nlog, identity, logit, probit, cloglog, inverse, 1/mu^2 and sqrt\n\n\n\n\nR glm() Link Functions; \\(\\eta = X\\beta = g(\\mu)\\)\n\n\n\n\n\n\n\n\n\nName\nDomain\nRange\nLink Function\nInverse Link Function\n\n\n\n\nidentity\n\\((-\\infty, \\infty)\\)\n\\((-\\infty, \\infty)\\)\n\\(\\eta = \\mu\\)\n\\(\\mu = \\eta\\)\n\n\nlog\n\\((0,\\infty)\\)\n\\((-\\infty, \\infty)\\)\n\\(\\eta = \\text{log}\\left\\{\\mu\\right\\}\\)\n\\(\\mu = \\text{exp}\\left\\{\\eta\\right\\}\\)\n\n\ninverse\n\\((0, \\infty)\\)\n\\((0,\\infty)\\)\n\\(\\eta = 1/\\mu\\)\n\\(\\mu = 1/\\eta\\)\n\n\nlogit\n\\((0,1)\\)\n\\((-\\infty, \\infty)\\)\n\\(\\eta = \\text{log}\\left\\{\\mu/(1-\\mu)\\right\\}\\)\n\\(\\mu = \\text{exp}\\left\\{\\eta\\right\\}/(1+\\text{exp}\\left\\{\\eta\\right\\})\\)\n\n\nprobit\n\\((0,1)\\)\n\\((-\\infty, \\infty)\\)\n\\(\\eta = \\Phi^{-1}(\\mu)\\)\n\\(\\mu = \\Phi(\\eta)\\)\n\n\ncloglog\n\\((0,1)\\)\n\\((-\\infty, \\infty)\\)\n\\(\\eta = \\text{log}\\left\\{-\\text{log}\\left\\{1-\\mu\\right\\}\\right\\}\\)\n\\(\\mu = {1-\\text{exp}\\left\\{-\\text{exp}\\left\\{\\eta\\right\\}\\right\\}}\\)\n\n\n1/mu^2\n\\((0,\\infty)\\)\n\\((0, \\infty)\\)\n\\(\\eta = 1/\\mu^2\\)\n\\(\\mu = 1/\\sqrt{\\eta}\\)\n\n\nsqrt\n\\((0,\\infty)\\)\n\\((0,\\infty)\\)\n\\(\\eta = \\sqrt{\\mu}\\)\n\\(\\mu = \\eta^2\\)", + "text": "This section is primarily adapted starting from the textbook “An Introduction to Generalized Linear Models” (4th edition, 2018) by Annette J. Dobson and Adrian G. Barnett:\nhttps://doi.org/10.1201/9781315182780\nThe type of predictive model one uses depends on several issues; one is the type of response.\n\nMeasured values such as quantity of a protein, age, weight usually can be handled in an ordinary linear regression model, possibly after a log transformation.\nPatient survival, which may be censored, calls for a different method (survival analysis, Cox regression).\nIf the response is binary, then can we use logistic regression models\nIf the response is a count, we can use Poisson regression\nIf the count has a higher variance than is consistent with the Poisson, we can use a negative binomial or over-dispersed Poisson\nOther forms of response can generate other types of generalized linear models\n\nWe need a linear predictor of the same form as in linear regression \\(\\beta x\\). In theory, such a linear predictor can generate any type of number as a prediction, positive, negative, or zero\nWe choose a suitable distribution for the type of data we are predicting (normal for any number, gamma for positive numbers, binomial for binary responses, Poisson for counts)\nWe create a link function which maps the mean of the distribution onto the set of all possible linear prediction results, which is the whole real line (\\(-\\infty, \\infty\\)). The inverse of the link function takes the linear predictor to the actual prediction.\n\nOrdinary linear regression has identity link (no transformation by the link function) and uses the normal distribution\nIf one is predicting an inherently positive quantity, one may want to use the log link since ex is always positive.\nAn alternative to using a generalized linear model with a log link, is to transform the data using the log. This is a device that works well with measurement data and may be usable in other cases, but it cannot be used for 0/1 data or for count data that may be 0.\n\n\nR glm() Families\n\n\n\n\n\n\nFamily\nLinks\n\n\n\n\ngaussian\nidentity, log, inverse\n\n\nbinomial\nlogit, probit, cauchit, log, cloglog\n\n\ngamma\ninverse, identity, log\n\n\ninverse.gaussian\n1/mu^2, inverse, identity, log\n\n\nPoisson\nlog, identity, sqrt\n\n\nquasi\nidentity, logit, probit, cloglog, inverse, log, 1/mu^2 and sqrt\n\n\nquasibinomial\nlogit, probit, identity, cloglog, inverse, log, 1/mu^2 and sqrt\n\n\nquasipoisson\nlog, identity, logit, probit, cloglog, inverse, 1/mu^2 and sqrt\n\n\n\n\nR glm() Link Functions; \\(\\eta = X\\beta = g(\\mu)\\)\n\n\n\n\n\n\n\n\n\nName\nDomain\nRange\nLink Function\nInverse Link Function\n\n\n\n\nidentity\n\\((-\\infty, \\infty)\\)\n\\((-\\infty, \\infty)\\)\n\\(\\eta = \\mu\\).\n\\(\\mu = \\eta\\)\n\n\nlog\n\\((0,\\infty)\\)\n\\((-\\infty, \\infty)\\)\n\\(\\eta = \\text{log}\\left\\{\\mu\\right\\}\\)\n\\(\\mu = \\text{exp}\\left\\{\\eta\\right\\}\\)\n\n\ninverse\n\\((0, \\infty)\\)\n\\((0,\\infty)\\)\n\\(\\eta = 1/\\mu\\)\n\\(\\mu = 1/\\eta\\)\n\n\nlogit\n\\((0,1)\\)\n\\((-\\infty, \\infty)\\)\n\\(\\eta = \\text{log}\\left\\{\\mu/(1-\\mu)\\right\\}\\)\n\\(\\mu = \\text{exp}\\left\\{\\eta\\right\\}/(1+\\text{exp}\\left\\{\\eta\\right\\})\\)\n\n\nprobit\n\\((0,1)\\)\n\\((-\\infty, \\infty)\\)\n\\(\\eta = \\Phi^{-1}(\\mu)\\)\n\\(\\mu = \\Phi(\\eta)\\)\n\n\ncloglog\n\\((0,1)\\)\n\\((-\\infty, \\infty)\\)\n\\(\\eta = \\text{log}\\left\\{-\\text{log}\\left\\{1-\\mu\\right\\}\\right\\}\\)\n\\(\\mu = {1-\\text{exp}\\left\\{-\\text{exp}\\left\\{\\eta\\right\\}\\right\\}}\\)\n\n\n1/mu^2\n\\((0,\\infty)\\)\n\\((0, \\infty)\\)\n\\(\\eta = 1/\\mu^2\\)\n\\(\\mu = 1/\\sqrt{\\eta}\\)\n\n\nsqrt\n\\((0,\\infty)\\)\n\\((0,\\infty)\\)\n\\(\\eta = \\sqrt{\\mu}\\)\n\\(\\mu = \\eta^2\\)", "crumbs": [ "Generalized Linear Models" ] @@ -116,7 +116,7 @@ "href": "Linear-models-overview.html#sec-understand-LMs", "title": "\n2  Linear (Gaussian) Models\n", "section": "\n2.2 Understanding Gaussian Linear Regression Models", - "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\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_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 2.4: 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 2.5: 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_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 2.6: 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#> -1773.32184 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", "crumbs": [ "Generalized Linear Models", "2  Linear (Gaussian) Models" @@ -127,7 +127,7 @@ "href": "Linear-models-overview.html#sec-est-LMs", "title": "\n2  Linear (Gaussian) Models\n", "section": "\n2.3 Estimating Linear Models via Maximum Likelihood", - "text": "2.3 Estimating Linear Models via Maximum Likelihood\n\n2.3.1 Likelihood, log-likelihood, and score functions for linear regression\n\nIn EPI 203 and Appendix E, we learned how to fit outcome-only models of the form \\(p(X=x|\\theta)\\) to iid data \\(\\tilde{x}= (x_1,…,x_n)\\) using maximum likelihood estimation.\nNow, we apply the same procedure to linear regression models:\n\n\\[\n\\mathcal L(\\tilde{y}|\\mathbf{x},\\beta, \\sigma^2) =\n\\prod_{i=1}^n (2\\pi\\sigma^2)^{-1/2}\n\\text{exp}\\left\\{-\\frac{1}{2\\sigma^2}(y_i - \\tilde{x_i}'\\beta)^2\\right\\}\n\\tag{2.4}\\]\n\\[\n\\ell(\\tilde{y}|\\mathbf{x},\\beta, \\sigma^2)\n= -\\frac{n}{2}\\text{log}\\left\\{\\sigma^2\\right\\} -\n\\frac{1}{2\\sigma^2}\\sum_{i=1}^n (y_i - \\tilde{x_i}' \\beta)^2\n\\tag{2.5}\\]\n\\[\n\\ell'_{\\beta}(\\tilde{y}|\\mathbf{x},\\beta, \\sigma^2)\n= -\n\\frac{1}{2\\sigma^2}\\frac{\\partial}{\\partial \\beta}\n\\left(\\sum_{i=1}^n (y_i - \\tilde{x_i}^{\\top} \\beta)^2\\right)\n\\tag{2.6}\\]\n\n\nLet’s switch to matrix-vector notation:\n\n\\[\n\\sum_{i=1}^n (y_i - \\tilde{x}_i^{\\top} \\tilde{\\beta})^2\n= (\\tilde{y}- \\mathbf{X}\\tilde{\\beta})'(\\tilde{y}- \\mathbf{X}\\tilde{\\beta})\n\\]\n\nSo\n\\[\n\\begin{aligned}\n(\\tilde{y}- \\mathbf{X}\\tilde{\\beta})'(\\tilde{y}- \\mathbf{X}\\tilde{\\beta})\n&= (\\tilde{y}' - \\tilde{\\beta}'X')(\\tilde{y}- \\mathbf{X}\\tilde{\\beta})\n\\\\ &= y'y - \\tilde{\\beta}'X'y - y'\\mathbf{X}\\tilde{\\beta}+\\tilde{\\beta}'\\mathbf{X}'\\mathbf{X}\\beta\n\\\\ &= y'y - 2y'\\mathbf{X}\\beta +\\beta'\\mathbf{X}'\\mathbf{X}\\beta\n\\end{aligned}\n\\]\n\n2.3.2 Deriving the linear regression score function\n\nWe will use some results from vector calculus:\n\n\\[\n\\begin{aligned}\n\\frac{\\partial}{\\partial \\beta}\\left(\\sum_{i=1}^n (y_i - x_i' \\beta)^2\\right)\n &= \\frac{\\partial}{\\partial \\beta}(\\tilde{y}- X\\beta)'(\\tilde{y}- X\\beta)\n\\\\ &= \\frac{\\partial}{\\partial \\beta} (y'y - 2y'X\\beta +\\beta'X'X\\beta)\n\\\\ &= (- 2X'y +2X'X\\beta)\n\\\\ &= - 2X'(y - X\\beta)\n\\\\ &= - 2X'(y - \\mathbb{E}[y])\n\\\\ &= - 2X' \\varepsilon(y)\n\\end{aligned}\n\\tag{2.7}\\]\n\nSo if \\(\\ell(\\beta,\\sigma^2) =0\\), then\n\\[\n\\begin{aligned}\n0 &= (- 2X'y +2X'X\\beta)\\\\\n2X'y &= 2X'X\\beta\\\\\nX'y &= X'X\\beta\\\\\n(X'X)^{-1}X'y &= \\beta\n\\end{aligned}\n\\]\n\nThe second derivative matrix \\(\\ell_{\\beta, \\beta'} ''(\\beta, \\sigma^2;\\mathbf X,\\tilde{y})\\) is negative definite at \\(\\beta = (X'X)^{-1}X'y\\), so \\(\\hat \\beta_{ML} = (X'X)^{-1}X'y\\) is the MLE for \\(\\beta\\).\n\nSimilarly (not shown):\n\\[\n\\hat\\sigma^2_{ML} = \\frac{1}{n} (Y-X\\hat\\beta)'(Y-X\\hat\\beta)\n\\]\nAnd\n\\[\n\\begin{aligned}\n\\mathcal I_{\\beta} &= E[-\\ell_{\\beta, \\beta'} ''(Y|X,\\beta, \\sigma^2)]\\\\\n&= \\frac{1}{\\sigma^2}X'X\n\\end{aligned}\n\\]\n\nSo:\n\\[\nVar(\\hat \\beta) \\approx (\\mathcal I_{\\beta})^{-1} = \\sigma^2 (X'X)^{-1}\n\\]\nand\n\\[\n\\hat\\beta \\dot \\sim N(\\beta, \\mathcal I_{\\beta}^{-1})\n\\]\n\nThese are all results you have hopefully seen before.\n\n\nIn the Gaussian linear regression case, we also have exact results:\n\\[\n\\frac{\\hat\\beta_j}{\\hat{\\text{se}}\\left(\\hat\\beta_j\\right)} \\ \\sim \\ t_{n-p}\n\\]\n\nIn our model 2 above, \\(\\hat{\\mathcal{I}}(\\beta)\\) is:\n\nShow R codebw_lm2 |> vcov()\n#> (Intercept) sexmale age sexmale:age\n#> (Intercept) 1353968 -1353968 -34871.0 34871.0\n#> sexmale -1353968 2596387 34871.0 -67211.0\n#> age -34871 34871 899.9 -899.9\n#> sexmale:age 34871 -67211 -899.9 1743.5\n\n\nIf we take the square roots of the diagonals, we get the standard errors listed in the model output:\n\nShow R code\nbw_lm2 |> vcov() |> diag() |> sqrt()\n#> (Intercept) sexmale age sexmale:age \n#> 1163.60 1611.33 30.00 41.76\n\n\n\nShow R codebw_lm2 |> parameters() |> print_md()\n\n\nTable 2.7: Estimated model for birthweight data with interaction term\n\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 (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\n\n\n\nSo we can do confidence intervals, hypothesis tests, and p-values exactly as in the one-variable case we looked at previously.\n\n2.3.3 Residual Standard Deviation\n\n\\(\\hat \\sigma\\) represents an estimate of the Residual Standard Deviation parameter, \\(\\sigma\\). We can extract \\(\\hat \\sigma\\) from the fitted model, using the sigma() function:\n\n\nShow R codesigma(bw_lm2)\n#> [1] 180.6\n\n\n\n\n\\(\\sigma\\) is NOT “Residual standard error”\n\nIn the summary.lm() output, this estimate is labeled as \"Residual standard error\":\n\n\nShow R codesummary(bw_lm2)\n#> \n#> Call:\n#> lm(formula = weight ~ sex + age + sex:age, data = bw)\n#> \n#> Residuals:\n#> Min 1Q Median 3Q Max \n#> -246.7 -138.1 -39.1 176.6 274.3 \n#> \n#> Coefficients:\n#> Estimate Std. Error t value Pr(>|t|) \n#> (Intercept) -2141.7 1163.6 -1.84 0.08057 . \n#> sexmale 873.0 1611.3 0.54 0.59395 \n#> age 130.4 30.0 4.35 0.00031 ***\n#> sexmale:age -18.4 41.8 -0.44 0.66389 \n#> ---\n#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n#> \n#> Residual standard error: 181 on 20 degrees of freedom\n#> Multiple R-squared: 0.643, Adjusted R-squared: 0.59 \n#> F-statistic: 12 on 3 and 20 DF, p-value: 0.000101\n\n\n\n\nHowever, this is a misnomer:\n\n\nShow R codelibrary(printr) # captures ? documentation\n?stats::sigma\n\n\n\n\nsigma\nR Documentation\n\nExtract Residual Standard Deviation 'Sigma'\n\nDescription\n\nExtract the estimated standard deviation of the errors, the\n“residual standard deviation” (misnamed also\n“residual standard error”, e.g., in\nsummary.lm()'s output, from a fitted model).\n\nMany classical statistical models have a scale parameter,\ntypically the standard deviation of a zero-mean normal (or Gaussian)\nrandom variable which is denoted as \\sigma.\nsigma(.) extracts the estimated parameter from a fitted\nmodel, i.e., \\hat\\sigma.\n\n\n\nNote\n\nThe misnomer “Residual standard error” has been part of\ntoo many R (and S) outputs to be easily changed there.", + "text": "2.3 Estimating Linear Models via Maximum Likelihood\n\n2.3.1 Likelihood, log-likelihood, and score functions for linear regression\n\nIn EPI 203 and Appendix E, we learned how to fit outcome-only models of the form \\(p(X=x|\\theta)\\) to iid data \\(\\tilde{x}= (x_1,…,x_n)\\) using maximum likelihood estimation.\nNow, we apply the same procedure to linear regression models:\n\n\\[\n\\mathcal L(\\tilde{y}|\\mathbf{x},\\beta, \\sigma^2) =\n\\prod_{i=1}^n (2\\pi\\sigma^2)^{-1/2}\n\\text{exp}\\left\\{-\\frac{1}{2\\sigma^2}(y_i - \\tilde{x_i}'\\beta)^2\\right\\}\n\\tag{2.4}\\]\n\\[\n\\ell(\\tilde{y}|\\mathbf{x},\\beta, \\sigma^2)\n= -\\frac{n}{2}\\text{log}\\left\\{\\sigma^2\\right\\} -\n\\frac{1}{2\\sigma^2}\\sum_{i=1}^n (y_i - \\tilde{x_i}' \\beta)^2\n\\tag{2.5}\\]\n\\[\n\\ell'_{\\beta}(\\tilde{y}|\\mathbf{x},\\beta, \\sigma^2)\n= -\n\\frac{1}{2\\sigma^2}\\frac{\\partial}{\\partial \\beta}\n\\left(\\sum_{i=1}^n (y_i - \\tilde{x_i}^{\\top} \\beta)^2\\right)\n\\tag{2.6}\\]\n\n\nLet’s switch to matrix-vector notation:\n\n\\[\n\\sum_{i=1}^n (y_i - \\tilde{x}_i^{\\top} \\tilde{\\beta})^2\n= (\\tilde{y}- \\mathbf{X}\\tilde{\\beta})'(\\tilde{y}- \\mathbf{X}\\tilde{\\beta})\n\\]\n\nSo\n\\[\n\\begin{aligned}\n(\\tilde{y}- \\mathbf{X}\\tilde{\\beta})'(\\tilde{y}- \\mathbf{X}\\tilde{\\beta})\n&= (\\tilde{y}' - \\tilde{\\beta}'X')(\\tilde{y}- \\mathbf{X}\\tilde{\\beta})\n\\\\ &= y'y - \\tilde{\\beta}'X'y - y'\\mathbf{X}\\tilde{\\beta}+\\tilde{\\beta}'\\mathbf{X}'\\mathbf{X}\\beta\n\\\\ &= y'y - 2y'\\mathbf{X}\\beta +\\beta'\\mathbf{X}'\\mathbf{X}\\beta\n\\end{aligned}\n\\]\n\n2.3.2 Deriving the linear regression score function\n\nWe will use some results from vector calculus:\n\n\\[\n\\begin{aligned}\n\\frac{\\partial}{\\partial \\beta}\\left(\\sum_{i=1}^n (y_i - x_i' \\beta)^2\\right)\n &= \\frac{\\partial}{\\partial \\beta}(\\tilde{y}- X\\beta)'(\\tilde{y}- X\\beta)\n\\\\ &= \\frac{\\partial}{\\partial \\beta} (y'y - 2y'X\\beta +\\beta'X'X\\beta)\n\\\\ &= (- 2X'y +2X'X\\beta)\n\\\\ &= - 2X'(y - X\\beta)\n\\\\ &= - 2X'(y - \\mathbb{E}[y])\n\\\\ &= - 2X' \\varepsilon(y)\n\\end{aligned}\n\\tag{2.7}\\]\n\nSo if \\(\\ell(\\beta,\\sigma^2) =0\\), then\n\\[\n\\begin{aligned}\n0 &= (- 2X'y +2X'X\\beta)\\\\\n2X'y &= 2X'X\\beta\\\\\nX'y &= X'X\\beta\\\\\n(X'X)^{-1}X'y &= \\beta\n\\end{aligned}\n\\]\n\nThe second derivative matrix \\(\\ell_{\\beta, \\beta'} ''(\\beta, \\sigma^2;\\mathbf X,\\tilde{y})\\) is negative definite at \\(\\beta = (X'X)^{-1}X'y\\), so \\(\\hat \\beta_{ML} = (X'X)^{-1}X'y\\) is the MLE for \\(\\beta\\).\n\nSimilarly (not shown):\n\\[\n\\hat\\sigma^2_{ML} = \\frac{1}{n} (Y-X\\hat\\beta)'(Y-X\\hat\\beta)\n\\]\nAnd\n\\[\n\\begin{aligned}\n\\mathcal I_{\\beta} &= E[-\\ell_{\\beta, \\beta'} ''(Y|X,\\beta, \\sigma^2)]\\\\\n&= \\frac{1}{\\sigma^2}X'X\n\\end{aligned}\n\\]\n\nSo:\n\\[\nVar(\\hat \\beta) \\approx (\\mathcal I_{\\beta})^{-1} = \\sigma^2 (X'X)^{-1}\n\\]\nand\n\\[\n\\hat\\beta \\dot \\sim N(\\beta, \\mathcal I_{\\beta}^{-1})\n\\]\n\nThese are all results you have hopefully seen before.\n\n\nIn the Gaussian linear regression case, we also have exact results:\n\\[\n\\frac{\\hat\\beta_j}{\\hat{\\text{se}}\\left(\\hat\\beta_j\\right)} \\ \\sim \\ t_{n-p}\n\\]\n\nIn our model 2 above, \\(\\hat{\\mathcal{I}}(\\beta)\\) is:\n\nShow R codebw_lm2 |> vcov()\n#> (Intercept) sexmale age sexmale:age\n#> (Intercept) 1353968 -1353968 -34871.0 34871.0\n#> sexmale -1353968 2596387 34871.0 -67211.0\n#> age -34871 34871 899.9 -899.9\n#> sexmale:age 34871 -67211 -899.9 1743.5\n\n\nIf we take the square roots of the diagonals, we get the standard errors listed in the model output:\n\nShow R code\nbw_lm2 |> vcov() |> diag() |> sqrt()\n#> (Intercept) sexmale age sexmale:age \n#> 1163.60 1611.33 30.00 41.76\n\n\n\nShow R codebw_lm2 |> parameters() |> print_md()\n\n\nTable 2.8: Estimated model for birthweight data with interaction term\n\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 (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\n\n\n\nSo we can do confidence intervals, hypothesis tests, and p-values exactly as in the one-variable case we looked at previously.\n\n2.3.3 Residual Standard Deviation\n\n\\(\\hat \\sigma\\) represents an estimate of the Residual Standard Deviation parameter, \\(\\sigma\\). We can extract \\(\\hat \\sigma\\) from the fitted model, using the sigma() function:\n\n\nShow R codesigma(bw_lm2)\n#> [1] 180.6\n\n\n\n\n\\(\\sigma\\) is NOT “Residual standard error”\n\nIn the summary.lm() output, this estimate is labeled as \"Residual standard error\":\n\n\nShow R codesummary(bw_lm2)\n#> \n#> Call:\n#> lm(formula = weight ~ sex + age + sex:age, data = bw)\n#> \n#> Residuals:\n#> Min 1Q Median 3Q Max \n#> -246.7 -138.1 -39.1 176.6 274.3 \n#> \n#> Coefficients:\n#> Estimate Std. Error t value Pr(>|t|) \n#> (Intercept) -2141.7 1163.6 -1.84 0.08057 . \n#> sexmale 873.0 1611.3 0.54 0.59395 \n#> age 130.4 30.0 4.35 0.00031 ***\n#> sexmale:age -18.4 41.8 -0.44 0.66389 \n#> ---\n#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n#> \n#> Residual standard error: 181 on 20 degrees of freedom\n#> Multiple R-squared: 0.643, Adjusted R-squared: 0.59 \n#> F-statistic: 12 on 3 and 20 DF, p-value: 0.000101\n\n\n\n\nHowever, this is a misnomer:\n\n\nShow R codelibrary(printr) # captures ? documentation\n?stats::sigma\n\n\n\n\nsigma\nR Documentation\n\nExtract Residual Standard Deviation 'Sigma'\n\nDescription\n\nExtract the estimated standard deviation of the errors, the\n“residual standard deviation” (misnamed also\n“residual standard error”, e.g., in\nsummary.lm()'s output, from a fitted model).\n\nMany classical statistical models have a scale parameter,\ntypically the standard deviation of a zero-mean normal (or Gaussian)\nrandom variable which is denoted as \\sigma.\nsigma(.) extracts the estimated parameter from a fitted\nmodel, i.e., \\hat\\sigma.\n\n\n\nNote\n\nThe misnomer “Residual standard error” has been part of\ntoo many R (and S) outputs to be easily changed there.", "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}| > |-18.4172| \\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", + "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", "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 2.8).\n\nShow R codebw.X.unique\n\n\nTable 2.8: 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 2.8).\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 2.9: 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.", + "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.", "crumbs": [ "Generalized Linear Models", "2  Linear (Gaussian) Models" @@ -182,7 +182,7 @@ "href": "Linear-models-overview.html#sec-diagnose-LMs", "title": "\n2  Linear (Gaussian) Models\n", "section": "\n2.8 Diagnostics", - "text": "2.8 Diagnostics\n\n\n\n\n\n\nTip\n\n\n\nThis section is adapted from Dobson and Barnett (2018, secs. 6.2–6.3) and Dunn, Smyth, et al. (2018) Chapter 3.\n\n\n\n2.8.1 Assumptions in linear regression models\n\\[Y|\\tilde{X}\\ \\sim_{⫫}\\ N(\\tilde{X}'\\beta,\\sigma^2)\\]\n\nNormality: The distribution conditional on a given \\(X\\) value is normal\nCorrect Functional Form: The conditional means have the structure\n\n\\[E[Y|\\tilde{X} = \\tilde{x}] = \\tilde{x}'\\beta\\] 3. Homoskedasticity: The variance \\(\\sigma^2\\) is constant (with respect to \\(\\tilde{x}\\))\n\nIndependence: The observations are statistically independent\n\n2.8.2 Direct visualization\n\nThe most direct way to examine the fit of a model is to compare it to the raw observed data.\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 2.6: Birthweight model with interaction term\n\n\n\n\n\n\n\n\nIt’s not easy to assess these assumptions from this model. If there are multiple continuous covariates, it becomes even harder to visualize the raw data.\n\n\n2.8.3 Residuals\n\nMaybe we can transform the data and model in some way to make it easier to inspect.\n\n\nDefinition 2.4 (Residual noise) The residual noise in a probabilistic model \\(p(Y)\\) is the difference between an observed value \\(y\\) and its distributional mean:\n\\[\\varepsilon(y) \\stackrel{\\text{def}}{=}y - \\mathbb{E}\\left[Y\\right] \\tag{2.8}\\]\n\n\nWe use the same notation for residual noise that we used for errors. \\(\\mathbb{E}\\left[Y\\right]\\) can be viewed as an estimate of \\(Y\\), before \\(y\\) is observed. Conversely, each observation \\(y\\) can be viewed as an estimate of \\(\\mathbb{E}\\left[Y\\right]\\) (albeit an imprecise one, individually, since \\(n=1\\)).\n\nWe can rearrange Equation 2.8 to view \\(y\\) as the sum of its mean plus the residual noise:\n\\[y = \\mathbb{E}\\left[Y\\right] + \\varepsilon{y}\\]\n\n\nTheorem 2.1 (Residuals in Gaussian models) If \\(Y\\) has a Gaussian distribution, then \\(\\varepsilon{Y}\\) also has a Gaussian distribution, and vice versa.\n\n\nProof. Left to the reader.\n\n\n\nDefinition 2.5 (Residual errors of a fitted model value) The residual of a fitted value \\(\\hat y\\) (shorthand: “residual”) is its error: \\[\n\\begin{aligned}\ne(\\hat y) &\\stackrel{\\text{def}}{=}\\varepsilon\\left(\\hat y\\right)\n\\\\&= y - \\hat y\n\\end{aligned}\n\\]\n\n\\(e(\\hat y)\\) can be seen as the maximum likelihood estimate of the residual noise:\n\\[\n\\begin{aligned}\ne(\\hat y) &= y - \\hat y\n\\\\ &= \\hat\\varepsilon_{ML}\n\\end{aligned}\n\\]\n\nGeneral characteristics of residuals\n\nTheorem 2.2 For unbiased estimators \\(\\hat\\theta\\):\n\\[\\mathbb{E}\\left[e(y)\\right] = 0 \\tag{2.9}\\] \\[\\text{Var}\\left(e(y)\\right) \\approx \\sigma^2 \\tag{2.10}\\]\n\n\nProof.  \nEquation 2.9:\n\\[\n\\begin{aligned}\n\\mathbb{E}\\left[e(y)\\right] &= \\mathbb{E}\\left[y - \\hat y\\right]\n\\\\ &= \\mathbb{E}\\left[y\\right] - \\mathbb{E}\\left[\\hat y\\right]\n\\\\ &= \\mathbb{E}\\left[y\\right] - \\mathbb{E}\\left[y\\right]\n\\\\ &= 0\n\\end{aligned}\n\\]\nEquation 2.10:\n\\[\n\\begin{aligned}\n\\text{Var}\\left(e(y)\\right) &= \\text{Var}\\left(y - \\hat y\\right)\n\\\\ &= \\text{Var}\\left(y\\right) + \\text{Var}\\left(\\hat y\\right) - 2 \\text{Cov}\\left(y, \\hat y\\right)\n\\\\ &{\\dot{\\approx}} \\text{Var}\\left(y\\right) + 0 - 2 \\cdot 0\n\\\\ &= \\text{Var}\\left(y\\right)\n\\\\ &= \\sigma^2\n\\end{aligned}\n\\]\n\nCharacteristics of residuals in Gaussian models\nWith enough data and a correct model, the residuals will be approximately Guassian distributed, with variance \\(\\sigma^2\\), which we can estimate using \\(\\hat\\sigma^2\\): that is:\n\\[\ne_i \\ \\sim_{\\text{iid}}\\ N(0, \\hat\\sigma^2)\n\\]\n\n\nExample 2.2 (residuals in birthweight data) R provides a function for residuals:\n\nShow R coderesid(bw_lm2)\n#> 1 2 3 4 5 6 7 8 9 10 \n#> 176.27 -140.73 -144.13 -59.53 177.47 -126.93 -68.93 242.67 -139.33 51.67 \n#> 11 12 13 14 15 16 17 18 19 20 \n#> 156.67 -125.13 274.28 -137.71 -27.69 -246.69 -191.67 189.33 -11.67 -242.64 \n#> 21 22 23 24 \n#> -47.64 262.36 210.36 -30.62\n\n\n\n\nExercise 2.8 Check R’s output by computing the residuals directly.\n\n\nSolution.  \n\nShow R codebw$weight - fitted(bw_lm2)\n#> 1 2 3 4 5 6 7 8 9 10 \n#> 176.27 -140.73 -144.13 -59.53 177.47 -126.93 -68.93 242.67 -139.33 51.67 \n#> 11 12 13 14 15 16 17 18 19 20 \n#> 156.67 -125.13 274.28 -137.71 -27.69 -246.69 -191.67 189.33 -11.67 -242.64 \n#> 21 22 23 24 \n#> -47.64 262.36 210.36 -30.62\n\n\nThis matches R’s output!\n\nGraph the residuals\n\nShow R codebw = bw |> \n mutate(resids_intxn = \n weight - fitted(bw_lm2))\n\nplot_bw_resid =\n bw |> \n ggplot(aes(\n x = age, \n y = resids_intxn,\n linetype = sex,\n shape = sex,\n col = sex)) +\n theme_bw() +\n xlab(\"Gestational age (weeks)\") +\n ylab(\"residuals (grams)\") +\n theme(legend.position = \"bottom\") +\n # expand_limits(y = 0, x = 0) +\n geom_point(alpha = .7)\nprint(plot_bw_resid + facet_wrap(~ sex))\n\n\n\nFigure 2.7: Residuals of interaction model for birthweight data\n\n\n\n\n\n\n\n\n\nDefinition 2.6 (Standardized residuals) \\[r_i = \\frac{e_i}{\\widehat{SD}(e_i)}\\]\n\nHence, with enough data and a correct model, the standardized residuals will be approximately standard Gaussian; that is,\n\\[\nr_i \\ \\sim_{\\text{iid}}\\ N(0,1)\n\\]\n\n2.8.4 Marginal distributions of residuals\nTo look for problems with our model, we can check whether the residuals \\(e_i\\) and standardized residuals \\(r_i\\) look like they have the distributions that they are supposed to have, according to the model.\n\nStandardized residuals in R\n\nShow R code\nrstandard(bw_lm2)\n#> 1 2 3 4 5 6 7 8 \n#> 1.15982 -0.92601 -0.87479 -0.34723 1.03507 -0.73473 -0.39901 1.43752 \n#> 9 10 11 12 13 14 15 16 \n#> -0.82539 0.30606 0.92807 -0.87616 1.91428 -0.86559 -0.16430 -1.46376 \n#> 17 18 19 20 21 22 23 24 \n#> -1.11016 1.09658 -0.06761 -1.46159 -0.28696 1.58040 1.26717 -0.19805\nresid(bw_lm2)/sigma(bw_lm2)\n#> 1 2 3 4 5 6 7 8 \n#> 0.97593 -0.77920 -0.79802 -0.32962 0.98258 -0.70279 -0.38166 1.34357 \n#> 9 10 11 12 13 14 15 16 \n#> -0.77144 0.28606 0.86741 -0.69282 1.51858 -0.76244 -0.15331 -1.36584 \n#> 17 18 19 20 21 22 23 24 \n#> -1.06123 1.04825 -0.06463 -1.34341 -0.26376 1.45262 1.16471 -0.16954\n\n\n\nThese are not quite the same, because R is doing something more complicated and precise to get the standard errors. Let’s not worry about those details for now; the difference is pretty small in this case:\n\n\n\nShow R code\nrstandard_compare_plot = \n tibble(\n x = resid(bw_lm2)/sigma(bw_lm2), \n y = rstandard(bw_lm2)) |> \n ggplot(aes(x = x, y = y)) +\n geom_point() + \n theme_bw() +\n coord_equal() + \n xlab(\"resid(bw_lm2)/sigma(bw_lm2)\") +\n ylab(\"rstandard(bw_lm2)\") +\n geom_abline(\n aes(\n intercept = 0,\n slope = 1, \n col = \"x=y\")) +\n labs(colour=\"\") +\n scale_colour_manual(values=\"red\")\n\nprint(rstandard_compare_plot)\n\n\n\n\n\n\n\n\nLet’s add these residuals to the tibble of our dataset:\n\nShow R code\nbw = \n bw |> \n mutate(\n fitted_lm2 = fitted(bw_lm2),\n \n resid_lm2 = resid(bw_lm2),\n # resid_lm2 = weight - fitted_lm2,\n \n std_resid_lm2 = rstandard(bw_lm2),\n # std_resid_lm2 = resid_lm2 / sigma(bw_lm2)\n )\n\nbw |> \n select(\n sex,\n age,\n weight,\n fitted_lm2,\n resid_lm2,\n std_resid_lm2\n )\n\n\n\nsex\nage\nweight\nfitted_lm2\nresid_lm2\nstd_resid_lm2\n\n\n\nfemale\n36\n2729\n2553\n176.27\n1.1598\n\n\nfemale\n36\n2412\n2553\n-140.73\n-0.9260\n\n\nfemale\n37\n2539\n2683\n-144.13\n-0.8748\n\n\nfemale\n38\n2754\n2814\n-59.53\n-0.3472\n\n\nfemale\n38\n2991\n2814\n177.47\n1.0351\n\n\nfemale\n39\n2817\n2944\n-126.93\n-0.7347\n\n\nfemale\n39\n2875\n2944\n-68.93\n-0.3990\n\n\nfemale\n40\n3317\n3074\n242.67\n1.4375\n\n\nfemale\n40\n2935\n3074\n-139.33\n-0.8254\n\n\nfemale\n40\n3126\n3074\n51.67\n0.3061\n\n\nfemale\n40\n3231\n3074\n156.67\n0.9281\n\n\nfemale\n42\n3210\n3335\n-125.13\n-0.8762\n\n\nmale\n35\n2925\n2651\n274.28\n1.9143\n\n\nmale\n36\n2625\n2763\n-137.71\n-0.8656\n\n\nmale\n37\n2847\n2875\n-27.69\n-0.1643\n\n\nmale\n37\n2628\n2875\n-246.69\n-1.4638\n\n\nmale\n38\n2795\n2987\n-191.67\n-1.1102\n\n\nmale\n38\n3176\n2987\n189.33\n1.0966\n\n\nmale\n38\n2975\n2987\n-11.67\n-0.0676\n\n\nmale\n40\n2968\n3211\n-242.64\n-1.4616\n\n\nmale\n40\n3163\n3211\n-47.64\n-0.2870\n\n\nmale\n40\n3473\n3211\n262.36\n1.5804\n\n\nmale\n40\n3421\n3211\n210.36\n1.2672\n\n\nmale\n41\n3292\n3323\n-30.62\n-0.1981\n\n\n\n\n\n\n\nNow let’s build histograms:\n\n\nShow R coderesid_marginal_hist = \n bw |> \n ggplot(aes(x = resid_lm2)) +\n geom_histogram()\n\nprint(resid_marginal_hist)\n\n\n\nFigure 2.8: Marginal distribution of (nonstandardized) residuals\n\n\n\n\n\n\n\n\nHard to tell with this small amount of data, but I’m a bit concerned that the histogram doesn’t show a bell-curve shape.\n\n\n\nShow R codestd_resid_marginal_hist = \n bw |> \n ggplot(aes(x = std_resid_lm2)) +\n geom_histogram()\n\nprint(std_resid_marginal_hist)\n\n\n\nFigure 2.9: Marginal distribution of standardized residuals\n\n\n\n\n\n\n\n\nThis looks similar, although the scale of the x-axis got narrower, because we divided by \\(\\hat\\sigma\\) (roughly speaking).\nStill hard to tell if the distribution is Gaussian.\n\n\n2.8.5 QQ plot of standardized residuals\n\nAnother way to assess normality is the QQ plot of the standardized residuals versus normal quantiles:\n\n\nShow R code\nlibrary(ggfortify) \n# needed to make ggplot2::autoplot() work for `lm` objects\n\nqqplot_lm2_auto = \n bw_lm2 |> \n autoplot(\n which = 2, # options are 1:6; can do multiple at once\n ncol = 1) +\n theme_classic()\n\nprint(qqplot_lm2_auto)\n\n\n\n\n\n\n\n\nIf the Gaussian model were correct, these points should follow the dotted line.\nFig 2.4 panel (c) in Dobson and Barnett (2018) is a little different; they didn’t specify how they produced it, but other statistical analysis systems do things differently from R.\nSee also Dunn, Smyth, et al. (2018) §3.5.4.\n\n\nQQ plot - how it’s built\n\nLet’s construct it by hand:\n\n\nShow R code\nbw = bw |> \n mutate(\n p = (rank(std_resid_lm2) - 1/2)/n(), # \"Blom's method\"\n expected_quantiles_lm2 = qnorm(p)\n )\n\nqqplot_lm2 = \n bw |> \n ggplot(\n aes(\n x = expected_quantiles_lm2, \n y = std_resid_lm2, \n col = sex, \n shape = sex)\n ) + \n geom_point() +\n theme_classic() +\n theme(legend.position='none') + # removing the plot legend\n ggtitle(\"Normal Q-Q\") +\n xlab(\"Theoretical Quantiles\") + \n ylab(\"Standardized residuals\")\n\n# find the expected line:\n\nps <- c(.25, .75) # reference probabilities\na <- quantile(rstandard(bw_lm2), ps) # empirical quantiles\nb <- qnorm(ps) # theoretical quantiles\n\nqq_slope = diff(a)/diff(b)\nqq_intcpt = a[1] - b[1] * qq_slope\n\nqqplot_lm2 = \n qqplot_lm2 +\n geom_abline(slope = qq_slope, intercept = qq_intcpt)\n\nprint(qqplot_lm2)\n\n\n\n\n\n\n\n\n2.8.6 Conditional distributions of residuals\nIf our Gaussian linear regression model is correct, the residuals \\(e_i\\) and standardized residuals \\(r_i\\) should have:\n\nan approximately Gaussian distribution, with:\na mean of 0\na constant variance\n\nThis should be true for every value of \\(x\\).\n\nIf we didn’t correctly guess the functional form of linear component of the mean, \\[\\text{E}[Y|X=x] = \\beta_0 + \\beta_1 X_1 + ... + \\beta_p X_p\\]\nThen the the residuals might have nonzero mean.\nRegardless of whether we guessed the mean function correctly, ther the variance of the residuals might differ between values of \\(x\\).\n\nResiduals versus fitted values\n\nTo look for these issues, we can plot the residuals \\(e_i\\) against the fitted values \\(\\hat y_i\\) (Figure 2.10).\n\n\nShow R codeautoplot(bw_lm2, which = 1, ncol = 1) |> print()\n\n\n\nFigure 2.10: birthweight model (Equation 2.2): residuals versus fitted values\n\n\n\n\n\n\n\n\nIf the model is correct, the blue line should stay flat and close to 0, and the cloud of dots should have the same vertical spread regardless of the fitted value.\nIf not, we probably need to change the functional form of linear component of the mean, \\[\\text{E}[Y|X=x] = \\beta_0 + \\beta_1 X_1 + ... + \\beta_p X_p\\]\n\nExample: PLOS Medicine title length data\n(Adapted from Dobson and Barnett (2018), §6.7.1)\n\nShow R codedata(PLOS, package = \"dobson\")\nlibrary(ggplot2)\nfig1 = \n PLOS |> \n ggplot(\n aes(x = authors,\n y = nchar)\n ) +\n geom_point() +\n theme(legend.position = \"bottom\") +\n labs(col = \"\") +\n guides(col=guide_legend(ncol=3))\nfig1\n\n\n\nFigure 2.11: Number of authors versus title length in PLOS Medicine articles\n\n\n\n\n\n\n\n\nLinear fit\n\nShow R codelm_PLOS_linear = lm(\n formula = nchar ~ authors, \n data = PLOS)\n\n\nShow R codefig2 = fig1 +\n geom_smooth(\n method = \"lm\", \n fullrange = TRUE,\n aes(col = \"lm(y ~ x)\"))\nfig2\n\nlibrary(ggfortify)\nautoplot(lm_PLOS_linear, which = 1, ncol = 1)\n\n\nFigure 2.12: Number of authors versus title length in PLOS Medicine, with linear model fit\n\n\n\n\n\n(a) Data and fit\n\n\n\n\n\n\n\n\n\n(b) Residuals vs fitted\n\n\n\n\n\n\n\n\n\n\nQuadratic fit\n\nShow R codelm_PLOS_quad = lm(\n formula = nchar ~ authors + I(authors^2), \n data = PLOS)\n\n\nShow R codefig3 = \n fig2 + \ngeom_smooth(\n method = \"lm\",\n fullrange = TRUE,\n formula = y ~ x + I(x ^ 2),\n aes(col = \"lm(y ~ x + I(x^2))\")\n )\nfig3\n\nautoplot(lm_PLOS_quad, which = 1, ncol = 1)\n\n\nFigure 2.13: Number of authors versus title length in PLOS Medicine, with quadratic model fit\n\n\n\n\n\n(a) Data and fit\n\n\n\n\n\n\n\n\n\n(b) Residuals vs fitted\n\n\n\n\n\n\n\n\n\n\nLinear versus quadratic fits\nShow R codelibrary(ggfortify)\nautoplot(lm_PLOS_linear, which = 1, ncol = 1)\n\nautoplot(lm_PLOS_quad, which = 1, ncol = 1)\n\n\nFigure 2.14: Residuals versus fitted plot for linear and quadratic fits to PLOS data\n\n\n\n\n\n(a) Linear\n\n\n\n\n\n\n\n\n\n(b) Quadratic\n\n\n\n\n\n\n\n\n\n\nCubic fit\n\nShow R codelm_PLOS_cub = lm(\n formula = nchar ~ authors + I(authors^2) + I(authors^3), \n data = PLOS)\n\n\nShow R codefig4 = \n fig3 + \ngeom_smooth(\n method = \"lm\",\n fullrange = TRUE,\n formula = y ~ x + I(x ^ 2) + I(x ^ 3),\n aes(col = \"lm(y ~ x + I(x^2) + I(x ^ 3))\")\n )\nfig4\n\nautoplot(lm_PLOS_cub, which = 1, ncol = 1)\n\n\nFigure 2.15: Number of authors versus title length in PLOS Medicine, with cubic model fit\n\n\n\n\n\n(a) Data and fit\n\n\n\n\n\n\n\n\n\n(b) Residuals vs fitted\n\n\n\n\n\n\n\n\n\n\nLogarithmic fit\n\nShow R codelm_PLOS_log = lm(nchar ~ log(authors), data = PLOS)\n\n\nShow R codefig5 = fig4 + \n geom_smooth(\n method = \"lm\",\n fullrange = TRUE,\n formula = y ~ log(x),\n aes(col = \"lm(y ~ log(x))\")\n )\nfig5\n\nautoplot(lm_PLOS_log, which = 1, ncol = 1)\n\n\nFigure 2.16: logarithmic fit\n\n\n\n\n\n(a) Data and fit\n\n\n\n\n\n\n\n\n\n(b) Residuals vs fitted\n\n\n\n\n\n\n\n\n\n\nModel selection\n\nShow R codeanova(lm_PLOS_linear, lm_PLOS_quad)\n\n\nTable 2.10: linear vs quadratic\n\n\n\n\nRes.Df\nRSS\nDf\nSum of Sq\nF\nPr(>F)\n\n\n\n876\n947502\nNA\nNA\nNA\nNA\n\n\n875\n880950\n1\n66552\n66.1\n0\n\n\n\n\n\n\n\n\n\nShow R codeanova(lm_PLOS_quad, lm_PLOS_cub)\n\n\nTable 2.11: quadratic vs cubic\n\n\n\n\nRes.Df\nRSS\nDf\nSum of Sq\nF\nPr(>F)\n\n\n\n875\n880950\nNA\nNA\nNA\nNA\n\n\n874\n865933\n1\n15018\n15.16\n1e-04\n\n\n\n\n\n\n\n\nAIC/BIC\n\nShow R codeAIC(lm_PLOS_quad)\n#> [1] 8568\nAIC(lm_PLOS_cub)\n#> [1] 8555\n\n\n\nShow R codeAIC(lm_PLOS_cub)\n#> [1] 8555\nAIC(lm_PLOS_log)\n#> [1] 8544\n\n\n\nShow R codeBIC(lm_PLOS_cub)\n#> [1] 8578\nBIC(lm_PLOS_log)\n#> [1] 8558\n\n\nExtrapolation is dangerous\n\nShow R codefig_all = fig5 +\n xlim(0, 60)\nfig_all\n\n\n\nFigure 2.17: Number of authors versus title length in PLOS Medicine\n\n\n\n\n\n\n\nScale-location plot\n\nWe can also plot the square roots of the absolute values of the standardized residuals against the fitted values (Figure 2.18).\n\n\nShow R codeautoplot(bw_lm2, which = 3, ncol = 1) |> print()\n\n\n\nFigure 2.18: Scale-location plot of birthweight data\n\n\n\n\n\n\n\n\nHere, the blue line doesn’t need to be near 0, but it should be flat. If not, the residual variance \\(\\sigma^2\\) might not be constant, and we might need to transform our outcome \\(Y\\) (or use a model that allows non-constant variance).\n\nResiduals versus leverage\n\nWe can also plot our standardized residuals against “leverage”, which roughly speaking is a measure of how unusual each \\(x_i\\) value is. Very unusual \\(x_i\\) values can have extreme effects on the model fit, so we might want to remove those observations as outliers, particularly if they have large residuals.\n\n\nShow R codeautoplot(bw_lm2, which = 5, ncol = 1) |> print()\n\n\nbirthweight model with interactions (Equation 2.2): residuals versus leverage\n\n\n\n\n\nThe blue line should be relatively flat and close to 0 here.\n\n\n2.8.7 Diagnostics constructed by hand\n\nShow R code\nbw = \n bw |> \n mutate(\n predlm2 = predict(bw_lm2),\n residlm2 = weight - predlm2,\n std_resid = residlm2 / sigma(bw_lm2),\n # std_resid_builtin = rstandard(bw_lm2), # uses leverage\n sqrt_abs_std_resid = std_resid |> abs() |> sqrt()\n \n )\n\n\nResiduals vs fitted\n\nShow R code\nresid_vs_fit = bw |> \n ggplot(\n aes(x = predlm2, y = residlm2, col = sex, shape = sex)\n ) + \n geom_point() +\n theme_classic() +\n geom_hline(yintercept = 0)\n\n\n\nShow R codeprint(resid_vs_fit)\n\n\n\n\n\n\n\nStandardized residuals vs fitted\n\nShow R codebw |> \n ggplot(\n aes(x = predlm2, y = std_resid, col = sex, shape = sex)\n ) + \n geom_point() +\n theme_classic() +\n geom_hline(yintercept = 0)\n\n\n\n\n\n\n\nStandardized residuals vs gestational age\n\nShow R code\nbw |> \n ggplot(\n aes(x = age, y = std_resid, col = sex, shape = sex)\n ) + \n geom_point() +\n theme_classic() +\n geom_hline(yintercept = 0)\n\n\n\n\n\n\n\n\nsqrt(abs(rstandard())) vs fitted\nCompare with autoplot(bw_lm2, 3)\n\nShow R code\n\nbw |> \n ggplot(\n aes(x = predlm2, y = sqrt_abs_std_resid, col = sex, shape = sex)\n ) + \n geom_point() +\n theme_classic() +\n geom_hline(yintercept = 0)", + "text": "2.8 Diagnostics\n\n\n\n\n\n\nTip\n\n\n\nThis section is adapted from Dobson and Barnett (2018, secs. 6.2–6.3) and Dunn, Smyth, et al. (2018) Chapter 3.\n\n\n\n2.8.1 Assumptions in linear regression models\n\\[Y|\\tilde{X}\\ \\sim_{⫫}\\ N(\\tilde{X}'\\beta,\\sigma^2)\\]\n\nNormality: The distribution conditional on a given \\(X\\) value is normal\nCorrect Functional Form: The conditional means have the structure\n\n\\[E[Y|\\tilde{X} = \\tilde{x}] = \\tilde{x}'\\beta\\] 3. Homoskedasticity: The variance \\(\\sigma^2\\) is constant (with respect to \\(\\tilde{x}\\))\n\nIndependence: The observations are statistically independent\n\n2.8.2 Direct visualization\n\nThe most direct way to examine the fit of a model is to compare it to the raw observed data.\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 2.6: Birthweight model with interaction term\n\n\n\n\n\n\n\n\nIt’s not easy to assess these assumptions from this model. If there are multiple continuous covariates, it becomes even harder to visualize the raw data.\n\n\n2.8.3 Residuals\n\nMaybe we can transform the data and model in some way to make it easier to inspect.\n\n\nDefinition 2.4 (Residual noise) The residual noise in a probabilistic model \\(p(Y)\\) is the difference between an observed value \\(y\\) and its distributional mean:\n\\[\\varepsilon(y) \\stackrel{\\text{def}}{=}y - \\mathbb{E}\\left[Y\\right] \\tag{2.8}\\]\n\n\nWe use the same notation for residual noise that we used for errors. \\(\\mathbb{E}\\left[Y\\right]\\) can be viewed as an estimate of \\(Y\\), before \\(y\\) is observed. Conversely, each observation \\(y\\) can be viewed as an estimate of \\(\\mathbb{E}\\left[Y\\right]\\) (albeit an imprecise one, individually, since \\(n=1\\)).\n\nWe can rearrange Equation 2.8 to view \\(y\\) as the sum of its mean plus the residual noise:\n\\[y = \\mathbb{E}\\left[Y\\right] + \\varepsilon{y}\\]\n\n\nTheorem 2.1 (Residuals in Gaussian models) If \\(Y\\) has a Gaussian distribution, then \\(\\varepsilon{Y}\\) also has a Gaussian distribution, and vice versa.\n\n\nProof. Left to the reader.\n\n\n\nDefinition 2.5 (Residual errors of a fitted model value) The residual of a fitted value \\(\\hat y\\) (shorthand: “residual”) is its error: \\[\n\\begin{aligned}\ne(\\hat y) &\\stackrel{\\text{def}}{=}\\varepsilon\\left(\\hat y\\right)\n\\\\&= y - \\hat y\n\\end{aligned}\n\\]\n\n\\(e(\\hat y)\\) can be seen as the maximum likelihood estimate of the residual noise:\n\\[\n\\begin{aligned}\ne(\\hat y) &= y - \\hat y\n\\\\ &= \\hat\\varepsilon_{ML}\n\\end{aligned}\n\\]\n\nGeneral characteristics of residuals\n\nTheorem 2.2 For unbiased estimators \\(\\hat\\theta\\):\n\\[\\mathbb{E}\\left[e(y)\\right] = 0 \\tag{2.9}\\] \\[\\text{Var}\\left(e(y)\\right) \\approx \\sigma^2 \\tag{2.10}\\]\n\n\nProof.  \nEquation 2.9:\n\\[\n\\begin{aligned}\n\\mathbb{E}\\left[e(y)\\right] &= \\mathbb{E}\\left[y - \\hat y\\right]\n\\\\ &= \\mathbb{E}\\left[y\\right] - \\mathbb{E}\\left[\\hat y\\right]\n\\\\ &= \\mathbb{E}\\left[y\\right] - \\mathbb{E}\\left[y\\right]\n\\\\ &= 0\n\\end{aligned}\n\\]\nEquation 2.10:\n\\[\n\\begin{aligned}\n\\text{Var}\\left(e(y)\\right) &= \\text{Var}\\left(y - \\hat y\\right)\n\\\\ &= \\text{Var}\\left(y\\right) + \\text{Var}\\left(\\hat y\\right) - 2 \\text{Cov}\\left(y, \\hat y\\right)\n\\\\ &{\\dot{\\approx}} \\text{Var}\\left(y\\right) + 0 - 2 \\cdot 0\n\\\\ &= \\text{Var}\\left(y\\right)\n\\\\ &= \\sigma^2\n\\end{aligned}\n\\]\n\nCharacteristics of residuals in Gaussian models\nWith enough data and a correct model, the residuals will be approximately Guassian distributed, with variance \\(\\sigma^2\\), which we can estimate using \\(\\hat\\sigma^2\\): that is:\n\\[\ne_i \\ \\sim_{\\text{iid}}\\ N(0, \\hat\\sigma^2)\n\\]\n\n\nExample 2.2 (residuals in birthweight data) R provides a function for residuals:\n\nShow R coderesid(bw_lm2)\n#> 1 2 3 4 5 6 7 8 9 10 \n#> 176.27 -140.73 -144.13 -59.53 177.47 -126.93 -68.93 242.67 -139.33 51.67 \n#> 11 12 13 14 15 16 17 18 19 20 \n#> 156.67 -125.13 274.28 -137.71 -27.69 -246.69 -191.67 189.33 -11.67 -242.64 \n#> 21 22 23 24 \n#> -47.64 262.36 210.36 -30.62\n\n\n\n\nExercise 2.8 Check R’s output by computing the residuals directly.\n\n\nSolution.  \n\nShow R codebw$weight - fitted(bw_lm2)\n#> 1 2 3 4 5 6 7 8 9 10 \n#> 176.27 -140.73 -144.13 -59.53 177.47 -126.93 -68.93 242.67 -139.33 51.67 \n#> 11 12 13 14 15 16 17 18 19 20 \n#> 156.67 -125.13 274.28 -137.71 -27.69 -246.69 -191.67 189.33 -11.67 -242.64 \n#> 21 22 23 24 \n#> -47.64 262.36 210.36 -30.62\n\n\nThis matches R’s output!\n\nGraph the residuals\n\nShow R codebw = bw |> \n mutate(resids_intxn = \n weight - fitted(bw_lm2))\n\nplot_bw_resid =\n bw |> \n ggplot(aes(\n x = age, \n y = resids_intxn,\n linetype = sex,\n shape = sex,\n col = sex)) +\n theme_bw() +\n xlab(\"Gestational age (weeks)\") +\n ylab(\"residuals (grams)\") +\n theme(legend.position = \"bottom\") +\n # expand_limits(y = 0, x = 0) +\n geom_point(alpha = .7)\nprint(plot_bw_resid + facet_wrap(~ sex))\n\n\n\nFigure 2.7: Residuals of interaction model for birthweight data\n\n\n\n\n\n\n\n\n\nDefinition 2.6 (Standardized residuals) \\[r_i = \\frac{e_i}{\\widehat{SD}(e_i)}\\]\n\nHence, with enough data and a correct model, the standardized residuals will be approximately standard Gaussian; that is,\n\\[\nr_i \\ \\sim_{\\text{iid}}\\ N(0,1)\n\\]\n\n2.8.4 Marginal distributions of residuals\nTo look for problems with our model, we can check whether the residuals \\(e_i\\) and standardized residuals \\(r_i\\) look like they have the distributions that they are supposed to have, according to the model.\n\nStandardized residuals in R\n\nShow R code\nrstandard(bw_lm2)\n#> 1 2 3 4 5 6 7 8 \n#> 1.15982 -0.92601 -0.87479 -0.34723 1.03507 -0.73473 -0.39901 1.43752 \n#> 9 10 11 12 13 14 15 16 \n#> -0.82539 0.30606 0.92807 -0.87616 1.91428 -0.86559 -0.16430 -1.46376 \n#> 17 18 19 20 21 22 23 24 \n#> -1.11016 1.09658 -0.06761 -1.46159 -0.28696 1.58040 1.26717 -0.19805\nresid(bw_lm2)/sigma(bw_lm2)\n#> 1 2 3 4 5 6 7 8 \n#> 0.97593 -0.77920 -0.79802 -0.32962 0.98258 -0.70279 -0.38166 1.34357 \n#> 9 10 11 12 13 14 15 16 \n#> -0.77144 0.28606 0.86741 -0.69282 1.51858 -0.76244 -0.15331 -1.36584 \n#> 17 18 19 20 21 22 23 24 \n#> -1.06123 1.04825 -0.06463 -1.34341 -0.26376 1.45262 1.16471 -0.16954\n\n\n\nThese are not quite the same, because R is doing something more complicated and precise to get the standard errors. Let’s not worry about those details for now; the difference is pretty small in this case:\n\n\n\nShow R code\nrstandard_compare_plot = \n tibble(\n x = resid(bw_lm2)/sigma(bw_lm2), \n y = rstandard(bw_lm2)) |> \n ggplot(aes(x = x, y = y)) +\n geom_point() + \n theme_bw() +\n coord_equal() + \n xlab(\"resid(bw_lm2)/sigma(bw_lm2)\") +\n ylab(\"rstandard(bw_lm2)\") +\n geom_abline(\n aes(\n intercept = 0,\n slope = 1, \n col = \"x=y\")) +\n labs(colour=\"\") +\n scale_colour_manual(values=\"red\")\n\nprint(rstandard_compare_plot)\n\n\n\n\n\n\n\n\nLet’s add these residuals to the tibble of our dataset:\n\nShow R code\nbw = \n bw |> \n mutate(\n fitted_lm2 = fitted(bw_lm2),\n \n resid_lm2 = resid(bw_lm2),\n # resid_lm2 = weight - fitted_lm2,\n \n std_resid_lm2 = rstandard(bw_lm2),\n # std_resid_lm2 = resid_lm2 / sigma(bw_lm2)\n )\n\nbw |> \n select(\n sex,\n age,\n weight,\n fitted_lm2,\n resid_lm2,\n std_resid_lm2\n )\n\n\n\nsex\nage\nweight\nfitted_lm2\nresid_lm2\nstd_resid_lm2\n\n\n\nfemale\n36\n2729\n2553\n176.27\n1.1598\n\n\nfemale\n36\n2412\n2553\n-140.73\n-0.9260\n\n\nfemale\n37\n2539\n2683\n-144.13\n-0.8748\n\n\nfemale\n38\n2754\n2814\n-59.53\n-0.3472\n\n\nfemale\n38\n2991\n2814\n177.47\n1.0351\n\n\nfemale\n39\n2817\n2944\n-126.93\n-0.7347\n\n\nfemale\n39\n2875\n2944\n-68.93\n-0.3990\n\n\nfemale\n40\n3317\n3074\n242.67\n1.4375\n\n\nfemale\n40\n2935\n3074\n-139.33\n-0.8254\n\n\nfemale\n40\n3126\n3074\n51.67\n0.3061\n\n\nfemale\n40\n3231\n3074\n156.67\n0.9281\n\n\nfemale\n42\n3210\n3335\n-125.13\n-0.8762\n\n\nmale\n35\n2925\n2651\n274.28\n1.9143\n\n\nmale\n36\n2625\n2763\n-137.71\n-0.8656\n\n\nmale\n37\n2847\n2875\n-27.69\n-0.1643\n\n\nmale\n37\n2628\n2875\n-246.69\n-1.4638\n\n\nmale\n38\n2795\n2987\n-191.67\n-1.1102\n\n\nmale\n38\n3176\n2987\n189.33\n1.0966\n\n\nmale\n38\n2975\n2987\n-11.67\n-0.0676\n\n\nmale\n40\n2968\n3211\n-242.64\n-1.4616\n\n\nmale\n40\n3163\n3211\n-47.64\n-0.2870\n\n\nmale\n40\n3473\n3211\n262.36\n1.5804\n\n\nmale\n40\n3421\n3211\n210.36\n1.2672\n\n\nmale\n41\n3292\n3323\n-30.62\n-0.1981\n\n\n\n\n\n\n\nNow let’s build histograms:\n\n\nShow R coderesid_marginal_hist = \n bw |> \n ggplot(aes(x = resid_lm2)) +\n geom_histogram()\n\nprint(resid_marginal_hist)\n\n\n\nFigure 2.8: Marginal distribution of (nonstandardized) residuals\n\n\n\n\n\n\n\n\nHard to tell with this small amount of data, but I’m a bit concerned that the histogram doesn’t show a bell-curve shape.\n\n\n\nShow R codestd_resid_marginal_hist = \n bw |> \n ggplot(aes(x = std_resid_lm2)) +\n geom_histogram()\n\nprint(std_resid_marginal_hist)\n\n\n\nFigure 2.9: Marginal distribution of standardized residuals\n\n\n\n\n\n\n\n\nThis looks similar, although the scale of the x-axis got narrower, because we divided by \\(\\hat\\sigma\\) (roughly speaking).\nStill hard to tell if the distribution is Gaussian.\n\n\n2.8.5 QQ plot of standardized residuals\n\nAnother way to assess normality is the QQ plot of the standardized residuals versus normal quantiles:\n\n\nShow R code\nlibrary(ggfortify) \n# needed to make ggplot2::autoplot() work for `lm` objects\n\nqqplot_lm2_auto = \n bw_lm2 |> \n autoplot(\n which = 2, # options are 1:6; can do multiple at once\n ncol = 1) +\n theme_classic()\n\nprint(qqplot_lm2_auto)\n\n\n\n\n\n\n\n\nIf the Gaussian model were correct, these points should follow the dotted line.\nFig 2.4 panel (c) in Dobson and Barnett (2018) is a little different; they didn’t specify how they produced it, but other statistical analysis systems do things differently from R.\nSee also Dunn, Smyth, et al. (2018) §3.5.4.\n\n\nQQ plot - how it’s built\n\nLet’s construct it by hand:\n\n\nShow R code\nbw = bw |> \n mutate(\n p = (rank(std_resid_lm2) - 1/2)/n(), # \"Blom's method\"\n expected_quantiles_lm2 = qnorm(p)\n )\n\nqqplot_lm2 = \n bw |> \n ggplot(\n aes(\n x = expected_quantiles_lm2, \n y = std_resid_lm2, \n col = sex, \n shape = sex)\n ) + \n geom_point() +\n theme_classic() +\n theme(legend.position='none') + # removing the plot legend\n ggtitle(\"Normal Q-Q\") +\n xlab(\"Theoretical Quantiles\") + \n ylab(\"Standardized residuals\")\n\n# find the expected line:\n\nps <- c(.25, .75) # reference probabilities\na <- quantile(rstandard(bw_lm2), ps) # empirical quantiles\nb <- qnorm(ps) # theoretical quantiles\n\nqq_slope = diff(a)/diff(b)\nqq_intcpt = a[1] - b[1] * qq_slope\n\nqqplot_lm2 = \n qqplot_lm2 +\n geom_abline(slope = qq_slope, intercept = qq_intcpt)\n\nprint(qqplot_lm2)\n\n\n\n\n\n\n\n\n2.8.6 Conditional distributions of residuals\nIf our Gaussian linear regression model is correct, the residuals \\(e_i\\) and standardized residuals \\(r_i\\) should have:\n\nan approximately Gaussian distribution, with:\na mean of 0\na constant variance\n\nThis should be true for every value of \\(x\\).\n\nIf we didn’t correctly guess the functional form of the linear component of the mean, \\[\\text{E}[Y|X=x] = \\beta_0 + \\beta_1 X_1 + ... + \\beta_p X_p\\]\nThen the the residuals might have nonzero mean.\nRegardless of whether we guessed the mean function correctly, ther the variance of the residuals might differ between values of \\(x\\).\n\nResiduals versus fitted values\n\nTo look for these issues, we can plot the residuals \\(e_i\\) against the fitted values \\(\\hat y_i\\) (Figure 2.10).\n\n\nShow R codeautoplot(bw_lm2, which = 1, ncol = 1) |> print()\n\n\n\nFigure 2.10: birthweight model (Equation 2.2): residuals versus fitted values\n\n\n\n\n\n\n\n\nIf the model is correct, the blue line should stay flat and close to 0, and the cloud of dots should have the same vertical spread regardless of the fitted value.\nIf not, we probably need to change the functional form of linear component of the mean, \\[\\text{E}[Y|X=x] = \\beta_0 + \\beta_1 X_1 + ... + \\beta_p X_p\\]\n\nExample: PLOS Medicine title length data\n(Adapted from Dobson and Barnett (2018), §6.7.1)\n\nShow R codedata(PLOS, package = \"dobson\")\nlibrary(ggplot2)\nfig1 = \n PLOS |> \n ggplot(\n aes(x = authors,\n y = nchar)\n ) +\n geom_point() +\n theme(legend.position = \"bottom\") +\n labs(col = \"\") +\n guides(col=guide_legend(ncol=3))\nfig1\n\n\n\nFigure 2.11: Number of authors versus title length in PLOS Medicine articles\n\n\n\n\n\n\n\n\nLinear fit\n\nShow R codelm_PLOS_linear = lm(\n formula = nchar ~ authors, \n data = PLOS)\n\n\nShow R codefig2 = fig1 +\n geom_smooth(\n method = \"lm\", \n fullrange = TRUE,\n aes(col = \"lm(y ~ x)\"))\nfig2\n\nlibrary(ggfortify)\nautoplot(lm_PLOS_linear, which = 1, ncol = 1)\n\n\nFigure 2.12: Number of authors versus title length in PLOS Medicine, with linear model fit\n\n\n\n\n\n(a) Data and fit\n\n\n\n\n\n\n\n\n\n(b) Residuals vs fitted\n\n\n\n\n\n\n\n\n\n\nQuadratic fit\n\nShow R codelm_PLOS_quad = lm(\n formula = nchar ~ authors + I(authors^2), \n data = PLOS)\n\n\nShow R codefig3 = \n fig2 + \ngeom_smooth(\n method = \"lm\",\n fullrange = TRUE,\n formula = y ~ x + I(x ^ 2),\n aes(col = \"lm(y ~ x + I(x^2))\")\n )\nfig3\n\nautoplot(lm_PLOS_quad, which = 1, ncol = 1)\n\n\nFigure 2.13: Number of authors versus title length in PLOS Medicine, with quadratic model fit\n\n\n\n\n\n(a) Data and fit\n\n\n\n\n\n\n\n\n\n(b) Residuals vs fitted\n\n\n\n\n\n\n\n\n\n\nLinear versus quadratic fits\nShow R codelibrary(ggfortify)\nautoplot(lm_PLOS_linear, which = 1, ncol = 1)\n\nautoplot(lm_PLOS_quad, which = 1, ncol = 1)\n\n\nFigure 2.14: Residuals versus fitted plot for linear and quadratic fits to PLOS data\n\n\n\n\n\n(a) Linear\n\n\n\n\n\n\n\n\n\n(b) Quadratic\n\n\n\n\n\n\n\n\n\n\nCubic fit\n\nShow R codelm_PLOS_cub = lm(\n formula = nchar ~ authors + I(authors^2) + I(authors^3), \n data = PLOS)\n\n\nShow R codefig4 = \n fig3 + \ngeom_smooth(\n method = \"lm\",\n fullrange = TRUE,\n formula = y ~ x + I(x ^ 2) + I(x ^ 3),\n aes(col = \"lm(y ~ x + I(x^2) + I(x ^ 3))\")\n )\nfig4\n\nautoplot(lm_PLOS_cub, which = 1, ncol = 1)\n\n\nFigure 2.15: Number of authors versus title length in PLOS Medicine, with cubic model fit\n\n\n\n\n\n(a) Data and fit\n\n\n\n\n\n\n\n\n\n(b) Residuals vs fitted\n\n\n\n\n\n\n\n\n\n\nLogarithmic fit\n\nShow R codelm_PLOS_log = lm(nchar ~ log(authors), data = PLOS)\n\n\nShow R codefig5 = fig4 + \n geom_smooth(\n method = \"lm\",\n fullrange = TRUE,\n formula = y ~ log(x),\n aes(col = \"lm(y ~ log(x))\")\n )\nfig5\n\nautoplot(lm_PLOS_log, which = 1, ncol = 1)\n\n\nFigure 2.16: logarithmic fit\n\n\n\n\n\n(a) Data and fit\n\n\n\n\n\n\n\n\n\n(b) Residuals vs fitted\n\n\n\n\n\n\n\n\n\n\nModel selection\n\nShow R codeanova(lm_PLOS_linear, lm_PLOS_quad)\n\n\nTable 2.11: linear vs quadratic\n\n\n\n\nRes.Df\nRSS\nDf\nSum of Sq\nF\nPr(>F)\n\n\n\n876\n947502\nNA\nNA\nNA\nNA\n\n\n875\n880950\n1\n66552\n66.1\n0\n\n\n\n\n\n\n\n\n\nShow R codeanova(lm_PLOS_quad, lm_PLOS_cub)\n\n\nTable 2.12: quadratic vs cubic\n\n\n\n\nRes.Df\nRSS\nDf\nSum of Sq\nF\nPr(>F)\n\n\n\n875\n880950\nNA\nNA\nNA\nNA\n\n\n874\n865933\n1\n15018\n15.16\n1e-04\n\n\n\n\n\n\n\n\nAIC/BIC\n\nShow R codeAIC(lm_PLOS_quad)\n#> [1] 8568\nAIC(lm_PLOS_cub)\n#> [1] 8555\n\n\n\nShow R codeAIC(lm_PLOS_cub)\n#> [1] 8555\nAIC(lm_PLOS_log)\n#> [1] 8544\n\n\n\nShow R codeBIC(lm_PLOS_cub)\n#> [1] 8578\nBIC(lm_PLOS_log)\n#> [1] 8558\n\n\nExtrapolation is dangerous\n\nShow R codefig_all = fig5 +\n xlim(0, 60)\nfig_all\n\n\n\nFigure 2.17: Number of authors versus title length in PLOS Medicine\n\n\n\n\n\n\n\nScale-location plot\n\nWe can also plot the square roots of the absolute values of the standardized residuals against the fitted values (Figure 2.18).\n\n\nShow R codeautoplot(bw_lm2, which = 3, ncol = 1) |> print()\n\n\n\nFigure 2.18: Scale-location plot of birthweight data\n\n\n\n\n\n\n\n\nHere, the blue line doesn’t need to be near 0, but it should be flat. If not, the residual variance \\(\\sigma^2\\) might not be constant, and we might need to transform our outcome \\(Y\\) (or use a model that allows non-constant variance).\n\nResiduals versus leverage\n\nWe can also plot our standardized residuals against “leverage”, which roughly speaking is a measure of how unusual each \\(x_i\\) value is. Very unusual \\(x_i\\) values can have extreme effects on the model fit, so we might want to remove those observations as outliers, particularly if they have large residuals.\n\n\nShow R codeautoplot(bw_lm2, which = 5, ncol = 1) |> print()\n\n\nbirthweight model with interactions (Equation 2.2): residuals versus leverage\n\n\n\n\n\nThe blue line should be relatively flat and close to 0 here.\n\n\n2.8.7 Diagnostics constructed by hand\n\nShow R code\nbw = \n bw |> \n mutate(\n predlm2 = predict(bw_lm2),\n residlm2 = weight - predlm2,\n std_resid = residlm2 / sigma(bw_lm2),\n # std_resid_builtin = rstandard(bw_lm2), # uses leverage\n sqrt_abs_std_resid = std_resid |> abs() |> sqrt()\n \n )\n\n\nResiduals vs fitted\n\nShow R code\nresid_vs_fit = bw |> \n ggplot(\n aes(x = predlm2, y = residlm2, col = sex, shape = sex)\n ) + \n geom_point() +\n theme_classic() +\n geom_hline(yintercept = 0)\n\n\n\nShow R codeprint(resid_vs_fit)\n\n\n\n\n\n\n\nStandardized residuals vs fitted\n\nShow R codebw |> \n ggplot(\n aes(x = predlm2, y = std_resid, col = sex, shape = sex)\n ) + \n geom_point() +\n theme_classic() +\n geom_hline(yintercept = 0)\n\n\n\n\n\n\n\nStandardized residuals vs gestational age\n\nShow R code\nbw |> \n ggplot(\n aes(x = age, y = std_resid, col = sex, shape = sex)\n ) + \n geom_point() +\n theme_classic() +\n geom_hline(yintercept = 0)\n\n\n\n\n\n\n\n\nsqrt(abs(rstandard())) vs fitted\nCompare with autoplot(bw_lm2, 3)\n\nShow R code\n\nbw |> \n ggplot(\n aes(x = predlm2, y = sqrt_abs_std_resid, col = sex, shape = sex)\n ) + \n geom_point() +\n theme_classic() +\n geom_hline(yintercept = 0)", "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\n6.721\n0.4805\n0.3831\n5.956\n-61.84\n\n\nreduced\n6.576\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$carbohydrate\nx = carbohydrate |> \n select(age, weight, protein) |> \n as.matrix()\nfit = glmnet(x,y)\n\n\n\n\nShow R codeautoplot(fit, xvar = 'lambda')\n\n\n\nFigure 2.19: Lasso selection\n\n\n\n\n\n\n\n\n\nShow R codecvfit = cv.glmnet(x,y)\nplot(cvfit)\n\n\n\n\n\n\n\n\n\nShow R codecoef(cvfit, s = \"lambda.1se\")\n#> 4 x 1 sparse Matrix of class \"dgCMatrix\"\n#> s1\n#> (Intercept) 34.2044\n#> age . \n#> weight -0.0926\n#> protein 0.8582", + "text": "2.9 Model selection\n(adapted from Dobson and Barnett (2018) §6.3.3; for more information on prediction, see James et al. (2013) and Harrell (2015)).\n\nIf we have a lot of covariates in our dataset, we might want to choose a small subset to use in our model.\nThere are a few possible metrics to consider for choosing a “best” model.\n\n\n2.9.1 Mean squared error\nWe might want to minimize the mean squared error, \\(\\text E[(y-\\hat y)^2]\\), for new observations that weren’t in our data set when we fit the model.\nUnfortunately, \\[\\frac{1}{n}\\sum_{i=1}^n (y_i-\\hat y_i)^2\\] gives a biased estimate of \\(\\text E[(y-\\hat y)^2]\\) for new data. If we want an unbiased estimate, we will have to be clever.\n\nCross-validation\n\nShow R codedata(\"carbohydrate\", package = \"dobson\")\nlibrary(cvTools)\nfull_model <- lm(carbohydrate ~ ., data = carbohydrate)\ncv_full = \n full_model |> cvFit(\n data = carbohydrate, K = 5, R = 10,\n y = carbohydrate$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", "crumbs": [ "Generalized Linear Models", "2  Linear (Gaussian) Models" @@ -204,7 +204,7 @@ "href": "Linear-models-overview.html#categorical-covariates-with-more-than-two-levels", "title": "\n2  Linear (Gaussian) Models\n", "section": "\n2.10 Categorical covariates with more than two levels", - "text": "2.10 Categorical covariates with more than two levels\n\n2.10.1 Example: birthweight\n\nIn the birthweight example, the variable sex had only two observed values:\n\nShow R codeunique(bw$sex)\n#> [1] female male \n#> Levels: female male\n\n\nIf there are more than two observed values, we can’t just use a single variable with 0s and 1s.\n\n2.10.2 \n\nFor example, Table 2.12 shows the (in)famous iris data (Anderson (1935)), and Table 2.13 provides summary statistics. The data include three species: “setosa”, “versicolor”, and “virginica”.\n\n\nShow R codehead(iris)\n\n\nTable 2.12: The iris data\n\n\n\n\nSepal.Length\nSepal.Width\nPetal.Length\nPetal.Width\nSpecies\n\n\n\n5.1\n3.5\n1.4\n0.2\nsetosa\n\n\n4.9\n3.0\n1.4\n0.2\nsetosa\n\n\n4.7\n3.2\n1.3\n0.2\nsetosa\n\n\n4.6\n3.1\n1.5\n0.2\nsetosa\n\n\n5.0\n3.6\n1.4\n0.2\nsetosa\n\n\n5.4\n3.9\n1.7\n0.4\nsetosa\n\n\n\n\n\n\n\n\n\nShow R codelibrary(table1)\ntable1(\n x = ~ . | Species,\n data = iris,\n overall = FALSE\n)\n\n\nTable 2.13: Summary statistics for the iris data\n\n\n\n\n\n\n\n\n\n\n\n\n\nsetosa(N=50)\nversicolor(N=50)\nvirginica(N=50)\n\n\n\nSepal.Length\n\n\n\n\n\nMean (SD)\n5.01 (0.352)\n5.94 (0.516)\n6.59 (0.636)\n\n\nMedian [Min, Max]\n5.00 [4.30, 5.80]\n5.90 [4.90, 7.00]\n6.50 [4.90, 7.90]\n\n\nSepal.Width\n\n\n\n\n\nMean (SD)\n3.43 (0.379)\n2.77 (0.314)\n2.97 (0.322)\n\n\nMedian [Min, Max]\n3.40 [2.30, 4.40]\n2.80 [2.00, 3.40]\n3.00 [2.20, 3.80]\n\n\nPetal.Length\n\n\n\n\n\nMean (SD)\n1.46 (0.174)\n4.26 (0.470)\n5.55 (0.552)\n\n\nMedian [Min, Max]\n1.50 [1.00, 1.90]\n4.35 [3.00, 5.10]\n5.55 [4.50, 6.90]\n\n\nPetal.Width\n\n\n\n\n\nMean (SD)\n0.246 (0.105)\n1.33 (0.198)\n2.03 (0.275)\n\n\nMedian [Min, Max]\n0.200 [0.100, 0.600]\n1.30 [1.00, 1.80]\n2.00 [1.40, 2.50]\n\n\n\n\n\n\n\n\n\n\n\nIf we want to model Sepal.Length by species, we could create a variable \\(X\\) that represents “setosa” as \\(X=1\\), “virginica” as \\(X=2\\), and “versicolor” as \\(X=3\\).\n\nShow R codedata(iris) # this step is not always necessary, but ensures you're starting \n# from the original version of a dataset stored in a loaded package\n\niris = \n iris |> \n tibble() |>\n mutate(\n X = case_when(\n Species == \"setosa\" ~ 1,\n Species == \"virginica\" ~ 2,\n Species == \"versicolor\" ~ 3\n )\n )\n\niris |> \n distinct(Species, X)\n\n\nTable 2.14: iris data with numeric coding of species\n\n\n\n\nSpecies\nX\n\n\n\nsetosa\n1\n\n\nversicolor\n3\n\n\nvirginica\n2\n\n\n\n\n\n\n\n\nThen we could fit a model like:\n\nShow R codeiris_lm1 = lm(Sepal.Length ~ X, data = iris)\niris_lm1 |> parameters() |> print_md()\n\n\nTable 2.15: Model of iris data with numeric coding of Species\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(148)\np\n\n\n\n(Intercept)\n4.91\n0.16\n(4.60, 5.23)\n30.83\n< .001\n\n\nX\n0.47\n0.07\n(0.32, 0.61)\n6.30\n< .001\n\n\n\n\n\n\n\n\n\n2.10.3 Let’s see how that model looks:\n\nShow R codeiris_plot1 = iris |> \n ggplot(\n aes(\n x = X, \n y = Sepal.Length)\n ) +\n geom_point(alpha = .1) +\n geom_abline(\n intercept = coef(iris_lm1)[1], \n slope = coef(iris_lm1)[2]) +\n theme_bw(base_size = 18)\nprint(iris_plot1)\n\n\n\nFigure 2.20: Model of iris data with numeric coding of Species\n\n\n\n\n\n\n\nWe have forced the model to use a straight line for the three estimated means. Maybe not a good idea?\n\n2.10.4 Let’s see what R does with categorical variables by default:\n\nShow R codeiris_lm2 = lm(Sepal.Length ~ Species, data = iris)\niris_lm2 |> parameters() |> print_md()\n\n\nTable 2.16: Model of iris data with Species as a categorical variable\n\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(147)\np\n\n\n\n(Intercept)\n5.01\n0.07\n(4.86, 5.15)\n68.76\n< .001\n\n\nSpecies (versicolor)\n0.93\n0.10\n(0.73, 1.13)\n9.03\n< .001\n\n\nSpecies (virginica)\n1.58\n0.10\n(1.38, 1.79)\n15.37\n< .001\n\n\n\n\n\n\n\n\n\n2.10.5 Re-parametrize with no intercept\nIf you don’t want the default and offset option, you can use “-1” like we’ve seen previously:\n\nShow R codeiris.lm2b = lm(Sepal.Length ~ Species - 1, data = iris)\niris.lm2b |> parameters() |> print_md()\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(147)\np\n\n\n\nSpecies (setosa)\n5.01\n0.07\n(4.86, 5.15)\n68.76\n< .001\n\n\nSpecies (versicolor)\n5.94\n0.07\n(5.79, 6.08)\n81.54\n< .001\n\n\nSpecies (virginica)\n6.59\n0.07\n(6.44, 6.73)\n90.49\n< .001\n\n\n\n\n\n\n2.10.6 Let’s see what these new models look like:\n\nShow R codeiris_plot2 = \n iris |> \n mutate(\n predlm2 = predict(iris_lm2)) |> \n arrange(X) |> \n ggplot(aes(x = X, y = Sepal.Length)) +\n geom_point(alpha = .1) +\n geom_line(aes(y = predlm2), col = \"red\") +\n geom_abline(\n intercept = coef(iris_lm1)[1], \n slope = coef(iris_lm1)[2]) + \n theme_bw(base_size = 18)\n\nprint(iris_plot2)\n\n\n\nFigure 2.21\n\n\n\n\n\n\n\n\n2.10.7 Let’s see how R did that:\n\nShow R codeformula(iris_lm2)\n#> Sepal.Length ~ Species\nmodel.matrix(iris_lm2) |> as_tibble() |> unique()\n\n\n\n(Intercept)\nSpeciesversicolor\nSpeciesvirginica\n\n\n\n1\n0\n0\n\n\n1\n1\n0\n\n\n1\n0\n1\n\n\n\n\n\nThis is called a “corner point parametrization”.\n\nShow R codeformula(iris.lm2b)\n#> Sepal.Length ~ Species - 1\nmodel.matrix(iris.lm2b) |> as_tibble() |> unique()\n\n\n\nSpeciessetosa\nSpeciesversicolor\nSpeciesvirginica\n\n\n\n1\n0\n0\n\n\n0\n1\n0\n\n\n0\n0\n1\n\n\n\n\n\nThis can be called a “group point parametrization”.\nThere are more options; see Dobson and Barnett (2018) §6.4.1 and the codingMatrices package vignette (Venables (2023)).", + "text": "2.10 Categorical covariates with more than two levels\n\n2.10.1 Example: birthweight\n\nIn the birthweight example, the variable sex had only two observed values:\n\nShow R codeunique(bw$sex)\n#> [1] female male \n#> Levels: female male\n\n\nIf there are more than two observed values, we can’t just use a single variable with 0s and 1s.\n\n2.10.2 \n\nFor example, Table 2.13 shows the (in)famous iris data (Anderson (1935)), and Table 2.14 provides summary statistics. The data include three species: “setosa”, “versicolor”, and “virginica”.\n\n\nShow R codehead(iris)\n\n\nTable 2.13: The iris data\n\n\n\n\nSepal.Length\nSepal.Width\nPetal.Length\nPetal.Width\nSpecies\n\n\n\n5.1\n3.5\n1.4\n0.2\nsetosa\n\n\n4.9\n3.0\n1.4\n0.2\nsetosa\n\n\n4.7\n3.2\n1.3\n0.2\nsetosa\n\n\n4.6\n3.1\n1.5\n0.2\nsetosa\n\n\n5.0\n3.6\n1.4\n0.2\nsetosa\n\n\n5.4\n3.9\n1.7\n0.4\nsetosa\n\n\n\n\n\n\n\n\n\nShow R codelibrary(table1)\ntable1(\n x = ~ . | Species,\n data = iris,\n overall = FALSE\n)\n\n\nTable 2.14: Summary statistics for the iris data\n\n\n\n\n\n\n\n\n\n\n\n\n\nsetosa(N=50)\nversicolor(N=50)\nvirginica(N=50)\n\n\n\nSepal.Length\n\n\n\n\n\nMean (SD)\n5.01 (0.352)\n5.94 (0.516)\n6.59 (0.636)\n\n\nMedian [Min, Max]\n5.00 [4.30, 5.80]\n5.90 [4.90, 7.00]\n6.50 [4.90, 7.90]\n\n\nSepal.Width\n\n\n\n\n\nMean (SD)\n3.43 (0.379)\n2.77 (0.314)\n2.97 (0.322)\n\n\nMedian [Min, Max]\n3.40 [2.30, 4.40]\n2.80 [2.00, 3.40]\n3.00 [2.20, 3.80]\n\n\nPetal.Length\n\n\n\n\n\nMean (SD)\n1.46 (0.174)\n4.26 (0.470)\n5.55 (0.552)\n\n\nMedian [Min, Max]\n1.50 [1.00, 1.90]\n4.35 [3.00, 5.10]\n5.55 [4.50, 6.90]\n\n\nPetal.Width\n\n\n\n\n\nMean (SD)\n0.246 (0.105)\n1.33 (0.198)\n2.03 (0.275)\n\n\nMedian [Min, Max]\n0.200 [0.100, 0.600]\n1.30 [1.00, 1.80]\n2.00 [1.40, 2.50]\n\n\n\n\n\n\n\n\n\n\n\nIf we want to model Sepal.Length by species, we could create a variable \\(X\\) that represents “setosa” as \\(X=1\\), “virginica” as \\(X=2\\), and “versicolor” as \\(X=3\\).\n\nShow R codedata(iris) # this step is not always necessary, but ensures you're starting \n# from the original version of a dataset stored in a loaded package\n\niris = \n iris |> \n tibble() |>\n mutate(\n X = case_when(\n Species == \"setosa\" ~ 1,\n Species == \"virginica\" ~ 2,\n Species == \"versicolor\" ~ 3\n )\n )\n\niris |> \n distinct(Species, X)\n\n\nTable 2.15: iris data with numeric coding of species\n\n\n\n\nSpecies\nX\n\n\n\nsetosa\n1\n\n\nversicolor\n3\n\n\nvirginica\n2\n\n\n\n\n\n\n\n\nThen we could fit a model like:\n\nShow R codeiris_lm1 = lm(Sepal.Length ~ X, data = iris)\niris_lm1 |> parameters() |> print_md()\n\n\nTable 2.16: Model of iris data with numeric coding of Species\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(148)\np\n\n\n\n(Intercept)\n4.91\n0.16\n(4.60, 5.23)\n30.83\n< .001\n\n\nX\n0.47\n0.07\n(0.32, 0.61)\n6.30\n< .001\n\n\n\n\n\n\n\n\n\n2.10.3 Let’s see how that model looks:\n\nShow R codeiris_plot1 = iris |> \n ggplot(\n aes(\n x = X, \n y = Sepal.Length)\n ) +\n geom_point(alpha = .1) +\n geom_abline(\n intercept = coef(iris_lm1)[1], \n slope = coef(iris_lm1)[2]) +\n theme_bw(base_size = 18)\nprint(iris_plot1)\n\n\n\nFigure 2.20: Model of iris data with numeric coding of Species\n\n\n\n\n\n\n\nWe have forced the model to use a straight line for the three estimated means. Maybe not a good idea?\n\n2.10.4 Let’s see what R does with categorical variables by default:\n\nShow R codeiris_lm2 = lm(Sepal.Length ~ Species, data = iris)\niris_lm2 |> parameters() |> print_md()\n\n\nTable 2.17: Model of iris data with Species as a categorical variable\n\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(147)\np\n\n\n\n(Intercept)\n5.01\n0.07\n(4.86, 5.15)\n68.76\n< .001\n\n\nSpecies (versicolor)\n0.93\n0.10\n(0.73, 1.13)\n9.03\n< .001\n\n\nSpecies (virginica)\n1.58\n0.10\n(1.38, 1.79)\n15.37\n< .001\n\n\n\n\n\n\n\n\n\n2.10.5 Re-parametrize with no intercept\nIf you don’t want the default and offset option, you can use “-1” like we’ve seen previously:\n\nShow R codeiris.lm2b = lm(Sepal.Length ~ Species - 1, data = iris)\niris.lm2b |> parameters() |> print_md()\n\n\n\n\n\n\n\n\n\n\n\nParameter\nCoefficient\nSE\n95% CI\nt(147)\np\n\n\n\nSpecies (setosa)\n5.01\n0.07\n(4.86, 5.15)\n68.76\n< .001\n\n\nSpecies (versicolor)\n5.94\n0.07\n(5.79, 6.08)\n81.54\n< .001\n\n\nSpecies (virginica)\n6.59\n0.07\n(6.44, 6.73)\n90.49\n< .001\n\n\n\n\n\n\n2.10.6 Let’s see what these new models look like:\n\nShow R codeiris_plot2 = \n iris |> \n mutate(\n predlm2 = predict(iris_lm2)) |> \n arrange(X) |> \n ggplot(aes(x = X, y = Sepal.Length)) +\n geom_point(alpha = .1) +\n geom_line(aes(y = predlm2), col = \"red\") +\n geom_abline(\n intercept = coef(iris_lm1)[1], \n slope = coef(iris_lm1)[2]) + \n theme_bw(base_size = 18)\n\nprint(iris_plot2)\n\n\n\nFigure 2.21\n\n\n\n\n\n\n\n\n2.10.7 Let’s see how R did that:\n\nShow R codeformula(iris_lm2)\n#> Sepal.Length ~ Species\nmodel.matrix(iris_lm2) |> as_tibble() |> unique()\n\n\n\n(Intercept)\nSpeciesversicolor\nSpeciesvirginica\n\n\n\n1\n0\n0\n\n\n1\n1\n0\n\n\n1\n0\n1\n\n\n\n\n\nThis is called a “corner point parametrization”.\n\nShow R codeformula(iris.lm2b)\n#> Sepal.Length ~ Species - 1\nmodel.matrix(iris.lm2b) |> as_tibble() |> unique()\n\n\n\nSpeciessetosa\nSpeciesversicolor\nSpeciesvirginica\n\n\n\n1\n0\n0\n\n\n0\n1\n0\n\n\n0\n0\n1\n\n\n\n\n\nThis can be called a “group point parametrization”.\nThere are more options; see Dobson and Barnett (2018) §6.4.1 and the codingMatrices package vignette (Venables (2023)).", "crumbs": [ "Generalized Linear Models", "2  Linear (Gaussian) Models" @@ -215,7 +215,7 @@ "href": "Linear-models-overview.html#ordinal-covariates", "title": "\n2  Linear (Gaussian) Models\n", "section": "\n2.11 Ordinal covariates", - "text": "2.11 Ordinal covariates\n(c.f. Dobson and Barnett (2018) §2.4.4)\n\n\nWe can create ordinal variables in R using the ordered() function4.\n\n\nExample 2.3  \n\nShow R codeurl = paste0(\n \"https://regression.ucsf.edu/sites/g/files/tkssra6706/\",\n \"f/wysiwyg/home/data/hersdata.dta\")\nlibrary(haven)\nhers = read_dta(url)\n\n\n\nShow R codehers |> head()\n\n\nTable 2.17: HERS dataset\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nHT\nage\nraceth\nnonwhite\nsmoking\ndrinkany\nexercise\nphysact\nglobrat\npoorfair\nmedcond\nhtnmeds\nstatins\ndiabetes\ndmpills\ninsulin\nweight\nBMI\nwaist\nWHR\nglucose\nweight1\nBMI1\nwaist1\nWHR1\nglucose1\ntchol\nLDL\nHDL\nTG\ntchol1\nLDL1\nHDL1\nTG1\nSBP\nDBP\nage10\n\n\n\n0\n70\n2\n1\n0\n0\n0\n5\n3\n0\n0\n1\n1\n0\n0\n0\n73.8\n23.69\n96.0\n0.932\n84\n73.6\n23.63\n93.0\n0.912\n94\n189\n122.4\n52\n73\n201\n137.6\n48\n77\n138\n78\n7.0\n\n\n0\n62\n2\n1\n0\n0\n0\n1\n3\n0\n1\n1\n0\n0\n0\n0\n70.9\n28.62\n93.0\n0.964\n111\n73.4\n28.89\n95.0\n0.964\n78\n307\n241.6\n44\n107\n216\n150.6\n48\n87\n118\n70\n6.2\n\n\n1\n69\n1\n0\n0\n0\n0\n3\n3\n0\n0\n1\n0\n1\n0\n0\n102.0\n42.51\n110.2\n0.782\n114\n96.1\n40.73\n103.0\n0.774\n98\n254\n166.2\n57\n154\n254\n156.0\n66\n160\n134\n78\n6.9\n\n\n0\n64\n1\n0\n1\n1\n0\n1\n3\n0\n1\n1\n0\n0\n0\n0\n64.4\n24.39\n87.0\n0.877\n94\n58.6\n22.52\n77.0\n0.802\n93\n204\n116.2\n56\n159\n207\n122.6\n57\n137\n152\n72\n6.4\n\n\n0\n65\n1\n0\n0\n0\n0\n2\n3\n0\n0\n0\n0\n0\n0\n0\n57.9\n21.90\n77.0\n0.794\n101\n58.9\n22.28\n76.5\n0.757\n92\n214\n150.6\n42\n107\n235\n172.2\n35\n139\n175\n95\n6.5\n\n\n1\n68\n2\n1\n0\n1\n0\n3\n3\n0\n0\n0\n0\n0\n0\n0\n60.9\n29.05\n96.0\n1.000\n116\n57.7\n27.52\n86.0\n0.910\n115\n212\n137.8\n52\n111\n202\n126.6\n53\n112\n174\n98\n6.8\n\n\n\n\n\n\n\n\n\n\n\nShow R code\n# C(contr = codingMatrices::contr.diff)\n\n\n\n\n\n\n\n\nAnderson, Edgar. 1935. “The Irises of the Gaspe Peninsula.” Bulletin of American Iris Society 59: 2–5.\n\n\nChatterjee, Samprit, and Ali S Hadi. 2015. Regression Analysis by Example. John Wiley & Sons. https://www.wiley.com/en-us/Regression+Analysis+by+Example%2C+4th+Edition-p-9780470055458.\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\nDunn, Peter K, Gordon K Smyth, et al. 2018. Generalized Linear Models with Examples in r. Vol. 53. Springer. https://link.springer.com/book/10.1007/978-1-4419-0118-7.\n\n\nHarrell, Frank E. 2015. Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis. 2nd ed. Springer. https://doi.org/10.1007/978-3-319-19425-7.\n\n\nJames, Gareth, Daniela Witten, Trevor Hastie, Robert Tibshirani, et al. 2013. An Introduction to Statistical Learning. Vol. 112. Springer. https://www.statlearning.com/.\n\n\nKleinbaum, David G, and Mitchel Klein. 2010. Logistic Regression. 3rd ed. Springer. https://link.springer.com/book/10.1007/978-1-4419-1742-3.\n\n\n———. 2012. Survival Analysis a Self-Learning Text. 3rd ed. Springer. https://link.springer.com/book/10.1007/978-1-4419-6646-9.\n\n\nKleinbaum, David G, Lawrence L Kupper, Azhar Nizam, K Muller, and ES Rosenberg. 2014. Applied Regression Analysis and Other Multivariable Methods. 5th ed. Cengage Learning. https://www.cengage.com/c/applied-regression-analysis-and-other-multivariable-methods-5e-kleinbaum/9781285051086/.\n\n\nKutner, Michael H, Christopher J Nachtsheim, John Neter, and William Li. 2005. Applied Linear Statistical Models. McGraw-Hill.\n\n\nPolin, Richard A, William W Fox, and Steven H Abman. 2011. Fetal and Neonatal Physiology. 4th ed. Elsevier health sciences.\n\n\nSeber, George AF, and Alan J Lee. 2012. Linear Regression Analysis. 2nd ed. John Wiley & Sons. https://www.wiley.com/en-us/Linear+Regression+Analysis%2C+2nd+Edition-p-9781118274422.\n\n\nVenables, Bill. 2023. codingMatrices: Alternative Factor Coding Matrices for Linear Model Formulae. https://CRAN.R-project.org/package=codingMatrices.\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.", + "text": "2.11 Ordinal covariates\n(c.f. Dobson and Barnett (2018) §2.4.4)\n\n\nWe can create ordinal variables in R using the ordered() function4.\n\n\nExample 2.3  \n\nShow R codeurl = paste0(\n \"https://regression.ucsf.edu/sites/g/files/tkssra6706/\",\n \"f/wysiwyg/home/data/hersdata.dta\")\nlibrary(haven)\nhers = read_dta(url)\n\n\n\nShow R codehers |> head()\n\n\nTable 2.18: HERS dataset\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nHT\nage\nraceth\nnonwhite\nsmoking\ndrinkany\nexercise\nphysact\nglobrat\npoorfair\nmedcond\nhtnmeds\nstatins\ndiabetes\ndmpills\ninsulin\nweight\nBMI\nwaist\nWHR\nglucose\nweight1\nBMI1\nwaist1\nWHR1\nglucose1\ntchol\nLDL\nHDL\nTG\ntchol1\nLDL1\nHDL1\nTG1\nSBP\nDBP\nage10\n\n\n\n0\n70\n2\n1\n0\n0\n0\n5\n3\n0\n0\n1\n1\n0\n0\n0\n73.8\n23.69\n96.0\n0.932\n84\n73.6\n23.63\n93.0\n0.912\n94\n189\n122.4\n52\n73\n201\n137.6\n48\n77\n138\n78\n7.0\n\n\n0\n62\n2\n1\n0\n0\n0\n1\n3\n0\n1\n1\n0\n0\n0\n0\n70.9\n28.62\n93.0\n0.964\n111\n73.4\n28.89\n95.0\n0.964\n78\n307\n241.6\n44\n107\n216\n150.6\n48\n87\n118\n70\n6.2\n\n\n1\n69\n1\n0\n0\n0\n0\n3\n3\n0\n0\n1\n0\n1\n0\n0\n102.0\n42.51\n110.2\n0.782\n114\n96.1\n40.73\n103.0\n0.774\n98\n254\n166.2\n57\n154\n254\n156.0\n66\n160\n134\n78\n6.9\n\n\n0\n64\n1\n0\n1\n1\n0\n1\n3\n0\n1\n1\n0\n0\n0\n0\n64.4\n24.39\n87.0\n0.877\n94\n58.6\n22.52\n77.0\n0.802\n93\n204\n116.2\n56\n159\n207\n122.6\n57\n137\n152\n72\n6.4\n\n\n0\n65\n1\n0\n0\n0\n0\n2\n3\n0\n0\n0\n0\n0\n0\n0\n57.9\n21.90\n77.0\n0.794\n101\n58.9\n22.28\n76.5\n0.757\n92\n214\n150.6\n42\n107\n235\n172.2\n35\n139\n175\n95\n6.5\n\n\n1\n68\n2\n1\n0\n1\n0\n3\n3\n0\n0\n0\n0\n0\n0\n0\n60.9\n29.05\n96.0\n1.000\n116\n57.7\n27.52\n86.0\n0.910\n115\n212\n137.8\n52\n111\n202\n126.6\n53\n112\n174\n98\n6.8\n\n\n\n\n\n\n\n\n\n\n\nShow R code\n# C(contr = codingMatrices::contr.diff)\n\n\n\n\n\n\n\n\nAnderson, Edgar. 1935. “The Irises of the Gaspe Peninsula.” Bulletin of American Iris Society 59: 2–5.\n\n\nChatterjee, Samprit, and Ali S Hadi. 2015. Regression Analysis by Example. John Wiley & Sons. https://www.wiley.com/en-us/Regression+Analysis+by+Example%2C+4th+Edition-p-9780470055458.\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\nDunn, Peter K, Gordon K Smyth, et al. 2018. Generalized Linear Models with Examples in r. Vol. 53. Springer. https://link.springer.com/book/10.1007/978-1-4419-0118-7.\n\n\nHarrell, Frank E. 2015. Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis. 2nd ed. Springer. https://doi.org/10.1007/978-3-319-19425-7.\n\n\nJames, Gareth, Daniela Witten, Trevor Hastie, Robert Tibshirani, et al. 2013. An Introduction to Statistical Learning. Vol. 112. Springer. https://www.statlearning.com/.\n\n\nKleinbaum, David G, and Mitchel Klein. 2010. Logistic Regression. 3rd ed. Springer. https://link.springer.com/book/10.1007/978-1-4419-1742-3.\n\n\n———. 2012. Survival Analysis a Self-Learning Text. 3rd ed. Springer. https://link.springer.com/book/10.1007/978-1-4419-6646-9.\n\n\nKleinbaum, David G, Lawrence L Kupper, Azhar Nizam, K Muller, and ES Rosenberg. 2014. Applied Regression Analysis and Other Multivariable Methods. 5th ed. Cengage Learning. https://www.cengage.com/c/applied-regression-analysis-and-other-multivariable-methods-5e-kleinbaum/9781285051086/.\n\n\nKutner, Michael H, Christopher J Nachtsheim, John Neter, and William Li. 2005. Applied Linear Statistical Models. McGraw-Hill.\n\n\nPolin, Richard A, William W Fox, and Steven H Abman. 2011. Fetal and Neonatal Physiology. 4th ed. Elsevier health sciences.\n\n\nSeber, George AF, and Alan J Lee. 2012. Linear Regression Analysis. 2nd ed. John Wiley & Sons. https://www.wiley.com/en-us/Linear+Regression+Analysis%2C+2nd+Edition-p-9781118274422.\n\n\nVenables, Bill. 2023. codingMatrices: Alternative Factor Coding Matrices for Linear Model Formulae. https://CRAN.R-project.org/package=codingMatrices.\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", "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&= 3.7143\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\\]", + "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\\]", "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) = 7.0049\\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.", + "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.", "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\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.", + "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.", "crumbs": [ "Generalized Linear Models", "3  Models for Binary Outcomes" @@ -314,7 +314,7 @@ "href": "logistic-regression.html#multiple-logistic-regression", "title": "3  Models for Binary Outcomes", "section": "3.7 Multiple logistic regression", - "text": "3.7 Multiple logistic regression\n\n3.7.1 Coronary heart disease (WCGS) study data\n\nLet’s use the data from the Western Collaborative Group Study (WCGS) (Rosenman et al. (1975)) to explore multiple logistic regression:\n\nFrom Vittinghoff et al. (2012):\n“The Western Collaborative Group Study (WCGS) was a large epidemiological study designed to investigate the association between the”type A” behavior pattern and coronary heart disease (CHD)“.\nFrom Wikipedia, “Type A and Type B personality theory”:\n“The hypothesis describes Type A individuals as outgoing, ambitious, rigidly organized, highly status-conscious, impatient, anxious, proactive, and concerned with time management….\nThe hypothesis describes Type B individuals as a contrast to those of Type A. Type B personalities, by definition, are noted to live at lower stress levels. They typically work steadily and may enjoy achievement, although they have a greater tendency to disregard physical or mental stress when they do not achieve.”\n\n\nStudy design\nfrom ?faraway::wcgs:\n3154 healthy young men aged 39-59 from the San Francisco area were assessed for their personality type. All were free from coronary heart disease at the start of the research. Eight and a half years later change in CHD status was recorded.\nDetails (from faraway::wcgs)\nThe WCGS began in 1960 with 3,524 male volunteers who were employed by 11 California companies. Subjects were 39 to 59 years old and free of heart disease as determined by electrocardiogram. After the initial screening, the study population dropped to 3,154 and the number of companies to 10 because of various exclusions. The cohort comprised both blue- and white-collar employees.\n\n\n\n\n3.7.2 Baseline data collection\nsocio-demographic characteristics:\n\nage\neducation\nmarital status\nincome\noccupation\nphysical and physiological including:\nheight\nweight\nblood pressure\nelectrocardiogram\ncorneal arcus;\n\n\nbiochemical measurements:\n\ncholesterol and lipoprotein fractions;\nmedical and family history and use of medications;\n\n\nbehavioral data:\n\nType A interview,\nsmoking,\nexercise\nalcohol use.\n\n\nLater surveys added data on:\n\nanthropometry\ntriglycerides\nJenkins Activity Survey\ncaffeine use\n\nAverage follow-up continued for 8.5 years with repeat examinations.\n\n\n3.7.3 Load the data\nHere, I load the data:\n\n\nShow R code\n### load the data directly from a UCSF website:\nlibrary(haven)\nurl = paste0(\n # I'm breaking up the url into two chunks for readability\n \"https://regression.ucsf.edu/sites/g/files/\",\n \"tkssra6706/f/wysiwyg/home/data/wcgs.dta\")\nwcgs = haven::read_dta(url)\n\n\n\n\nShow R code\nwcgs |> head()\n\n\n\n\nTable 3.6: wcgs data\n\n\n\n\n \n\n\n\n\n\n\n\n\n3.7.4 Data cleaning\n\nNow let’s do some data cleaning\n\n\n\nShow R code\nlibrary(arsenal) # provides `set_labels()`\nlibrary(forcats) # provides `as_factor()`\nlibrary(haven)\nlibrary(plotly)\nwcgs = wcgs |> \n mutate(\n age = age |> \n arsenal::set_labels(\"Age (years)\"),\n \n arcus = \n arcus |> \n as.logical() |> \n arsenal::set_labels(\"Arcus Senilis\"),\n \n time169 = \n time169 |> \n as.numeric() |> \n arsenal::set_labels(\"Observation (follow up) time (days)\"),\n \n dibpat =\n dibpat |> \n as_factor() |> \n relevel(ref = \"Type A\") |> \n arsenal::set_labels(\"Behavioral Pattern\"),\n \n typchd69 = typchd69 |> \n labelled(\n label = \"Type of CHD Event\",\n labels = \n c(\n \"None\" = 0, \n \"infdeath\" = 1,\n \"silent\" = 2,\n \"angina\" = 3)),\n \n # turn stata-style labelled variables in to R-style factors:\n across(\n where(is.labelled), \n haven::as_factor)\n )\n\n\n\n\n3.7.5 What’s in the data\nHere’s a table of the data:\n\nShow R code\nwcgs |>\n select(-c(id, uni, t1)) |>\n tableby(chd69 ~ ., data = _) |>\n summary(\n pfootnote = TRUE,\n title =\n \"Baseline characteristics by CHD status at end of follow-up\")\n\n\nBaseline characteristics by CHD status at end of follow-up\n\n\n\n\n\n\n\n\n\n\nNo (N=2897)\nYes (N=257)\nTotal (N=3154)\np value\n\n\n\n\nAge (years)\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n46.082 (5.457)\n48.490 (5.801)\n46.279 (5.524)\n\n\n\n   Range\n39.000 - 59.000\n39.000 - 59.000\n39.000 - 59.000\n\n\n\nArcus Senilis\n\n\n\n< 0.0012\n\n\n   N-Miss\n0\n2\n2\n\n\n\n   FALSE\n2058 (71.0%)\n153 (60.0%)\n2211 (70.1%)\n\n\n\n   TRUE\n839 (29.0%)\n102 (40.0%)\n941 (29.9%)\n\n\n\nBehavioral Pattern\n\n\n\n< 0.0012\n\n\n   A1\n234 (8.1%)\n30 (11.7%)\n264 (8.4%)\n\n\n\n   A2\n1177 (40.6%)\n148 (57.6%)\n1325 (42.0%)\n\n\n\n   B3\n1155 (39.9%)\n61 (23.7%)\n1216 (38.6%)\n\n\n\n   B4\n331 (11.4%)\n18 (7.0%)\n349 (11.1%)\n\n\n\nBody Mass Index (kg/m2)\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n24.471 (2.561)\n25.055 (2.579)\n24.518 (2.567)\n\n\n\n   Range\n11.191 - 37.653\n19.225 - 38.947\n11.191 - 38.947\n\n\n\nTotal Cholesterol\n\n\n\n< 0.0011\n\n\n   N-Miss\n12\n0\n12\n\n\n\n   Mean (SD)\n224.261 (42.217)\n250.070 (49.396)\n226.372 (43.420)\n\n\n\n   Range\n103.000 - 400.000\n155.000 - 645.000\n103.000 - 645.000\n\n\n\nDiastolic Blood Pressure\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n81.723 (9.621)\n85.315 (10.311)\n82.016 (9.727)\n\n\n\n   Range\n58.000 - 150.000\n64.000 - 122.000\n58.000 - 150.000\n\n\n\nBehavioral Pattern\n\n\n\n< 0.0012\n\n\n   Type A\n1411 (48.7%)\n178 (69.3%)\n1589 (50.4%)\n\n\n\n   Type B\n1486 (51.3%)\n79 (30.7%)\n1565 (49.6%)\n\n\n\nHeight (inches)\n\n\n\n0.2901\n\n\n   Mean (SD)\n69.764 (2.539)\n69.938 (2.410)\n69.778 (2.529)\n\n\n\n   Range\n60.000 - 78.000\n63.000 - 77.000\n60.000 - 78.000\n\n\n\nLn of Systolic Blood Pressure\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n4.846 (0.110)\n4.900 (0.125)\n4.850 (0.112)\n\n\n\n   Range\n4.585 - 5.438\n4.605 - 5.298\n4.585 - 5.438\n\n\n\nLn of Weight\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n5.126 (0.123)\n5.155 (0.118)\n5.128 (0.123)\n\n\n\n   Range\n4.357 - 5.670\n4.868 - 5.768\n4.357 - 5.768\n\n\n\nCigarettes per day\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n11.151 (14.329)\n16.665 (15.657)\n11.601 (14.518)\n\n\n\n   Range\n0.000 - 99.000\n0.000 - 60.000\n0.000 - 99.000\n\n\n\nSystolic Blood Pressure\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n128.034 (14.746)\n135.385 (17.473)\n128.633 (15.118)\n\n\n\n   Range\n98.000 - 230.000\n100.000 - 200.000\n98.000 - 230.000\n\n\n\nCurrent smoking\n\n\n\n< 0.0012\n\n\n   No\n1554 (53.6%)\n98 (38.1%)\n1652 (52.4%)\n\n\n\n   Yes\n1343 (46.4%)\n159 (61.9%)\n1502 (47.6%)\n\n\n\nObservation (follow up) time (days)\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n2775.158 (562.205)\n1654.700 (859.297)\n2683.859 (666.524)\n\n\n\n   Range\n238.000 - 3430.000\n18.000 - 3229.000\n18.000 - 3430.000\n\n\n\nType of CHD Event\n\n\n\n\n\n\n   None\n0 (0.0%)\n0 (0.0%)\n0 (0.0%)\n\n\n\n   infdeath\n2897 (100.0%)\n0 (0.0%)\n2897 (91.9%)\n\n\n\n   silent\n0 (0.0%)\n135 (52.5%)\n135 (4.3%)\n\n\n\n   angina\n0 (0.0%)\n71 (27.6%)\n71 (2.3%)\n\n\n\n   4\n0 (0.0%)\n51 (19.8%)\n51 (1.6%)\n\n\n\nWeight (lbs)\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n169.554 (21.010)\n174.463 (21.573)\n169.954 (21.096)\n\n\n\n   Range\n78.000 - 290.000\n130.000 - 320.000\n78.000 - 320.000\n\n\n\nWeight Category\n\n\n\n< 0.0012\n\n\n   < 140\n217 (7.5%)\n15 (5.8%)\n232 (7.4%)\n\n\n\n   140-170\n1440 (49.7%)\n98 (38.1%)\n1538 (48.8%)\n\n\n\n   170-200\n1049 (36.2%)\n122 (47.5%)\n1171 (37.1%)\n\n\n\n   > 200\n191 (6.6%)\n22 (8.6%)\n213 (6.8%)\n\n\n\nRECODE of age (Age)\n\n\n\n< 0.0012\n\n\n   35-40\n512 (17.7%)\n31 (12.1%)\n543 (17.2%)\n\n\n\n   41-45\n1036 (35.8%)\n55 (21.4%)\n1091 (34.6%)\n\n\n\n   46-50\n680 (23.5%)\n70 (27.2%)\n750 (23.8%)\n\n\n\n   51-55\n463 (16.0%)\n65 (25.3%)\n528 (16.7%)\n\n\n\n   56-60\n206 (7.1%)\n36 (14.0%)\n242 (7.7%)\n\n\n\n\n\nLinear Model ANOVA\nPearson’s Chi-squared test\n\n\n\n3.7.6 Data by age and personality type\nFor now, we will look at the interaction between age and personality type (dibpat). To make it easier to visualize the data, we summarize the event rates for each combination of age:\n\n\nShow R code\nchd_grouped_data = \n wcgs |> \n summarize(\n .by = c(age, dibpat),\n n = n(),\n `p(chd)` = mean(chd69 == \"Yes\") |> \n labelled(label = \"CHD Event by 1969\"),\n `odds(chd)` = `p(chd)`/(1-`p(chd)`),\n `logit(chd)` = log(`odds(chd)`)\n )\n\nchd_grouped_data\n\n\n\n \n\n\n\n\n\n3.7.7 Graphical exploration\n\n\nShow R code\nlibrary(ggplot2)\nlibrary(ggeasy)\nlibrary(scales)\nchd_plot_probs = \n chd_grouped_data |> \n ggplot(\n aes(\n x = age, \n y = `p(chd)`, \n col = dibpat)\n ) +\n geom_point(aes(size = n), alpha = .7) + \n scale_size(range = c(1,4)) +\n geom_line() +\n theme_bw() +\n ylab(\"P(CHD Event by 1969)\") +\n scale_y_continuous(\n labels = scales::label_percent(),\n sec.axis = sec_axis(\n ~ odds(.),\n name = \"odds(CHD Event by 1969)\")) +\n ggeasy::easy_labs() +\n theme(legend.position = \"bottom\")\n\nprint(chd_plot_probs)\n\n\n\n\n\nFigure 3.9: CHD rates by age group, probability scale\n\n\n\n\n\n\n\n\n\n\nOdds scale\n\n\nShow R code\ntrans_odds = trans_new(\n name = \"odds\", \n transform = odds, \n inverse = odds_inv)\n\nchd_plot_odds = chd_plot_probs + \n scale_y_continuous(\n trans = trans_odds, # this line changes the vertical spacing\n name = chd_plot_probs$labels$y,\n sec.axis = sec_axis(\n ~ odds(.),\n name = \"odds(CHD Event by 1969)\"))\n\nprint(chd_plot_odds)\n\n\n\n\n\nFigure 3.10: CHD rates by age group, odds spacing\n\n\n\n\n\n\n\n\n\n\n\nLog-odds (logit) scale\n\n\nShow R code\nchd_plot_logit = \n chd_grouped_data |> \n ggplot(\n aes(\n x = age, \n y = `logit(chd)`, \n col = dibpat)\n ) +\n geom_point(aes(size = n), alpha = .7) + \n scale_size(range = c(1,4)) +\n geom_line() +\n theme_bw() +\n ylab(\"log{odds(CHD Event by 1969)}\") +\n ggeasy::easy_labs()\n\nprint(chd_plot_logit)\n\n\n\n\n\nFigure 3.11: CHD data (logit-scale)\n\n\n\n\n\n\n\n\n\n\n\n3.7.8 Logistic regression models for CHD data\nHere, we fit stratified models for CHD by personality type.\n\n\nShow R code\nchd_glm_strat = glm(\n \"formula\" = chd69 == \"Yes\" ~ dibpat + dibpat:age - 1, \n \"data\" = wcgs,\n \"family\" = binomial(link = \"logit\")\n)\n\nchd_glm_strat |> parameters() |> print_md()\n\n\n\n\nTable 3.7: CHD model, stratified parametrization\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nParameter\nLog-Odds\nSE\n95% CI\nz\np\n\n\n\n\ndibpat (Type A)\n-5.50\n0.67\n(-6.83, -4.19)\n-8.18\n< .001\n\n\ndibpat (Type B)\n-5.80\n0.98\n(-7.73, -3.90)\n-5.95\n< .001\n\n\ndibpat (Type A) × age\n0.07\n0.01\n(0.05, 0.10)\n5.24\n< .001\n\n\ndibpat (Type B) × age\n0.06\n0.02\n(0.02, 0.10)\n3.01\n0.003\n\n\n\n\n\n\n\n\nWe can get the corresponding odds ratios (\\(e^{\\beta}\\)s) by passing exponentiate = TRUE to parameters():\n\n\nShow R code\nchd_glm_strat |> \n parameters(exponentiate = TRUE) |> \n print_md()\n\n\n\n\nTable 3.8: Odds ratio estimates for CHD model\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nParameter\nOdds Ratio\nSE\n95% CI\nz\np\n\n\n\n\ndibpat (Type A)\n4.09e-03\n2.75e-03\n(1.08e-03, 0.02)\n-8.18\n< .001\n\n\ndibpat (Type B)\n3.02e-03\n2.94e-03\n(4.40e-04, 0.02)\n-5.95\n< .001\n\n\ndibpat (Type A) × age\n1.07\n0.01\n(1.05, 1.10)\n5.24\n< .001\n\n\ndibpat (Type B) × age\n1.06\n0.02\n(1.02, 1.11)\n3.01\n0.003\n\n\n\n\n\n\n\n\n\n\n3.7.9 Models superimposed on data\nWe can graph our fitted models on each scale (probability, odds, log-odds).\n\n\nprobability scale\n\n\nShow R code\n\ncurve_type_A = function(x) \n{\n chd_glm_strat |> predict(\n type = \"response\",\n newdata = tibble(age = x, dibpat = \"Type A\"))\n}\n\ncurve_type_B = function(x) \n{\n chd_glm_strat |> predict(\n type = \"response\",\n newdata = tibble(age = x, dibpat = \"Type B\"))\n}\n\nchd_plot_probs_2 =\n chd_plot_probs +\n geom_function(\n fun = curve_type_A,\n aes(col = \"Type A\")\n ) +\n geom_function(\n fun = curve_type_B,\n aes(col = \"Type B\")\n )\nprint(chd_plot_probs_2)\n\n\n\n\n\n\n\n\n\n\n\n\nodds scale\n\n\nShow R code\ncurve_type_A = function(x) \n{\n chd_glm_strat |> predict(\n type = \"link\",\n newdata = tibble(age = x, dibpat = \"Type A\")) |> exp()\n}\ncurve_type_B = function(x) \n{\n chd_glm_strat |> predict(\n type = \"link\",\n newdata = tibble(age = x, dibpat = \"Type B\")) |> exp()\n}\n\nchd_plot_odds_2 =\n chd_plot_odds +\n geom_function(\n fun = curve_type_A,\n aes(col = \"Type A\")\n ) +\n geom_function(\n fun = curve_type_B,\n aes(col = \"Type B\")\n )\nprint(chd_plot_odds_2)\n\n\n\n\n\nFigure 3.12\n\n\n\n\n\n\n\n\n\n\n\nlog-odds (logit) scale\n\n\nShow R code\ncurve_type_A = function(x) \n{\n chd_glm_strat |> predict(\n type = \"link\",\n newdata = tibble(age = x, dibpat = \"Type A\"))\n}\ncurve_type_B = function(x) \n{\n chd_glm_strat |> predict(\n type = \"link\",\n newdata = tibble(age = x, dibpat = \"Type B\"))\n}\n\nchd_plot_logit_2 =\n chd_plot_logit +\n geom_function(\n fun = curve_type_A,\n aes(col = \"Type A\")\n ) +\n geom_function(\n fun = curve_type_B,\n aes(col = \"Type B\")\n )\n\nprint(chd_plot_logit_2)\n\n\n\n\n\nFigure 3.13\n\n\n\n\n\n\n\n\n\n\n\n3.7.10 reference-group and contrast parametrization\nWe can also use the corner-point parametrization (with reference groups and contrasts):\n\n\nShow R code\nchd_glm_contrasts = \n wcgs |> \n glm(\n \"data\" = _,\n \"formula\" = chd69 == \"Yes\" ~ dibpat*age, \n \"family\" = binomial(link = \"logit\")\n )\n\nchd_glm_contrasts |> \n parameters() |> \n print_md()\n\n\n\n\nTable 3.9: CHD model (corner-point parametrization)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nParameter\nLog-Odds\nSE\n95% CI\nz\np\n\n\n\n\n(Intercept)\n-5.50\n0.67\n(-6.83, -4.19)\n-8.18\n< .001\n\n\ndibpat (Type B)\n-0.30\n1.18\n(-2.63, 2.02)\n-0.26\n0.797\n\n\nage\n0.07\n0.01\n(0.05, 0.10)\n5.24\n< .001\n\n\ndibpat (Type B) × age\n-0.01\n0.02\n(-0.06, 0.04)\n-0.42\n0.674\n\n\n\n\n\n\n\n\n\nCompare with Table 3.8:\n\n\n\nExercise 3.15 If I give you model 1, how would you get the coefficients of model 2?", + "text": "3.7 Multiple logistic regression\n\n3.7.1 Coronary heart disease (WCGS) study data\n\nLet’s use the data from the Western Collaborative Group Study (WCGS) (Rosenman et al. (1975)) to explore multiple logistic regression:\n\nFrom Vittinghoff et al. (2012):\n“The Western Collaborative Group Study (WCGS) was a large epidemiological study designed to investigate the association between the”type A” behavior pattern and coronary heart disease (CHD)“.\nFrom Wikipedia, “Type A and Type B personality theory”:\n“The hypothesis describes Type A individuals as outgoing, ambitious, rigidly organized, highly status-conscious, impatient, anxious, proactive, and concerned with time management….\nThe hypothesis describes Type B individuals as a contrast to those of Type A. Type B personalities, by definition, are noted to live at lower stress levels. They typically work steadily and may enjoy achievement, although they have a greater tendency to disregard physical or mental stress when they do not achieve.”\n\n\nStudy design\nfrom ?faraway::wcgs:\n3154 healthy young men aged 39-59 from the San Francisco area were assessed for their personality type. All were free from coronary heart disease at the start of the research. Eight and a half years later change in CHD status was recorded.\nDetails (from faraway::wcgs)\nThe WCGS began in 1960 with 3,524 male volunteers who were employed by 11 California companies. Subjects were 39 to 59 years old and free of heart disease as determined by electrocardiogram. After the initial screening, the study population dropped to 3,154 and the number of companies to 10 because of various exclusions. The cohort comprised both blue- and white-collar employees.\n\n\n\n\n3.7.2 Baseline data collection\nsocio-demographic characteristics:\n\nage\neducation\nmarital status\nincome\noccupation\nphysical and physiological including:\nheight\nweight\nblood pressure\nelectrocardiogram\ncorneal arcus;\n\n\nbiochemical measurements:\n\ncholesterol and lipoprotein fractions;\nmedical and family history and use of medications;\n\n\nbehavioral data:\n\nType A interview,\nsmoking,\nexercise\nalcohol use.\n\n\nLater surveys added data on:\n\nanthropometry\ntriglycerides\nJenkins Activity Survey\ncaffeine use\n\nAverage follow-up continued for 8.5 years with repeat examinations.\n\n\n3.7.3 Load the data\nHere, I load the data:\n\n\nShow R code\n### load the data directly from a UCSF website:\nlibrary(haven)\nurl = paste0(\n # I'm breaking up the url into two chunks for readability\n \"https://regression.ucsf.edu/sites/g/files/\",\n \"tkssra6706/f/wysiwyg/home/data/wcgs.dta\")\nwcgs = haven::read_dta(url)\n\n\n\n\nShow R code\nwcgs |> head()\n\n\n\n\nTable 3.6: wcgs data\n\n\n\n\n \n\n\n\n\n\n\n\n\n3.7.4 Data cleaning\n\nNow let’s do some data cleaning\n\n\n\nShow R code\nlibrary(arsenal) # provides `set_labels()`\nlibrary(forcats) # provides `as_factor()`\nlibrary(haven)\nlibrary(plotly)\nwcgs = wcgs |> \n mutate(\n age = age |> \n arsenal::set_labels(\"Age (years)\"),\n \n arcus = \n arcus |> \n as.logical() |> \n arsenal::set_labels(\"Arcus Senilis\"),\n \n time169 = \n time169 |> \n as.numeric() |> \n arsenal::set_labels(\"Observation (follow up) time (days)\"),\n \n dibpat =\n dibpat |> \n as_factor() |> \n relevel(ref = \"Type A\") |> \n arsenal::set_labels(\"Behavioral Pattern\"),\n \n typchd69 = typchd69 |> \n labelled(\n label = \"Type of CHD Event\",\n labels = \n c(\n \"None\" = 0, \n \"infdeath\" = 1,\n \"silent\" = 2,\n \"angina\" = 3)),\n \n # turn stata-style labelled variables in to R-style factors:\n across(\n where(is.labelled), \n haven::as_factor)\n )\n\n\n\n\n3.7.5 What’s in the data\nHere’s a table of the data:\n\nShow R code\nwcgs |>\n select(-c(id, uni, t1)) |>\n tableby(chd69 ~ ., data = _) |>\n summary(\n pfootnote = TRUE,\n title =\n \"Baseline characteristics by CHD status at end of follow-up\")\n\n\nBaseline characteristics by CHD status at end of follow-up\n\n\n\n\n\n\n\n\n\n\nNo (N=2897)\nYes (N=257)\nTotal (N=3154)\np value\n\n\n\n\nAge (years)\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n46.082 (5.457)\n48.490 (5.801)\n46.279 (5.524)\n\n\n\n   Range\n39.000 - 59.000\n39.000 - 59.000\n39.000 - 59.000\n\n\n\nArcus Senilis\n\n\n\n< 0.0012\n\n\n   N-Miss\n0\n2\n2\n\n\n\n   FALSE\n2058 (71.0%)\n153 (60.0%)\n2211 (70.1%)\n\n\n\n   TRUE\n839 (29.0%)\n102 (40.0%)\n941 (29.9%)\n\n\n\nBehavioral Pattern\n\n\n\n< 0.0012\n\n\n   A1\n234 (8.1%)\n30 (11.7%)\n264 (8.4%)\n\n\n\n   A2\n1177 (40.6%)\n148 (57.6%)\n1325 (42.0%)\n\n\n\n   B3\n1155 (39.9%)\n61 (23.7%)\n1216 (38.6%)\n\n\n\n   B4\n331 (11.4%)\n18 (7.0%)\n349 (11.1%)\n\n\n\nBody Mass Index (kg/m2)\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n24.471 (2.561)\n25.055 (2.579)\n24.518 (2.567)\n\n\n\n   Range\n11.191 - 37.653\n19.225 - 38.947\n11.191 - 38.947\n\n\n\nTotal Cholesterol\n\n\n\n< 0.0011\n\n\n   N-Miss\n12\n0\n12\n\n\n\n   Mean (SD)\n224.261 (42.217)\n250.070 (49.396)\n226.372 (43.420)\n\n\n\n   Range\n103.000 - 400.000\n155.000 - 645.000\n103.000 - 645.000\n\n\n\nDiastolic Blood Pressure\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n81.723 (9.621)\n85.315 (10.311)\n82.016 (9.727)\n\n\n\n   Range\n58.000 - 150.000\n64.000 - 122.000\n58.000 - 150.000\n\n\n\nBehavioral Pattern\n\n\n\n< 0.0012\n\n\n   Type A\n1411 (48.7%)\n178 (69.3%)\n1589 (50.4%)\n\n\n\n   Type B\n1486 (51.3%)\n79 (30.7%)\n1565 (49.6%)\n\n\n\nHeight (inches)\n\n\n\n0.2901\n\n\n   Mean (SD)\n69.764 (2.539)\n69.938 (2.410)\n69.778 (2.529)\n\n\n\n   Range\n60.000 - 78.000\n63.000 - 77.000\n60.000 - 78.000\n\n\n\nLn of Systolic Blood Pressure\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n4.846 (0.110)\n4.900 (0.125)\n4.850 (0.112)\n\n\n\n   Range\n4.585 - 5.438\n4.605 - 5.298\n4.585 - 5.438\n\n\n\nLn of Weight\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n5.126 (0.123)\n5.155 (0.118)\n5.128 (0.123)\n\n\n\n   Range\n4.357 - 5.670\n4.868 - 5.768\n4.357 - 5.768\n\n\n\nCigarettes per day\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n11.151 (14.329)\n16.665 (15.657)\n11.601 (14.518)\n\n\n\n   Range\n0.000 - 99.000\n0.000 - 60.000\n0.000 - 99.000\n\n\n\nSystolic Blood Pressure\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n128.034 (14.746)\n135.385 (17.473)\n128.633 (15.118)\n\n\n\n   Range\n98.000 - 230.000\n100.000 - 200.000\n98.000 - 230.000\n\n\n\nCurrent smoking\n\n\n\n< 0.0012\n\n\n   No\n1554 (53.6%)\n98 (38.1%)\n1652 (52.4%)\n\n\n\n   Yes\n1343 (46.4%)\n159 (61.9%)\n1502 (47.6%)\n\n\n\nObservation (follow up) time (days)\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n2775.158 (562.205)\n1654.700 (859.297)\n2683.859 (666.524)\n\n\n\n   Range\n238.000 - 3430.000\n18.000 - 3229.000\n18.000 - 3430.000\n\n\n\nType of CHD Event\n\n\n\n\n\n\n   None\n0 (0.0%)\n0 (0.0%)\n0 (0.0%)\n\n\n\n   infdeath\n2897 (100.0%)\n0 (0.0%)\n2897 (91.9%)\n\n\n\n   silent\n0 (0.0%)\n135 (52.5%)\n135 (4.3%)\n\n\n\n   angina\n0 (0.0%)\n71 (27.6%)\n71 (2.3%)\n\n\n\n   4\n0 (0.0%)\n51 (19.8%)\n51 (1.6%)\n\n\n\nWeight (lbs)\n\n\n\n< 0.0011\n\n\n   Mean (SD)\n169.554 (21.010)\n174.463 (21.573)\n169.954 (21.096)\n\n\n\n   Range\n78.000 - 290.000\n130.000 - 320.000\n78.000 - 320.000\n\n\n\nWeight Category\n\n\n\n< 0.0012\n\n\n   < 140\n217 (7.5%)\n15 (5.8%)\n232 (7.4%)\n\n\n\n   140-170\n1440 (49.7%)\n98 (38.1%)\n1538 (48.8%)\n\n\n\n   170-200\n1049 (36.2%)\n122 (47.5%)\n1171 (37.1%)\n\n\n\n   > 200\n191 (6.6%)\n22 (8.6%)\n213 (6.8%)\n\n\n\nRECODE of age (Age)\n\n\n\n< 0.0012\n\n\n   35-40\n512 (17.7%)\n31 (12.1%)\n543 (17.2%)\n\n\n\n   41-45\n1036 (35.8%)\n55 (21.4%)\n1091 (34.6%)\n\n\n\n   46-50\n680 (23.5%)\n70 (27.2%)\n750 (23.8%)\n\n\n\n   51-55\n463 (16.0%)\n65 (25.3%)\n528 (16.7%)\n\n\n\n   56-60\n206 (7.1%)\n36 (14.0%)\n242 (7.7%)\n\n\n\n\n\nLinear Model ANOVA\nPearson’s Chi-squared test\n\n\n\n3.7.6 Data by age and personality type\nFor now, we will look at the interaction between age and personality type (dibpat). To make it easier to visualize the data, we summarize the event rates for each combination of age:\n\n\nShow R code\nchd_grouped_data = \n wcgs |> \n summarize(\n .by = c(age, dibpat),\n n = sum(chd69 %in% c(\"Yes\", \"No\")),\n x = sum(chd69 == \"Yes\")) |> \n mutate(\n `n - x` = n - x,\n `p(chd)` = (x / n) |> \n labelled(label = \"CHD Event by 1969\"),\n `odds(chd)` = `p(chd)` / (1 - `p(chd)`), \n `logit(chd)` = log(`odds(chd)`)\n )\n\nchd_grouped_data\n\n\n\n \n\n\n\n\n\n3.7.7 Graphical exploration\n\n\nShow R code\nlibrary(ggplot2)\nlibrary(ggeasy)\nlibrary(scales)\nchd_plot_probs = \n chd_grouped_data |> \n ggplot(\n aes(\n x = age, \n y = `p(chd)`, \n col = dibpat)\n ) +\n geom_point(aes(size = n), alpha = .7) + \n scale_size(range = c(1,4)) +\n geom_line() +\n theme_bw() +\n ylab(\"P(CHD Event by 1969)\") +\n scale_y_continuous(\n labels = scales::label_percent(),\n sec.axis = sec_axis(\n ~ odds(.),\n name = \"odds(CHD Event by 1969)\")) +\n ggeasy::easy_labs() +\n theme(legend.position = \"bottom\")\n\nprint(chd_plot_probs)\n\n\n\n\n\nFigure 3.9: CHD rates by age group, probability scale\n\n\n\n\n\n\n\n\n\n\nOdds scale\n\n\nShow R code\ntrans_odds = trans_new(\n name = \"odds\", \n transform = odds, \n inverse = odds_inv)\n\nchd_plot_odds = chd_plot_probs + \n scale_y_continuous(\n trans = trans_odds, # this line changes the vertical spacing\n name = chd_plot_probs$labels$y,\n sec.axis = sec_axis(\n ~ odds(.),\n name = \"odds(CHD Event by 1969)\"))\n\nprint(chd_plot_odds)\n\n\n\n\n\nFigure 3.10: CHD rates by age group, odds spacing\n\n\n\n\n\n\n\n\n\n\n\nLog-odds (logit) scale\n\n\nShow R code\ntrans_logit = trans_new(\n name = \"logit\", \n transform = logit, \n inverse = expit)\n\nchd_plot_logit = \n chd_plot_probs + \n scale_y_continuous(\n trans = trans_logit, # this line changes the vertical spacing\n name = chd_plot_probs$labels$y,\n breaks = c(seq(.01, .1, by = .01), .15, .2),\n minor_breaks = NULL,\n sec.axis = sec_axis(\n ~ logit(.),\n name = \"log(odds(CHD Event by 1969))\"))\n\nprint(chd_plot_logit)\n\n\n\n\n\nFigure 3.11: CHD data (logit-scale)\n\n\n\n\n\n\n\n\n\n\n\n3.7.8 Logistic regression models for CHD data\nHere, we fit stratified models for CHD by personality type.\n\n\nShow R code\nchd_glm_strat = glm(\n \"formula\" = chd69 == \"Yes\" ~ dibpat + dibpat:age - 1, \n \"data\" = wcgs,\n \"family\" = binomial(link = \"logit\")\n)\n\nchd_glm_strat |> parameters() |> print_md()\n\n\n\n\nTable 3.7: CHD model, stratified parametrization\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nParameter\nLog-Odds\nSE\n95% CI\nz\np\n\n\n\n\ndibpat (Type A)\n-5.50\n0.67\n(-6.83, -4.19)\n-8.18\n< .001\n\n\ndibpat (Type B)\n-5.80\n0.98\n(-7.73, -3.90)\n-5.95\n< .001\n\n\ndibpat (Type A) × age\n0.07\n0.01\n(0.05, 0.10)\n5.24\n< .001\n\n\ndibpat (Type B) × age\n0.06\n0.02\n(0.02, 0.10)\n3.01\n0.003\n\n\n\n\n\n\n\n\nWe can get the corresponding odds ratios (\\(e^{\\beta}\\)s) by passing exponentiate = TRUE to parameters():\n\n\nShow R code\nchd_glm_strat |> \n parameters(exponentiate = TRUE) |> \n print_md()\n\n\n\n\nTable 3.8: Odds ratio estimates for CHD model\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nParameter\nOdds Ratio\nSE\n95% CI\nz\np\n\n\n\n\ndibpat (Type A)\n4.09e-03\n2.75e-03\n(1.08e-03, 0.02)\n-8.18\n< .001\n\n\ndibpat (Type B)\n3.02e-03\n2.94e-03\n(4.40e-04, 0.02)\n-5.95\n< .001\n\n\ndibpat (Type A) × age\n1.07\n0.01\n(1.05, 1.10)\n5.24\n< .001\n\n\ndibpat (Type B) × age\n1.06\n0.02\n(1.02, 1.11)\n3.01\n0.003\n\n\n\n\n\n\n\n\n\n\n3.7.9 Models superimposed on data\nWe can graph our fitted models on each scale (probability, odds, log-odds).\n\n\nprobability scale\n\n\nShow R code\n\ncurve_type_A = function(x) \n{\n chd_glm_strat |> predict(\n type = \"response\",\n newdata = tibble(age = x, dibpat = \"Type A\"))\n}\n\ncurve_type_B = function(x) \n{\n chd_glm_strat |> predict(\n type = \"response\",\n newdata = tibble(age = x, dibpat = \"Type B\"))\n}\n\nchd_plot_probs_2 =\n chd_plot_probs +\n geom_function(\n fun = curve_type_A,\n aes(col = \"Type A\")\n ) +\n geom_function(\n fun = curve_type_B,\n aes(col = \"Type B\")\n )\nprint(chd_plot_probs_2)\n\n\n\n\n\n\n\n\n\n\n\n\nodds scale\n\n\nShow R code\n# curve_type_A = function(x) \n# {\n# chd_glm_strat |> predict(\n# type = \"link\",\n# newdata = tibble(age = x, dibpat = \"Type A\")) |> exp()\n# }\n# curve_type_B = function(x) \n# {\n# chd_glm_strat |> predict(\n# type = \"link\",\n# newdata = tibble(age = x, dibpat = \"Type B\")) |> exp()\n# }\n\nchd_plot_odds_2 =\n chd_plot_odds +\n geom_function(\n fun = curve_type_A,\n aes(col = \"Type A\")\n ) +\n geom_function(\n fun = curve_type_B,\n aes(col = \"Type B\")\n )\nprint(chd_plot_odds_2)\n\n\n\n\n\nFigure 3.12\n\n\n\n\n\n\n\n\n\n\n\nlog-odds (logit) scale\n\n\nShow R code\n# curve_type_A = function(x) \n# {\n# chd_glm_strat |> predict(\n# type = \"link\",\n# newdata = tibble(age = x, dibpat = \"Type A\"))\n# }\n# curve_type_B = function(x) \n# {\n# chd_glm_strat |> predict(\n# type = \"link\",\n# newdata = tibble(age = x, dibpat = \"Type B\"))\n# }\n\nchd_plot_logit_2 =\n chd_plot_logit +\n geom_function(\n fun = curve_type_A,\n aes(col = \"Type A\")\n ) +\n geom_function(\n fun = curve_type_B,\n aes(col = \"Type B\")\n )\n\nprint(chd_plot_logit_2)\n\n\n\n\n\nFigure 3.13\n\n\n\n\n\n\n\n\n\n\n\n3.7.10 reference-group and contrast parametrization\nWe can also use the corner-point parametrization (with reference groups and contrasts):\n\n\nShow R code\nchd_glm_contrasts = \n wcgs |> \n glm(\n \"data\" = _,\n \"formula\" = chd69 == \"Yes\" ~ dibpat*age, \n \"family\" = binomial(link = \"logit\")\n )\n\nchd_glm_contrasts |> \n parameters() |> \n print_md()\n\n\n\n\nTable 3.9: CHD model (corner-point parametrization)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nParameter\nLog-Odds\nSE\n95% CI\nz\np\n\n\n\n\n(Intercept)\n-5.50\n0.67\n(-6.83, -4.19)\n-8.18\n< .001\n\n\ndibpat (Type B)\n-0.30\n1.18\n(-2.63, 2.02)\n-0.26\n0.797\n\n\nage\n0.07\n0.01\n(0.05, 0.10)\n5.24\n< .001\n\n\ndibpat (Type B) × age\n-0.01\n0.02\n(-0.06, 0.04)\n-0.42\n0.674\n\n\n\n\n\n\n\n\n\nCompare with Table 3.8.\n\n\n\nExercise 3.15 If I give you model 1, how would you get the coefficients of model 2?\n\n\n\nTheorem 3.15 For the logistic regression model:\n\n\\(Y_i|\\tilde{X}_i \\ \\sim_{⫫}\\ \\text{Ber}(\\pi(\\tilde{X}_i))\\)\n\\(\\pi(\\tilde{x}) = \\text{expit}\\left\\{\\tilde{x}'\\tilde{\\beta}\\right\\}\\)\n\nConsider two covariate patterns, \\(\\tilde{x}\\) and \\(\\tilde{x^*}\\).\nThe odds ratio comparing these covariate patterns is:\n\\[\n\\omega(\\tilde{x},\\tilde{x^*}) = \\text{exp}\\left\\{(\\tilde{x}-\\tilde{x^*})^{\\top} \\tilde{\\beta}\\right\\}\n\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n\\omega(\\tilde{x},\\tilde{x^*})\n&= \\frac {\\omega(Y=1 | \\tilde{X}= \\tilde{x})} {\\omega(Y=1 | \\tilde{X}= \\tilde{x^*})}\n\\\\ &= \\frac{\\text{exp}\\left\\{\\tilde{x}^{\\top}\\tilde{\\beta}\\right\\}}{\\text{exp}\\left\\{{\\tilde{x^*}}^{\\top} \\tilde{\\beta}\\right\\}}\n\\\\ &= \\text{exp}\\left\\{\\tilde{x}^{\\top}\\tilde{\\beta}- {\\tilde{x^*}}^{\\top} \\tilde{\\beta}\\right\\}\n\\\\ &= \\text{exp}\\left\\{(\\tilde{x}^{\\top} - {\\tilde{x^*}}^{\\top}) \\tilde{\\beta}\\right\\}\n\\\\ &= \\text{exp}\\left\\{{(\\tilde{x}- \\tilde{x^*})}^{\\top} \\tilde{\\beta}\\right\\}\n\\end{aligned}\n\\]", "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\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]", + "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]", "crumbs": [ "Generalized Linear Models", "3  Models for Binary Outcomes" @@ -358,7 +358,7 @@ "href": "logistic-regression.html#other-link-functions-for-bernoulli-outcomes", "title": "3  Models for Binary Outcomes", "section": "3.11 Other link functions for Bernoulli outcomes", - "text": "3.11 Other link functions for Bernoulli outcomes\nIf you want risk ratios, you can sometimes get them by changing the link function:\n\n\nShow R code\n\ndata(anthers, package = \"dobson\")\nanthers.sum<-aggregate(\n anthers[c(\"n\",\"y\")], \n by=anthers[c(\"storage\")],FUN=sum) \n\nanthers_glm_log = glm(\n formula = cbind(y,n-y)~storage,\n data=anthers.sum, \n family=binomial(link=\"log\"))\n\nanthers_glm_log |> parameters() |> print_md()\n\n\n\n\n\nParameter\nLog-Risk\nSE\n95% CI\nz\np\n\n\n\n\n(Intercept)\n-0.80\n0.12\n(-1.04, -0.58)\n-6.81\n< .001\n\n\nstorage\n0.17\n0.07\n(0.02, 0.31)\n2.31\n0.021\n\n\n\n\n\nNow \\(\\text{exp}\\left\\{\\beta\\right\\}\\) gives us risk ratios instead of odds ratios:\n\n\nShow R code\nanthers_glm_log |> parameters(exponentiate = TRUE) |> print_md()\n\n\n\n\n\nParameter\nRisk Ratio\nSE\n95% CI\nz\np\n\n\n\n\n(Intercept)\n0.45\n0.05\n(0.35, 0.56)\n-6.81\n< .001\n\n\nstorage\n1.18\n0.09\n(1.03, 1.36)\n2.31\n0.021\n\n\n\n\n\nLet’s compare this model with a logistic model:\n\n\nShow R code\n\nanthers_glm_logit = glm(\n formula = cbind(y, n - y) ~ storage,\n data = anthers.sum,\n family = binomial(link = \"logit\"))\n\nanthers_glm_logit |> parameters(exponentiate = TRUE) |> print_md()\n\n\n\n\n\nParameter\nOdds Ratio\nSE\n95% CI\nz\np\n\n\n\n\n(Intercept)\n0.76\n0.20\n(0.45, 1.27)\n-1.05\n0.296\n\n\nstorage\n1.49\n0.26\n(1.06, 2.10)\n2.29\n0.022\n\n\n\n\n\n[to add: fitted plots on each outcome scale]\nWhen I try to use link =\"log\" in practice, I often get errors about not finding good starting values for the estimation procedure. This is likely because the model is producing fitted probabilities greater than 1.\nWhen this happens, you can try to fit Poisson regression models instead (we will see those soon!). But then the outcome distribution isn’t quite right, and you won’t get warnings about fitted probabilities greater than 1. In my opinion, the Poisson model for binary outcomes is confusing and not very appealing.", + "text": "3.11 Other link functions for Bernoulli outcomes\nIf you want risk ratios, you can sometimes get them by changing the link function:\n\n\nShow R code\n\ndata(anthers, package = \"dobson\")\nanthers.sum<-aggregate(\n anthers[c(\"n\",\"y\")], \n by=anthers[c(\"storage\")],FUN=sum) \n\nanthers_glm_log = glm(\n formula = cbind(y,n-y)~storage,\n data=anthers.sum, \n family=binomial(link=\"log\"))\n\nanthers_glm_log |> parameters() |> print_md()\n\n\n\n\n\nParameter\nLog-Risk\nSE\n95% CI\nz\np\n\n\n\n\n(Intercept)\n-0.80\n0.12\n(-1.04, -0.58)\n-6.81\n< .001\n\n\nstorage\n0.17\n0.07\n(0.02, 0.31)\n2.31\n0.021\n\n\n\n\n\nNow \\(\\text{exp}\\left\\{\\beta\\right\\}\\) gives us risk ratios instead of odds ratios:\n\n\nShow R code\nanthers_glm_log |> parameters(exponentiate = TRUE) |> print_md()\n\n\n\n\n\nParameter\nRisk Ratio\nSE\n95% CI\nz\np\n\n\n\n\n(Intercept)\n0.45\n0.05\n(0.35, 0.56)\n-6.81\n< .001\n\n\nstorage\n1.18\n0.09\n(1.03, 1.36)\n2.31\n0.021\n\n\n\n\n\nLet’s compare this model with a logistic model:\n\n\nShow R code\n\nanthers_glm_logit = glm(\n formula = cbind(y, n - y) ~ storage,\n data = anthers.sum,\n family = binomial(link = \"logit\"))\n\nanthers_glm_logit |> parameters(exponentiate = TRUE) |> print_md()\n\n\n\n\n\nParameter\nOdds Ratio\nSE\n95% CI\nz\np\n\n\n\n\n(Intercept)\n0.76\n0.20\n(0.45, 1.27)\n-1.05\n0.296\n\n\nstorage\n1.49\n0.26\n(1.06, 2.10)\n2.29\n0.022\n\n\n\n\n\n[to add: fitted plots on each outcome scale]\nWhen I try to use link =\"log\" in practice, I often get errors about not finding good starting values for the estimation procedure. This is likely because the model is producing fitted probabilities greater than 1.\nWhen this happens, you can try to fit Poisson regression models instead (we will see those soon!). But then the outcome distribution isn’t quite right, and you won’t get warnings about fitted probabilities greater than 1. In my opinion, the Poisson model for binary outcomes is confusing and not very appealing.\n\n3.11.1 WCGS: link functions\n\n\nShow R code\nwcgs_glm_logit_link = \n chd_grouped_data |> \n mutate(type = dibpat |> relevel(ref = \"Type B\")) |> \n glm(\n \"formula\" = cbind(x, `n - x`) ~ dibpat * age, \n \"data\" = _,\n \"family\" = binomial(link = \"logit\")\n )\n\nwcgs_glm_identity_link = \n chd_grouped_data |> \n mutate(type = dibpat |> relevel(ref = \"Type B\")) |> \n glm(\n \"formula\" = cbind(x, `n - x`) ~ dibpat * age, \n \"data\" = _,\n \"family\" = binomial(link = \"identity\")\n )\nwcgs_glm_identity_link |> coef() |> pander()\n\n\n\n\n\n\n\n\n\n\n\n(Intercept)\ndibpatType B\nage\ndibpatType B:age\n\n\n\n\n-0.2199\n0.1374\n0.0071\n-0.004194\n\n\n\n\n\n\nShow R code\nlibrary(ggfortify)\nwcgs_glm_logit_link |> autoplot(which = c(1), ncol = 1) + facet_wrap(~dibpat)\nwcgs_glm_identity_link |> autoplot(which = c(1), ncol = 1) + facet_wrap(~dibpat)\n\n\n\n\nFigure 3.15: Residuals vs Fitted plot for wcgs models\n\n\n\n\n\n\n\n(a) Logistic link\n\n\n\n\n\n\n\n\n\n\n\n(b) Identity link\n\n\n\n\n\n\n\n\n\n\n\n\nShow R code\nbeetles_lm = \n beetles_long |> \n lm(formula = died ~ dose)\n\nbeetles = \n beetles |> mutate(\n resid_logit = beetles_glm_grouped |> resid(type = \"response\"))\nbeetles_glm_grouped |> autoplot(which = c(1), ncol = 1)\nbeetles_lm |> autoplot(which = c(1), ncol = 1)\n\n\n\n\nFigure 3.16: Residuals vs Fitted plot for BeetleMortality models\n\n\n\n\n\n\n\n(a) Logistic link\n\n\n\n\n\n\n\n\n\n\n\n(b) Identity link", "crumbs": [ "Generalized Linear Models", "3  Models for Binary Outcomes" @@ -369,7 +369,18 @@ "href": "logistic-regression.html#quasibinomial", "title": "3  Models for Binary Outcomes", "section": "3.12 Quasibinomial", - "text": "3.12 Quasibinomial\nSee Hua Zhou’s lecture notes\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\nRosenman, Ray H, Richard J Brand, C David Jenkins, Meyer Friedman, Reuben Straus, and Moses Wurm. 1975. “Coronary Heart Disease in the Western Collaborative Group Study: Final Follow-up Experience of 8 1/2 Years.” JAMA 233 (8): 872–77. https://doi.org/10.1001/jama.1975.03260080034016.\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.", + "text": "3.12 Quasibinomial\nSee Hua Zhou’s lecture notes", + "crumbs": [ + "Generalized Linear Models", + "3  Models for Binary Outcomes" + ] + }, + { + "objectID": "logistic-regression.html#further-reading", + "href": "logistic-regression.html#further-reading", + "title": "3  Models for Binary Outcomes", + "section": "3.13 Further reading", + "text": "3.13 Further reading\n\nHosmer Jr, Lemeshow, and Sturdivant (2013) is a classic textbook on logistic regression\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\nHosmer Jr, David W, Stanley Lemeshow, and Rodney X Sturdivant. 2013. Applied Logistic Regression. John Wiley & Sons.\n\n\nRosenman, Ray H, Richard J Brand, C David Jenkins, Meyer Friedman, Reuben Straus, and Moses Wurm. 1975. “Coronary Heart Disease in the Western Collaborative Group Study: Final Follow-up Experience of 8 1/2 Years.” JAMA 233 (8): 872–77. https://doi.org/10.1001/jama.1975.03260080034016.\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", "3  Models for Binary Outcomes" @@ -402,7 +413,7 @@ "href": "count-regression.html#introduction", "title": "\n4  Models for Count Outcomes\n", "section": "\n4.1 Introduction", - "text": "4.1 Introduction\n\n4.1.1 Examples of count outcomes\n\nCyclones 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\n\n\n\n\nNote\n\n\n\nIn many count outcomes, there is some sense of “exposure magnitude” or “duration of observation”: person-year, time at risk, session, miles driven, etc.\n\n\n\n4.1.2 Poisson distribution\n\\[P(Y=y) = \\frac{\\mu^y e^{-\\mu}}{y!}\\]\nProperties\n\n\\(\\mathbb{E}[Y] = \\mu\\)\n\\(\\text{Var}[Y] = \\mu\\)\n\n4.1.3 Accounting for exposure\nIf the exposures/observation durations, denoted \\(T=t\\), are not all equal, we model \\[\\mu = \\lambda 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\n4.1.4 Adding covariates\nWith covariates, \\(\\lambda\\) becomes a function of the covariates \\(\\tilde X = (X_1, \\dots,X_n)\\), with a \\(\\text{log}\\left\\{\\right\\}\\) link function (and thus an \\(\\text{exp}\\left\\{\\right\\}\\) inverse-link). That is:\n\\[\n\\begin{aligned}\n\\mathbb E[Y | \\tilde X = \\tilde x,T=t]\n&= \\mu(\\tilde x,t)\\\\\n\\mu(\\tilde x,t)\n&= \\lambda(\\tilde x)\\cdot t\\\\\n\\lambda(\\tilde x)\n&= \\text{exp}\\left\\{\\eta(\\tilde x)\\right\\}\\\\\n\\eta(\\tilde x)\n&= \\tilde x'\\tilde \\beta = \\beta_0 + \\beta_1 x_1 + \\dots + \\beta_p x_p\n\\end{aligned}\n\\]\nTherefore, \\[\n\\begin{aligned}\n\\text{log}\\left\\{\\{\\mathbb E[Y | \\tilde X = \\tilde x,T=t] \\}\\right\\}\n&= \\text{log}\\left\\{\\{\\mu(\\tilde x)\\}\\right\\}\\\\\n&=\\text{log}\\left\\{\\{\\lambda(\\tilde x) \\cdot t \\}\\right\\}\\\\\n&=\\text{log}\\left\\{\\lambda(\\tilde x)\\right\\} + \\text{log}\\left\\{t\\right\\}\\\\\n&=\\text{log}\\left\\{\\text{exp}\\left\\{\\eta(\\tilde x)\\right\\}\\right\\} + \\text{log}\\left\\{t\\right\\}\\\\\n&=\\eta(\\tilde x) + \\text{log}\\left\\{t\\right\\}\\\\\n&=\\tilde x'\\tilde\\beta + \\text{log}\\left\\{t\\right\\}\\\\\n&=(\\beta_0 +\\beta_1 x_1+\\dots + \\beta_p x_p) + \\text{log}\\left\\{t\\right\\}\\\\\n\\end{aligned}\n\\]\nIn contrast with the \\(X\\)s, \\(T\\) enters this expression with a \\(\\text{log}\\left\\{\\right\\}\\) transformation and without a corresponding \\(\\beta\\) coefficient.\n\n\n\n\n\n\nNote\n\n\n\nTerms that enter the linear component of a model without a coefficient, such as \\(\\text{log}\\left\\{t\\right\\}\\) here, are called offsets.\n\n\n\n4.1.5 Rate ratios\nDifferences on the log-rate scale become ratios on the rate scale.\n\n\n\n\n\n\nTip\n\n\n\n\\[\\text{exp}\\left\\{a-b\\right\\} = \\frac{\\text{exp}\\left\\{a\\right\\}}{\\text{exp}\\left\\{b\\right\\}}\\]\n\n\nTherefore, according to this model, differences of \\(\\delta\\) in covariate \\(x_j\\) correspond to rate ratios of \\(\\text{exp}\\left\\{\\beta_j \\cdot \\delta\\right\\}\\).\nThat is, letting \\(\\tilde X_{-j}\\) denote vector \\(\\tilde X\\) with element \\(j\\) removed:\n\\[\n\\begin{aligned}\n&{\n\\left\\{\n \\text{log}\\left\\{\\mathbb E[Y |{\\color{red}{X_j = a}}, \\tilde X_{-j}=\\tilde x_{-j},T=t]\\right\\}\n \\atop\n {-\\text{log}\\left\\{\\mathbb E[Y |{\\color{red}{X_j = b}}, \\tilde X_{-j}=\\tilde x_{-j},T=t]\\right\\}}\n \\right\\}\n}\\\\\n&=\n{\\left\\{\n\\text{log}\\left\\{t\\right\\} + \\beta_0 + \\beta_1 x_1 + ... + {\\color{red}{\\beta_j (a)}} + ...+\\beta_p x_p\n\\atop\n{-\\text{log}\\left\\{t\\right\\} + \\beta_0 + \\beta_1 x_1 + ... + {\\color{red}{\\beta_j (b)}} + ...+\\beta_p x_p}\n\\right\\}}\\\\\n&= \\color{red}{\\beta_j(a-b)}\n\\end{aligned}\n\\]\nAnd accordingly,\n\\[\n\\begin{aligned}\n\\frac\n{\\mathbb{E}[Y |{\\color{red}{X_j = a}}, \\tilde X_{-j} = \\tilde x_{-j}, T = t]\n}\n{\n\\mathbb E[Y |{\\color{red}{X_j = b}}, \\tilde X_{-j}=\\tilde x_{-j},T=t]\n}\n=\n\\text{exp}\\left\\{{\\color{red}{\\beta_j(a-b)}}\\right\\}\n\\end{aligned}\n\\]", + "text": "4.1 Introduction\n\n4.1.1 Examples of count outcomes\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\n\n\n\n\nNote\n\n\n\nIn many count outcomes, there is some sense of “exposure magnitude” or “duration of observation”: person-year, time at risk, session, miles driven, etc.\n\n\n\n4.1.2 Probability distributions for count outcomes\n\nPoisson distribution\nNegative binomial distribution\n\n4.1.3 Adding covariates\nWith covariates, \\(\\lambda\\) becomes a function of the covariates \\(\\tilde X = (X_1, \\dots,X_n)\\), with a \\(\\text{log}\\left\\{\\right\\}\\) link function (and thus an \\(\\text{exp}\\left\\{\\right\\}\\) inverse-link). That is:\n\\[\n\\begin{aligned}\n\\mathbb E[Y | \\tilde X = \\tilde x,T=t]\n&= \\mu(\\tilde x,t)\\\\\n\\mu(\\tilde x,t)\n&= \\lambda(\\tilde x)\\cdot t\\\\\n\\lambda(\\tilde x)\n&= \\text{exp}\\left\\{\\eta(\\tilde x)\\right\\}\\\\\n\\eta(\\tilde x)\n&= \\tilde x'\\tilde \\beta = \\beta_0 + \\beta_1 x_1 + \\dots + \\beta_p x_p\n\\end{aligned}\n\\]\nTherefore, \\[\n\\begin{aligned}\n\\text{log}\\left\\{\\{\\mathbb E[Y | \\tilde X = \\tilde x,T=t] \\}\\right\\}\n&= \\text{log}\\left\\{\\{\\mu(\\tilde x)\\}\\right\\}\\\\\n&=\\text{log}\\left\\{\\{\\lambda(\\tilde x) \\cdot t \\}\\right\\}\\\\\n&=\\text{log}\\left\\{\\lambda(\\tilde x)\\right\\} + \\text{log}\\left\\{t\\right\\}\\\\\n&=\\text{log}\\left\\{\\text{exp}\\left\\{\\eta(\\tilde x)\\right\\}\\right\\} + \\text{log}\\left\\{t\\right\\}\\\\\n&=\\eta(\\tilde x) + \\text{log}\\left\\{t\\right\\}\\\\\n&=\\tilde x'\\tilde\\beta + \\text{log}\\left\\{t\\right\\}\\\\\n&=(\\beta_0 +\\beta_1 x_1+\\dots + \\beta_p x_p) + \\text{log}\\left\\{t\\right\\}\\\\\n\\end{aligned}\n\\]\nIn contrast with the \\(X\\)s, \\(T\\) enters this expression with a \\(\\text{log}\\left\\{\\right\\}\\) transformation and without a corresponding \\(\\beta\\) coefficient.\n\n\n\n\n\n\nNote\n\n\n\nTerms that enter the linear component of a model without a coefficient, such as \\(\\text{log}\\left\\{t\\right\\}\\) here, are called offsets.\n\n\n\n4.1.4 Rate ratios\n\nDifferences on the log-rate scale become ratios on the rate scale, because\n\n\\[\\text{exp}\\left\\{a-b\\right\\} = \\frac{\\text{exp}\\left\\{a\\right\\}}{\\text{exp}\\left\\{b\\right\\}}\\]\n[math-preqs.qmd#cor-exp-sum]\nTherefore, according to this model, differences of \\(\\delta\\) in covariate \\(x_j\\) correspond to rate ratios of \\(\\text{exp}\\left\\{\\beta_j \\cdot \\delta\\right\\}\\).\nThat is, letting \\(\\tilde X_{-j}\\) denote vector \\(\\tilde X\\) with element \\(j\\) removed:\n\\[\n\\begin{aligned}\n&{\n\\left\\{\n \\text{log}\\left\\{\\mathbb E[Y |{\\color{red}{X_j = a}}, \\tilde X_{-j}=\\tilde x_{-j},T=t]\\right\\}\n \\atop\n {-\\text{log}\\left\\{\\mathbb E[Y |{\\color{red}{X_j = b}}, \\tilde X_{-j}=\\tilde x_{-j},T=t]\\right\\}}\n \\right\\}\n}\\\\\n&=\n{\\left\\{\n\\text{log}\\left\\{t\\right\\} + \\beta_0 + \\beta_1 x_1 + ... + {\\color{red}{\\beta_j (a)}} + ...+\\beta_p x_p\n\\atop\n{-\\text{log}\\left\\{t\\right\\} + \\beta_0 + \\beta_1 x_1 + ... + {\\color{red}{\\beta_j (b)}} + ...+\\beta_p x_p}\n\\right\\}}\\\\\n&= \\color{red}{\\beta_j(a-b)}\n\\end{aligned}\n\\]\nAnd accordingly,\n\\[\n\\begin{aligned}\n\\frac\n{\\mathbb{E}[Y |{\\color{red}{X_j = a}}, \\tilde X_{-j} = \\tilde x_{-j}, T = t]\n}\n{\n\\mathbb E[Y |{\\color{red}{X_j = b}}, \\tilde X_{-j}=\\tilde x_{-j},T=t]\n}\n=\n\\text{exp}\\left\\{{\\color{red}{\\beta_j(a-b)}}\\right\\}\n\\end{aligned}\n\\]", "crumbs": [ "Generalized Linear Models", "4  Models for Count Outcomes" @@ -457,7 +468,7 @@ "href": "count-regression.html#over-dispersion", "title": "\n4  Models for Count Outcomes\n", "section": "\n4.6 Over-dispersion", - "text": "4.6 Over-dispersion\n\n4.6.1 Negative binomial models\n\nThe Poisson distribution model forces the variance to equal the mean. In practice, many count distributions will have a variance substantially larger than the mean (or occasionally smaller).\nWhen we encounter this, we can try to reduce the residual variance by adding more covariates. However, there are also alternatives to the Poisson model.\nMost notably, the negative binomial model:\n\n\\[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}\\]\nwhere \\(\\rho\\) is an overdispersion parameter and \\(\\Gamma(x) = (x-1)!\\) for integers \\(x\\).\nYou don’t need to memorize or understand this expression, but 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.\nFor this distribution, \\(\\mathbb E[Y] = \\mu\\) and \\(\\text{Var}(Y) = \\mu + \\frac{\\mu^2}{\\rho} > \\mu\\).\nWe can still model \\(\\mu\\) as a function of \\(X\\) and \\(T\\) as before, and we can combine this model with zero-inflation by using it in place of the Poisson distribution for \\(P(Y=y|Z=0,X=x,T=t)\\).\n\n4.6.2 Quasipoisson\nAn alternative to Negative binomial is the “quasipoisson” distribution. I’ve never used it, but it seems to be a method-of-moments type approach rather than maximum likelihood. It models the variance as \\(\\text{Var}(Y) = \\mu\\theta\\), and estimates \\(\\theta\\) accordingly.\nSee ?quasipoisson in R for more.\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.", + "text": "4.6 Over-dispersion\n\n4.6.1 Negative binomial models\n\nThe Poisson distribution model forces the variance to equal the mean. In practice, many count distributions will have a variance substantially larger than the mean (or occasionally smaller).\nWhen we encounter this, we can try to reduce the residual variance by adding more covariates. However, there are also alternatives to the Poisson model.\nMost notably, the negative binomial model.\n\nWe can still model \\(\\mu\\) as a function of \\(X\\) and \\(T\\) as before, and we can combine this model with zero-inflation by using it in place of the Poisson distribution for \\(\\text{P}(Y=y|Z=0,X=x,T=t)\\).\n\n4.6.2 Quasipoisson\nAn alternative to Negative binomial is the “quasipoisson” distribution. I’ve never used it, but it seems to be a method-of-moments type approach rather than maximum likelihood. It models the variance as \\(\\text{Var}\\left(Y\\right) = \\mu\\theta\\), and estimates \\(\\theta\\) accordingly.\nSee ?quasipoisson in R for more.\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" @@ -687,7 +698,7 @@ "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) = 0.1266 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.", + "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.", "crumbs": [ "Time to Event Models", "7  Building Cox Proportional Hazards models" @@ -808,7 +819,7 @@ "href": "references.html", "title": "References", "section": "", - "text": "Anderson, Edgar. 1935. “The Irises of the Gaspe Peninsula.”\nBulletin of American Iris Society 59: 2–5.\n\n\nBache, Stefan Milton, and Hadley Wickham. 2022. Magrittr: A\nForward-Pipe Operator for r. https://CRAN.R-project.org/package=magrittr.\n\n\nBanerjee, Sudipto, and Anindya Roy. 2014. Linear Algebra and Matrix\nAnalysis for Statistics. Vol. 181. Crc Press Boca Raton.\n\n\nCanchola, Alison J, Susan L Stewart, Leslie Bernstein, Dee W West,\nRonald K Ross, Dennis Deapen, Richard Pinder, et al. 2003. “Cox\nRegression Using Different Time-Scales.” Western Users of SAS\nSoftware. https://www.lexjansen.com/wuss/2003/DataAnalysis/i-cox_time_scales.pdf.\n\n\nCasella, George, and Roger Berger. 2002. Statistical Inference.\n2nd ed. Cengage Learning. https://www.cengage.com/c/statistical-inference-2e-casella-berger/9780534243128/.\n\n\nChang, Winston. 2024. R Graphics Cookbook: Practical Recipes for\nVisualizing Data. O’Reilly Media. https://r-graphics.org/.\n\n\nChatterjee, Samprit, and Ali S Hadi. 2015. Regression Analysis by\nExample. John Wiley & Sons. https://www.wiley.com/en-us/Regression+Analysis+by+Example%2C+4th+Edition-p-9780470055458.\n\n\nDalgaard, Peter. 2008. Introductory Statistics with r. New\nYork, NY: Springer New York. https://link.springer.com/book/10.1007/978-0-387-79054-1.\n\n\nDobson, Annette J, and Adrian G Barnett. 2018. An Introduction to\nGeneralized Linear Models. 4th ed. CRC press. https://doi.org/10.1201/9781315182780.\n\n\nDunn, Peter K, Gordon K Smyth, et al. 2018. Generalized Linear\nModels with Examples in r. Vol. 53. Springer. https://link.springer.com/book/10.1007/978-1-4419-0118-7.\n\n\nEfron, Bradley, and David V Hinkley. 1978. “Assessing the Accuracy\nof the Maximum Likelihood Estimator: Observed Versus Expected Fisher\nInformation.” Biometrika 65 (3): 457–83.\n\n\nFaraway, Julian J. 2016. Extending the Linear Model with r:\nGeneralized Linear, Mixed Effects and Nonparametric Regression\nModels. 2nd ed. Chapman; Hall/CRC. https://doi.org/10.1201/9781315382722.\n\n\nFay, Colin, Sébastien Rochette, Vincent Guyader, and Cervan Girard.\n2021. Engineering Production-Grade Shiny Apps. Chapman;\nHall/CRC. https://engineering-shiny.org/.\n\n\nFieller, Nick. 2016. Basics of Matrix Algebra for Statistics with\nr. Chapman; Hall/CRC. https://doi.org/10.1201/9781315370200.\n\n\nFox, John. 2015. Applied Regression Analysis and Generalized Linear\nModels. Sage publications.\n\n\nGrambsch, Patricia M, and Terry M Therneau. 1994. “Proportional\nHazards Tests and Diagnostics Based on Weighted Residuals.”\nBiometrika 81 (3): 515–26. https://doi.org/10.1093/biomet/81.3.515.\n\n\nHarrell, Frank E. 2015. Regression Modeling Strategies: With\nApplications to Linear Models, Logistic Regression, and Survival\nAnalysis. 2nd ed. Springer. https://doi.org/10.1007/978-3-319-19425-7.\n\n\nJames, Gareth, Daniela Witten, Trevor Hastie, Robert Tibshirani, et al.\n2013. An Introduction to Statistical Learning. Vol. 112.\nSpringer. https://www.statlearning.com/.\n\n\nKhuri, André I. 2003. Advanced Calculus with Applications in\nStatistics. John Wiley & Sons. https://www.wiley.com/en-us/Advanced+Calculus+with+Applications+in+Statistics%2C+2nd+Edition-p-9780471391043.\n\n\nKlein, John P, Melvin L Moeschberger, et al. 2003. Survival\nAnalysis: Techniques for Censored and Truncated Data. Vol. 1230.\nSpringer. https://link.springer.com/book/10.1007/b97377.\n\n\nKleinbaum, David G, and Mitchel Klein. 2010. Logistic\nRegression. 3rd ed. Springer. https://link.springer.com/book/10.1007/978-1-4419-1742-3.\n\n\n———. 2012. Survival Analysis a Self-Learning Text. 3rd ed.\nSpringer. https://link.springer.com/book/10.1007/978-1-4419-6646-9.\n\n\nKleinbaum, David G, Lawrence L Kupper, Azhar Nizam, K Muller, and ES\nRosenberg. 2014. Applied Regression Analysis and Other Multivariable\nMethods. 5th ed. Cengage Learning. https://www.cengage.com/c/applied-regression-analysis-and-other-multivariable-methods-5e-kleinbaum/9781285051086/.\n\n\nKleinman, Ken, and Nicholas J Horton. 2009. SAS and r: Data\nManagement, Statistical Analysis, and Graphics. Chapman; Hall/CRC.\nhttps://www.routledge.com/SAS-and-R-Data-Management-Statistical-Analysis-and-Graphics-Second-Edition/Kleinman-Horton/p/book/9781466584495.\n\n\nKuhn, Max, and Julia Silge. 2022. Tidy Modeling with r. \"\nO’Reilly Media, Inc.\". https://www.tmwr.org/.\n\n\nKutner, Michael H, Christopher J Nachtsheim, John Neter, and William Li.\n2005. Applied Linear Statistical Models. McGraw-Hill.\n\n\nLawrance, Rachael, Evgeny Degtyarev, Philip Griffiths, Peter Trask,\nHelen Lau, Denise D’Alessio, Ingolf Griebsch, Gudrun Wallenstein, Kim\nCocks, and Kaspar Rufibach. 2020. “What Is an Estimand, and How\nDoes It Relate to Quantifying the Effect of Treatment on\nPatient-Reported Quality of Life Outcomes in Clinical Trials?”\nJournal of Patient-Reported Outcomes 4 (1): 1–8. https://link.springer.com/article/10.1186/s41687-020-00218-5.\n\n\nMcCullagh, Peter, and J. A. Nelder. 1989. Generalized Linear\nModels. 2nd ed. Routledge. https://www.utstat.toronto.edu/~brunner/oldclass/2201s11/readings/glmbook.pdf.\n\n\nMcLachlan, Geoffrey J, and Thriyambakam Krishnan. 2007. The EM\nAlgorithm and Extensions. 2nd ed. John Wiley & Sons. https://doi.org/10.1002/9780470191613.\n\n\nMoore, Dirk F. 2016. Applied Survival Analysis Using r. Vol.\n473. Springer.\n\n\nNahhas, Ramzi W. n.d. Introduction to Regression Methods for Public\nHealth Using r. CRC Press. https://www.bookdown.org/rwnahhas/RMPH/.\n\n\nPebesma, Edzer, and Roger Bivand. 2023. Spatial Data Science: With\nApplications in R. Boca Raton: Chapman; Hall/CRC. https://doi.org/10.1201/9780429459016.\n\n\nPohl, Moritz, Lukas Baumann, Rouven Behnisch, Marietta Kirchner,\nJohannes Krisam, and Anja Sander. 2021. “Estimands—A Basic Element for Clinical\nTrials.” Deutsches Ärzteblatt\nInternational 118 (51-52): 883–88. https://doi.org/10.3238/arztebl.m2021.0373.\n\n\nPolin, Richard A, William W Fox, and Steven H Abman. 2011. Fetal and\nNeonatal Physiology. 4th ed. Elsevier health sciences.\n\n\nRosenman, Ray H, Richard J Brand, C David Jenkins, Meyer Friedman,\nReuben Straus, and Moses Wurm. 1975. “Coronary Heart Disease in\nthe Western Collaborative Group Study: Final Follow-up Experience of 8\n1/2 Years.” JAMA 233 (8): 872–77. https://doi.org/10.1001/jama.1975.03260080034016.\n\n\nSearle, Shayle R, and Andre I Khuri. 2017. Matrix Algebra Useful for\nStatistics. John Wiley & Sons.\n\n\nSeber, George AF, and Alan J Lee. 2012. Linear Regression\nAnalysis. 2nd ed. John Wiley & Sons. https://www.wiley.com/en-us/Linear+Regression+Analysis%2C+2nd+Edition-p-9781118274422.\n\n\nVan Buuren, Stef. 2018. Flexible Imputation of Missing Data.\nCRC press. https://stefvanbuuren.name/fimd/.\n\n\nVenables, Bill. 2023. codingMatrices: Alternative Factor Coding\nMatrices for Linear Model Formulae. https://CRAN.R-project.org/package=codingMatrices.\n\n\nVittinghoff, Eric, David V Glidden, Stephen C Shiboski, and Charles E\nMcCulloch. 2012. Regression Methods in Biostatistics: Linear,\nLogistic, Survival, and Repeated Measures Models. 2nd ed. Springer.\nhttps://doi.org/10.1007/978-1-4614-1353-0.\n\n\nWickham, Hadley. 2019. Advanced r. Chapman; Hall/CRC. https://adv-r.hadley.nz/index.html.\n\n\n———. 2021. Mastering Shiny. \" O’Reilly Media, Inc.\". https://mastering-shiny.org/.\n\n\nWickham, Hadley, Mara Averick, Jennifer Bryan, Winston Chang, Lucy\nD’Agostino McGowan, Romain François, Garrett Grolemund, et al. 2019.\n“Welcome to the tidyverse.”\nJournal of Open Source Software 4 (43): 1686. https://doi.org/10.21105/joss.01686.\n\n\nWickham, Hadley, and Jennifer Bryan. 2023. R Packages. \"\nO’Reilly Media, Inc.\". https://r-pkgs.org/.\n\n\nWickham, Hadley, Mine Çetinkaya-Rundel, and Garrett Grolemund. 2023.\nR for Data Science. \" O’Reilly Media, Inc.\". https://r4ds.hadley.nz/.", + "text": "Anderson, Edgar. 1935. “The Irises of the Gaspe Peninsula.”\nBulletin of American Iris Society 59: 2–5.\n\n\nBache, Stefan Milton, and Hadley Wickham. 2022. Magrittr: A\nForward-Pipe Operator for r. https://CRAN.R-project.org/package=magrittr.\n\n\nBanerjee, Sudipto, and Anindya Roy. 2014. Linear Algebra and Matrix\nAnalysis for Statistics. Vol. 181. Crc Press Boca Raton.\n\n\nCanchola, Alison J, Susan L Stewart, Leslie Bernstein, Dee W West,\nRonald K Ross, Dennis Deapen, Richard Pinder, et al. 2003. “Cox\nRegression Using Different Time-Scales.” Western Users of SAS\nSoftware. https://www.lexjansen.com/wuss/2003/DataAnalysis/i-cox_time_scales.pdf.\n\n\nCasella, George, and Roger Berger. 2002. Statistical Inference.\n2nd ed. Cengage Learning. https://www.cengage.com/c/statistical-inference-2e-casella-berger/9780534243128/.\n\n\nChang, Winston. 2024. R Graphics Cookbook: Practical Recipes for\nVisualizing Data. O’Reilly Media. https://r-graphics.org/.\n\n\nChatterjee, Samprit, and Ali S Hadi. 2015. Regression Analysis by\nExample. John Wiley & Sons. https://www.wiley.com/en-us/Regression+Analysis+by+Example%2C+4th+Edition-p-9780470055458.\n\n\nDalgaard, Peter. 2008. Introductory Statistics with r. New\nYork, NY: Springer New York. https://link.springer.com/book/10.1007/978-0-387-79054-1.\n\n\nDobson, Annette J, and Adrian G Barnett. 2018. An Introduction to\nGeneralized Linear Models. 4th ed. CRC press. https://doi.org/10.1201/9781315182780.\n\n\nDunn, Peter K, Gordon K Smyth, et al. 2018. Generalized Linear\nModels with Examples in r. Vol. 53. Springer. https://link.springer.com/book/10.1007/978-1-4419-0118-7.\n\n\nEfron, Bradley, and David V Hinkley. 1978. “Assessing the Accuracy\nof the Maximum Likelihood Estimator: Observed Versus Expected Fisher\nInformation.” Biometrika 65 (3): 457–83.\n\n\nFaraway, Julian J. 2016. Extending the Linear Model with r:\nGeneralized Linear, Mixed Effects and Nonparametric Regression\nModels. 2nd ed. Chapman; Hall/CRC. https://doi.org/10.1201/9781315382722.\n\n\nFay, Colin, Sébastien Rochette, Vincent Guyader, and Cervan Girard.\n2021. Engineering Production-Grade Shiny Apps. Chapman;\nHall/CRC. https://engineering-shiny.org/.\n\n\nFieller, Nick. 2016. Basics of Matrix Algebra for Statistics with\nr. Chapman; Hall/CRC. https://doi.org/10.1201/9781315370200.\n\n\nFox, John. 2015. Applied Regression Analysis and Generalized Linear\nModels. Sage publications.\n\n\nGrambsch, Patricia M, and Terry M Therneau. 1994. “Proportional\nHazards Tests and Diagnostics Based on Weighted Residuals.”\nBiometrika 81 (3): 515–26. https://doi.org/10.1093/biomet/81.3.515.\n\n\nHarrell, Frank E. 2015. Regression Modeling Strategies: With\nApplications to Linear Models, Logistic Regression, and Survival\nAnalysis. 2nd ed. Springer. https://doi.org/10.1007/978-3-319-19425-7.\n\n\nHosmer Jr, David W, Stanley Lemeshow, and Rodney X Sturdivant. 2013.\nApplied Logistic Regression. John Wiley & Sons.\n\n\nJames, Gareth, Daniela Witten, Trevor Hastie, Robert Tibshirani, et al.\n2013. An Introduction to Statistical Learning. Vol. 112.\nSpringer. https://www.statlearning.com/.\n\n\nKhuri, André I. 2003. Advanced Calculus with Applications in\nStatistics. John Wiley & Sons. https://www.wiley.com/en-us/Advanced+Calculus+with+Applications+in+Statistics%2C+2nd+Edition-p-9780471391043.\n\n\nKlein, John P, Melvin L Moeschberger, et al. 2003. Survival\nAnalysis: Techniques for Censored and Truncated Data. Vol. 1230.\nSpringer. https://link.springer.com/book/10.1007/b97377.\n\n\nKleinbaum, David G, and Mitchel Klein. 2010. Logistic\nRegression. 3rd ed. Springer. https://link.springer.com/book/10.1007/978-1-4419-1742-3.\n\n\n———. 2012. Survival Analysis a Self-Learning Text. 3rd ed.\nSpringer. https://link.springer.com/book/10.1007/978-1-4419-6646-9.\n\n\nKleinbaum, David G, Lawrence L Kupper, Azhar Nizam, K Muller, and ES\nRosenberg. 2014. Applied Regression Analysis and Other Multivariable\nMethods. 5th ed. Cengage Learning. https://www.cengage.com/c/applied-regression-analysis-and-other-multivariable-methods-5e-kleinbaum/9781285051086/.\n\n\nKleinman, Ken, and Nicholas J Horton. 2009. SAS and r: Data\nManagement, Statistical Analysis, and Graphics. Chapman; Hall/CRC.\nhttps://www.routledge.com/SAS-and-R-Data-Management-Statistical-Analysis-and-Graphics-Second-Edition/Kleinman-Horton/p/book/9781466584495.\n\n\nKuhn, Max, and Julia Silge. 2022. Tidy Modeling with r. \"\nO’Reilly Media, Inc.\". https://www.tmwr.org/.\n\n\nKutner, Michael H, Christopher J Nachtsheim, John Neter, and William Li.\n2005. Applied Linear Statistical Models. McGraw-Hill.\n\n\nLawrance, Rachael, Evgeny Degtyarev, Philip Griffiths, Peter Trask,\nHelen Lau, Denise D’Alessio, Ingolf Griebsch, Gudrun Wallenstein, Kim\nCocks, and Kaspar Rufibach. 2020. “What Is an Estimand, and How\nDoes It Relate to Quantifying the Effect of Treatment on\nPatient-Reported Quality of Life Outcomes in Clinical Trials?”\nJournal of Patient-Reported Outcomes 4 (1): 1–8. https://link.springer.com/article/10.1186/s41687-020-00218-5.\n\n\nMcCullagh, Peter, and J. A. Nelder. 1989. Generalized Linear\nModels. 2nd ed. Routledge. https://www.utstat.toronto.edu/~brunner/oldclass/2201s11/readings/glmbook.pdf.\n\n\nMcLachlan, Geoffrey J, and Thriyambakam Krishnan. 2007. The EM\nAlgorithm and Extensions. 2nd ed. John Wiley & Sons. https://doi.org/10.1002/9780470191613.\n\n\nMoore, Dirk F. 2016. Applied Survival Analysis Using r. Vol.\n473. Springer.\n\n\nNahhas, Ramzi W. n.d. Introduction to Regression Methods for Public\nHealth Using r. CRC Press. https://www.bookdown.org/rwnahhas/RMPH/.\n\n\nPebesma, Edzer, and Roger Bivand. 2023. Spatial Data Science: With\nApplications in R. Boca Raton: Chapman; Hall/CRC. https://doi.org/10.1201/9780429459016.\n\n\nPohl, Moritz, Lukas Baumann, Rouven Behnisch, Marietta Kirchner,\nJohannes Krisam, and Anja Sander. 2021. “Estimands—A Basic Element for Clinical\nTrials.” Deutsches Ärzteblatt\nInternational 118 (51-52): 883–88. https://doi.org/10.3238/arztebl.m2021.0373.\n\n\nPolin, Richard A, William W Fox, and Steven H Abman. 2011. Fetal and\nNeonatal Physiology. 4th ed. Elsevier health sciences.\n\n\nRosenman, Ray H, Richard J Brand, C David Jenkins, Meyer Friedman,\nReuben Straus, and Moses Wurm. 1975. “Coronary Heart Disease in\nthe Western Collaborative Group Study: Final Follow-up Experience of 8\n1/2 Years.” JAMA 233 (8): 872–77. https://doi.org/10.1001/jama.1975.03260080034016.\n\n\nSearle, Shayle R, and Andre I Khuri. 2017. Matrix Algebra Useful for\nStatistics. John Wiley & Sons.\n\n\nSeber, George AF, and Alan J Lee. 2012. Linear Regression\nAnalysis. 2nd ed. John Wiley & Sons. https://www.wiley.com/en-us/Linear+Regression+Analysis%2C+2nd+Edition-p-9781118274422.\n\n\nSelvin, Steve. 2001. Epidemiologic Analysis: A Case-Oriented\nApproach. Oxford University Press.\n\n\nVan Buuren, Stef. 2018. Flexible Imputation of Missing Data.\nCRC press. https://stefvanbuuren.name/fimd/.\n\n\nVenables, Bill. 2023. codingMatrices: Alternative Factor Coding\nMatrices for Linear Model Formulae. https://CRAN.R-project.org/package=codingMatrices.\n\n\nVittinghoff, Eric, David V Glidden, Stephen C Shiboski, and Charles E\nMcCulloch. 2012. Regression Methods in Biostatistics: Linear,\nLogistic, Survival, and Repeated Measures Models. 2nd ed. Springer.\nhttps://doi.org/10.1007/978-1-4614-1353-0.\n\n\nWickham, Hadley. 2019. Advanced r. Chapman; Hall/CRC. https://adv-r.hadley.nz/index.html.\n\n\n———. 2021. Mastering Shiny. \" O’Reilly Media, Inc.\". https://mastering-shiny.org/.\n\n\nWickham, Hadley, Mara Averick, Jennifer Bryan, Winston Chang, Lucy\nD’Agostino McGowan, Romain François, Garrett Grolemund, et al. 2019.\n“Welcome to the tidyverse.”\nJournal of Open Source Software 4 (43): 1686. https://doi.org/10.21105/joss.01686.\n\n\nWickham, Hadley, and Jennifer Bryan. 2023. R Packages. \"\nO’Reilly Media, Inc.\". https://r-pkgs.org/.\n\n\nWickham, Hadley, Mine Çetinkaya-Rundel, and Garrett Grolemund. 2023.\nR for Data Science. \" O’Reilly Media, Inc.\". https://r4ds.hadley.nz/.", "crumbs": [ "References" ] @@ -840,7 +851,7 @@ "href": "math-prereqs.html#logarithms-and-exponentials", "title": "Appendix A — Mathematics", "section": "A.2 Logarithms and Exponentials", - "text": "A.2 Logarithms and Exponentials\n\nTheorem A.4 (logarithm of a product is the sum of the logs of the factors) \\[\n\\text{log}\\left\\{a\\cdot b\\right\\} = \\text{log}\\left\\{a\\right\\} + \\text{log}\\left\\{b\\right\\}\n\\]\n\n\nCorollary A.1 (logarithm of a quotient)  \n\nThe logarithm of a quotient is equal to the difference of the logs of the factors:\n\n\\[\\text{log}\\left\\{\\frac{a}{b}\\right\\} = \\text{log}\\left\\{a\\right\\} - \\text{log}\\left\\{b\\right\\}\\]\n\n\nTheorem A.5 (logarithm of an exponential function) \\[\n\\text{log}\\left\\{a^b\\right\\} = b \\cdot\\text{log}\\left\\{a\\right\\}\n\\]\n\n\nTheorem A.6 (exponential of a sum)  \n\nThe exponential of a sum is equal to the product of the exponentials of the addends:\n\n\\[\\text{exp}\\left\\{a+b\\right\\} = \\text{exp}\\left\\{a\\right\\} \\cdot\\text{exp}\\left\\{b\\right\\}\\]\n\n\nTheorem A.7 (derivative of logarithm) \\[\\text{log}'\\left\\{x\\right\\} = \\frac{1}{x} = x^{-1}\\]\n\n\nTheorem A.8 (derivative of exponential) \\[\\text{exp}'\\left\\{x\\right\\} = \\text{exp}\\left\\{x\\right\\}\\]\n\n\n\nCorollary A.2 (Chain rule for logarithms) \\[\n\\frac{\\partial}{\\partial x}\\text{log}\\left\\{f(x)\\right\\} = \\frac{f'(x)}{f(x)}\n\\]\n\n\nProof. Apply Theorem A.3 and Theorem A.7.", + "text": "A.2 Logarithms and Exponentials\n\nTheorem A.4 (logarithm of a product is the sum of the logs of the factors) \\[\n\\text{log}\\left\\{a\\cdot b\\right\\} = \\text{log}\\left\\{a\\right\\} + \\text{log}\\left\\{b\\right\\}\n\\]\n\n\nCorollary A.1 (logarithm of a quotient)  \n\nThe logarithm of a quotient is equal to the difference of the logs of the factors:\n\n\\[\\text{log}\\left\\{\\frac{a}{b}\\right\\} = \\text{log}\\left\\{a\\right\\} - \\text{log}\\left\\{b\\right\\}\\]\n\n\nTheorem A.5 (logarithm of an exponential function) \\[\n\\text{log}\\left\\{a^b\\right\\} = b \\cdot\\text{log}\\left\\{a\\right\\}\n\\]\n\n\nTheorem A.6 (exponential of a sum)  \n\nThe exponential of a sum is equal to the product of the exponentials of the addends:\n\n\\[\\text{exp}\\left\\{a+b\\right\\} = \\text{exp}\\left\\{a\\right\\} \\cdot\\text{exp}\\left\\{b\\right\\}\\]\n\n\nCorollary A.2 (exponential of a difference)  \n\nThe exponential of a difference is equal to the quotient of the exponentials of the addends:\n\n\\[\\text{exp}\\left\\{a-b\\right\\} = \\frac{\\text{exp}\\left\\{a\\right\\}}{\\text{exp}\\left\\{b\\right\\}}\\]\n\n\nTheorem A.7 (derivative of logarithm) \\[\\text{log}'\\left\\{x\\right\\} = \\frac{1}{x} = x^{-1}\\]\n\n\nTheorem A.8 (derivative of exponential) \\[\\text{exp}'\\left\\{x\\right\\} = \\text{exp}\\left\\{x\\right\\}\\]\n\n\n\nCorollary A.3 (Chain rule for logarithms) \\[\n\\frac{\\partial}{\\partial x}\\text{log}\\left\\{f(x)\\right\\} = \\frac{f'(x)}{f(x)}\n\\]\n\n\nProof. Apply Theorem A.3 and Theorem A.7.", "crumbs": [ "Appendices", "A  Mathematics" @@ -884,7 +895,7 @@ "href": "probability.html#random-variables", "title": "Appendix B — Probability", "section": "\nB.1 Random variables", - "text": "B.1 Random variables\n\nB.1.1 Types of random 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)", + "text": "B.1 Random variables\n\nB.1.1 Types of random 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\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\n\nDefinition B.3 (Exposure) In many count outcomes, there is some sense of exposure magnitude or duration of observation:\n\nperson*year\ntime at risk\nlength of toothbrushing session\nmiles driven\nhours driven\nsize of pond\n\nThis concept is similar to the number of trials in a binomial distribution, except that in non-binomial count outcomes, there can be more than one event per trial.\nWe can use \\(t\\) to represent continuous-valued exposures/observation durations, and \\(n\\) to represent discrete-valued exposures.", "crumbs": [ "Appendices", "B  Probability" @@ -895,7 +906,7 @@ "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.2 (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\nDefinition B.3 (Poisson distribution) \\[P(Y=y) = \\frac{\\lambda^{y} e^{-\\lambda}}{y!}\\]", + "text": "B.2 Key probability distributions\n\nB.2.1 The Bernoulli distribution\n\nDefinition B.4 (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.5 (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.1}\\]\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.2}\\]\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.1 (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.1.\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\\), 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.2 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.6 (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\nDefinition B.7 (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.3 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\\)", "crumbs": [ "Appendices", "B  Probability" @@ -906,7 +917,7 @@ "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.4 (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.5 (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.6 (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.1 (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.7 (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.2 (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.8 (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.9 (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.10 (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.3 \\[\\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.11 (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.4 (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.5 (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.5 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.12 (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.13 (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.13).\n\n\n\n\n\nDefinition B.14 (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.15 (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.16 (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.17 (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.18 (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.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\\).", "crumbs": [ "Appendices", "B  Probability" @@ -1005,7 +1016,7 @@ "href": "intro-MLEs.html#overview-of-maximum-likelihood-estimation", "title": "Appendix E — Introduction to Maximum Likelihood Inference", "section": "\nE.1 Overview of maximum likelihood estimation", - "text": "E.1 Overview of maximum likelihood estimation\n\nE.1.1 The likelihood function\n\nDefinition E.1 (Likelihood of a single observation) Let \\(X\\) be a random variable and let \\(x\\) be \\(X\\)’s observed data value. Let \\(\\text{p}_{\\Theta}(X=x)\\) be a probability model for the distribution of \\(X\\), with parameter vector \\(\\Theta\\).\nThen the likelihood of parameter value \\(\\theta\\), for model \\(\\text{p}_{\\Theta}(X=x)\\) and data \\(X = x\\), is simply the probability of the event \\(X=x\\) given \\(\\Theta= \\theta\\):\n\\[\n\\begin{aligned}\n\\mathcal{L}(\\theta) &\\stackrel{\\text{def}}{=}\\text{P}_{\\theta}(X = x)\n\\end{aligned}\n\\]\n\n\n\nDefinition E.2 (Likelihood of a dataset) Let \\(\\tilde{x}\\) be a dataset with corresponding random variable \\(\\tilde{X}\\). Let \\(\\text{p}_{\\Theta}(\\tilde{X})\\) be a probability model for the distribution of \\(\\tilde{X}\\) with unknown parameter vector \\(\\Theta\\).\nThen the likelihood of parameter value \\(\\theta\\), for model \\(\\text{p}_{\\Theta}(X)\\) and data \\(\\tilde{X}= \\tilde{x}\\), is the joint probability of \\(\\tilde{X}= \\tilde{x}\\) given \\(\\Theta= \\theta\\):\n\\[\n\\begin{aligned}\n\\mathcal{L}(\\theta) &\\stackrel{\\text{def}}{=}p_{\\theta}(\\tilde{X}= \\tilde{x})\n\\\\&=p_{\\theta}(X_1=x_1, ..., X_n = x_n)\n\\end{aligned}\n\\]\n\n\n\n\n\n\n\nNotation for the likelihood function\n\n\n\nThe likelihood function can be written as:\n\n\\(\\mathcal{L}(\\theta)\\)\n\\(\\mathcal{L}(\\tilde{x};\\theta)\\)\n\\(\\mathcal{L}(\\theta; \\tilde{x})\\)\n\\(\\mathcal{L}_{\\tilde{x}}(\\theta)\\)\n\\(\\mathcal{L}_{\\theta}(\\tilde{x})\\)\n\\(\\mathcal{L}(\\tilde{x} | \\theta)\\)\n\nAll of these notations mean the same thing.\n\n\n\nThe likelihood is a function that takes \\(\\theta\\) (and implicitly, \\(\\tilde{X}\\)) as inputs and outputs a single number, the joint probability of \\(\\tilde{x}\\) for model \\(p_\\Theta(\\tilde{X}=\\tilde{x})\\) with \\(\\Theta = \\theta\\).\n\n\n\nTheorem E.1 (Likelihood of an independent sample) For mutually independent data \\(X_1, ..., X_n\\):\n\\[\\mathcal{L}(\\tilde{x}|\\theta) = \\prod_{i=1}^n \\text{p}(X_i=x_i|\\theta) \\tag{E.1}\\]\n\n\nProof. \\[\n\\begin{aligned}\n\\mathcal{L}(\\tilde{x}|\\theta)\n&\\stackrel{\\text{def}}{=}\\text{p}(X_1 = x_1, …,X_n =x_n|\\theta)\n\\\\&= \\prod_{i=1}^n \\text{p}(X_i=x_i|\\theta)\n\\end{aligned}\n\\] The second equality is by the definition of statistical independence.\n\n\n\nDefinition E.3 (Likelihood components) Given an \\(\\text{iid}\\) dataset \\(\\tilde{x}\\), the likelihood component or likelihood factor of observation \\(X_i=x_i\\) is the marginal likelihood of \\(X_i=x_i\\):\n\\[\\mathcal{L}_i(\\theta) = \\text{P}(X_i=x_i)\\]\n\n\n\nTheorem E.2 For \\(\\text{iid}\\) data \\(\\tilde{x}\\stackrel{\\text{def}}{=}x_1, \\ldots, x_n\\), the likelihood of the dataset is equal to the product of the observation-specific likelihood factors:\n\\[\\mathcal{L}(\\theta) = \\prod_{i=1}^n\\mathcal{L}_i(\\theta)\\]\n\n\nE.1.2 The maximum likelihood estimate\n\nDefinition E.4 (Maximum likelihood estimate) The maximum likelihood estimate of a parameter vector \\(\\Theta\\), denoted \\(\\hat\\theta_{\\text{ML}}\\), is the value of \\(\\Theta\\) that maximizes the likelihood:\n\\[\n\\hat\\theta_{\\text{ML}}\\stackrel{\\text{def}}{=}\\arg \\max_\\Theta\\mathcal{L}(\\Theta)\n\\tag{E.2}\\]\n\n\nE.1.3 Finding the maximum of a function\nRecall from calculus: the maxima of a continuous function \\(f(x)\\) over a range of input values \\(\\mathcal{R}(x)\\) can be found either:\n\nat the edges of the range of input values, OR:\nwhere the function is flat (i.e. where the gradient function \\(f'(x) = 0\\)) AND the second derivative is negative definite (\\(f''(x) < 0\\)).\n\nE.1.4 Directly maximizing the likelihood function for iid data\nTo find the maximizer(s) of the likelihood function, we need to solve \\(\\mathcal{L}'(\\theta) = 0\\) for \\(\\theta\\). However, even for mutually independent data, we quickly run into a problem:\n\\[\n\\begin{aligned}\n\\mathcal{L}'(\\theta)\n&= \\frac{\\partial}{\\partial \\theta} \\mathcal{L}(\\theta)\n\\\\ &= \\frac{\\partial}{\\partial \\theta} \\prod_{i=1}^n p(X_i=x_i|\\theta)\n\\end{aligned}\n\\tag{E.3}\\]\nThe derivative of the likelihood of independent data is the derivative of a product. We will have to perform a massive application of the product rule to evaluate this derivative.\n\nE.1.5 The log-likelihood function\nIt is typically easier to work with the log of the likelihood function:\n\nDefinition E.5 (Log-likelihood) The log-likelihood of parameter value \\(\\theta\\), for model \\(\\text{p}_{\\Theta}(\\tilde{X})\\) and data \\(\\tilde{X}= \\tilde{x}\\), is the natural logarithm of the likelihood1:\n\\[\\ell(\\theta) \\stackrel{\\text{def}}{=}\\text{log}\\left\\{\\mathcal{L}(\\theta)\\right\\}\\]\n\n\n\nDefinition E.6 (Log-likelihood components) Given a dataset \\(\\tilde{X}= \\tilde{x}\\), the log-likelihood component of observation \\(X_i=x_i\\) is the natural logarithm of the likelihood component:\n\\[\\ell_i(\\theta) \\stackrel{\\text{def}}{=}\\text{log}\\left\\{\\mathcal{L}_i(\\theta)\\right\\}\\]\n\n\n\nTheorem E.3 The likelihood and log-likelihood have the same maximizer:\n\\[\n\\arg \\max_\\theta\\mathcal{L}(\\theta) = \\arg \\max_\\theta\\ell(\\theta)\n\\]\n\n\nProof. Left to the reader.\n\n\n\nTheorem E.4 (Log-likelihood of an \\(\\text{iid}\\) sample) For \\(\\text{iid}\\) data \\(X_1, ..., X_n\\) with shared distribution \\(\\text{p}(X=x)\\):\n\\[\\ell(x|\\theta) = \\sum_{i=1}^n \\text{log}\\left\\{p(X=x_i|\\theta)\\right\\} \\tag{E.4}\\]\n\n\nProof. \\[\n\\begin{aligned}\n\\ell(x|\\theta)\n&\\stackrel{\\text{def}}{=}\\text{log}\\left\\{\\mathcal{L}(\\tilde{x}|\\theta)\\right\\}\n\\\\&= \\text{log}\\left\\{\\prod_{i=1}^n \\text{p}(X_i=x_i|\\theta)\\right\\}\n\\\\&= \\sum_{i=1}^n \\text{log}\\left\\{p(X=x_i|\\theta)\\right\\}\n\\end{aligned}\n\\]\n\n\n\nFor \\(\\text{iid}\\) data, we will have a much easier time taking the derivative of the log-likelihood:\n\n\nTheorem E.5 (Derivative of the log-likelihood function for \\(\\text{iid}\\) data) For \\(\\text{iid}\\) data:\n\\[\\ell'(\\theta) = \\sum_{i=1}^n\\frac{\\partial}{\\partial \\theta} \\text{log}\\left\\{\\text{p}(X=x_i|\\theta)\\right\\} \\tag{E.5}\\]\n\n\nProof. \\[\n\\begin{aligned}\n\\ell'(\\theta)\n&= \\frac{\\partial}{\\partial \\theta} \\ell(\\theta)\n\\\\ &= \\frac{\\partial}{\\partial \\theta} \\sum_{i=1}^n \\text{log}\\left\\{\\text{p}(X=x_i|\\theta)\\right\\}\n\\\\ &= \\sum_{i=1}^n \\frac{\\partial}{\\partial \\theta} \\text{log}\\left\\{\\text{p}(X=x_i|\\theta)\\right\\}\n\\end{aligned}\n\\]\n\n\nE.1.6 The score function\nThe first derivative2 of the log-likelihood, \\(\\ell'(\\theta)\\), is important enough to have its own name: the score function.\n\nDefinition E.7 (Score function) The score function of a statistical model \\(\\text{p}(\\tilde{X}=\\tilde{x})\\) is the gradient (i.e., first derivative) of the log-likelihood of that model:\n\\[\\ell'(\\theta) \\stackrel{\\text{def}}{=}\\frac{\\partial}{\\partial \\theta} \\ell(\\theta)\\]\n\n\nWe often skip writing the arguments \\(x\\) and/or \\(\\theta)\\), so \\(\\ell' \\stackrel{\\text{def}}{=}\\ell'(\\tilde{x};\\theta) \\stackrel{\\text{def}}{=}\\ell'(\\theta)\\).3 Some statisticians use \\(U\\) or \\(S\\) instead of \\(\\ell'\\). I prefer \\(\\ell'\\). Why use up extra letters?\n\n\nE.1.7 Asymptotic distribution of the maximum likelihood estimate\n\nWe learned how to quantify our uncertainty about these maximum likelihood estimates; with sufficient sample size, \\(\\hat\\theta_{\\text{ML}}\\) has the approximate distribution:\n\n\\[\n\\hat\\theta_{ML} \\dot \\sim N(\\theta,\\mathcal I(\\theta)^{-1})\n\\]\nRecall:\n\n\\(\\mathcal{I}(\\theta) \\stackrel{\\text{def}}{=}\\mathbb{E}\\left[I(\\tilde{X};\\theta)\\right]\\)\n\\(I(\\tilde{X},\\theta) \\stackrel{\\text{def}}{=}-\\ell''(\\tilde{X};\\theta)\\)\n\nWe can estimate \\(\\mathcal{I}(\\theta)\\) using either \\(\\mathcal{I}(\\hat\\theta_{\\text{ML}})\\) or \\(I(\\tilde{x}; \\hat\\theta_{\\text{ML}})\\).\nSo we can estimate the standard error of \\(\\hat\\theta_k\\) as:\n\\[\n\\widehat{\\text{SE}}\\left(\\hat\\theta_k\\right) = \\sqrt{\\left[\\left(\\hat{\\mathcal{I}}\\left(\\hat\\theta_{\\text{ML}}\\right)\\right)^{-1}\\right]_{kk}}\n\\]\n\nE.1.8 The (Fisher) (expected) information matrix\nThe variance of \\(\\ell'(x,\\theta)\\), \\({Cov}\\left\\{ \\ell'(x,\\theta) \\right\\}\\), is also very important; we call it the “expected information matrix”, “Fisher information matrix”, or just “information matrix”, and we represent it using the symbol \\(\\mathfrak{I}\\) (\\frakturI in Unicode, \\mathfrak{I} in LaTeX).\n\\[\\mathfrak{I \\stackrel{\\text{def}}{=}I(}\\theta) \\stackrel{\\text{def}}{=}{Cov}\\left( \\ell'|\\theta \\right) = E[ \\ell'{\\ell'}^{\\top} ] - E[ \\ell' ]\\ E[ \\ell' ]^{\\top}\n\\]\nThe elements of \\(\\mathfrak{I}\\) are:\n\\[\\left\\{ \\mathfrak{I}_{ij} \\stackrel{\\text{def}}{=}{Cov}\\left( {\\ell'}_{i},{\\ell'}_{j} \\right) = E[ \\ell_{i}'\\ell_{j}' ] - E[ {\\ell'}_{i} ] E[ {\\ell'}_{j} ] \\right\\}\\]\nHere,\n\\[\n\\begin{aligned}\n\\mathbb{E}\\left[\\ell'\\right]\n&\\stackrel{\\text{def}}{=}\\int_{x \\in \\mathcal{R}(x)}\n{\n\\ell'(x,\\theta) \\text{p}(X = x | \\theta) dx\n}\n\\\\ &= \\int_{x \\in \\mathcal{R}(X)}\n{\n\\left(\n\\frac{\\partial}{\\partial \\theta}\n\\text{log}\\left\\{\\text{p}(X = x | \\theta)\\right\\}\n\\right)\n\\text{p}(X = x | \\theta) dx\n}\n\\\\ &=\n\\int_{x \\in \\mathcal{R}(X)}\n{\n\\frac\n{\\frac{\\partial}{\\partial \\theta} \\text{p}(X = x | \\theta)}\n{\\text{p}(X = x | \\theta)}\n\\text{p}(X = x | \\theta) dx\n}\n\\\\ &=\n\\int_{x \\in \\mathcal{R}(X)}\n{\n\\frac{\\partial}{\\partial \\theta} \\text{p}(X = x | \\theta) dx\n}\n\\end{aligned}\n\\]\nAnd similarly\n\\[\n\\mathbb{E}\\left[\\ell' \\ell'^{\\top}\\right]\n\\stackrel{\\text{def}}{=}\n\\int_{x \\in R(x)}\n{\\ell'(x,\\theta)\\ell'(x,\\theta)^{\\top}\\\n\\text{p}\\left(X = x | \\theta\\right)\\ dx}\n\\]\nNote that \\(\\mathbb{E}\\left[\\ell'\\right]\\) and \\(\\mathbb{E}\\left[\\ell'{\\ell'}^{\\top}\\right]\\) are functions of \\(\\theta\\) but not of \\(x\\); the expectation operator removed \\(x\\).\nAlso note that for most of the distributions you are familiar with (including Gaussian, binomial, Poisson, exponential):\n\\[\\mathbb{E}\\left[\\ell'\\right] = 0\\]\nSo\n\\[\\mathcal{I}\\left(\\theta\\right) = \\mathbb{E}\\left[\\ell'{\\ell'}^{\\top} \\right]\\]\nMoreover, for those distributions (called the “exponential family”), we have:\n\\[\n\\mathfrak{I} = -\\mathbb{E}\\left[\\ell''\\right]\n= \\mathbb{E}\\left[- \\ell''\\right]\n\\]\n(see Dobson and Barnett (2018), §3.17), where\n\\[\\ell'' \\stackrel{\\text{def}}{=}\\frac{\\partial}{\\partial \\theta}\\ell^{'(x,\\theta)^{\\top}} = \\frac{\\partial}{\\partial \\theta}\\frac{\\partial}{\\partial \\theta^{\\top}}\\ell(x,\\theta)\\]\nis the \\(p \\times p\\) matrix whose elements are:\n\\[\\ell_{ij}'' \\stackrel{\\text{def}}{=}\\frac{\\partial}{\\partial \\theta_{i}}\\frac{\\partial}{\\partial \\theta_{j}}\\text{log}\\left\\{ p\\left( X = x \\mid \\theta \\right)\\right\\}\\]\n\\(\\ell''\\) is called the “Hessian”4 of the log-likelihood function.\nSometimes, we use \\(I(\\theta;x) \\stackrel{\\text{def}}{=}- \\ell''\\) (note the standard-font “I” here). \\(I(\\theta;x)\\) is the observed information, precision, or concentration matrix (Negative Hessian).\n\n\n\n\n\n\nKey point\n\n\n\nThe asymptotics of MLEs gives us \\({\\widehat{\\theta}}_{ML} \\sim N\\left( \\theta,\\mathfrak{I}^{- 1}(\\theta) \\right)\\), approximately, for large sample sizes.\n\n\nWe can estimate \\(\\mathcal{I}^{- 1}(\\theta)\\) by working out \\(\\mathbb{E}\\left[-\\ell''\\right]\\) or \\(\\mathbb{E}\\left[\\ell'{\\ell'}^{\\top}\\right]\\) and plugging in \\(\\hat\\theta_{\\text{ML}}\\), but sometimes we instead use \\(I(\\hat\\theta_{\\text{ML}},\\tilde{x})\\) for convenience; there are some cases where it’s provably better according to some criteria (Efron and Hinkley (1978)).\n\nE.1.9 Iterative maximization\n(c.f., Dobson and Barnett (2018), Chapter 4)\n\nLater, when we are trying to find MLEs for likelihoods which we can’t easily differentiate, we will “hill-climb” using the Newton-Raphson algorithm:\n\n\\[\n\\begin{aligned}\n{\\widehat{\\theta}}^*\n&\\leftarrow {\\widehat{\\theta}}^* + \\left(I\\left(\\tilde{y};{\\widehat{\\theta}}^*\\right)\\right)^{-1}\n\\ell'\\left(\\tilde{y};{\\widehat{\\theta}}^*\\right)\n\\\\ &= {\\widehat{\\theta}}^* - \\left(\\ell''\\left(\\tilde{y};{\\widehat{\\theta}}^*\\right)\\right)^{-1}\n\\ell'\\left(\\tilde{y};{\\widehat{\\theta}}^*\\right)\n\\end{aligned}\n\\]\n\n\nThe reasoning for this algorithm is that we can approximate the the score function using the first-order Taylor polynomial:\n\n\\[\n\\begin{aligned}\n\\ell'(\\theta)\n&\\approx \\ell'^*(\\theta)\n\\\\ &\\stackrel{\\text{def}}{=}\\ell'({\\widehat{\\theta}}^*) + \\ell''({\\widehat{\\theta}}^*)(\\theta- {\\widehat{\\theta}}^*)\n\\end{aligned}\n\\]\n\n\nThe approximate score function, \\(\\ell'^*(\\theta)\\), is a linear function of \\(\\theta\\), so it is easy to solve the corresponding approximate score equation, \\(\\ell'^*(\\theta) = 0\\), for \\(\\theta\\):\n\n\\[\n\\begin{aligned}\n\\theta\n&= {\\widehat{\\theta}}^* - \\ell'({\\widehat{\\theta}}^*) \\cdot\\left(\\ell''({\\widehat{\\theta}}^*)\\right)^{-1}\n\\end{aligned}\n\\]\n\nFor computational simplicity, we will sometimes use \\(\\mathfrak{I}^{- 1}(\\theta)\\) in place of \\(I\\left( \\widehat{\\theta},y \\right)\\); doing so is called “Fisher scoring” or the “method of scoring”. Note that this is the opposite of the substitution that we are making for estimating the variance of the MLE; this time we should technically use the observed information but we use the expected information instead.\n\nThere’s also an “empirical information matrix” (see McLachlan and Krishnan (2007)):\n\\[I_{e}(\\theta,y) \\stackrel{\\text{def}}{=}\\sum_{i = 1}^{n}{\\ell_{i}'\\ {\\ell_{i}'}^{\\top}} - \\frac{1}{n}\\ell'{\\ell'}^{\\top}\\]\nwhere \\(\\ell_{i}\\) is the log-likelihood of the ith observation. Note that \\(\\ell' = \\sum_{i = 1}^{n}\\ell_{i}'\\).\n\\(\\frac{1}{n}I_{e}(\\theta,y)\\) is the sample equivalent of\n\\[\\mathfrak{I \\stackrel{\\text{def}}{=}I(}\\theta) \\stackrel{\\text{def}}{=}{Cov}\\left( \\ell'|\\theta \\right) = E[ \\ell'{\\ell'}^{\\top} ] - E[ \\ell' ]\\ E[ \\ell' ]^{\\top}\\]\n\\[\\left\\{ \\mathfrak{I}_{jk} \\stackrel{\\text{def}}{=}{Cov}\\left( {\\ell'}_{j},{\\ell'}_{k} \\right) = E[ \\ell_{j}'\\ell_{k}' ] - E[ {\\ell'}_{j} ] E[ {\\ell'}_{k} ] \\right\\}\\]\n\\(I_{e}(\\theta,y)\\) is sometimes computationally easier to compute for Newton-Raphson-type maximization algorithms.\nc.f. https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization\n\nE.1.10 Quantifying (un)certainty of MLEs\nConfidence intervals for MLEs\nAn asymptotic approximation of a 95% confidence interval for \\(\\theta_k\\) is\n\\[\n\\hat\\theta_{\\text{ML}}\\pm z_{0.975} \\times \\widehat{\\text{SE}}\\left(\\hat\\theta_k\\right)\n\\]\nwhere \\(z_\\beta\\) the \\(\\beta\\) quantile of the standard Gaussian distribution.\np-values and hypothesis tests for MLEs\n(to add)\nLikelihood ratio tests for MLEs\nlog(likelihood ratio) tests (c.f. Dobson and Barnett 2018, sec. 5.7):\n\\[-2\\ell_{0} \\sim \\chi^{2}(p - q)\\]\nSee also https://online.stat.psu.edu/stat504/book/export/html/657\nPrediction intervals for MLEs\n\\[\\overline{X} \\in [ \\widehat{\\mu} \\pm z_{1 - \\alpha\\text{/}2}\\frac{\\sigma}{m} ]\\]\nWhere \\(m\\) is the sample size of the new data to be predicted (typically 1, except for binary outcomes, where it needs to be bigger for prediction intervals to make sense)", + "text": "E.1 Overview of maximum likelihood estimation\n\nE.1.1 The likelihood function\n\nDefinition E.1 (Likelihood of a single observation) Let \\(X\\) be a random variable and let \\(x\\) be \\(X\\)’s observed data value. Let \\(\\text{p}_{\\Theta}(X=x)\\) be a probability model for the distribution of \\(X\\), with parameter vector \\(\\Theta\\).\nThen the likelihood of parameter value \\(\\theta\\), for model \\(\\text{p}_{\\Theta}(X=x)\\) and data \\(X = x\\), is simply the probability of the event \\(X=x\\) given \\(\\Theta= \\theta\\):\n\\[\n\\begin{aligned}\n\\mathcal{L}(\\theta) &\\stackrel{\\text{def}}{=}\\text{P}_{\\theta}(X = x)\n\\end{aligned}\n\\]\n\n\n\nDefinition E.2 (Likelihood of a dataset) Let \\(\\tilde{x}\\) be a dataset with corresponding random variable \\(\\tilde{X}\\). Let \\(\\text{p}_{\\Theta}(\\tilde{X})\\) be a probability model for the distribution of \\(\\tilde{X}\\) with unknown parameter vector \\(\\Theta\\).\nThen the likelihood of parameter value \\(\\theta\\), for model \\(\\text{p}_{\\Theta}(X)\\) and data \\(\\tilde{X}= \\tilde{x}\\), is the joint probability of \\(\\tilde{X}= \\tilde{x}\\) given \\(\\Theta= \\theta\\):\n\\[\n\\begin{aligned}\n\\mathcal{L}(\\theta) &\\stackrel{\\text{def}}{=}p_{\\theta}(\\tilde{X}= \\tilde{x})\n\\\\&=p_{\\theta}(X_1=x_1, ..., X_n = x_n)\n\\end{aligned}\n\\]\n\n\n\n\n\n\n\nNotation for the likelihood function\n\n\n\nThe likelihood function can be written as:\n\n\\(\\mathcal{L}(\\theta)\\)\n\\(\\mathcal{L}(\\tilde{x};\\theta)\\)\n\\(\\mathcal{L}(\\theta; \\tilde{x})\\)\n\\(\\mathcal{L}_{\\tilde{x}}(\\theta)\\)\n\\(\\mathcal{L}_{\\theta}(\\tilde{x})\\)\n\\(\\mathcal{L}(\\tilde{x} | \\theta)\\)\n\nAll of these notations mean the same thing.\n\n\n\nThe likelihood is a function that takes \\(\\theta\\) (and implicitly, \\(\\tilde{X}\\)) as inputs and outputs a single number, the joint probability of \\(\\tilde{x}\\) for model \\(p_\\Theta(\\tilde{X}=\\tilde{x})\\) with \\(\\Theta = \\theta\\).\n\n\n\nTheorem E.1 (Likelihood of an independent sample) For mutually independent data \\(X_1, ..., X_n\\):\n\\[\\mathcal{L}(\\tilde{x}|\\theta) = \\prod_{i=1}^n \\text{p}(X_i=x_i|\\theta) \\tag{E.1}\\]\n\n\nProof. \\[\n\\begin{aligned}\n\\mathcal{L}(\\tilde{x}|\\theta)\n&\\stackrel{\\text{def}}{=}\\text{p}(X_1 = x_1, …,X_n =x_n|\\theta)\n\\\\&= \\prod_{i=1}^n \\text{p}(X_i=x_i|\\theta)\n\\end{aligned}\n\\] The second equality is by the definition of statistical independence.\n\n\n\nDefinition E.3 (Likelihood components) Given an \\(\\text{iid}\\) dataset \\(\\tilde{x}\\), the likelihood component or likelihood factor of observation \\(X_i=x_i\\) is the marginal likelihood of \\(X_i=x_i\\):\n\\[\\mathcal{L}_i(\\theta) = \\text{P}(X_i=x_i)\\]\n\n\n\nTheorem E.2 For \\(\\text{iid}\\) data \\(\\tilde{x}\\stackrel{\\text{def}}{=}x_1, \\ldots, x_n\\), the likelihood of the dataset is equal to the product of the observation-specific likelihood factors:\n\\[\\mathcal{L}(\\theta) = \\prod_{i=1}^n\\mathcal{L}_i(\\theta)\\]\n\n\nE.1.2 The maximum likelihood estimate\n\nDefinition E.4 (Maximum likelihood estimate) The maximum likelihood estimate of a parameter vector \\(\\Theta\\), denoted \\(\\hat\\theta_{\\text{ML}}\\), is the value of \\(\\Theta\\) that maximizes the likelihood:\n\\[\n\\hat\\theta_{\\text{ML}}\\stackrel{\\text{def}}{=}\\arg \\max_\\Theta\\mathcal{L}(\\Theta)\n\\tag{E.2}\\]\n\n\nE.1.3 Finding the maximum of a function\nRecall from calculus: the maxima of a continuous function \\(f(x)\\) over a range of input values \\(\\mathcal{R}(x)\\) can be found either:\n\nat the edges of the range of input values, OR:\nwhere the function is flat (i.e. where the gradient function \\(f'(x) = 0\\)) AND the second derivative is negative definite (\\(f''(x) < 0\\)).\n\nE.1.4 Directly maximizing the likelihood function for iid data\nTo find the maximizer(s) of the likelihood function, we need to solve \\(\\mathcal{L}'(\\theta) = 0\\) for \\(\\theta\\). However, even for mutually independent data, we quickly run into a problem:\n\\[\n\\begin{aligned}\n\\mathcal{L}'(\\theta)\n&= \\frac{\\partial}{\\partial \\theta} \\mathcal{L}(\\theta)\n\\\\ &= \\frac{\\partial}{\\partial \\theta} \\prod_{i=1}^n p(X_i=x_i|\\theta)\n\\end{aligned}\n\\tag{E.3}\\]\nThe derivative of the likelihood of independent data is the derivative of a product. We will have to perform a massive application of the product rule to evaluate this derivative.\n\nE.1.5 The log-likelihood function\nIt is typically easier to work with the log of the likelihood function:\n\nDefinition E.5 (Log-likelihood) The log-likelihood of parameter value \\(\\theta\\), for model \\(\\text{p}_{\\Theta}(\\tilde{X})\\) and data \\(\\tilde{X}= \\tilde{x}\\), is the natural logarithm of the likelihood1:\n\\[\\ell(\\theta) \\stackrel{\\text{def}}{=}\\text{log}\\left\\{\\mathcal{L}(\\theta)\\right\\}\\]\n\n\n\nDefinition E.6 (Log-likelihood components) Given a dataset \\(\\tilde{X}= \\tilde{x}\\), the log-likelihood component of observation \\(X_i=x_i\\) is the natural logarithm of the likelihood component:\n\\[\\ell_i(\\theta) \\stackrel{\\text{def}}{=}\\text{log}\\left\\{\\mathcal{L}_i(\\theta)\\right\\}\\]\n\n\n\nTheorem E.3 The likelihood and log-likelihood have the same maximizer:\n\\[\n\\arg \\max_\\theta\\mathcal{L}(\\theta) = \\arg \\max_\\theta\\ell(\\theta)\n\\]\n\n\nProof. Left to the reader.\n\n\n\nTheorem E.4 (Log-likelihood of an \\(\\text{iid}\\) sample) For \\(\\text{iid}\\) data \\(X_1, ..., X_n\\) with shared distribution \\(\\text{p}(X=x)\\):\n\\[\\ell(x|\\theta) = \\sum_{i=1}^n \\text{log}\\left\\{p(X=x_i|\\theta)\\right\\} \\tag{E.4}\\]\n\n\nProof. \\[\n\\begin{aligned}\n\\ell(x|\\theta)\n&\\stackrel{\\text{def}}{=}\\text{log}\\left\\{\\mathcal{L}(\\tilde{x}|\\theta)\\right\\}\n\\\\&= \\text{log}\\left\\{\\prod_{i=1}^n \\text{p}(X_i=x_i|\\theta)\\right\\}\n\\\\&= \\sum_{i=1}^n \\text{log}\\left\\{p(X=x_i|\\theta)\\right\\}\n\\end{aligned}\n\\]\n\n\n\nFor \\(\\text{iid}\\) data, we will have a much easier time taking the derivative of the log-likelihood:\n\n\nTheorem E.5 (Derivative of the log-likelihood function for \\(\\text{iid}\\) data) For \\(\\text{iid}\\) data:\n\\[\\ell'(\\theta) = \\sum_{i=1}^n\\frac{\\partial}{\\partial \\theta} \\text{log}\\left\\{\\text{p}(X=x_i|\\theta)\\right\\} \\tag{E.5}\\]\n\n\nProof. \\[\n\\begin{aligned}\n\\ell'(\\theta)\n&= \\frac{\\partial}{\\partial \\theta} \\ell(\\theta)\n\\\\ &= \\frac{\\partial}{\\partial \\theta} \\sum_{i=1}^n \\text{log}\\left\\{\\text{p}(X=x_i|\\theta)\\right\\}\n\\\\ &= \\sum_{i=1}^n \\frac{\\partial}{\\partial \\theta} \\text{log}\\left\\{\\text{p}(X=x_i|\\theta)\\right\\}\n\\end{aligned}\n\\]\n\n\nE.1.6 The score function\nThe first derivative2 of the log-likelihood, \\(\\ell'(\\theta)\\), is important enough to have its own name: the score function.\n\nDefinition E.7 (Score function) The score function of a statistical model \\(\\text{p}(\\tilde{X}=\\tilde{x})\\) is the gradient (i.e., first derivative) of the log-likelihood of that model:\n\\[\\ell'(\\theta) \\stackrel{\\text{def}}{=}\\frac{\\partial}{\\partial \\theta} \\ell(\\theta)\\]\n\n\nWe often skip writing the arguments \\(x\\) and/or \\(\\theta)\\), so \\(\\ell' \\stackrel{\\text{def}}{=}\\ell'(\\tilde{x};\\theta) \\stackrel{\\text{def}}{=}\\ell'(\\theta)\\).3 Some statisticians use \\(U\\) or \\(S\\) instead of \\(\\ell'\\). I prefer \\(\\ell'\\). Why use up extra letters?\n\n\nE.1.7 Asymptotic distribution of the maximum likelihood estimate\n\nWe learned how to quantify our uncertainty about these maximum likelihood estimates; with sufficient sample size, \\(\\hat\\theta_{\\text{ML}}\\) has the approximate distribution:\n\n\\[\n\\hat\\theta_{ML} \\dot \\sim N(\\theta,\\mathcal I(\\theta)^{-1})\n\\]\nRecall:\n\n\\(\\mathcal{I}(\\theta) \\stackrel{\\text{def}}{=}\\mathbb{E}\\left[I(\\tilde{X};\\theta)\\right]\\)\n\\(I(\\tilde{X},\\theta) \\stackrel{\\text{def}}{=}-\\ell''(\\tilde{X};\\theta)\\)\n\nWe can estimate \\(\\mathcal{I}(\\theta)\\) using either \\(\\mathcal{I}(\\hat\\theta_{\\text{ML}})\\) or \\(I(\\tilde{x}; \\hat\\theta_{\\text{ML}})\\).\nSo we can estimate the standard error of \\(\\hat\\theta_k\\) as:\n\\[\n\\widehat{\\text{SE}}\\left(\\hat\\theta_k\\right) = \\sqrt{\\left[\\left(\\hat{\\mathcal{I}}\\left(\\hat\\theta_{\\text{ML}}\\right)\\right)^{-1}\\right]_{kk}}\n\\]\n\nE.1.8 The (Fisher) (expected) information matrix\nThe variance of \\(\\ell'(x,\\theta)\\), \\({Cov}\\left\\{ \\ell'(x,\\theta) \\right\\}\\), is also very important; we call it the “expected information matrix”, “Fisher information matrix”, or just “information matrix”, and we represent it using the symbol \\(\\mathcal{I}\\left(I\\right)\\) (\\frakturI in Unicode, \\mathfrak{I} in LaTeX).\n\\[\n\\begin{aligned}\n\\mathcal{I}\n\\stackrel{\\text{def}}{=}\\mathcal{I}(\\theta)\n\\\\ &\\stackrel{\\text{def}}{=}\\text{Cov}\\left(\\ell'|\\theta\\right)\n\\\\ &= \\mathbb{E}[ \\ell'{\\ell'}^{\\top} ] - \\mathbb{E}[ \\ell' ] \\ \\mathbb{E}[ \\ell' ]^{\\top}\n\\end{aligned}\n\\]\nThe elements of \\(\\mathfrak{I}\\) are:\n\\[\n\\begin{aligned}\n\\mathfrak{I}_{ij}\n&\\stackrel{\\text{def}}{=}\\text{Cov}\\left({\\ell'}_{i},{\\ell'}_{j}\\right)\n\\\\ &= \\mathbb{E}[ \\ell_{i}'\\ell_{j}' ] - \\mathbb{E}[ {\\ell'}_{i} ] \\mathbb{E}[ {\\ell'}_{j} ]\n\\end{aligned}\n\\]\nHere,\n\\[\n\\begin{aligned}\n\\mathbb{E}\\left[\\ell'\\right]\n&\\stackrel{\\text{def}}{=}\\int_{x \\in \\mathcal{R}(x)}\n{\n\\ell'(x,\\theta) \\text{p}(X = x | \\theta) dx\n}\n\\\\ &= \\int_{x \\in \\mathcal{R}(X)}\n{\n\\left(\n\\frac{\\partial}{\\partial \\theta}\n\\text{log}\\left\\{\\text{p}(X = x | \\theta)\\right\\}\n\\right)\n\\text{p}(X = x | \\theta) dx\n}\n\\\\ &=\n\\int_{x \\in \\mathcal{R}(X)}\n{\n\\frac\n{\\frac{\\partial}{\\partial \\theta} \\text{p}(X = x | \\theta)}\n{\\text{p}(X = x | \\theta)}\n\\text{p}(X = x | \\theta) dx\n}\n\\\\ &=\n\\int_{x \\in \\mathcal{R}(X)}\n{\n\\frac{\\partial}{\\partial \\theta} \\text{p}(X = x | \\theta) dx\n}\n\\end{aligned}\n\\]\nAnd similarly\n\\[\n\\mathbb{E}\\left[\\ell' \\ell'^{\\top}\\right]\n\\stackrel{\\text{def}}{=}\n\\int_{x \\in R(x)}\n{\\ell'(x,\\theta)\\ell'(x,\\theta)^{\\top}\\\n\\text{p}\\left(X = x | \\theta\\right)\\ dx}\n\\]\nNote that \\(\\mathbb{E}\\left[\\ell'\\right]\\) and \\(\\mathbb{E}\\left[\\ell'{\\ell'}^{\\top}\\right]\\) are functions of \\(\\theta\\) but not of \\(x\\); the expectation operator removed \\(x\\).\nAlso note that for most of the distributions you are familiar with (including Gaussian, binomial, Poisson, exponential):\n\\[\\mathbb{E}\\left[\\ell'\\right] = 0\\]\nSo\n\\[\\mathcal{I}\\left(\\theta\\right) = \\mathbb{E}\\left[\\ell'{\\ell'}^{\\top} \\right]\\]\nMoreover, for those distributions (called the “exponential family”), we have:\n\\[\n\\mathfrak{I} = -\\mathbb{E}\\left[\\ell''\\right]\n= \\mathbb{E}\\left[- \\ell''\\right]\n\\]\n(see Dobson and Barnett (2018), §3.17), where\n\\[\\ell'' \\stackrel{\\text{def}}{=}\\frac{\\partial}{\\partial \\theta}\\ell^{'(x,\\theta)^{\\top}} = \\frac{\\partial}{\\partial \\theta}\\frac{\\partial}{\\partial \\theta^{\\top}}\\ell(x,\\theta)\\]\nis the \\(p \\times p\\) matrix whose elements are:\n\\[\\ell_{ij}'' \\stackrel{\\text{def}}{=}\\frac{\\partial}{\\partial \\theta_{i}}\\frac{\\partial}{\\partial \\theta_{j}}\\text{log}\\left\\{ p\\left( X = x \\mid \\theta \\right)\\right\\}\\]\n\\(\\ell''\\) is called the “Hessian”4 of the log-likelihood function.\nSometimes, we use \\(I(\\theta;x) \\stackrel{\\text{def}}{=}- \\ell''\\) (note the standard-font “I” here). \\(I(\\theta;x)\\) is the observed information, precision, or concentration matrix (Negative Hessian).\n\n\n\n\n\n\nKey point\n\n\n\nThe asymptotics of MLEs gives us \\({\\widehat{\\theta}}_{ML} \\sim N\\left( \\theta,\\mathfrak{I}^{- 1}(\\theta) \\right)\\), approximately, for large sample sizes.\n\n\nWe can estimate \\(\\mathcal{I}^{- 1}(\\theta)\\) by working out \\(\\mathbb{E}\\left[-\\ell''\\right]\\) or \\(\\mathbb{E}\\left[\\ell'{\\ell'}^{\\top}\\right]\\) and plugging in \\(\\hat\\theta_{\\text{ML}}\\), but sometimes we instead use \\(I(\\hat\\theta_{\\text{ML}},\\tilde{x})\\) for convenience; there are some cases where it’s provably better according to some criteria (Efron and Hinkley (1978)).\n\nE.1.9 Iterative maximization\n(c.f., Dobson and Barnett (2018), Chapter 4)\n\nLater, when we are trying to find MLEs for likelihoods which we can’t easily differentiate, we will “hill-climb” using the Newton-Raphson algorithm:\n\n\\[\n\\begin{aligned}\n{\\widehat{\\theta}}^*\n&\\leftarrow {\\widehat{\\theta}}^* + \\left(I\\left(\\tilde{y};{\\widehat{\\theta}}^*\\right)\\right)^{-1}\n\\ell'\\left(\\tilde{y};{\\widehat{\\theta}}^*\\right)\n\\\\ &= {\\widehat{\\theta}}^* - \\left(\\ell''\\left(\\tilde{y};{\\widehat{\\theta}}^*\\right)\\right)^{-1}\n\\ell'\\left(\\tilde{y};{\\widehat{\\theta}}^*\\right)\n\\end{aligned}\n\\]\n\n\nThe reasoning for this algorithm is that we can approximate the the score function using the first-order Taylor polynomial:\n\n\\[\n\\begin{aligned}\n\\ell'(\\theta)\n&\\approx \\ell'^*(\\theta)\n\\\\ &\\stackrel{\\text{def}}{=}\\ell'({\\widehat{\\theta}}^*) + \\ell''({\\widehat{\\theta}}^*)(\\theta- {\\widehat{\\theta}}^*)\n\\end{aligned}\n\\]\n\n\nThe approximate score function, \\(\\ell'^*(\\theta)\\), is a linear function of \\(\\theta\\), so it is easy to solve the corresponding approximate score equation, \\(\\ell'^*(\\theta) = 0\\), for \\(\\theta\\):\n\n\\[\n\\begin{aligned}\n\\theta\n&= {\\widehat{\\theta}}^* - \\ell'({\\widehat{\\theta}}^*) \\cdot\\left(\\ell''({\\widehat{\\theta}}^*)\\right)^{-1}\n\\end{aligned}\n\\]\n\nFor computational simplicity, we will sometimes use \\(\\mathfrak{I}^{- 1}(\\theta)\\) in place of \\(I\\left( \\widehat{\\theta},y \\right)\\); doing so is called “Fisher scoring” or the “method of scoring”. Note that this is the opposite of the substitution that we are making for estimating the variance of the MLE; this time we should technically use the observed information but we use the expected information instead.\n\nThere’s also an “empirical information matrix” (see McLachlan and Krishnan (2007)):\n\\[I_{e}(\\theta,y) \\stackrel{\\text{def}}{=}\\sum_{i = 1}^{n}{\\ell_{i}'\\ {\\ell_{i}'}^{\\top}} - \\frac{1}{n}\\ell'{\\ell'}^{\\top}\\]\nwhere \\(\\ell_{i}\\) is the log-likelihood of the ith observation. Note that \\(\\ell' = \\sum_{i = 1}^{n}\\ell_{i}'\\).\n\\(\\frac{1}{n}I_{e}(\\theta,y)\\) is the sample equivalent of\n\\[\\mathfrak{I \\stackrel{\\text{def}}{=}I(}\\theta) \\stackrel{\\text{def}}{=}{Cov}\\left( \\ell'|\\theta \\right) = E[ \\ell'{\\ell'}^{\\top} ] - E[ \\ell' ]\\ E[ \\ell' ]^{\\top}\\]\n\\[\\left\\{ \\mathfrak{I}_{jk} \\stackrel{\\text{def}}{=}{Cov}\\left( {\\ell'}_{j},{\\ell'}_{k} \\right) = E[ \\ell_{j}'\\ell_{k}' ] - E[ {\\ell'}_{j} ] E[ {\\ell'}_{k} ] \\right\\}\\]\n\\(I_{e}(\\theta,y)\\) is sometimes computationally easier to compute for Newton-Raphson-type maximization algorithms.\nc.f. https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization\n\nE.1.10 Quantifying (un)certainty of MLEs\nConfidence intervals for MLEs\nAn asymptotic approximation of a 95% confidence interval for \\(\\theta_k\\) is\n\\[\n\\hat\\theta_{\\text{ML}}\\pm z_{0.975} \\times \\widehat{\\text{SE}}\\left(\\hat\\theta_k\\right)\n\\]\nwhere \\(z_\\beta\\) the \\(\\beta\\) quantile of the standard Gaussian distribution.\np-values and hypothesis tests for MLEs\n(to add)\nLikelihood ratio tests for MLEs\nlog(likelihood ratio) tests (c.f. Dobson and Barnett 2018, sec. 5.7):\n\\[-2\\ell_{0} \\sim \\chi^{2}(p - q)\\]\nSee also https://online.stat.psu.edu/stat504/book/export/html/657\nPrediction intervals for MLEs\n\\[\\overline{X} \\in [ \\widehat{\\mu} \\pm z_{1 - \\alpha\\text{/}2}\\frac{\\sigma}{m} ]\\]\nWhere \\(m\\) is the sample size of the new data to be predicted (typically 1, except for binary outcomes, where it needs to be bigger for prediction intervals to make sense)", "crumbs": [ "Appendices", "E  Introduction to Maximum Likelihood Inference" @@ -1016,7 +1027,7 @@ "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$number)\n\n\n\\[\\hat{\\lambda}_{\\text{ML}} = \\bar x = 5.5385\\]\n\n\n\nExercise E.18 Graph the log-likelihood with the MLE superimposed.\n\nSolution. \n\nShow R codelibrary(dplyr)\n\nmle_data = tibble(x = mle, y = loglik(mle))\nll_plot + geom_point(data = mle_data, aes(x = x, y = y), col = 'red')\n\n\n\nFigure E.7: log-likelihood of Dobson cyclone data with MLE\n\n\n\n\n\n\n\n\n\nInformation matrices\n\nShow R codeobs_inf = function(...) -hessian(...)\nggplot() +\ngeom_function(fun = obs_inf, n = 1001) +\nxlim(min(cyclones$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", + "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", "crumbs": [ "Appendices", "E  Introduction to Maximum Likelihood Inference" @@ -1088,6 +1099,17 @@ "F  Common Mistakes" ] }, + { + "objectID": "common-mistakes.html#latex", + "href": "common-mistakes.html#latex", + "title": "Appendix F — Common Mistakes", + "section": "\nF.3 LaTeX", + "text": "F.3 LaTeX\nDouble superscript issues: https://www.overleaf.com/learn/latex/Errors/Double_superscript", + "crumbs": [ + "Appendices", + "F  Common Mistakes" + ] + }, { "objectID": "notation.html", "href": "notation.html", @@ -1143,12 +1165,23 @@ "H  Statistical computing in R" ] }, + { + "objectID": "intro-to-R.html#data.frames-and-tibbles", + "href": "intro-to-R.html#data.frames-and-tibbles", + "title": "Appendix H — Statistical computing in R", + "section": "H.2 data.frames and tibbles", + "text": "H.2 data.frames and tibbles\n\nH.2.1 Displaying tibbles\nSee vignette(\"digits\", package = \"tibble\")", + "crumbs": [ + "Appendices", + "H  Statistical computing in R" + ] + }, { "objectID": "intro-to-R.html#the-tidyverse", "href": "intro-to-R.html#the-tidyverse", "title": "Appendix H — Statistical computing in R", - "section": "H.2 The tidyverse", - "text": "H.2 The tidyverse\n\nThe tidyverse is an opinionated collection of R packages designed for data science. All packages share an underlying design philosophy, grammar, and data structures.\n\n\nhttps://www.tidyverse.org/\n\nThese packages are being actively developed by Hadley Wickham and his colleagues at posit1.\nDetails:\n\nWickham et al. (2019)\nWickham, Çetinkaya-Rundel, and Grolemund (2023)\nKuhn and Silge (2022)", + "section": "H.3 The tidyverse", + "text": "H.3 The tidyverse\n\nThe tidyverse is an opinionated collection of R packages designed for data science. All packages share an underlying design philosophy, grammar, and data structures.\n\n\nhttps://www.tidyverse.org/\n\nThese packages are being actively developed by Hadley Wickham and his colleagues at posit1.\nDetails:\n\nWickham et al. (2019)\nWickham, Çetinkaya-Rundel, and Grolemund (2023)\nKuhn and Silge (2022)", "crumbs": [ "Appendices", "H  Statistical computing in R" @@ -1158,8 +1191,8 @@ "objectID": "intro-to-R.html#piping", "href": "intro-to-R.html#piping", "title": "Appendix H — Statistical computing in R", - "section": "H.3 Piping", - "text": "H.3 Piping\nSee Wickham, Çetinkaya-Rundel, and Grolemund (2023) for details.\nThere are currently (2024) two commonly-used pipe operators in R:\n\n%>%: the “magrittr pipe”, from the magrittr package (Bache and Wickham (2022); re-exported by dplyr and others) .\n|>: the “native pipe”, from base R (≥4.1.0)\n\n\nH.3.1 Which pipe should I use?\nWickham, Çetinkaya-Rundel, and Grolemund (2023) recommends the native pipe:\n\nFor simple cases, |> and %>% behave identically. So why do we recommend the base pipe? Firstly, because it’s part of base R, it’s always available for you to use, even when you’re not using the tidyverse. Secondly, |> is quite a bit simpler than %>%: in the time between the invention of %>% in 2014 and the inclusion of |> in R 4.1.0 in 2021, we gained a better understanding of the pipe. This allowed the base implementation to jettison infrequently used and less important features.\n\n\n\nH.3.2 Why doesn’t ggplot2 use piping?\nHere’s tidyverse creator Hadley Wickham’s answer (from 2018):\n\nI think it’s worth unpacking this question into a few smaller pieces:\n\nShould ggplot2 use the pipe? IMO, yes.\nCould ggplot2 support both the pipe and plus? No\nWould it be worth it to create a ggplot3 that uses the pipe? No.\n\n\nhttps://forum.posit.co/t/why-cant-ggplot2-use/4372/7", + "section": "H.4 Piping", + "text": "H.4 Piping\nSee Wickham, Çetinkaya-Rundel, and Grolemund (2023) for details.\nThere are currently (2024) two commonly-used pipe operators in R:\n\n%>%: the “magrittr pipe”, from the magrittr package (Bache and Wickham (2022); re-exported by dplyr and others) .\n|>: the “native pipe”, from base R (≥4.1.0)\n\n\nH.4.1 Which pipe should I use?\nWickham, Çetinkaya-Rundel, and Grolemund (2023) recommends the native pipe:\n\nFor simple cases, |> and %>% behave identically. So why do we recommend the base pipe? Firstly, because it’s part of base R, it’s always available for you to use, even when you’re not using the tidyverse. Secondly, |> is quite a bit simpler than %>%: in the time between the invention of %>% in 2014 and the inclusion of |> in R 4.1.0 in 2021, we gained a better understanding of the pipe. This allowed the base implementation to jettison infrequently used and less important features.\n\n\n\nH.4.2 Why doesn’t ggplot2 use piping?\nHere’s tidyverse creator Hadley Wickham’s answer (from 2018):\n\nI think it’s worth unpacking this question into a few smaller pieces:\n\nShould ggplot2 use the pipe? IMO, yes.\nCould ggplot2 support both the pipe and plus? No\nWould it be worth it to create a ggplot3 that uses the pipe? No.\n\n\nhttps://forum.posit.co/t/why-cant-ggplot2-use/4372/7", "crumbs": [ "Appendices", "H  Statistical computing in R" @@ -1169,8 +1202,8 @@ "objectID": "intro-to-R.html#quarto", "href": "intro-to-R.html#quarto", "title": "Appendix H — Statistical computing in R", - "section": "H.4 Quarto", - "text": "H.4 Quarto\nQuarto is a system for writing documents with embedded R code and/or results:\n\nRead this ASAP: https://r4ds.hadley.nz/communicate\nThen use this for reference: https://quarto.org/docs/reference/", + "section": "H.5 Quarto", + "text": "H.5 Quarto\nQuarto is a system for writing documents with embedded R code and/or results:\n\nRead this ASAP: https://r4ds.hadley.nz/communicate\nThen use this for reference: https://quarto.org/docs/reference/", "crumbs": [ "Appendices", "H  Statistical computing in R" @@ -1180,8 +1213,8 @@ "objectID": "intro-to-R.html#packages", "href": "intro-to-R.html#packages", "title": "Appendix H — Statistical computing in R", - "section": "H.5 Packages", - "text": "H.5 Packages\n\nThis book espouses our philosophy of package development: anything that can be automated, should be automated. Do as little as possible by hand. Do as much as possible with functions. The goal is to spend your time thinking about what you want your package to do rather than thinking about the minutiae of package structure.\n\n\nhttps://r-pkgs.org/introduction.html#:~:text=This%20book%20espouses,of%20package%20structure.\nRead this ASAP: https://r-pkgs.org/whole-game.html\nUse the rest of Wickham and Bryan (2023) as a reference", + "section": "H.6 Packages", + "text": "H.6 Packages\n\nThis book espouses our philosophy of package development: anything that can be automated, should be automated. Do as little as possible by hand. Do as much as possible with functions. The goal is to spend your time thinking about what you want your package to do rather than thinking about the minutiae of package structure.\n\n\nhttps://r-pkgs.org/introduction.html#:~:text=This%20book%20espouses,of%20package%20structure.\nRead this ASAP: https://r-pkgs.org/whole-game.html\nUse the rest of Wickham and Bryan (2023) as a reference", "crumbs": [ "Appendices", "H  Statistical computing in R" @@ -1191,8 +1224,8 @@ "objectID": "intro-to-R.html#git", "href": "intro-to-R.html#git", "title": "Appendix H — Statistical computing in R", - "section": "H.6 Git", - "text": "H.6 Git\n94% of respondents to a 2022 Stack Overflow survey reported using git for version control link\nMore details\n\nhttps://happygitwithr.com/\nhttps://usethis.r-lib.org/articles/pr-functions.html", + "section": "H.7 Git", + "text": "H.7 Git\n94% of respondents to a 2022 Stack Overflow survey reported using git for version control link\nMore details\n\nhttps://happygitwithr.com/\nhttps://usethis.r-lib.org/articles/pr-functions.html", "crumbs": [ "Appendices", "H  Statistical computing in R" @@ -1202,8 +1235,8 @@ "objectID": "intro-to-R.html#spatial-data-science", "href": "intro-to-R.html#spatial-data-science", "title": "Appendix H — Statistical computing in R", - "section": "H.7 Spatial data science", - "text": "H.7 Spatial data science\n\nPebesma and Bivand (2023)", + "section": "H.8 Spatial data science", + "text": "H.8 Spatial data science\n\nPebesma and Bivand (2023)", "crumbs": [ "Appendices", "H  Statistical computing in R" @@ -1213,8 +1246,8 @@ "objectID": "intro-to-R.html#shiny-apps", "href": "intro-to-R.html#shiny-apps", "title": "Appendix H — Statistical computing in R", - "section": "H.8 Shiny apps", - "text": "H.8 Shiny apps\n\nRead this first: Wickham (2021)\nUse this as a reference: Fay et al. (2021)\n\n\n\n\n\n\n\nBache, Stefan Milton, and Hadley Wickham. 2022. Magrittr: A Forward-Pipe Operator for r. https://CRAN.R-project.org/package=magrittr.\n\n\nChang, Winston. 2024. R Graphics Cookbook: Practical Recipes for Visualizing Data. O’Reilly Media. https://r-graphics.org/.\n\n\nFay, Colin, Sébastien Rochette, Vincent Guyader, and Cervan Girard. 2021. Engineering Production-Grade Shiny Apps. Chapman; Hall/CRC. https://engineering-shiny.org/.\n\n\nKleinman, Ken, and Nicholas J Horton. 2009. SAS and r: Data Management, Statistical Analysis, and Graphics. Chapman; Hall/CRC. https://www.routledge.com/SAS-and-R-Data-Management-Statistical-Analysis-and-Graphics-Second-Edition/Kleinman-Horton/p/book/9781466584495.\n\n\nKuhn, Max, and Julia Silge. 2022. Tidy Modeling with r. \" O’Reilly Media, Inc.\". https://www.tmwr.org/.\n\n\nPebesma, Edzer, and Roger Bivand. 2023. Spatial Data Science: With Applications in R. Boca Raton: Chapman; Hall/CRC. https://doi.org/10.1201/9780429459016.\n\n\nWickham, Hadley. 2019. Advanced r. Chapman; Hall/CRC. https://adv-r.hadley.nz/index.html.\n\n\n———. 2021. Mastering Shiny. \" O’Reilly Media, Inc.\". https://mastering-shiny.org/.\n\n\nWickham, Hadley, Mara Averick, Jennifer Bryan, Winston Chang, Lucy D’Agostino McGowan, Romain François, Garrett Grolemund, et al. 2019. “Welcome to the tidyverse.” Journal of Open Source Software 4 (43): 1686. https://doi.org/10.21105/joss.01686.\n\n\nWickham, Hadley, and Jennifer Bryan. 2023. R Packages. \" O’Reilly Media, Inc.\". https://r-pkgs.org/.\n\n\nWickham, Hadley, Mine Çetinkaya-Rundel, and Garrett Grolemund. 2023. R for Data Science. \" O’Reilly Media, Inc.\". https://r4ds.hadley.nz/.", + "section": "H.9 Shiny apps", + "text": "H.9 Shiny apps\n\nRead this first: Wickham (2021)\nUse this as a reference: Fay et al. (2021)\n\n\n\n\n\n\n\nBache, Stefan Milton, and Hadley Wickham. 2022. Magrittr: A Forward-Pipe Operator for r. https://CRAN.R-project.org/package=magrittr.\n\n\nChang, Winston. 2024. R Graphics Cookbook: Practical Recipes for Visualizing Data. O’Reilly Media. https://r-graphics.org/.\n\n\nFay, Colin, Sébastien Rochette, Vincent Guyader, and Cervan Girard. 2021. Engineering Production-Grade Shiny Apps. Chapman; Hall/CRC. https://engineering-shiny.org/.\n\n\nKleinman, Ken, and Nicholas J Horton. 2009. SAS and r: Data Management, Statistical Analysis, and Graphics. Chapman; Hall/CRC. https://www.routledge.com/SAS-and-R-Data-Management-Statistical-Analysis-and-Graphics-Second-Edition/Kleinman-Horton/p/book/9781466584495.\n\n\nKuhn, Max, and Julia Silge. 2022. Tidy Modeling with r. \" O’Reilly Media, Inc.\". https://www.tmwr.org/.\n\n\nPebesma, Edzer, and Roger Bivand. 2023. Spatial Data Science: With Applications in R. Boca Raton: Chapman; Hall/CRC. https://doi.org/10.1201/9780429459016.\n\n\nWickham, Hadley. 2019. Advanced r. Chapman; Hall/CRC. https://adv-r.hadley.nz/index.html.\n\n\n———. 2021. Mastering Shiny. \" O’Reilly Media, Inc.\". https://mastering-shiny.org/.\n\n\nWickham, Hadley, Mara Averick, Jennifer Bryan, Winston Chang, Lucy D’Agostino McGowan, Romain François, Garrett Grolemund, et al. 2019. “Welcome to the tidyverse.” Journal of Open Source Software 4 (43): 1686. https://doi.org/10.21105/joss.01686.\n\n\nWickham, Hadley, and Jennifer Bryan. 2023. R Packages. \" O’Reilly Media, Inc.\". https://r-pkgs.org/.\n\n\nWickham, Hadley, Mine Çetinkaya-Rundel, and Garrett Grolemund. 2023. R for Data Science. \" O’Reilly Media, Inc.\". https://r4ds.hadley.nz/.", "crumbs": [ "Appendices", "H  Statistical computing in R"