From bd3b5f7731ecd7127c437ac797fdcae88da0e851 Mon Sep 17 00:00:00 2001
From: Douglas Ezra Morrison <demorrison@ucdavis.edu>
Date: Mon, 27 May 2024 13:02:27 -0700
Subject: [PATCH] Built site for gh-pages

---
 .nojekyll                                     |    2 +-
 Linear-models-overview.html                   |    4 +-
 .../figure-html/unnamed-chunk-104-1.png       |  Bin 86790 -> 86939 bytes
 .../figure-html/unnamed-chunk-98-1.png        |  Bin 44699 -> 44503 bytes
 Regression-Models-for-Epidemiology.pdf        |  Bin 3013154 -> 3013802 bytes
 index.html                                    |    2 +-
 intro-MLEs.html                               |    2 +-
 intro-to-survival-analysis.html               | 3242 +++++++++--------
 logistic-regression.html                      |   12 +-
 search.json                                   |   14 +-
 10 files changed, 1662 insertions(+), 1616 deletions(-)

diff --git a/.nojekyll b/.nojekyll
index f38b1f54a..796047ecf 100644
--- a/.nojekyll
+++ b/.nojekyll
@@ -1 +1 @@
-fb006b97
\ No newline at end of file
+55f3eddd
\ No newline at end of file
diff --git a/Linear-models-overview.html b/Linear-models-overview.html
index 981677402..94bd64fe8 100644
--- a/Linear-models-overview.html
+++ b/Linear-models-overview.html
@@ -3926,7 +3926,7 @@ <h3 class="anchored">Note</h3>
 <tbody>
 <tr class="odd">
 <td style="text-align: left;">full</td>
-<td style="text-align: right;">6.900</td>
+<td style="text-align: right;">7.071</td>
 <td style="text-align: right;">0.4805</td>
 <td style="text-align: right;">0.3831</td>
 <td style="text-align: right;">5.956</td>
@@ -3934,7 +3934,7 @@ <h3 class="anchored">Note</h3>
 </tr>
 <tr class="even">
 <td style="text-align: left;">reduced</td>
-<td style="text-align: right;">6.581</td>
+<td style="text-align: right;">6.767</td>
 <td style="text-align: right;">0.4454</td>
 <td style="text-align: right;">0.3802</td>
 <td style="text-align: right;">5.971</td>
diff --git a/Linear-models-overview_files/figure-html/unnamed-chunk-104-1.png b/Linear-models-overview_files/figure-html/unnamed-chunk-104-1.png
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index f3fe81a93..f54d12209 100644
--- a/index.html
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@@ -395,7 +395,7 @@ <h2 id="toc-title">Table of contents</h2>
     <div>
     <div class="quarto-title-meta-heading">Published</div>
     <div class="quarto-title-meta-contents">
-      <p class="date">Last modified: 2024-05-27: 11:49:27 (AM)</p>
+      <p class="date">Last modified: 2024-05-27: 12:56:57 (PM)</p>
     </div>
   </div>
   
diff --git a/intro-MLEs.html b/intro-MLEs.html
index 7b18107a8..828b10ef8 100644
--- a/intro-MLEs.html
+++ b/intro-MLEs.html
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diff --git a/intro-to-survival-analysis.html b/intro-to-survival-analysis.html
index 3ce1203d8..57348c59b 100644
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+++ b/intro-to-survival-analysis.html
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     <div>
     <div class="quarto-title-meta-heading">Published</div>
     <div class="quarto-title-meta-contents">
-      <p class="date">Last modified: 2024-05-27: 11:49:27 (AM)</p>
+      <p class="date">Last modified: 2024-05-27: 12:56:57 (PM)</p>
     </div>
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@@ -638,8 +638,7 @@
 <div class="notes">
 <p>The survival function <span class="math inline">\(S(t)\)</span> is the probability that the event time is later than <span class="math inline">\(t\)</span>. If the event in a clinical trial is death, then <span class="math inline">\(S(t)\)</span> is the expected fraction of the original population at time 0 who have survived up to time <span class="math inline">\(t\)</span> and are still alive at time <span class="math inline">\(t\)</span>; that is:</p>
 </div>
-<p>If <span class="math inline">\(X_t\)</span> represents survival status at time <span class="math inline">\(t\)</span>, with <span class="math inline">\(X_t = 1\)</span> denoting alive at time <span class="math inline">\(t\)</span> and <span class="math inline">\(X_t = 0\)</span> denoting deceased at time <span class="math inline">\(t\)</span>, then:</p>
-<p><span class="math display">\[S(t) = \Pr(X_t=1) = \mathbb{E}[X_t]\]</span></p>
+<p><span id="eq-def-surv"><span class="math display">\[S(t) \stackrel{\text{def}}{=}\Pr(T &gt; t) \tag{5.1}\]</span></span></p>
 </div>
 <hr>
 <div id="exm-exp-survfn" class="theorem example">
@@ -662,6 +661,20 @@
 </div>
 </div>
 </div>
+<hr>
+<div id="thm-surv-fn-as-mean-status" class="theorem">
+<p><span class="theorem-title"><strong>Theorem 5.1</strong></span> If <span class="math inline">\(A_t\)</span> represents survival status at time <span class="math inline">\(t\)</span>, with <span class="math inline">\(A_t = 1\)</span> denoting alive at time <span class="math inline">\(t\)</span> and <span class="math inline">\(A_t = 0\)</span> denoting deceased at time <span class="math inline">\(t\)</span>, then:</p>
+<p><span class="math display">\[S(t) = \Pr(A_t=1) = \mathbb{E}[A_t]\]</span></p>
+</div>
+<hr>
+<div id="thm-surv-and-mean" class="theorem">
+<p><span class="theorem-title"><strong>Theorem 5.2</strong></span> If <span class="math inline">\(T\)</span> is a nonnegative random variable, then:</p>
+<p><span class="math display">\[\mathbb{E}[T] = \int_{t=0}^{\infty} S(t)dt\]</span></p>
+</div>
+<hr>
+<div class="proof">
+<p><span class="proof-title"><em>Proof</em>. </span>See <a href="https://statproofbook.github.io/P/mean-nnrvar.html" class="uri">https://statproofbook.github.io/P/mean-nnrvar.html</a> or</p>
+</div>
 </section><section id="the-hazard-function" class="level3" data-number="5.3.4"><h3 data-number="5.3.4" class="anchored" data-anchor-id="the-hazard-function">
 <span class="header-section-number">5.3.4</span> The Hazard Function</h3>
 <p>Another important quantity is the <strong>hazard function</strong>:</p>
@@ -672,7 +685,7 @@
 </div>
 <hr>
 <div class="notes">
-<p>The hazard function has an important relationship to the density and survival functions, which we can use to derive the hazard function for a given probability distribution (<a href="#thm-hazard1" class="quarto-xref">Theorem&nbsp;<span>5.1</span></a>).</p>
+<p>The hazard function has an important relationship to the density and survival functions, which we can use to derive the hazard function for a given probability distribution (<a href="#thm-hazard1" class="quarto-xref">Theorem&nbsp;<span>5.3</span></a>).</p>
 </div>
 <div id="lem-joint-prob-same-var" class="theorem lemma">
 <p><span class="theorem-title"><strong>Lemma 5.1 (Joint probability of a variable with itself)</strong></span> <span class="math display">\[p(T=t, T\ge t) = p(T=t)\]</span></p>
@@ -682,7 +695,7 @@
 </div>
 <hr>
 <div id="thm-hazard1" class="theorem">
-<p><span class="theorem-title"><strong>Theorem 5.1</strong></span> <span class="math display">\[h(t)=\frac{f(t)}{S(t)}\]</span></p>
+<p><span class="theorem-title"><strong>Theorem 5.3</strong></span> <span class="math display">\[h(t)=\frac{f(t)}{S(t)}\]</span></p>
 </div>
 <hr>
 <div class="proof">
@@ -723,7 +736,7 @@
 <hr>
 <p>We can also view the hazard function as the derivative of the negative of the logarithm of the survival function:</p>
 <div id="thm-h-logS" class="theorem">
-<p><span class="theorem-title"><strong>Theorem 5.2 (transform survival to hazard)</strong></span> <span class="math display">\[h(t) = \frac{\partial}{\partial t}\left\{-\text{log}\left\{S(t)\right\}\right\}\]</span></p>
+<p><span class="theorem-title"><strong>Theorem 5.4 (transform survival to hazard)</strong></span> <span class="math display">\[h(t) = \frac{\partial}{\partial t}\left\{-\text{log}\left\{S(t)\right\}\right\}\]</span></p>
 </div>
 <hr>
 <div class="proof">
@@ -740,13 +753,13 @@
 </div>
 </section><section id="the-cumulative-hazard-function" class="level3" data-number="5.3.5"><h3 data-number="5.3.5" class="anchored" data-anchor-id="the-cumulative-hazard-function">
 <span class="header-section-number">5.3.5</span> The Cumulative Hazard Function</h3>
-<p>Since <span class="math inline">\(h(t) = \frac{\partial}{\partial t}\left\{-\text{log}\left\{S(t)\right\}\right\}\)</span> (see <a href="#thm-h-logS" class="quarto-xref">Theorem&nbsp;<span>5.2</span></a>), we also have:</p>
+<p>Since <span class="math inline">\(h(t) = \frac{\partial}{\partial t}\left\{-\text{log}\left\{S(t)\right\}\right\}\)</span> (see <a href="#thm-h-logS" class="quarto-xref">Theorem&nbsp;<span>5.4</span></a>), we also have:</p>
 <div id="cor-surv-int-haz" class="theorem corollary">
-<p><span class="theorem-title"><strong>Corollary 5.1</strong></span> <span id="eq-surv-int-haz"><span class="math display">\[S(t) = \text{exp}\left\{-\int_{u=0}^t h(u)du\right\} \tag{5.1}\]</span></span></p>
+<p><span class="theorem-title"><strong>Corollary 5.1</strong></span> <span id="eq-surv-int-haz"><span class="math display">\[S(t) = \text{exp}\left\{-\int_{u=0}^t h(u)du\right\} \tag{5.2}\]</span></span></p>
 </div>
 <hr>
 <div class="notes">
-<p>The integral in <a href="#eq-surv-int-haz" class="quarto-xref">Equation&nbsp;<span>5.1</span></a> is important enough to have its own name: <strong>cumulative hazard</strong>.</p>
+<p>The integral in <a href="#eq-surv-int-haz" class="quarto-xref">Equation&nbsp;<span>5.2</span></a> is important enough to have its own name: <strong>cumulative hazard</strong>.</p>
 </div>
 <div id="def-cumhaz" class="theorem definition">
 <p><span class="theorem-title"><strong>Definition 5.3 (cumulative hazard)</strong></span> The <strong>cumulative hazard function</strong> <span class="math inline">\(H(t)\)</span> is defined as:</p>
@@ -1104,10 +1117,10 @@
 \]</span></p>
 <hr>
 <div id="thm-mle-exp" class="theorem">
-<p><span class="theorem-title"><strong>Theorem 5.3</strong></span> Let <span class="math inline">\(T=\sum t_i\)</span> and <span class="math inline">\(U=\sum u_j\)</span>. Then:</p>
+<p><span class="theorem-title"><strong>Theorem 5.5</strong></span> Let <span class="math inline">\(T=\sum t_i\)</span> and <span class="math inline">\(U=\sum u_j\)</span>. Then:</p>
 <p><span id="eq-mle-exp"><span class="math display">\[
-\ell(\lambda) = \frac{m}{T+U}
-\tag{5.2}\]</span></span></p>
+\hat{\lambda}_{ML} = \frac{m}{T+U}
+\tag{5.3}\]</span></span></p>
 </div>
 <hr>
 <div class="proof">
@@ -1266,7 +1279,7 @@
 <span class="header-section-number">5.6.5</span> Exponential model</h3>
 <div class="notes">
 <ul>
-<li>We <em>can</em> compute the hazard rate, assuming an exponential model: number of relapses divided by the sum of the exit times (<a href="#eq-mle-exp" class="quarto-xref">Equation&nbsp;<span>5.2</span></a>).</li>
+<li>We <em>can</em> compute the hazard rate, assuming an exponential model: number of relapses divided by the sum of the exit times (<a href="#eq-mle-exp" class="quarto-xref">Equation&nbsp;<span>5.3</span></a>).</li>
 </ul>
 </div>
 <p><span class="math display">\[\hat\lambda = \frac{\sum_{i=1}^nD_i}{\sum_{i=1}^nY_i}\]</span></p>
@@ -1321,11 +1334,12 @@
 <hr>
 <div id="def-KM-estimator" class="theorem definition">
 <p><span class="theorem-title"><strong>Definition 5.4 (Kaplan-Meier Product-Limit Estimator of Survival Function)</strong></span> If the event times are <span class="math inline">\(t_i\)</span> with events per time of <span class="math inline">\(d_i\)</span> (<span class="math inline">\(1\le i \le k\)</span>), then the <strong>Kaplan-Meier Product-Limit Estimator</strong> of the <a href="@def-surv-fn">survival function</a> is:</p>
-<p><span id="eq-km-surv-est"><span class="math display">\[\hat S(t) = \prod_{t_i &lt; t} \left[\frac{1-d_i}{Y_i}\right] \tag{5.3}\]</span></span></p>
+<p><span id="eq-km-surv-est"><span class="math display">\[\hat S(t) = \prod_{t_i &lt; t} \left[\frac{1-d_i}{Y_i}\right] \tag{5.4}\]</span></span></p>
 <p>where <span class="math inline">\(Y_i\)</span> is the set of observations whose time (event or censored) is <span class="math inline">\(\ge t_i\)</span>, the group at risk at time <span class="math inline">\(t_i\)</span>.</p>
 </div>
 <hr>
-<p>If there are no censored data, and there are <span class="math inline">\(n\)</span> data points, then just after (say) the third event time</p>
+<div id="thm-KM-est-no-cens" class="theorem">
+<p><span class="theorem-title"><strong>Theorem 5.6 (Kaplan-Meier Estimate with No Censored Observations)</strong></span> If there are no censored data, and there are <span class="math inline">\(n\)</span> data points, then just after (say) the third event time</p>
 <p><span class="math display">\[
 \begin{aligned}
 \hat S(t)
@@ -1337,6 +1351,7 @@
 \end{aligned}
 \]</span></p>
 <p>where <span class="math inline">\(\hat F(t)\)</span> is the usual empirical CDF estimate.</p>
+</div>
 </section><section id="kaplan-meier-curve-for-drug6mp-data" class="level3" data-number="5.7.3"><h3 data-number="5.7.3" class="anchored" data-anchor-id="kaplan-meier-curve-for-drug6mp-data">
 <span class="header-section-number">5.7.3</span> Kaplan-Meier curve for <code>drug6mp</code> data</h3>
 <p>Here is the Kaplan-Meier estimated survival curve for the patients who received 6-MP in the <code>drug6mp</code> dataset (we will see code to produce figures like this one shortly):</p>
@@ -2015,11 +2030,11 @@
 <p><span class="math display">\[\hat S(t) = \prod_{t_i &lt; t}\left[1-\frac{d_i}{Y_i}\right]\]</span> where <span class="math inline">\(Y_i\)</span> is the group at risk at time <span class="math inline">\(t_i\)</span>.</p>
 <p>The estimated variance of <span class="math inline">\(\hat S(t)\)</span> is:</p>
 <div id="thm-greenwood" class="theorem">
-<p><span class="theorem-title"><strong>Theorem 5.4 (Greenwood’s estimator for variance of Kaplan-Meier survival estimator)</strong></span> <span id="eq-var-est-surv"><span class="math display">\[
+<p><span class="theorem-title"><strong>Theorem 5.7 (Greenwood’s estimator for variance of Kaplan-Meier survival estimator)</strong></span> <span id="eq-var-est-surv"><span class="math display">\[
 \widehat{\text{Var}}\left(\hat S(t)\right) = \hat S(t)^2\sum_{t_i &lt;t}\frac{d_i}{Y_i(Y_i-d_i)}
-\tag{5.4}\]</span></span></p>
+\tag{5.5}\]</span></span></p>
 </div>
-<p>We can use <a href="#eq-var-est-surv" class="quarto-xref">Equation&nbsp;<span>5.4</span></a> for confidence intervals for a survival function or a difference of survival functions.</p>
+<p>We can use <a href="#eq-var-est-surv" class="quarto-xref">Equation&nbsp;<span>5.5</span></a> for confidence intervals for a survival function or a difference of survival functions.</p>
 <hr>
 <section id="kaplan-meier-survival-curves" class="level5"><h5 class="anchored" data-anchor-id="kaplan-meier-survival-curves">Kaplan-Meier survival curves</h5>
 <div class="cell">
@@ -2164,11 +2179,11 @@
 <div class="notes">
 <p>The point hazard at time <span class="math inline">\(t_i\)</span> can be estimated by <span class="math inline">\(d_i/Y_i\)</span>, which leads to the <strong>Nelson-Aalen estimator of the cumulative hazard</strong>:</p>
 </div>
-<p><span id="eq-NA-cuhaz-est"><span class="math display">\[\hat H_{NA}(t) \stackrel{\text{def}}{=}\sum_{t_i &lt; t}\frac{d_i}{Y_i} \tag{5.5}\]</span></span></p>
+<p><span id="eq-NA-cuhaz-est"><span class="math display">\[\hat H_{NA}(t) \stackrel{\text{def}}{=}\sum_{t_i &lt; t}\frac{d_i}{Y_i} \tag{5.6}\]</span></span></p>
 </div>
 <hr>
 <div id="thm-var-NA-est" class="theorem">
-<p><span class="theorem-title"><strong>Theorem 5.5 (Variance of Nelson-Aalen estimator)</strong></span> &nbsp;</p>
+<p><span class="theorem-title"><strong>Theorem 5.8 (Variance of Nelson-Aalen estimator)</strong></span> &nbsp;</p>
 <div class="notes">
 <p>The variance of this estimator is approximately:</p>
 </div>
@@ -2178,7 +2193,7 @@
 &amp;= \sum_{t_i &lt;t}\frac{(d_i/Y_i)(1-d_i/Y_i)}{Y_i}\\
 &amp;\approx \sum_{t_i &lt;t}\frac{d_i}{Y_i^2}
 \end{aligned}
-\tag{5.6}\]</span></span></p>
+\tag{5.7}\]</span></span></p>
 </div>
 <hr>
 <p>Since <span class="math inline">\(S(t)=\text{exp}\left\{-H(t)\right\}\)</span>, the Nelson-Aalen cumulative hazard estimate can be converted into an alternate estimate of the survival function:</p>
@@ -3076,1698 +3091,1729 @@
 <span id="cb31-205"><a href="#cb31-205" aria-hidden="true" tabindex="-1"></a>the expected fraction of the original population at time 0 who have survived up to time $t$ and are still alive at time $t$; that is:</span>
 <span id="cb31-206"><a href="#cb31-206" aria-hidden="true" tabindex="-1"></a>::::</span>
 <span id="cb31-207"><a href="#cb31-207" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-208"><a href="#cb31-208" aria-hidden="true" tabindex="-1"></a>If $X_t$ represents survival status at time $t$, with $X_t = 1$ denoting alive at time $t$ and $X_t = 0$ denoting deceased at time $t$, then:</span>
+<span id="cb31-208"><a href="#cb31-208" aria-hidden="true" tabindex="-1"></a>$$S(t) \eqdef \Pr(T &gt; t)$${#eq-def-surv}</span>
 <span id="cb31-209"><a href="#cb31-209" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-210"><a href="#cb31-210" aria-hidden="true" tabindex="-1"></a>$$S(t) = \Pr(X_t=1) = \Expp<span class="co">[</span><span class="ot">X_t</span><span class="co">]</span>$$</span>
+<span id="cb31-210"><a href="#cb31-210" aria-hidden="true" tabindex="-1"></a>:::</span>
 <span id="cb31-211"><a href="#cb31-211" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-212"><a href="#cb31-212" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-212"><a href="#cb31-212" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-213"><a href="#cb31-213" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-214"><a href="#cb31-214" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-215"><a href="#cb31-215" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-216"><a href="#cb31-216" aria-hidden="true" tabindex="-1"></a>:::{#exm-exp-survfn}</span>
-<span id="cb31-217"><a href="#cb31-217" aria-hidden="true" tabindex="-1"></a><span class="fu">##### exponential distribution</span></span>
-<span id="cb31-218"><a href="#cb31-218" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-219"><a href="#cb31-219" aria-hidden="true" tabindex="-1"></a>Since $S(t) = 1 - F(t)$, the survival function of the exponential</span>
-<span id="cb31-220"><a href="#cb31-220" aria-hidden="true" tabindex="-1"></a>distribution family of models is:</span>
-<span id="cb31-221"><a href="#cb31-221" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-222"><a href="#cb31-222" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-223"><a href="#cb31-223" aria-hidden="true" tabindex="-1"></a>P(T&gt; t) = \left<span class="sc">\{</span> {{\text{e}^{-\lambda t}, t\ge0} \atop {1, t \le 0}}\right. </span>
-<span id="cb31-224"><a href="#cb31-224" aria-hidden="true" tabindex="-1"></a>$$ where $\lambda &gt; 0$.</span>
+<span id="cb31-214"><a href="#cb31-214" aria-hidden="true" tabindex="-1"></a>:::{#exm-exp-survfn}</span>
+<span id="cb31-215"><a href="#cb31-215" aria-hidden="true" tabindex="-1"></a><span class="fu">##### exponential distribution</span></span>
+<span id="cb31-216"><a href="#cb31-216" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-217"><a href="#cb31-217" aria-hidden="true" tabindex="-1"></a>Since $S(t) = 1 - F(t)$, the survival function of the exponential</span>
+<span id="cb31-218"><a href="#cb31-218" aria-hidden="true" tabindex="-1"></a>distribution family of models is:</span>
+<span id="cb31-219"><a href="#cb31-219" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-220"><a href="#cb31-220" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-221"><a href="#cb31-221" aria-hidden="true" tabindex="-1"></a>P(T&gt; t) = \left<span class="sc">\{</span> {{\text{e}^{-\lambda t}, t\ge0} \atop {1, t \le 0}}\right. </span>
+<span id="cb31-222"><a href="#cb31-222" aria-hidden="true" tabindex="-1"></a>$$ where $\lambda &gt; 0$.</span>
+<span id="cb31-223"><a href="#cb31-223" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-224"><a href="#cb31-224" aria-hidden="true" tabindex="-1"></a>@fig-exp-survfuns shows some examples of exponential survival functions.</span>
 <span id="cb31-225"><a href="#cb31-225" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-226"><a href="#cb31-226" aria-hidden="true" tabindex="-1"></a>@fig-exp-survfuns shows some examples of exponential survival functions.</span>
+<span id="cb31-226"><a href="#cb31-226" aria-hidden="true" tabindex="-1"></a>:::</span>
 <span id="cb31-227"><a href="#cb31-227" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-228"><a href="#cb31-228" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-228"><a href="#cb31-228" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-229"><a href="#cb31-229" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-230"><a href="#cb31-230" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-231"><a href="#cb31-231" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-232"><a href="#cb31-232" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, echo = FALSE}</span></span>
-<span id="cb31-233"><a href="#cb31-233" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "Exponential Survival Functions"</span></span>
-<span id="cb31-234"><a href="#cb31-234" aria-hidden="true" tabindex="-1"></a><span class="in">#| label: fig-exp-survfuns</span></span>
-<span id="cb31-235"><a href="#cb31-235" aria-hidden="true" tabindex="-1"></a><span class="in">library(ggplot2)</span></span>
-<span id="cb31-236"><a href="#cb31-236" aria-hidden="true" tabindex="-1"></a><span class="in">ggplot() +</span></span>
-<span id="cb31-237"><a href="#cb31-237" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_function(</span></span>
-<span id="cb31-238"><a href="#cb31-238" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "0.5"),</span></span>
-<span id="cb31-239"><a href="#cb31-239" aria-hidden="true" tabindex="-1"></a><span class="in">    fun = \(x) pexp(x, lower = FALSE, rate = 0.5)) +</span></span>
-<span id="cb31-240"><a href="#cb31-240" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_function(</span></span>
-<span id="cb31-241"><a href="#cb31-241" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 1"),</span></span>
-<span id="cb31-242"><a href="#cb31-242" aria-hidden="true" tabindex="-1"></a><span class="in">    fun = \(x) pexp(x, lower = FALSE, rate = 1)) +</span></span>
-<span id="cb31-243"><a href="#cb31-243" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_function(</span></span>
-<span id="cb31-244"><a href="#cb31-244" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 1.5"),</span></span>
-<span id="cb31-245"><a href="#cb31-245" aria-hidden="true" tabindex="-1"></a><span class="in">    fun = \(x) pexp(x, lower = FALSE, rate = 1.5)) +</span></span>
-<span id="cb31-246"><a href="#cb31-246" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_function(</span></span>
-<span id="cb31-247"><a href="#cb31-247" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 5"),</span></span>
-<span id="cb31-248"><a href="#cb31-248" aria-hidden="true" tabindex="-1"></a><span class="in">    fun = \(x) pexp(x, lower = FALSE, rate = 5)) +</span></span>
-<span id="cb31-249"><a href="#cb31-249" aria-hidden="true" tabindex="-1"></a><span class="in">  theme_bw() + </span></span>
-<span id="cb31-250"><a href="#cb31-250" aria-hidden="true" tabindex="-1"></a><span class="in">  ylab("S(t)") +</span></span>
-<span id="cb31-251"><a href="#cb31-251" aria-hidden="true" tabindex="-1"></a><span class="in">  guides(col = guide_legend(title = expr(lambda))) +</span></span>
-<span id="cb31-252"><a href="#cb31-252" aria-hidden="true" tabindex="-1"></a><span class="in">  xlab("Time (t)") +</span></span>
-<span id="cb31-253"><a href="#cb31-253" aria-hidden="true" tabindex="-1"></a><span class="in">  xlim(0, 2.5) +</span></span>
-<span id="cb31-254"><a href="#cb31-254" aria-hidden="true" tabindex="-1"></a><span class="in">  theme(</span></span>
-<span id="cb31-255"><a href="#cb31-255" aria-hidden="true" tabindex="-1"></a><span class="in">    axis.title.x = </span></span>
-<span id="cb31-256"><a href="#cb31-256" aria-hidden="true" tabindex="-1"></a><span class="in">      element_text(</span></span>
-<span id="cb31-257"><a href="#cb31-257" aria-hidden="true" tabindex="-1"></a><span class="in">        angle = 0, </span></span>
-<span id="cb31-258"><a href="#cb31-258" aria-hidden="true" tabindex="-1"></a><span class="in">        vjust = 1, </span></span>
-<span id="cb31-259"><a href="#cb31-259" aria-hidden="true" tabindex="-1"></a><span class="in">        hjust = 1),</span></span>
-<span id="cb31-260"><a href="#cb31-260" aria-hidden="true" tabindex="-1"></a><span class="in">    axis.title.y = </span></span>
-<span id="cb31-261"><a href="#cb31-261" aria-hidden="true" tabindex="-1"></a><span class="in">      element_text(</span></span>
-<span id="cb31-262"><a href="#cb31-262" aria-hidden="true" tabindex="-1"></a><span class="in">        angle = 0, </span></span>
-<span id="cb31-263"><a href="#cb31-263" aria-hidden="true" tabindex="-1"></a><span class="in">        vjust = 1, </span></span>
-<span id="cb31-264"><a href="#cb31-264" aria-hidden="true" tabindex="-1"></a><span class="in">        hjust = 1))</span></span>
-<span id="cb31-265"><a href="#cb31-265" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-230"><a href="#cb31-230" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, echo = FALSE}</span></span>
+<span id="cb31-231"><a href="#cb31-231" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "Exponential Survival Functions"</span></span>
+<span id="cb31-232"><a href="#cb31-232" aria-hidden="true" tabindex="-1"></a><span class="in">#| label: fig-exp-survfuns</span></span>
+<span id="cb31-233"><a href="#cb31-233" aria-hidden="true" tabindex="-1"></a><span class="in">library(ggplot2)</span></span>
+<span id="cb31-234"><a href="#cb31-234" aria-hidden="true" tabindex="-1"></a><span class="in">ggplot() +</span></span>
+<span id="cb31-235"><a href="#cb31-235" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_function(</span></span>
+<span id="cb31-236"><a href="#cb31-236" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "0.5"),</span></span>
+<span id="cb31-237"><a href="#cb31-237" aria-hidden="true" tabindex="-1"></a><span class="in">    fun = \(x) pexp(x, lower = FALSE, rate = 0.5)) +</span></span>
+<span id="cb31-238"><a href="#cb31-238" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_function(</span></span>
+<span id="cb31-239"><a href="#cb31-239" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 1"),</span></span>
+<span id="cb31-240"><a href="#cb31-240" aria-hidden="true" tabindex="-1"></a><span class="in">    fun = \(x) pexp(x, lower = FALSE, rate = 1)) +</span></span>
+<span id="cb31-241"><a href="#cb31-241" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_function(</span></span>
+<span id="cb31-242"><a href="#cb31-242" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 1.5"),</span></span>
+<span id="cb31-243"><a href="#cb31-243" aria-hidden="true" tabindex="-1"></a><span class="in">    fun = \(x) pexp(x, lower = FALSE, rate = 1.5)) +</span></span>
+<span id="cb31-244"><a href="#cb31-244" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_function(</span></span>
+<span id="cb31-245"><a href="#cb31-245" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 5"),</span></span>
+<span id="cb31-246"><a href="#cb31-246" aria-hidden="true" tabindex="-1"></a><span class="in">    fun = \(x) pexp(x, lower = FALSE, rate = 5)) +</span></span>
+<span id="cb31-247"><a href="#cb31-247" aria-hidden="true" tabindex="-1"></a><span class="in">  theme_bw() + </span></span>
+<span id="cb31-248"><a href="#cb31-248" aria-hidden="true" tabindex="-1"></a><span class="in">  ylab("S(t)") +</span></span>
+<span id="cb31-249"><a href="#cb31-249" aria-hidden="true" tabindex="-1"></a><span class="in">  guides(col = guide_legend(title = expr(lambda))) +</span></span>
+<span id="cb31-250"><a href="#cb31-250" aria-hidden="true" tabindex="-1"></a><span class="in">  xlab("Time (t)") +</span></span>
+<span id="cb31-251"><a href="#cb31-251" aria-hidden="true" tabindex="-1"></a><span class="in">  xlim(0, 2.5) +</span></span>
+<span id="cb31-252"><a href="#cb31-252" aria-hidden="true" tabindex="-1"></a><span class="in">  theme(</span></span>
+<span id="cb31-253"><a href="#cb31-253" aria-hidden="true" tabindex="-1"></a><span class="in">    axis.title.x = </span></span>
+<span id="cb31-254"><a href="#cb31-254" aria-hidden="true" tabindex="-1"></a><span class="in">      element_text(</span></span>
+<span id="cb31-255"><a href="#cb31-255" aria-hidden="true" tabindex="-1"></a><span class="in">        angle = 0, </span></span>
+<span id="cb31-256"><a href="#cb31-256" aria-hidden="true" tabindex="-1"></a><span class="in">        vjust = 1, </span></span>
+<span id="cb31-257"><a href="#cb31-257" aria-hidden="true" tabindex="-1"></a><span class="in">        hjust = 1),</span></span>
+<span id="cb31-258"><a href="#cb31-258" aria-hidden="true" tabindex="-1"></a><span class="in">    axis.title.y = </span></span>
+<span id="cb31-259"><a href="#cb31-259" aria-hidden="true" tabindex="-1"></a><span class="in">      element_text(</span></span>
+<span id="cb31-260"><a href="#cb31-260" aria-hidden="true" tabindex="-1"></a><span class="in">        angle = 0, </span></span>
+<span id="cb31-261"><a href="#cb31-261" aria-hidden="true" tabindex="-1"></a><span class="in">        vjust = 1, </span></span>
+<span id="cb31-262"><a href="#cb31-262" aria-hidden="true" tabindex="-1"></a><span class="in">        hjust = 1))</span></span>
+<span id="cb31-263"><a href="#cb31-263" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-264"><a href="#cb31-264" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-265"><a href="#cb31-265" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-266"><a href="#cb31-266" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-267"><a href="#cb31-267" aria-hidden="true" tabindex="-1"></a><span class="fu">### The Hazard Function</span></span>
+<span id="cb31-267"><a href="#cb31-267" aria-hidden="true" tabindex="-1"></a>:::{#thm-surv-fn-as-mean-status}</span>
 <span id="cb31-268"><a href="#cb31-268" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-269"><a href="#cb31-269" aria-hidden="true" tabindex="-1"></a>Another important quantity is the **hazard function**:</span>
+<span id="cb31-269"><a href="#cb31-269" aria-hidden="true" tabindex="-1"></a>If $A_t$ represents survival status at time $t$, with $A_t = 1$ denoting alive at time $t$ and $A_t = 0$ denoting deceased at time $t$, then:</span>
 <span id="cb31-270"><a href="#cb31-270" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-271"><a href="#cb31-271" aria-hidden="true" tabindex="-1"></a>:::{#def-hazard}</span>
+<span id="cb31-271"><a href="#cb31-271" aria-hidden="true" tabindex="-1"></a>$$S(t) = \Pr(A_t=1) = \Expp<span class="co">[</span><span class="ot">A_t</span><span class="co">]</span>$$</span>
 <span id="cb31-272"><a href="#cb31-272" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-273"><a href="#cb31-273" aria-hidden="true" tabindex="-1"></a>{{&lt; include _def-hazard.qmd &gt;}}</span>
+<span id="cb31-273"><a href="#cb31-273" aria-hidden="true" tabindex="-1"></a>:::</span>
 <span id="cb31-274"><a href="#cb31-274" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-275"><a href="#cb31-275" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-275"><a href="#cb31-275" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-276"><a href="#cb31-276" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-277"><a href="#cb31-277" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-277"><a href="#cb31-277" aria-hidden="true" tabindex="-1"></a>:::{#thm-surv-and-mean}</span>
 <span id="cb31-278"><a href="#cb31-278" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-279"><a href="#cb31-279" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-280"><a href="#cb31-280" aria-hidden="true" tabindex="-1"></a>The hazard function has an important relationship to the density and survival functions, </span>
-<span id="cb31-281"><a href="#cb31-281" aria-hidden="true" tabindex="-1"></a>which we can use to derive the hazard function for a given probability distribution (@thm-hazard1).</span>
-<span id="cb31-282"><a href="#cb31-282" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-283"><a href="#cb31-283" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-284"><a href="#cb31-284" aria-hidden="true" tabindex="-1"></a>:::::{#lem-joint-prob-same-var}</span>
-<span id="cb31-285"><a href="#cb31-285" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-286"><a href="#cb31-286" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Joint probability of a variable with itself</span></span>
-<span id="cb31-287"><a href="#cb31-287" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-288"><a href="#cb31-288" aria-hidden="true" tabindex="-1"></a>$$p(T=t, T\ge t) = p(T=t)$$</span>
-<span id="cb31-289"><a href="#cb31-289" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-290"><a href="#cb31-290" aria-hidden="true" tabindex="-1"></a>::::::{.proof}</span>
-<span id="cb31-291"><a href="#cb31-291" aria-hidden="true" tabindex="-1"></a>Recall from Epi 202: </span>
-<span id="cb31-292"><a href="#cb31-292" aria-hidden="true" tabindex="-1"></a>if $A$ and $B$ are statistical events and $A\subseteq B$, then $p(A, B) = p(A)$.</span>
-<span id="cb31-293"><a href="#cb31-293" aria-hidden="true" tabindex="-1"></a>In particular, $<span class="sc">\{</span>T=t<span class="sc">\}</span> \subseteq <span class="sc">\{</span>T\geq t<span class="sc">\}</span>$, so $p(T=t, T\ge t) = p(T=t)$.</span>
-<span id="cb31-294"><a href="#cb31-294" aria-hidden="true" tabindex="-1"></a>::::::</span>
-<span id="cb31-295"><a href="#cb31-295" aria-hidden="true" tabindex="-1"></a>:::::</span>
+<span id="cb31-279"><a href="#cb31-279" aria-hidden="true" tabindex="-1"></a>If $T$ is a nonnegative random variable, then:</span>
+<span id="cb31-280"><a href="#cb31-280" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-281"><a href="#cb31-281" aria-hidden="true" tabindex="-1"></a>$$\Expp<span class="co">[</span><span class="ot">T</span><span class="co">]</span> = \int_{t=0}^{\infty} S(t)dt$$</span>
+<span id="cb31-282"><a href="#cb31-282" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-283"><a href="#cb31-283" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-284"><a href="#cb31-284" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-285"><a href="#cb31-285" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-286"><a href="#cb31-286" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-287"><a href="#cb31-287" aria-hidden="true" tabindex="-1"></a>:::{.proof}</span>
+<span id="cb31-288"><a href="#cb31-288" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-289"><a href="#cb31-289" aria-hidden="true" tabindex="-1"></a>See <span class="ot">&lt;https://statproofbook.github.io/P/mean-nnrvar.html&gt;</span> or </span>
+<span id="cb31-290"><a href="#cb31-290" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-291"><a href="#cb31-291" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-292"><a href="#cb31-292" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-293"><a href="#cb31-293" aria-hidden="true" tabindex="-1"></a><span class="fu">### The Hazard Function</span></span>
+<span id="cb31-294"><a href="#cb31-294" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-295"><a href="#cb31-295" aria-hidden="true" tabindex="-1"></a>Another important quantity is the **hazard function**:</span>
 <span id="cb31-296"><a href="#cb31-296" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-297"><a href="#cb31-297" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-297"><a href="#cb31-297" aria-hidden="true" tabindex="-1"></a>:::{#def-hazard}</span>
 <span id="cb31-298"><a href="#cb31-298" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-299"><a href="#cb31-299" aria-hidden="true" tabindex="-1"></a>:::{#thm-hazard1}</span>
+<span id="cb31-299"><a href="#cb31-299" aria-hidden="true" tabindex="-1"></a>{{&lt; include _def-hazard.qmd &gt;}}</span>
 <span id="cb31-300"><a href="#cb31-300" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-301"><a href="#cb31-301" aria-hidden="true" tabindex="-1"></a>$$h(t)=\frac{f(t)}{S(t)}$$</span>
-<span id="cb31-302"><a href="#cb31-302" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-303"><a href="#cb31-303" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-304"><a href="#cb31-304" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-305"><a href="#cb31-305" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-306"><a href="#cb31-306" aria-hidden="true" tabindex="-1"></a>::::{.proof}</span>
-<span id="cb31-307"><a href="#cb31-307" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-308"><a href="#cb31-308" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-309"><a href="#cb31-309" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-310"><a href="#cb31-310" aria-hidden="true" tabindex="-1"></a>h(t) &amp;=p(T=t|T\ge t)<span class="sc">\\</span></span>
-<span id="cb31-311"><a href="#cb31-311" aria-hidden="true" tabindex="-1"></a>&amp;=\frac{p(T=t, T\ge t)}{p(T \ge t)}<span class="sc">\\</span></span>
-<span id="cb31-312"><a href="#cb31-312" aria-hidden="true" tabindex="-1"></a>&amp;=\frac{p(T=t)}{p(T \ge t)}<span class="sc">\\</span></span>
-<span id="cb31-313"><a href="#cb31-313" aria-hidden="true" tabindex="-1"></a>&amp;=\frac{f(t)}{S(t)}</span>
-<span id="cb31-314"><a href="#cb31-314" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-315"><a href="#cb31-315" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-316"><a href="#cb31-316" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-317"><a href="#cb31-317" aria-hidden="true" tabindex="-1"></a>::::</span>
-<span id="cb31-318"><a href="#cb31-318" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-319"><a href="#cb31-319" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-320"><a href="#cb31-320" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-321"><a href="#cb31-321" aria-hidden="true" tabindex="-1"></a>:::{#exm-exp-haz}</span>
-<span id="cb31-322"><a href="#cb31-322" aria-hidden="true" tabindex="-1"></a><span class="fu">##### exponential distribution</span></span>
-<span id="cb31-323"><a href="#cb31-323" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-324"><a href="#cb31-324" aria-hidden="true" tabindex="-1"></a>The hazard function of the exponential distribution family of models is:</span>
-<span id="cb31-325"><a href="#cb31-325" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-326"><a href="#cb31-326" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-327"><a href="#cb31-327" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-328"><a href="#cb31-328" aria-hidden="true" tabindex="-1"></a>P(T=t|T \ge t) </span>
-<span id="cb31-329"><a href="#cb31-329" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{f(t)}{S(t)}<span class="sc">\\</span></span>
-<span id="cb31-330"><a href="#cb31-330" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{\mathbb{1}_{t \ge 0}\cdot \lambda  \text{e}^{-\lambda t}}{\text{e}^{-\lambda t}}<span class="sc">\\</span></span>
-<span id="cb31-331"><a href="#cb31-331" aria-hidden="true" tabindex="-1"></a>&amp;=\mathbb{1}_{t \ge 0}\cdot \lambda</span>
-<span id="cb31-332"><a href="#cb31-332" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-333"><a href="#cb31-333" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-334"><a href="#cb31-334" aria-hidden="true" tabindex="-1"></a>@fig-exp-hazard shows some examples of exponential hazard functions.</span>
-<span id="cb31-335"><a href="#cb31-335" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-336"><a href="#cb31-336" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-337"><a href="#cb31-337" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-338"><a href="#cb31-338" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-339"><a href="#cb31-339" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-340"><a href="#cb31-340" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, echo = FALSE}</span></span>
-<span id="cb31-341"><a href="#cb31-341" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "Examples of hazard functions for exponential distributions"</span></span>
-<span id="cb31-342"><a href="#cb31-342" aria-hidden="true" tabindex="-1"></a><span class="in">#| label: fig-exp-hazard</span></span>
-<span id="cb31-343"><a href="#cb31-343" aria-hidden="true" tabindex="-1"></a><span class="in">library(ggplot2)</span></span>
-<span id="cb31-344"><a href="#cb31-344" aria-hidden="true" tabindex="-1"></a><span class="in">ggplot() +</span></span>
-<span id="cb31-345"><a href="#cb31-345" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_hline(</span></span>
-<span id="cb31-346"><a href="#cb31-346" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "0.5",yintercept = 0.5)) +</span></span>
-<span id="cb31-347"><a href="#cb31-347" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_hline(</span></span>
-<span id="cb31-348"><a href="#cb31-348" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 1", yintercept = 1)) +</span></span>
-<span id="cb31-349"><a href="#cb31-349" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_hline(</span></span>
-<span id="cb31-350"><a href="#cb31-350" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 1.5", yintercept = 1.5)) +</span></span>
-<span id="cb31-351"><a href="#cb31-351" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_hline(</span></span>
-<span id="cb31-352"><a href="#cb31-352" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 5", yintercept = 5)) +</span></span>
-<span id="cb31-353"><a href="#cb31-353" aria-hidden="true" tabindex="-1"></a><span class="in">  theme_bw() + </span></span>
-<span id="cb31-354"><a href="#cb31-354" aria-hidden="true" tabindex="-1"></a><span class="in">  ylab("h(t)") +</span></span>
-<span id="cb31-355"><a href="#cb31-355" aria-hidden="true" tabindex="-1"></a><span class="in">  ylim(0,5) +</span></span>
-<span id="cb31-356"><a href="#cb31-356" aria-hidden="true" tabindex="-1"></a><span class="in">  guides(col = guide_legend(title = expr(lambda))) +</span></span>
-<span id="cb31-357"><a href="#cb31-357" aria-hidden="true" tabindex="-1"></a><span class="in">  xlab("Time (t)") +</span></span>
-<span id="cb31-358"><a href="#cb31-358" aria-hidden="true" tabindex="-1"></a><span class="in">  xlim(0, 2.5) +</span></span>
-<span id="cb31-359"><a href="#cb31-359" aria-hidden="true" tabindex="-1"></a><span class="in">  theme(</span></span>
-<span id="cb31-360"><a href="#cb31-360" aria-hidden="true" tabindex="-1"></a><span class="in">    axis.title.x = </span></span>
-<span id="cb31-361"><a href="#cb31-361" aria-hidden="true" tabindex="-1"></a><span class="in">      element_text(</span></span>
-<span id="cb31-362"><a href="#cb31-362" aria-hidden="true" tabindex="-1"></a><span class="in">        angle = 0, </span></span>
-<span id="cb31-363"><a href="#cb31-363" aria-hidden="true" tabindex="-1"></a><span class="in">        vjust = 1, </span></span>
-<span id="cb31-364"><a href="#cb31-364" aria-hidden="true" tabindex="-1"></a><span class="in">        hjust = 1),</span></span>
-<span id="cb31-365"><a href="#cb31-365" aria-hidden="true" tabindex="-1"></a><span class="in">    axis.title.y = </span></span>
-<span id="cb31-366"><a href="#cb31-366" aria-hidden="true" tabindex="-1"></a><span class="in">      element_text(</span></span>
-<span id="cb31-367"><a href="#cb31-367" aria-hidden="true" tabindex="-1"></a><span class="in">        angle = 0, </span></span>
-<span id="cb31-368"><a href="#cb31-368" aria-hidden="true" tabindex="-1"></a><span class="in">        vjust = 1, </span></span>
-<span id="cb31-369"><a href="#cb31-369" aria-hidden="true" tabindex="-1"></a><span class="in">        hjust = 1))</span></span>
-<span id="cb31-370"><a href="#cb31-370" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-371"><a href="#cb31-371" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-372"><a href="#cb31-372" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-373"><a href="#cb31-373" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-374"><a href="#cb31-374" aria-hidden="true" tabindex="-1"></a>We can also view the hazard function as the derivative of the negative of the logarithm of the survival function:</span>
-<span id="cb31-375"><a href="#cb31-375" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-376"><a href="#cb31-376" aria-hidden="true" tabindex="-1"></a>:::{#thm-h-logS}</span>
-<span id="cb31-377"><a href="#cb31-377" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-378"><a href="#cb31-378" aria-hidden="true" tabindex="-1"></a><span class="fu">#### transform survival to hazard</span></span>
-<span id="cb31-379"><a href="#cb31-379" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-380"><a href="#cb31-380" aria-hidden="true" tabindex="-1"></a>$$h(t) = \deriv{t}\cb{-\log{S(t)}}$$</span>
-<span id="cb31-381"><a href="#cb31-381" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-382"><a href="#cb31-382" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-383"><a href="#cb31-383" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-384"><a href="#cb31-384" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-385"><a href="#cb31-385" aria-hidden="true" tabindex="-1"></a>::::{.proof}</span>
-<span id="cb31-386"><a href="#cb31-386" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-387"><a href="#cb31-387" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-388"><a href="#cb31-388" aria-hidden="true" tabindex="-1"></a>h(t)</span>
-<span id="cb31-389"><a href="#cb31-389" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{f(t)}{S(t)}<span class="sc">\\</span></span>
-<span id="cb31-390"><a href="#cb31-390" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{-S'(t)}{S(t)}<span class="sc">\\</span></span>
-<span id="cb31-391"><a href="#cb31-391" aria-hidden="true" tabindex="-1"></a>&amp;= -\frac{S'(t)}{S(t)}<span class="sc">\\</span></span>
-<span id="cb31-392"><a href="#cb31-392" aria-hidden="true" tabindex="-1"></a>&amp;=-\deriv{t}\log{S(t)}<span class="sc">\\</span></span>
-<span id="cb31-393"><a href="#cb31-393" aria-hidden="true" tabindex="-1"></a>&amp;=\deriv{t}\cb{-\log{S(t)}}</span>
-<span id="cb31-394"><a href="#cb31-394" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-395"><a href="#cb31-395" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-396"><a href="#cb31-396" aria-hidden="true" tabindex="-1"></a>::::</span>
+<span id="cb31-301"><a href="#cb31-301" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-302"><a href="#cb31-302" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-303"><a href="#cb31-303" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-304"><a href="#cb31-304" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-305"><a href="#cb31-305" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-306"><a href="#cb31-306" aria-hidden="true" tabindex="-1"></a>The hazard function has an important relationship to the density and survival functions, </span>
+<span id="cb31-307"><a href="#cb31-307" aria-hidden="true" tabindex="-1"></a>which we can use to derive the hazard function for a given probability distribution (@thm-hazard1).</span>
+<span id="cb31-308"><a href="#cb31-308" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-309"><a href="#cb31-309" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-310"><a href="#cb31-310" aria-hidden="true" tabindex="-1"></a>:::::{#lem-joint-prob-same-var}</span>
+<span id="cb31-311"><a href="#cb31-311" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-312"><a href="#cb31-312" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Joint probability of a variable with itself</span></span>
+<span id="cb31-313"><a href="#cb31-313" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-314"><a href="#cb31-314" aria-hidden="true" tabindex="-1"></a>$$p(T=t, T\ge t) = p(T=t)$$</span>
+<span id="cb31-315"><a href="#cb31-315" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-316"><a href="#cb31-316" aria-hidden="true" tabindex="-1"></a>::::::{.proof}</span>
+<span id="cb31-317"><a href="#cb31-317" aria-hidden="true" tabindex="-1"></a>Recall from Epi 202: </span>
+<span id="cb31-318"><a href="#cb31-318" aria-hidden="true" tabindex="-1"></a>if $A$ and $B$ are statistical events and $A\subseteq B$, then $p(A, B) = p(A)$.</span>
+<span id="cb31-319"><a href="#cb31-319" aria-hidden="true" tabindex="-1"></a>In particular, $<span class="sc">\{</span>T=t<span class="sc">\}</span> \subseteq <span class="sc">\{</span>T\geq t<span class="sc">\}</span>$, so $p(T=t, T\ge t) = p(T=t)$.</span>
+<span id="cb31-320"><a href="#cb31-320" aria-hidden="true" tabindex="-1"></a>::::::</span>
+<span id="cb31-321"><a href="#cb31-321" aria-hidden="true" tabindex="-1"></a>:::::</span>
+<span id="cb31-322"><a href="#cb31-322" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-323"><a href="#cb31-323" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-324"><a href="#cb31-324" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-325"><a href="#cb31-325" aria-hidden="true" tabindex="-1"></a>:::{#thm-hazard1}</span>
+<span id="cb31-326"><a href="#cb31-326" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-327"><a href="#cb31-327" aria-hidden="true" tabindex="-1"></a>$$h(t)=\frac{f(t)}{S(t)}$$</span>
+<span id="cb31-328"><a href="#cb31-328" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-329"><a href="#cb31-329" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-330"><a href="#cb31-330" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-331"><a href="#cb31-331" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-332"><a href="#cb31-332" aria-hidden="true" tabindex="-1"></a>::::{.proof}</span>
+<span id="cb31-333"><a href="#cb31-333" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-334"><a href="#cb31-334" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-335"><a href="#cb31-335" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-336"><a href="#cb31-336" aria-hidden="true" tabindex="-1"></a>h(t) &amp;=p(T=t|T\ge t)<span class="sc">\\</span></span>
+<span id="cb31-337"><a href="#cb31-337" aria-hidden="true" tabindex="-1"></a>&amp;=\frac{p(T=t, T\ge t)}{p(T \ge t)}<span class="sc">\\</span></span>
+<span id="cb31-338"><a href="#cb31-338" aria-hidden="true" tabindex="-1"></a>&amp;=\frac{p(T=t)}{p(T \ge t)}<span class="sc">\\</span></span>
+<span id="cb31-339"><a href="#cb31-339" aria-hidden="true" tabindex="-1"></a>&amp;=\frac{f(t)}{S(t)}</span>
+<span id="cb31-340"><a href="#cb31-340" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-341"><a href="#cb31-341" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-342"><a href="#cb31-342" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-343"><a href="#cb31-343" aria-hidden="true" tabindex="-1"></a>::::</span>
+<span id="cb31-344"><a href="#cb31-344" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-345"><a href="#cb31-345" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-346"><a href="#cb31-346" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-347"><a href="#cb31-347" aria-hidden="true" tabindex="-1"></a>:::{#exm-exp-haz}</span>
+<span id="cb31-348"><a href="#cb31-348" aria-hidden="true" tabindex="-1"></a><span class="fu">##### exponential distribution</span></span>
+<span id="cb31-349"><a href="#cb31-349" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-350"><a href="#cb31-350" aria-hidden="true" tabindex="-1"></a>The hazard function of the exponential distribution family of models is:</span>
+<span id="cb31-351"><a href="#cb31-351" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-352"><a href="#cb31-352" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-353"><a href="#cb31-353" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-354"><a href="#cb31-354" aria-hidden="true" tabindex="-1"></a>P(T=t|T \ge t) </span>
+<span id="cb31-355"><a href="#cb31-355" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{f(t)}{S(t)}<span class="sc">\\</span></span>
+<span id="cb31-356"><a href="#cb31-356" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{\mathbb{1}_{t \ge 0}\cdot \lambda  \text{e}^{-\lambda t}}{\text{e}^{-\lambda t}}<span class="sc">\\</span></span>
+<span id="cb31-357"><a href="#cb31-357" aria-hidden="true" tabindex="-1"></a>&amp;=\mathbb{1}_{t \ge 0}\cdot \lambda</span>
+<span id="cb31-358"><a href="#cb31-358" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-359"><a href="#cb31-359" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-360"><a href="#cb31-360" aria-hidden="true" tabindex="-1"></a>@fig-exp-hazard shows some examples of exponential hazard functions.</span>
+<span id="cb31-361"><a href="#cb31-361" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-362"><a href="#cb31-362" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-363"><a href="#cb31-363" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-364"><a href="#cb31-364" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-365"><a href="#cb31-365" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-366"><a href="#cb31-366" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, echo = FALSE}</span></span>
+<span id="cb31-367"><a href="#cb31-367" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "Examples of hazard functions for exponential distributions"</span></span>
+<span id="cb31-368"><a href="#cb31-368" aria-hidden="true" tabindex="-1"></a><span class="in">#| label: fig-exp-hazard</span></span>
+<span id="cb31-369"><a href="#cb31-369" aria-hidden="true" tabindex="-1"></a><span class="in">library(ggplot2)</span></span>
+<span id="cb31-370"><a href="#cb31-370" aria-hidden="true" tabindex="-1"></a><span class="in">ggplot() +</span></span>
+<span id="cb31-371"><a href="#cb31-371" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_hline(</span></span>
+<span id="cb31-372"><a href="#cb31-372" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "0.5",yintercept = 0.5)) +</span></span>
+<span id="cb31-373"><a href="#cb31-373" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_hline(</span></span>
+<span id="cb31-374"><a href="#cb31-374" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 1", yintercept = 1)) +</span></span>
+<span id="cb31-375"><a href="#cb31-375" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_hline(</span></span>
+<span id="cb31-376"><a href="#cb31-376" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 1.5", yintercept = 1.5)) +</span></span>
+<span id="cb31-377"><a href="#cb31-377" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_hline(</span></span>
+<span id="cb31-378"><a href="#cb31-378" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 5", yintercept = 5)) +</span></span>
+<span id="cb31-379"><a href="#cb31-379" aria-hidden="true" tabindex="-1"></a><span class="in">  theme_bw() + </span></span>
+<span id="cb31-380"><a href="#cb31-380" aria-hidden="true" tabindex="-1"></a><span class="in">  ylab("h(t)") +</span></span>
+<span id="cb31-381"><a href="#cb31-381" aria-hidden="true" tabindex="-1"></a><span class="in">  ylim(0,5) +</span></span>
+<span id="cb31-382"><a href="#cb31-382" aria-hidden="true" tabindex="-1"></a><span class="in">  guides(col = guide_legend(title = expr(lambda))) +</span></span>
+<span id="cb31-383"><a href="#cb31-383" aria-hidden="true" tabindex="-1"></a><span class="in">  xlab("Time (t)") +</span></span>
+<span id="cb31-384"><a href="#cb31-384" aria-hidden="true" tabindex="-1"></a><span class="in">  xlim(0, 2.5) +</span></span>
+<span id="cb31-385"><a href="#cb31-385" aria-hidden="true" tabindex="-1"></a><span class="in">  theme(</span></span>
+<span id="cb31-386"><a href="#cb31-386" aria-hidden="true" tabindex="-1"></a><span class="in">    axis.title.x = </span></span>
+<span id="cb31-387"><a href="#cb31-387" aria-hidden="true" tabindex="-1"></a><span class="in">      element_text(</span></span>
+<span id="cb31-388"><a href="#cb31-388" aria-hidden="true" tabindex="-1"></a><span class="in">        angle = 0, </span></span>
+<span id="cb31-389"><a href="#cb31-389" aria-hidden="true" tabindex="-1"></a><span class="in">        vjust = 1, </span></span>
+<span id="cb31-390"><a href="#cb31-390" aria-hidden="true" tabindex="-1"></a><span class="in">        hjust = 1),</span></span>
+<span id="cb31-391"><a href="#cb31-391" aria-hidden="true" tabindex="-1"></a><span class="in">    axis.title.y = </span></span>
+<span id="cb31-392"><a href="#cb31-392" aria-hidden="true" tabindex="-1"></a><span class="in">      element_text(</span></span>
+<span id="cb31-393"><a href="#cb31-393" aria-hidden="true" tabindex="-1"></a><span class="in">        angle = 0, </span></span>
+<span id="cb31-394"><a href="#cb31-394" aria-hidden="true" tabindex="-1"></a><span class="in">        vjust = 1, </span></span>
+<span id="cb31-395"><a href="#cb31-395" aria-hidden="true" tabindex="-1"></a><span class="in">        hjust = 1))</span></span>
+<span id="cb31-396"><a href="#cb31-396" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
 <span id="cb31-397"><a href="#cb31-397" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-398"><a href="#cb31-398" aria-hidden="true" tabindex="-1"></a><span class="fu">### The Cumulative Hazard Function</span></span>
+<span id="cb31-398"><a href="#cb31-398" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-399"><a href="#cb31-399" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-400"><a href="#cb31-400" aria-hidden="true" tabindex="-1"></a>Since $h(t) = \deriv{t}\cb{-\log{S(t)}}$ (see @thm-h-logS), we also have:</span>
+<span id="cb31-400"><a href="#cb31-400" aria-hidden="true" tabindex="-1"></a>We can also view the hazard function as the derivative of the negative of the logarithm of the survival function:</span>
 <span id="cb31-401"><a href="#cb31-401" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-402"><a href="#cb31-402" aria-hidden="true" tabindex="-1"></a>:::{#cor-surv-int-haz}</span>
-<span id="cb31-403"><a href="#cb31-403" aria-hidden="true" tabindex="-1"></a>$$S(t) = \exp{-\int_{u=0}^t h(u)du}$${#eq-surv-int-haz}</span>
-<span id="cb31-404"><a href="#cb31-404" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-402"><a href="#cb31-402" aria-hidden="true" tabindex="-1"></a>:::{#thm-h-logS}</span>
+<span id="cb31-403"><a href="#cb31-403" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-404"><a href="#cb31-404" aria-hidden="true" tabindex="-1"></a><span class="fu">#### transform survival to hazard</span></span>
 <span id="cb31-405"><a href="#cb31-405" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-406"><a href="#cb31-406" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-407"><a href="#cb31-407" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-408"><a href="#cb31-408" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-409"><a href="#cb31-409" aria-hidden="true" tabindex="-1"></a>The integral in @eq-surv-int-haz is important enough to have its own name: **cumulative hazard**.</span>
-<span id="cb31-410"><a href="#cb31-410" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-411"><a href="#cb31-411" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-412"><a href="#cb31-412" aria-hidden="true" tabindex="-1"></a>:::{#def-cumhaz}</span>
-<span id="cb31-413"><a href="#cb31-413" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-414"><a href="#cb31-414" aria-hidden="true" tabindex="-1"></a><span class="fu">##### cumulative hazard</span></span>
-<span id="cb31-415"><a href="#cb31-415" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-416"><a href="#cb31-416" aria-hidden="true" tabindex="-1"></a>The **cumulative hazard function** $H(t)$ is defined as:</span>
-<span id="cb31-417"><a href="#cb31-417" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-418"><a href="#cb31-418" aria-hidden="true" tabindex="-1"></a>$$H(t) \eqdef \int_{u=0}^t h(u) du$$</span>
-<span id="cb31-419"><a href="#cb31-419" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-420"><a href="#cb31-420" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-421"><a href="#cb31-421" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-422"><a href="#cb31-422" aria-hidden="true" tabindex="-1"></a>As we will see below, $H(t)$ is tractable to estimate, and we can then</span>
-<span id="cb31-423"><a href="#cb31-423" aria-hidden="true" tabindex="-1"></a>derive an estimate of the hazard function using an approximate derivative</span>
-<span id="cb31-424"><a href="#cb31-424" aria-hidden="true" tabindex="-1"></a>of the estimated cumulative hazard.</span>
+<span id="cb31-406"><a href="#cb31-406" aria-hidden="true" tabindex="-1"></a>$$h(t) = \deriv{t}\cb{-\log{S(t)}}$$</span>
+<span id="cb31-407"><a href="#cb31-407" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-408"><a href="#cb31-408" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-409"><a href="#cb31-409" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-410"><a href="#cb31-410" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-411"><a href="#cb31-411" aria-hidden="true" tabindex="-1"></a>::::{.proof}</span>
+<span id="cb31-412"><a href="#cb31-412" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-413"><a href="#cb31-413" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-414"><a href="#cb31-414" aria-hidden="true" tabindex="-1"></a>h(t)</span>
+<span id="cb31-415"><a href="#cb31-415" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{f(t)}{S(t)}<span class="sc">\\</span></span>
+<span id="cb31-416"><a href="#cb31-416" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{-S'(t)}{S(t)}<span class="sc">\\</span></span>
+<span id="cb31-417"><a href="#cb31-417" aria-hidden="true" tabindex="-1"></a>&amp;= -\frac{S'(t)}{S(t)}<span class="sc">\\</span></span>
+<span id="cb31-418"><a href="#cb31-418" aria-hidden="true" tabindex="-1"></a>&amp;=-\deriv{t}\log{S(t)}<span class="sc">\\</span></span>
+<span id="cb31-419"><a href="#cb31-419" aria-hidden="true" tabindex="-1"></a>&amp;=\deriv{t}\cb{-\log{S(t)}}</span>
+<span id="cb31-420"><a href="#cb31-420" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-421"><a href="#cb31-421" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-422"><a href="#cb31-422" aria-hidden="true" tabindex="-1"></a>::::</span>
+<span id="cb31-423"><a href="#cb31-423" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-424"><a href="#cb31-424" aria-hidden="true" tabindex="-1"></a><span class="fu">### The Cumulative Hazard Function</span></span>
 <span id="cb31-425"><a href="#cb31-425" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-426"><a href="#cb31-426" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-426"><a href="#cb31-426" aria-hidden="true" tabindex="-1"></a>Since $h(t) = \deriv{t}\cb{-\log{S(t)}}$ (see @thm-h-logS), we also have:</span>
 <span id="cb31-427"><a href="#cb31-427" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-428"><a href="#cb31-428" aria-hidden="true" tabindex="-1"></a>:::{#exm-exp-cumhaz}</span>
-<span id="cb31-429"><a href="#cb31-429" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-430"><a href="#cb31-430" aria-hidden="true" tabindex="-1"></a>The cumulative hazard function of the exponential distribution family of</span>
-<span id="cb31-431"><a href="#cb31-431" aria-hidden="true" tabindex="-1"></a>models is:</span>
-<span id="cb31-432"><a href="#cb31-432" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-433"><a href="#cb31-433" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-434"><a href="#cb31-434" aria-hidden="true" tabindex="-1"></a>H(t) = \mathbb{1}_{t \ge 0}\cdot \lambda t</span>
-<span id="cb31-435"><a href="#cb31-435" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-436"><a href="#cb31-436" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-437"><a href="#cb31-437" aria-hidden="true" tabindex="-1"></a>@fig-cuhaz-exp shows some examples of exponential cumulative hazard functions.</span>
-<span id="cb31-438"><a href="#cb31-438" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-439"><a href="#cb31-439" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-440"><a href="#cb31-440" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-441"><a href="#cb31-441" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-442"><a href="#cb31-442" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-443"><a href="#cb31-443" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, echo = FALSE}</span></span>
-<span id="cb31-444"><a href="#cb31-444" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "Examples of exponential cumulative hazard functions"</span></span>
-<span id="cb31-445"><a href="#cb31-445" aria-hidden="true" tabindex="-1"></a><span class="in">#| label: fig-cuhaz-exp</span></span>
-<span id="cb31-446"><a href="#cb31-446" aria-hidden="true" tabindex="-1"></a><span class="in">library(ggplot2)</span></span>
-<span id="cb31-447"><a href="#cb31-447" aria-hidden="true" tabindex="-1"></a><span class="in">ggplot() +</span></span>
-<span id="cb31-448"><a href="#cb31-448" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_abline(</span></span>
-<span id="cb31-449"><a href="#cb31-449" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "0.5",intercept = 0, slope = 0.5)) +</span></span>
-<span id="cb31-450"><a href="#cb31-450" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_abline(</span></span>
-<span id="cb31-451"><a href="#cb31-451" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 1", intercept = 0, slope = 1)) +</span></span>
-<span id="cb31-452"><a href="#cb31-452" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_abline(</span></span>
-<span id="cb31-453"><a href="#cb31-453" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 1.5", intercept = 0, slope = 1.5)) +</span></span>
-<span id="cb31-454"><a href="#cb31-454" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_abline(</span></span>
-<span id="cb31-455"><a href="#cb31-455" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 5", intercept = 0, slope = 5)) +</span></span>
-<span id="cb31-456"><a href="#cb31-456" aria-hidden="true" tabindex="-1"></a><span class="in">  theme_bw() + </span></span>
-<span id="cb31-457"><a href="#cb31-457" aria-hidden="true" tabindex="-1"></a><span class="in">  ylab("H(t)") +</span></span>
-<span id="cb31-458"><a href="#cb31-458" aria-hidden="true" tabindex="-1"></a><span class="in">  ylim(0,5) +</span></span>
-<span id="cb31-459"><a href="#cb31-459" aria-hidden="true" tabindex="-1"></a><span class="in">  guides(col = guide_legend(title = expr(lambda))) +</span></span>
-<span id="cb31-460"><a href="#cb31-460" aria-hidden="true" tabindex="-1"></a><span class="in">  xlab("Time (t)") +</span></span>
-<span id="cb31-461"><a href="#cb31-461" aria-hidden="true" tabindex="-1"></a><span class="in">  xlim(0, 2.5) +</span></span>
-<span id="cb31-462"><a href="#cb31-462" aria-hidden="true" tabindex="-1"></a><span class="in">  theme(</span></span>
-<span id="cb31-463"><a href="#cb31-463" aria-hidden="true" tabindex="-1"></a><span class="in">    axis.title.x = </span></span>
-<span id="cb31-464"><a href="#cb31-464" aria-hidden="true" tabindex="-1"></a><span class="in">      element_text(</span></span>
-<span id="cb31-465"><a href="#cb31-465" aria-hidden="true" tabindex="-1"></a><span class="in">        angle = 0, </span></span>
-<span id="cb31-466"><a href="#cb31-466" aria-hidden="true" tabindex="-1"></a><span class="in">        vjust = 1, </span></span>
-<span id="cb31-467"><a href="#cb31-467" aria-hidden="true" tabindex="-1"></a><span class="in">        hjust = 1),</span></span>
-<span id="cb31-468"><a href="#cb31-468" aria-hidden="true" tabindex="-1"></a><span class="in">    axis.title.y = </span></span>
-<span id="cb31-469"><a href="#cb31-469" aria-hidden="true" tabindex="-1"></a><span class="in">      element_text(</span></span>
-<span id="cb31-470"><a href="#cb31-470" aria-hidden="true" tabindex="-1"></a><span class="in">        angle = 0, </span></span>
-<span id="cb31-471"><a href="#cb31-471" aria-hidden="true" tabindex="-1"></a><span class="in">        vjust = 1, </span></span>
-<span id="cb31-472"><a href="#cb31-472" aria-hidden="true" tabindex="-1"></a><span class="in">        hjust = 1))</span></span>
-<span id="cb31-473"><a href="#cb31-473" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-474"><a href="#cb31-474" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-475"><a href="#cb31-475" aria-hidden="true" tabindex="-1"></a><span class="fu">### Some Key Mathematical Relationships among Survival Concepts</span></span>
-<span id="cb31-476"><a href="#cb31-476" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-477"><a href="#cb31-477" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Diagram:</span></span>
-<span id="cb31-478"><a href="#cb31-478" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-479"><a href="#cb31-479" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-480"><a href="#cb31-480" aria-hidden="true" tabindex="-1"></a>h(t) \xrightarrow[]{\int_{u=0}^t h(u)du} H(t) </span>
-<span id="cb31-481"><a href="#cb31-481" aria-hidden="true" tabindex="-1"></a>\xrightarrow[]{\exp{-H(t)}} S(t)</span>
-<span id="cb31-482"><a href="#cb31-482" aria-hidden="true" tabindex="-1"></a>\xrightarrow[]{1-S(t)} F(t)</span>
-<span id="cb31-483"><a href="#cb31-483" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-484"><a href="#cb31-484" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-485"><a href="#cb31-485" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-486"><a href="#cb31-486" aria-hidden="true" tabindex="-1"></a>h(t) \xleftarrow<span class="co">[</span><span class="ot">\deriv{t}H(t)</span><span class="co">]</span>{} H(t) </span>
-<span id="cb31-487"><a href="#cb31-487" aria-hidden="true" tabindex="-1"></a>\xleftarrow<span class="co">[</span><span class="ot">-\log{S(t)}</span><span class="co">]</span>{} S(t)</span>
-<span id="cb31-488"><a href="#cb31-488" aria-hidden="true" tabindex="-1"></a>\xleftarrow<span class="co">[</span><span class="ot">1-F(t)</span><span class="co">]</span>{} F(t)</span>
-<span id="cb31-489"><a href="#cb31-489" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-490"><a href="#cb31-490" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-491"><a href="#cb31-491" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-492"><a href="#cb31-492" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-493"><a href="#cb31-493" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-494"><a href="#cb31-494" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Identities:</span></span>
-<span id="cb31-495"><a href="#cb31-495" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-496"><a href="#cb31-496" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-497"><a href="#cb31-497" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-498"><a href="#cb31-498" aria-hidden="true" tabindex="-1"></a>S(t) &amp;= 1 - F(t)<span class="sc">\\</span></span>
-<span id="cb31-499"><a href="#cb31-499" aria-hidden="true" tabindex="-1"></a>&amp;= \text{exp}\left<span class="sc">\{</span>-H(t)\right<span class="sc">\}\\</span></span>
-<span id="cb31-500"><a href="#cb31-500" aria-hidden="true" tabindex="-1"></a>S'(t) &amp;= -f(t)<span class="sc">\\</span></span>
-<span id="cb31-501"><a href="#cb31-501" aria-hidden="true" tabindex="-1"></a>H(t) &amp;= -\text{log}\left<span class="sc">\{</span>S(t)\right<span class="sc">\}\\</span></span>
-<span id="cb31-502"><a href="#cb31-502" aria-hidden="true" tabindex="-1"></a>H'(t) &amp;= h(t)<span class="sc">\\</span></span>
-<span id="cb31-503"><a href="#cb31-503" aria-hidden="true" tabindex="-1"></a>h(t) &amp;= \frac{f(t)}{S(t)}<span class="sc">\\</span></span>
-<span id="cb31-504"><a href="#cb31-504" aria-hidden="true" tabindex="-1"></a> &amp;= -\deriv{t}\log{S(t)} <span class="sc">\\</span></span>
-<span id="cb31-505"><a href="#cb31-505" aria-hidden="true" tabindex="-1"></a>f(t) &amp;= h(t)\cdot S(t)<span class="sc">\\</span></span>
-<span id="cb31-506"><a href="#cb31-506" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-507"><a href="#cb31-507" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-508"><a href="#cb31-508" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-509"><a href="#cb31-509" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-428"><a href="#cb31-428" aria-hidden="true" tabindex="-1"></a>:::{#cor-surv-int-haz}</span>
+<span id="cb31-429"><a href="#cb31-429" aria-hidden="true" tabindex="-1"></a>$$S(t) = \exp{-\int_{u=0}^t h(u)du}$${#eq-surv-int-haz}</span>
+<span id="cb31-430"><a href="#cb31-430" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-431"><a href="#cb31-431" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-432"><a href="#cb31-432" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-433"><a href="#cb31-433" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-434"><a href="#cb31-434" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-435"><a href="#cb31-435" aria-hidden="true" tabindex="-1"></a>The integral in @eq-surv-int-haz is important enough to have its own name: **cumulative hazard**.</span>
+<span id="cb31-436"><a href="#cb31-436" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-437"><a href="#cb31-437" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-438"><a href="#cb31-438" aria-hidden="true" tabindex="-1"></a>:::{#def-cumhaz}</span>
+<span id="cb31-439"><a href="#cb31-439" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-440"><a href="#cb31-440" aria-hidden="true" tabindex="-1"></a><span class="fu">##### cumulative hazard</span></span>
+<span id="cb31-441"><a href="#cb31-441" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-442"><a href="#cb31-442" aria-hidden="true" tabindex="-1"></a>The **cumulative hazard function** $H(t)$ is defined as:</span>
+<span id="cb31-443"><a href="#cb31-443" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-444"><a href="#cb31-444" aria-hidden="true" tabindex="-1"></a>$$H(t) \eqdef \int_{u=0}^t h(u) du$$</span>
+<span id="cb31-445"><a href="#cb31-445" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-446"><a href="#cb31-446" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-447"><a href="#cb31-447" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-448"><a href="#cb31-448" aria-hidden="true" tabindex="-1"></a>As we will see below, $H(t)$ is tractable to estimate, and we can then</span>
+<span id="cb31-449"><a href="#cb31-449" aria-hidden="true" tabindex="-1"></a>derive an estimate of the hazard function using an approximate derivative</span>
+<span id="cb31-450"><a href="#cb31-450" aria-hidden="true" tabindex="-1"></a>of the estimated cumulative hazard.</span>
+<span id="cb31-451"><a href="#cb31-451" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-452"><a href="#cb31-452" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-453"><a href="#cb31-453" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-454"><a href="#cb31-454" aria-hidden="true" tabindex="-1"></a>:::{#exm-exp-cumhaz}</span>
+<span id="cb31-455"><a href="#cb31-455" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-456"><a href="#cb31-456" aria-hidden="true" tabindex="-1"></a>The cumulative hazard function of the exponential distribution family of</span>
+<span id="cb31-457"><a href="#cb31-457" aria-hidden="true" tabindex="-1"></a>models is:</span>
+<span id="cb31-458"><a href="#cb31-458" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-459"><a href="#cb31-459" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-460"><a href="#cb31-460" aria-hidden="true" tabindex="-1"></a>H(t) = \mathbb{1}_{t \ge 0}\cdot \lambda t</span>
+<span id="cb31-461"><a href="#cb31-461" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-462"><a href="#cb31-462" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-463"><a href="#cb31-463" aria-hidden="true" tabindex="-1"></a>@fig-cuhaz-exp shows some examples of exponential cumulative hazard functions.</span>
+<span id="cb31-464"><a href="#cb31-464" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-465"><a href="#cb31-465" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-466"><a href="#cb31-466" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-467"><a href="#cb31-467" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-468"><a href="#cb31-468" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-469"><a href="#cb31-469" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, echo = FALSE}</span></span>
+<span id="cb31-470"><a href="#cb31-470" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "Examples of exponential cumulative hazard functions"</span></span>
+<span id="cb31-471"><a href="#cb31-471" aria-hidden="true" tabindex="-1"></a><span class="in">#| label: fig-cuhaz-exp</span></span>
+<span id="cb31-472"><a href="#cb31-472" aria-hidden="true" tabindex="-1"></a><span class="in">library(ggplot2)</span></span>
+<span id="cb31-473"><a href="#cb31-473" aria-hidden="true" tabindex="-1"></a><span class="in">ggplot() +</span></span>
+<span id="cb31-474"><a href="#cb31-474" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_abline(</span></span>
+<span id="cb31-475"><a href="#cb31-475" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "0.5",intercept = 0, slope = 0.5)) +</span></span>
+<span id="cb31-476"><a href="#cb31-476" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_abline(</span></span>
+<span id="cb31-477"><a href="#cb31-477" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 1", intercept = 0, slope = 1)) +</span></span>
+<span id="cb31-478"><a href="#cb31-478" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_abline(</span></span>
+<span id="cb31-479"><a href="#cb31-479" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 1.5", intercept = 0, slope = 1.5)) +</span></span>
+<span id="cb31-480"><a href="#cb31-480" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_abline(</span></span>
+<span id="cb31-481"><a href="#cb31-481" aria-hidden="true" tabindex="-1"></a><span class="in">    aes(col = "p = 5", intercept = 0, slope = 5)) +</span></span>
+<span id="cb31-482"><a href="#cb31-482" aria-hidden="true" tabindex="-1"></a><span class="in">  theme_bw() + </span></span>
+<span id="cb31-483"><a href="#cb31-483" aria-hidden="true" tabindex="-1"></a><span class="in">  ylab("H(t)") +</span></span>
+<span id="cb31-484"><a href="#cb31-484" aria-hidden="true" tabindex="-1"></a><span class="in">  ylim(0,5) +</span></span>
+<span id="cb31-485"><a href="#cb31-485" aria-hidden="true" tabindex="-1"></a><span class="in">  guides(col = guide_legend(title = expr(lambda))) +</span></span>
+<span id="cb31-486"><a href="#cb31-486" aria-hidden="true" tabindex="-1"></a><span class="in">  xlab("Time (t)") +</span></span>
+<span id="cb31-487"><a href="#cb31-487" aria-hidden="true" tabindex="-1"></a><span class="in">  xlim(0, 2.5) +</span></span>
+<span id="cb31-488"><a href="#cb31-488" aria-hidden="true" tabindex="-1"></a><span class="in">  theme(</span></span>
+<span id="cb31-489"><a href="#cb31-489" aria-hidden="true" tabindex="-1"></a><span class="in">    axis.title.x = </span></span>
+<span id="cb31-490"><a href="#cb31-490" aria-hidden="true" tabindex="-1"></a><span class="in">      element_text(</span></span>
+<span id="cb31-491"><a href="#cb31-491" aria-hidden="true" tabindex="-1"></a><span class="in">        angle = 0, </span></span>
+<span id="cb31-492"><a href="#cb31-492" aria-hidden="true" tabindex="-1"></a><span class="in">        vjust = 1, </span></span>
+<span id="cb31-493"><a href="#cb31-493" aria-hidden="true" tabindex="-1"></a><span class="in">        hjust = 1),</span></span>
+<span id="cb31-494"><a href="#cb31-494" aria-hidden="true" tabindex="-1"></a><span class="in">    axis.title.y = </span></span>
+<span id="cb31-495"><a href="#cb31-495" aria-hidden="true" tabindex="-1"></a><span class="in">      element_text(</span></span>
+<span id="cb31-496"><a href="#cb31-496" aria-hidden="true" tabindex="-1"></a><span class="in">        angle = 0, </span></span>
+<span id="cb31-497"><a href="#cb31-497" aria-hidden="true" tabindex="-1"></a><span class="in">        vjust = 1, </span></span>
+<span id="cb31-498"><a href="#cb31-498" aria-hidden="true" tabindex="-1"></a><span class="in">        hjust = 1))</span></span>
+<span id="cb31-499"><a href="#cb31-499" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-500"><a href="#cb31-500" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-501"><a href="#cb31-501" aria-hidden="true" tabindex="-1"></a><span class="fu">### Some Key Mathematical Relationships among Survival Concepts</span></span>
+<span id="cb31-502"><a href="#cb31-502" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-503"><a href="#cb31-503" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Diagram:</span></span>
+<span id="cb31-504"><a href="#cb31-504" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-505"><a href="#cb31-505" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-506"><a href="#cb31-506" aria-hidden="true" tabindex="-1"></a>h(t) \xrightarrow[]{\int_{u=0}^t h(u)du} H(t) </span>
+<span id="cb31-507"><a href="#cb31-507" aria-hidden="true" tabindex="-1"></a>\xrightarrow[]{\exp{-H(t)}} S(t)</span>
+<span id="cb31-508"><a href="#cb31-508" aria-hidden="true" tabindex="-1"></a>\xrightarrow[]{1-S(t)} F(t)</span>
+<span id="cb31-509"><a href="#cb31-509" aria-hidden="true" tabindex="-1"></a>$$</span>
 <span id="cb31-510"><a href="#cb31-510" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-511"><a href="#cb31-511" aria-hidden="true" tabindex="-1"></a>Some proofs (others left as exercises):</span>
-<span id="cb31-512"><a href="#cb31-512" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-513"><a href="#cb31-513" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-514"><a href="#cb31-514" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-515"><a href="#cb31-515" aria-hidden="true" tabindex="-1"></a>S'(t) &amp;= \deriv{t}(1-F(t))<span class="sc">\\</span></span>
-<span id="cb31-516"><a href="#cb31-516" aria-hidden="true" tabindex="-1"></a>&amp;= -F'(t)<span class="sc">\\</span></span>
-<span id="cb31-517"><a href="#cb31-517" aria-hidden="true" tabindex="-1"></a>&amp;= -f(t)<span class="sc">\\</span></span>
-<span id="cb31-518"><a href="#cb31-518" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-519"><a href="#cb31-519" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-520"><a href="#cb31-520" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-521"><a href="#cb31-521" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-522"><a href="#cb31-522" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-523"><a href="#cb31-523" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-524"><a href="#cb31-524" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-525"><a href="#cb31-525" aria-hidden="true" tabindex="-1"></a>\deriv{t}\log{S(t)} </span>
-<span id="cb31-526"><a href="#cb31-526" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{S'(t)}{S(t)}<span class="sc">\\</span></span>
-<span id="cb31-527"><a href="#cb31-527" aria-hidden="true" tabindex="-1"></a>&amp;= -\frac{f(t)}{S(t)}<span class="sc">\\</span></span>
-<span id="cb31-528"><a href="#cb31-528" aria-hidden="true" tabindex="-1"></a>&amp;= -h(t)<span class="sc">\\</span></span>
-<span id="cb31-529"><a href="#cb31-529" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-530"><a href="#cb31-530" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-531"><a href="#cb31-531" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-532"><a href="#cb31-532" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-533"><a href="#cb31-533" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-534"><a href="#cb31-534" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-535"><a href="#cb31-535" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-536"><a href="#cb31-536" aria-hidden="true" tabindex="-1"></a>H(t)</span>
-<span id="cb31-537"><a href="#cb31-537" aria-hidden="true" tabindex="-1"></a>&amp;\eqdef \int_{u=0}^t h(u) du<span class="sc">\\</span></span>
-<span id="cb31-538"><a href="#cb31-538" aria-hidden="true" tabindex="-1"></a>&amp;= \int_0^t -\deriv{u}\text{log}\left<span class="sc">\{</span>S(u)\right<span class="sc">\}</span> du<span class="sc">\\</span></span>
-<span id="cb31-539"><a href="#cb31-539" aria-hidden="true" tabindex="-1"></a>&amp;= \left<span class="co">[</span><span class="ot">-\text{log}\left\{S(u)\right\}\right</span><span class="co">]</span>_{u=0}^{u=t}<span class="sc">\\</span></span>
-<span id="cb31-540"><a href="#cb31-540" aria-hidden="true" tabindex="-1"></a>&amp;= \left<span class="co">[</span><span class="ot">\text{log}\left\{S(u)\right\}\right</span><span class="co">]</span>_{u=t}^{u=0}<span class="sc">\\</span></span>
-<span id="cb31-541"><a href="#cb31-541" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>S(0)\right<span class="sc">\}</span> - \text{log}\left<span class="sc">\{</span>S(t)\right<span class="sc">\}\\</span></span>
-<span id="cb31-542"><a href="#cb31-542" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>1\right<span class="sc">\}</span> - \text{log}\left<span class="sc">\{</span>S(t)\right<span class="sc">\}\\</span></span>
-<span id="cb31-543"><a href="#cb31-543" aria-hidden="true" tabindex="-1"></a>&amp;= 0 - \text{log}\left<span class="sc">\{</span>S(t)\right<span class="sc">\}\\</span></span>
-<span id="cb31-544"><a href="#cb31-544" aria-hidden="true" tabindex="-1"></a>&amp;=-\text{log}\left<span class="sc">\{</span>S(t)\right<span class="sc">\}</span></span>
-<span id="cb31-545"><a href="#cb31-545" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-546"><a href="#cb31-546" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-547"><a href="#cb31-547" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-548"><a href="#cb31-548" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-549"><a href="#cb31-549" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-550"><a href="#cb31-550" aria-hidden="true" tabindex="-1"></a>Corollary:</span>
-<span id="cb31-551"><a href="#cb31-551" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-552"><a href="#cb31-552" aria-hidden="true" tabindex="-1"></a>$$S(t) = \text{exp}\left<span class="sc">\{</span>-H(t)\right<span class="sc">\}</span>$$</span>
-<span id="cb31-553"><a href="#cb31-553" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-554"><a href="#cb31-554" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-555"><a href="#cb31-555" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-556"><a href="#cb31-556" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Example: Time to death the US in 2004</span></span>
+<span id="cb31-511"><a href="#cb31-511" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-512"><a href="#cb31-512" aria-hidden="true" tabindex="-1"></a>h(t) \xleftarrow<span class="co">[</span><span class="ot">\deriv{t}H(t)</span><span class="co">]</span>{} H(t) </span>
+<span id="cb31-513"><a href="#cb31-513" aria-hidden="true" tabindex="-1"></a>\xleftarrow<span class="co">[</span><span class="ot">-\log{S(t)}</span><span class="co">]</span>{} S(t)</span>
+<span id="cb31-514"><a href="#cb31-514" aria-hidden="true" tabindex="-1"></a>\xleftarrow<span class="co">[</span><span class="ot">1-F(t)</span><span class="co">]</span>{} F(t)</span>
+<span id="cb31-515"><a href="#cb31-515" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-516"><a href="#cb31-516" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-517"><a href="#cb31-517" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-518"><a href="#cb31-518" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-519"><a href="#cb31-519" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-520"><a href="#cb31-520" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Identities:</span></span>
+<span id="cb31-521"><a href="#cb31-521" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-522"><a href="#cb31-522" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-523"><a href="#cb31-523" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-524"><a href="#cb31-524" aria-hidden="true" tabindex="-1"></a>S(t) &amp;= 1 - F(t)<span class="sc">\\</span></span>
+<span id="cb31-525"><a href="#cb31-525" aria-hidden="true" tabindex="-1"></a>&amp;= \text{exp}\left<span class="sc">\{</span>-H(t)\right<span class="sc">\}\\</span></span>
+<span id="cb31-526"><a href="#cb31-526" aria-hidden="true" tabindex="-1"></a>S'(t) &amp;= -f(t)<span class="sc">\\</span></span>
+<span id="cb31-527"><a href="#cb31-527" aria-hidden="true" tabindex="-1"></a>H(t) &amp;= -\text{log}\left<span class="sc">\{</span>S(t)\right<span class="sc">\}\\</span></span>
+<span id="cb31-528"><a href="#cb31-528" aria-hidden="true" tabindex="-1"></a>H'(t) &amp;= h(t)<span class="sc">\\</span></span>
+<span id="cb31-529"><a href="#cb31-529" aria-hidden="true" tabindex="-1"></a>h(t) &amp;= \frac{f(t)}{S(t)}<span class="sc">\\</span></span>
+<span id="cb31-530"><a href="#cb31-530" aria-hidden="true" tabindex="-1"></a> &amp;= -\deriv{t}\log{S(t)} <span class="sc">\\</span></span>
+<span id="cb31-531"><a href="#cb31-531" aria-hidden="true" tabindex="-1"></a>f(t) &amp;= h(t)\cdot S(t)<span class="sc">\\</span></span>
+<span id="cb31-532"><a href="#cb31-532" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-533"><a href="#cb31-533" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-534"><a href="#cb31-534" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-535"><a href="#cb31-535" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-536"><a href="#cb31-536" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-537"><a href="#cb31-537" aria-hidden="true" tabindex="-1"></a>Some proofs (others left as exercises):</span>
+<span id="cb31-538"><a href="#cb31-538" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-539"><a href="#cb31-539" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-540"><a href="#cb31-540" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-541"><a href="#cb31-541" aria-hidden="true" tabindex="-1"></a>S'(t) &amp;= \deriv{t}(1-F(t))<span class="sc">\\</span></span>
+<span id="cb31-542"><a href="#cb31-542" aria-hidden="true" tabindex="-1"></a>&amp;= -F'(t)<span class="sc">\\</span></span>
+<span id="cb31-543"><a href="#cb31-543" aria-hidden="true" tabindex="-1"></a>&amp;= -f(t)<span class="sc">\\</span></span>
+<span id="cb31-544"><a href="#cb31-544" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-545"><a href="#cb31-545" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-546"><a href="#cb31-546" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-547"><a href="#cb31-547" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-548"><a href="#cb31-548" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-549"><a href="#cb31-549" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-550"><a href="#cb31-550" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-551"><a href="#cb31-551" aria-hidden="true" tabindex="-1"></a>\deriv{t}\log{S(t)} </span>
+<span id="cb31-552"><a href="#cb31-552" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{S'(t)}{S(t)}<span class="sc">\\</span></span>
+<span id="cb31-553"><a href="#cb31-553" aria-hidden="true" tabindex="-1"></a>&amp;= -\frac{f(t)}{S(t)}<span class="sc">\\</span></span>
+<span id="cb31-554"><a href="#cb31-554" aria-hidden="true" tabindex="-1"></a>&amp;= -h(t)<span class="sc">\\</span></span>
+<span id="cb31-555"><a href="#cb31-555" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-556"><a href="#cb31-556" aria-hidden="true" tabindex="-1"></a>$$</span>
 <span id="cb31-557"><a href="#cb31-557" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-558"><a href="#cb31-558" aria-hidden="true" tabindex="-1"></a>The first day is the most dangerous:</span>
+<span id="cb31-558"><a href="#cb31-558" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-559"><a href="#cb31-559" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-560"><a href="#cb31-560" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, echo = FALSE}</span></span>
-<span id="cb31-561"><a href="#cb31-561" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "Daily Hazard Rates in 2004 for US Females"</span></span>
-<span id="cb31-562"><a href="#cb31-562" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-pos: "H"</span></span>
-<span id="cb31-563"><a href="#cb31-563" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-height: 6</span></span>
-<span id="cb31-564"><a href="#cb31-564" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-565"><a href="#cb31-565" aria-hidden="true" tabindex="-1"></a><span class="in"># download `survexp.rda` from: </span></span>
-<span id="cb31-566"><a href="#cb31-566" aria-hidden="true" tabindex="-1"></a><span class="in"># paste0(</span></span>
-<span id="cb31-567"><a href="#cb31-567" aria-hidden="true" tabindex="-1"></a><span class="in"># "https://github.com/therneau/survival/raw/",</span></span>
-<span id="cb31-568"><a href="#cb31-568" aria-hidden="true" tabindex="-1"></a><span class="in"># "f3ac93704949ff26e07720b56f2b18ffa8066470/",</span></span>
-<span id="cb31-569"><a href="#cb31-569" aria-hidden="true" tabindex="-1"></a><span class="in"># "data/survexp.rda")</span></span>
-<span id="cb31-570"><a href="#cb31-570" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-571"><a href="#cb31-571" aria-hidden="true" tabindex="-1"></a><span class="in">#(newer versions of `survival` don't have the first-year breakdown; see:</span></span>
-<span id="cb31-572"><a href="#cb31-572" aria-hidden="true" tabindex="-1"></a><span class="in"># https://cran.r-project.org/web/packages/survival/news.html)</span></span>
+<span id="cb31-560"><a href="#cb31-560" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-561"><a href="#cb31-561" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-562"><a href="#cb31-562" aria-hidden="true" tabindex="-1"></a>H(t)</span>
+<span id="cb31-563"><a href="#cb31-563" aria-hidden="true" tabindex="-1"></a>&amp;\eqdef \int_{u=0}^t h(u) du<span class="sc">\\</span></span>
+<span id="cb31-564"><a href="#cb31-564" aria-hidden="true" tabindex="-1"></a>&amp;= \int_0^t -\deriv{u}\text{log}\left<span class="sc">\{</span>S(u)\right<span class="sc">\}</span> du<span class="sc">\\</span></span>
+<span id="cb31-565"><a href="#cb31-565" aria-hidden="true" tabindex="-1"></a>&amp;= \left<span class="co">[</span><span class="ot">-\text{log}\left\{S(u)\right\}\right</span><span class="co">]</span>_{u=0}^{u=t}<span class="sc">\\</span></span>
+<span id="cb31-566"><a href="#cb31-566" aria-hidden="true" tabindex="-1"></a>&amp;= \left<span class="co">[</span><span class="ot">\text{log}\left\{S(u)\right\}\right</span><span class="co">]</span>_{u=t}^{u=0}<span class="sc">\\</span></span>
+<span id="cb31-567"><a href="#cb31-567" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>S(0)\right<span class="sc">\}</span> - \text{log}\left<span class="sc">\{</span>S(t)\right<span class="sc">\}\\</span></span>
+<span id="cb31-568"><a href="#cb31-568" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>1\right<span class="sc">\}</span> - \text{log}\left<span class="sc">\{</span>S(t)\right<span class="sc">\}\\</span></span>
+<span id="cb31-569"><a href="#cb31-569" aria-hidden="true" tabindex="-1"></a>&amp;= 0 - \text{log}\left<span class="sc">\{</span>S(t)\right<span class="sc">\}\\</span></span>
+<span id="cb31-570"><a href="#cb31-570" aria-hidden="true" tabindex="-1"></a>&amp;=-\text{log}\left<span class="sc">\{</span>S(t)\right<span class="sc">\}</span></span>
+<span id="cb31-571"><a href="#cb31-571" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-572"><a href="#cb31-572" aria-hidden="true" tabindex="-1"></a>$$</span>
 <span id="cb31-573"><a href="#cb31-573" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-574"><a href="#cb31-574" aria-hidden="true" tabindex="-1"></a><span class="in">fs::path(</span></span>
-<span id="cb31-575"><a href="#cb31-575" aria-hidden="true" tabindex="-1"></a><span class="in">  here::here(),</span></span>
-<span id="cb31-576"><a href="#cb31-576" aria-hidden="true" tabindex="-1"></a><span class="in">  "data",</span></span>
-<span id="cb31-577"><a href="#cb31-577" aria-hidden="true" tabindex="-1"></a><span class="in">  "survexp.rda") |&gt;</span></span>
-<span id="cb31-578"><a href="#cb31-578" aria-hidden="true" tabindex="-1"></a><span class="in">  load()</span></span>
-<span id="cb31-579"><a href="#cb31-579" aria-hidden="true" tabindex="-1"></a><span class="in">s1 &lt;- survexp.us[,"female","2004"]</span></span>
-<span id="cb31-580"><a href="#cb31-580" aria-hidden="true" tabindex="-1"></a><span class="in">age1 &lt;- c(</span></span>
-<span id="cb31-581"><a href="#cb31-581" aria-hidden="true" tabindex="-1"></a><span class="in">  0.5/365.25,</span></span>
-<span id="cb31-582"><a href="#cb31-582" aria-hidden="true" tabindex="-1"></a><span class="in">  4/365.25,</span></span>
-<span id="cb31-583"><a href="#cb31-583" aria-hidden="true" tabindex="-1"></a><span class="in">  17.5/365.25,</span></span>
-<span id="cb31-584"><a href="#cb31-584" aria-hidden="true" tabindex="-1"></a><span class="in">  196.6/365.25,</span></span>
-<span id="cb31-585"><a href="#cb31-585" aria-hidden="true" tabindex="-1"></a><span class="in">  1:109+0.5)</span></span>
-<span id="cb31-586"><a href="#cb31-586" aria-hidden="true" tabindex="-1"></a><span class="in">s2 &lt;- 365.25*s1[5:113]</span></span>
-<span id="cb31-587"><a href="#cb31-587" aria-hidden="true" tabindex="-1"></a><span class="in">s2 &lt;- c(s1[1], 6*s1[2], 22*s1[3], 337.25*s1[4], s2)</span></span>
-<span id="cb31-588"><a href="#cb31-588" aria-hidden="true" tabindex="-1"></a><span class="in">cols &lt;- rep(1,113)</span></span>
-<span id="cb31-589"><a href="#cb31-589" aria-hidden="true" tabindex="-1"></a><span class="in">cols[1] &lt;- 2</span></span>
-<span id="cb31-590"><a href="#cb31-590" aria-hidden="true" tabindex="-1"></a><span class="in">cols[2] &lt;- 3</span></span>
-<span id="cb31-591"><a href="#cb31-591" aria-hidden="true" tabindex="-1"></a><span class="in">cols[3] &lt;- 4</span></span>
-<span id="cb31-592"><a href="#cb31-592" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-593"><a href="#cb31-593" aria-hidden="true" tabindex="-1"></a><span class="in">plot(age1,s1,type="b",lwd=2,xlab="Age",ylab="Daily Hazard Rate",col=cols)</span></span>
-<span id="cb31-594"><a href="#cb31-594" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-595"><a href="#cb31-595" aria-hidden="true" tabindex="-1"></a><span class="in">text(10,.003,"First Day",col=2)</span></span>
-<span id="cb31-596"><a href="#cb31-596" aria-hidden="true" tabindex="-1"></a><span class="in">text(18,.00030,"Rest of First Week",col=3)</span></span>
-<span id="cb31-597"><a href="#cb31-597" aria-hidden="true" tabindex="-1"></a><span class="in">text(18,.00015,"Rest of First month",col=4)</span></span>
-<span id="cb31-598"><a href="#cb31-598" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-574"><a href="#cb31-574" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-575"><a href="#cb31-575" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-576"><a href="#cb31-576" aria-hidden="true" tabindex="-1"></a>Corollary:</span>
+<span id="cb31-577"><a href="#cb31-577" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-578"><a href="#cb31-578" aria-hidden="true" tabindex="-1"></a>$$S(t) = \text{exp}\left<span class="sc">\{</span>-H(t)\right<span class="sc">\}</span>$$</span>
+<span id="cb31-579"><a href="#cb31-579" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-580"><a href="#cb31-580" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-581"><a href="#cb31-581" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-582"><a href="#cb31-582" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Example: Time to death the US in 2004</span></span>
+<span id="cb31-583"><a href="#cb31-583" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-584"><a href="#cb31-584" aria-hidden="true" tabindex="-1"></a>The first day is the most dangerous:</span>
+<span id="cb31-585"><a href="#cb31-585" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-586"><a href="#cb31-586" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, echo = FALSE}</span></span>
+<span id="cb31-587"><a href="#cb31-587" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "Daily Hazard Rates in 2004 for US Females"</span></span>
+<span id="cb31-588"><a href="#cb31-588" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-pos: "H"</span></span>
+<span id="cb31-589"><a href="#cb31-589" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-height: 6</span></span>
+<span id="cb31-590"><a href="#cb31-590" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-591"><a href="#cb31-591" aria-hidden="true" tabindex="-1"></a><span class="in"># download `survexp.rda` from: </span></span>
+<span id="cb31-592"><a href="#cb31-592" aria-hidden="true" tabindex="-1"></a><span class="in"># paste0(</span></span>
+<span id="cb31-593"><a href="#cb31-593" aria-hidden="true" tabindex="-1"></a><span class="in"># "https://github.com/therneau/survival/raw/",</span></span>
+<span id="cb31-594"><a href="#cb31-594" aria-hidden="true" tabindex="-1"></a><span class="in"># "f3ac93704949ff26e07720b56f2b18ffa8066470/",</span></span>
+<span id="cb31-595"><a href="#cb31-595" aria-hidden="true" tabindex="-1"></a><span class="in"># "data/survexp.rda")</span></span>
+<span id="cb31-596"><a href="#cb31-596" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-597"><a href="#cb31-597" aria-hidden="true" tabindex="-1"></a><span class="in">#(newer versions of `survival` don't have the first-year breakdown; see:</span></span>
+<span id="cb31-598"><a href="#cb31-598" aria-hidden="true" tabindex="-1"></a><span class="in"># https://cran.r-project.org/web/packages/survival/news.html)</span></span>
 <span id="cb31-599"><a href="#cb31-599" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-600"><a href="#cb31-600" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-601"><a href="#cb31-601" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-602"><a href="#cb31-602" aria-hidden="true" tabindex="-1"></a>Exercise: hypothesize why these curves differ where they do?</span>
-<span id="cb31-603"><a href="#cb31-603" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-604"><a href="#cb31-604" aria-hidden="true" tabindex="-1"></a><span class="in">```{r,echo = FALSE}</span></span>
-<span id="cb31-605"><a href="#cb31-605" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "Daily Hazard Rates in 2004 for US Males and Females 1-40"</span></span>
-<span id="cb31-606"><a href="#cb31-606" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-pos: "H"</span></span>
-<span id="cb31-607"><a href="#cb31-607" aria-hidden="true" tabindex="-1"></a><span class="in">yrs=1:40</span></span>
-<span id="cb31-608"><a href="#cb31-608" aria-hidden="true" tabindex="-1"></a><span class="in">s1 &lt;- survexp.us[5:113,"male","2004"]</span></span>
-<span id="cb31-609"><a href="#cb31-609" aria-hidden="true" tabindex="-1"></a><span class="in">s2 &lt;- survexp.us[5:113,"female","2004"]</span></span>
-<span id="cb31-610"><a href="#cb31-610" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-611"><a href="#cb31-611" aria-hidden="true" tabindex="-1"></a><span class="in">age1 &lt;- 1:109</span></span>
-<span id="cb31-612"><a href="#cb31-612" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-613"><a href="#cb31-613" aria-hidden="true" tabindex="-1"></a><span class="in">plot(age1[yrs],s1[yrs],type="l",lwd=2,xlab="Age",ylab="Daily Hazard Rate")</span></span>
-<span id="cb31-614"><a href="#cb31-614" aria-hidden="true" tabindex="-1"></a><span class="in">lines(age1[yrs],s2[yrs],col=2,lwd=2)</span></span>
-<span id="cb31-615"><a href="#cb31-615" aria-hidden="true" tabindex="-1"></a><span class="in">legend(5,5e-6,c("Males","Females"),col=1:2,lwd=2)</span></span>
-<span id="cb31-616"><a href="#cb31-616" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-617"><a href="#cb31-617" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-600"><a href="#cb31-600" aria-hidden="true" tabindex="-1"></a><span class="in">fs::path(</span></span>
+<span id="cb31-601"><a href="#cb31-601" aria-hidden="true" tabindex="-1"></a><span class="in">  here::here(),</span></span>
+<span id="cb31-602"><a href="#cb31-602" aria-hidden="true" tabindex="-1"></a><span class="in">  "data",</span></span>
+<span id="cb31-603"><a href="#cb31-603" aria-hidden="true" tabindex="-1"></a><span class="in">  "survexp.rda") |&gt;</span></span>
+<span id="cb31-604"><a href="#cb31-604" aria-hidden="true" tabindex="-1"></a><span class="in">  load()</span></span>
+<span id="cb31-605"><a href="#cb31-605" aria-hidden="true" tabindex="-1"></a><span class="in">s1 &lt;- survexp.us[,"female","2004"]</span></span>
+<span id="cb31-606"><a href="#cb31-606" aria-hidden="true" tabindex="-1"></a><span class="in">age1 &lt;- c(</span></span>
+<span id="cb31-607"><a href="#cb31-607" aria-hidden="true" tabindex="-1"></a><span class="in">  0.5/365.25,</span></span>
+<span id="cb31-608"><a href="#cb31-608" aria-hidden="true" tabindex="-1"></a><span class="in">  4/365.25,</span></span>
+<span id="cb31-609"><a href="#cb31-609" aria-hidden="true" tabindex="-1"></a><span class="in">  17.5/365.25,</span></span>
+<span id="cb31-610"><a href="#cb31-610" aria-hidden="true" tabindex="-1"></a><span class="in">  196.6/365.25,</span></span>
+<span id="cb31-611"><a href="#cb31-611" aria-hidden="true" tabindex="-1"></a><span class="in">  1:109+0.5)</span></span>
+<span id="cb31-612"><a href="#cb31-612" aria-hidden="true" tabindex="-1"></a><span class="in">s2 &lt;- 365.25*s1[5:113]</span></span>
+<span id="cb31-613"><a href="#cb31-613" aria-hidden="true" tabindex="-1"></a><span class="in">s2 &lt;- c(s1[1], 6*s1[2], 22*s1[3], 337.25*s1[4], s2)</span></span>
+<span id="cb31-614"><a href="#cb31-614" aria-hidden="true" tabindex="-1"></a><span class="in">cols &lt;- rep(1,113)</span></span>
+<span id="cb31-615"><a href="#cb31-615" aria-hidden="true" tabindex="-1"></a><span class="in">cols[1] &lt;- 2</span></span>
+<span id="cb31-616"><a href="#cb31-616" aria-hidden="true" tabindex="-1"></a><span class="in">cols[2] &lt;- 3</span></span>
+<span id="cb31-617"><a href="#cb31-617" aria-hidden="true" tabindex="-1"></a><span class="in">cols[3] &lt;- 4</span></span>
 <span id="cb31-618"><a href="#cb31-618" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-619"><a href="#cb31-619" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-619"><a href="#cb31-619" aria-hidden="true" tabindex="-1"></a><span class="in">plot(age1,s1,type="b",lwd=2,xlab="Age",ylab="Daily Hazard Rate",col=cols)</span></span>
 <span id="cb31-620"><a href="#cb31-620" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-621"><a href="#cb31-621" aria-hidden="true" tabindex="-1"></a>Exercise: compare and contrast this curve with the corresponding hazard</span>
-<span id="cb31-622"><a href="#cb31-622" aria-hidden="true" tabindex="-1"></a>curve.</span>
-<span id="cb31-623"><a href="#cb31-623" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-624"><a href="#cb31-624" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, echo = FALSE}</span></span>
-<span id="cb31-625"><a href="#cb31-625" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "Survival Curve in 2004 for US Females"</span></span>
-<span id="cb31-626"><a href="#cb31-626" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-pos: "H"</span></span>
+<span id="cb31-621"><a href="#cb31-621" aria-hidden="true" tabindex="-1"></a><span class="in">text(10,.003,"First Day",col=2)</span></span>
+<span id="cb31-622"><a href="#cb31-622" aria-hidden="true" tabindex="-1"></a><span class="in">text(18,.00030,"Rest of First Week",col=3)</span></span>
+<span id="cb31-623"><a href="#cb31-623" aria-hidden="true" tabindex="-1"></a><span class="in">text(18,.00015,"Rest of First month",col=4)</span></span>
+<span id="cb31-624"><a href="#cb31-624" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-625"><a href="#cb31-625" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-626"><a href="#cb31-626" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-627"><a href="#cb31-627" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-628"><a href="#cb31-628" aria-hidden="true" tabindex="-1"></a><span class="in">s1 &lt;- survexp.us[,"female","2004"]</span></span>
+<span id="cb31-628"><a href="#cb31-628" aria-hidden="true" tabindex="-1"></a>Exercise: hypothesize why these curves differ where they do?</span>
 <span id="cb31-629"><a href="#cb31-629" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-630"><a href="#cb31-630" aria-hidden="true" tabindex="-1"></a><span class="in">s2 &lt;- 365.25*s1[5:113]</span></span>
-<span id="cb31-631"><a href="#cb31-631" aria-hidden="true" tabindex="-1"></a><span class="in">s2 &lt;- c(s1[1], 6*s1[2], 21*s1[3], 337.25*s1[4], s2)</span></span>
-<span id="cb31-632"><a href="#cb31-632" aria-hidden="true" tabindex="-1"></a><span class="in">cs2 &lt;- cumsum(s2)</span></span>
-<span id="cb31-633"><a href="#cb31-633" aria-hidden="true" tabindex="-1"></a><span class="in">age2 &lt;- c(1/365.25, 7/365.25, 28/365.25, 1:110)</span></span>
-<span id="cb31-634"><a href="#cb31-634" aria-hidden="true" tabindex="-1"></a><span class="in">plot(age2,exp(-cs2),type="l",lwd=2,xlab="Age",ylab="Survival")</span></span>
-<span id="cb31-635"><a href="#cb31-635" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-636"><a href="#cb31-636" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-637"><a href="#cb31-637" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-638"><a href="#cb31-638" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-639"><a href="#cb31-639" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-640"><a href="#cb31-640" aria-hidden="true" tabindex="-1"></a><span class="fu">### Likelihood with censoring</span></span>
-<span id="cb31-641"><a href="#cb31-641" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-642"><a href="#cb31-642" aria-hidden="true" tabindex="-1"></a>If an event time $T$ is observed exactly as $T=t$, then the likelihood</span>
-<span id="cb31-643"><a href="#cb31-643" aria-hidden="true" tabindex="-1"></a>of that observation is just its probability density function:</span>
+<span id="cb31-630"><a href="#cb31-630" aria-hidden="true" tabindex="-1"></a><span class="in">```{r,echo = FALSE}</span></span>
+<span id="cb31-631"><a href="#cb31-631" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "Daily Hazard Rates in 2004 for US Males and Females 1-40"</span></span>
+<span id="cb31-632"><a href="#cb31-632" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-pos: "H"</span></span>
+<span id="cb31-633"><a href="#cb31-633" aria-hidden="true" tabindex="-1"></a><span class="in">yrs=1:40</span></span>
+<span id="cb31-634"><a href="#cb31-634" aria-hidden="true" tabindex="-1"></a><span class="in">s1 &lt;- survexp.us[5:113,"male","2004"]</span></span>
+<span id="cb31-635"><a href="#cb31-635" aria-hidden="true" tabindex="-1"></a><span class="in">s2 &lt;- survexp.us[5:113,"female","2004"]</span></span>
+<span id="cb31-636"><a href="#cb31-636" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-637"><a href="#cb31-637" aria-hidden="true" tabindex="-1"></a><span class="in">age1 &lt;- 1:109</span></span>
+<span id="cb31-638"><a href="#cb31-638" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-639"><a href="#cb31-639" aria-hidden="true" tabindex="-1"></a><span class="in">plot(age1[yrs],s1[yrs],type="l",lwd=2,xlab="Age",ylab="Daily Hazard Rate")</span></span>
+<span id="cb31-640"><a href="#cb31-640" aria-hidden="true" tabindex="-1"></a><span class="in">lines(age1[yrs],s2[yrs],col=2,lwd=2)</span></span>
+<span id="cb31-641"><a href="#cb31-641" aria-hidden="true" tabindex="-1"></a><span class="in">legend(5,5e-6,c("Males","Females"),col=1:2,lwd=2)</span></span>
+<span id="cb31-642"><a href="#cb31-642" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-643"><a href="#cb31-643" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
 <span id="cb31-644"><a href="#cb31-644" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-645"><a href="#cb31-645" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-646"><a href="#cb31-646" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-647"><a href="#cb31-647" aria-hidden="true" tabindex="-1"></a>\mathcal L(t) </span>
-<span id="cb31-648"><a href="#cb31-648" aria-hidden="true" tabindex="-1"></a>&amp;= p(T=t)<span class="sc">\\</span> </span>
-<span id="cb31-649"><a href="#cb31-649" aria-hidden="true" tabindex="-1"></a>&amp;\eqdef  f_T(t)<span class="sc">\\</span></span>
-<span id="cb31-650"><a href="#cb31-650" aria-hidden="true" tabindex="-1"></a>&amp;= h_T(t)S_T(t)<span class="sc">\\</span></span>
-<span id="cb31-651"><a href="#cb31-651" aria-hidden="true" tabindex="-1"></a>\ell(t) </span>
-<span id="cb31-652"><a href="#cb31-652" aria-hidden="true" tabindex="-1"></a>&amp;\eqdef \text{log}\left<span class="sc">\{</span>\mathcal L(t)\right<span class="sc">\}\\</span></span>
-<span id="cb31-653"><a href="#cb31-653" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>h_T(t)S_T(t)\right<span class="sc">\}\\</span></span>
-<span id="cb31-654"><a href="#cb31-654" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>h_T(t)\right<span class="sc">\}</span> + \text{log}\left<span class="sc">\{</span>S_T(t)\right<span class="sc">\}\\</span></span>
-<span id="cb31-655"><a href="#cb31-655" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>h_T(t)\right<span class="sc">\}</span> - H_T(t)<span class="sc">\\</span></span>
-<span id="cb31-656"><a href="#cb31-656" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-657"><a href="#cb31-657" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-658"><a href="#cb31-658" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-659"><a href="#cb31-659" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-660"><a href="#cb31-660" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-661"><a href="#cb31-661" aria-hidden="true" tabindex="-1"></a>If instead the event time $T$ is censored and only known to be after</span>
-<span id="cb31-662"><a href="#cb31-662" aria-hidden="true" tabindex="-1"></a>time $y$, then the likelihood of that censored observation is instead</span>
-<span id="cb31-663"><a href="#cb31-663" aria-hidden="true" tabindex="-1"></a>the survival function evaluated at the censoring time:</span>
-<span id="cb31-664"><a href="#cb31-664" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-665"><a href="#cb31-665" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-666"><a href="#cb31-666" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-667"><a href="#cb31-667" aria-hidden="true" tabindex="-1"></a>\mathcal L(y) </span>
-<span id="cb31-668"><a href="#cb31-668" aria-hidden="true" tabindex="-1"></a>&amp;=p_T(T&gt;y)<span class="sc">\\</span></span>
-<span id="cb31-669"><a href="#cb31-669" aria-hidden="true" tabindex="-1"></a>&amp;\eqdef  S_T(y)<span class="sc">\\</span></span>
-<span id="cb31-670"><a href="#cb31-670" aria-hidden="true" tabindex="-1"></a>\ell(y)</span>
-<span id="cb31-671"><a href="#cb31-671" aria-hidden="true" tabindex="-1"></a>&amp;\eqdef \text{log}\left<span class="sc">\{</span>\mathcal L(y)\right<span class="sc">\}\\</span></span>
-<span id="cb31-672"><a href="#cb31-672" aria-hidden="true" tabindex="-1"></a>&amp;=\text{log}\left<span class="sc">\{</span>S(y)\right<span class="sc">\}\\</span></span>
-<span id="cb31-673"><a href="#cb31-673" aria-hidden="true" tabindex="-1"></a>&amp;=-H(y)<span class="sc">\\</span></span>
-<span id="cb31-674"><a href="#cb31-674" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-675"><a href="#cb31-675" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-676"><a href="#cb31-676" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-677"><a href="#cb31-677" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-678"><a href="#cb31-678" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-679"><a href="#cb31-679" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-680"><a href="#cb31-680" aria-hidden="true" tabindex="-1"></a>What's written above is incomplete. We also observed whether or not the</span>
-<span id="cb31-681"><a href="#cb31-681" aria-hidden="true" tabindex="-1"></a>observation was censored. Let $C$ denote the time when censoring would</span>
-<span id="cb31-682"><a href="#cb31-682" aria-hidden="true" tabindex="-1"></a>occur (if the event did not occur first); let $f_C(y)$ and $S_C(y)$ be</span>
-<span id="cb31-683"><a href="#cb31-683" aria-hidden="true" tabindex="-1"></a>the corresponding density and survival functions for the censoring</span>
-<span id="cb31-684"><a href="#cb31-684" aria-hidden="true" tabindex="-1"></a>event.</span>
-<span id="cb31-685"><a href="#cb31-685" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-686"><a href="#cb31-686" aria-hidden="true" tabindex="-1"></a>Let $Y$ denote the time when observation ended (either by censoring or</span>
-<span id="cb31-687"><a href="#cb31-687" aria-hidden="true" tabindex="-1"></a>by the event of interest occurring), and let $D$ be an indicator</span>
-<span id="cb31-688"><a href="#cb31-688" aria-hidden="true" tabindex="-1"></a>variable for the event occurring at $Y$ (so $D=0$ represents a censored</span>
-<span id="cb31-689"><a href="#cb31-689" aria-hidden="true" tabindex="-1"></a>observation and $D=1$ represents an uncensored observation). In other</span>
-<span id="cb31-690"><a href="#cb31-690" aria-hidden="true" tabindex="-1"></a>words, let $Y \eqdef  \min(T,C)$ and</span>
-<span id="cb31-691"><a href="#cb31-691" aria-hidden="true" tabindex="-1"></a>$D \eqdef  \mathbb 1{<span class="sc">\{</span>T&lt;=C<span class="sc">\}</span>}$.</span>
-<span id="cb31-692"><a href="#cb31-692" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-693"><a href="#cb31-693" aria-hidden="true" tabindex="-1"></a>Then the complete likelihood of the observed data $(Y,D)$ is:</span>
-<span id="cb31-694"><a href="#cb31-694" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-695"><a href="#cb31-695" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-696"><a href="#cb31-696" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-697"><a href="#cb31-697" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-698"><a href="#cb31-698" aria-hidden="true" tabindex="-1"></a>\mathcal L(y,d) </span>
-<span id="cb31-699"><a href="#cb31-699" aria-hidden="true" tabindex="-1"></a>&amp;= p(Y=y, D=d)<span class="sc">\\</span></span>
-<span id="cb31-700"><a href="#cb31-700" aria-hidden="true" tabindex="-1"></a>&amp;= \left<span class="co">[</span><span class="ot">p(T=y,C&gt; y)\right</span><span class="co">]</span>^d \cdot </span>
-<span id="cb31-701"><a href="#cb31-701" aria-hidden="true" tabindex="-1"></a>\left<span class="co">[</span><span class="ot">p(T&gt;y,C=y)\right</span><span class="co">]</span>^{1-d}<span class="sc">\\</span></span>
-<span id="cb31-702"><a href="#cb31-702" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-703"><a href="#cb31-703" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-645"><a href="#cb31-645" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-646"><a href="#cb31-646" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-647"><a href="#cb31-647" aria-hidden="true" tabindex="-1"></a>Exercise: compare and contrast this curve with the corresponding hazard</span>
+<span id="cb31-648"><a href="#cb31-648" aria-hidden="true" tabindex="-1"></a>curve.</span>
+<span id="cb31-649"><a href="#cb31-649" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-650"><a href="#cb31-650" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, echo = FALSE}</span></span>
+<span id="cb31-651"><a href="#cb31-651" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "Survival Curve in 2004 for US Females"</span></span>
+<span id="cb31-652"><a href="#cb31-652" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-pos: "H"</span></span>
+<span id="cb31-653"><a href="#cb31-653" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-654"><a href="#cb31-654" aria-hidden="true" tabindex="-1"></a><span class="in">s1 &lt;- survexp.us[,"female","2004"]</span></span>
+<span id="cb31-655"><a href="#cb31-655" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-656"><a href="#cb31-656" aria-hidden="true" tabindex="-1"></a><span class="in">s2 &lt;- 365.25*s1[5:113]</span></span>
+<span id="cb31-657"><a href="#cb31-657" aria-hidden="true" tabindex="-1"></a><span class="in">s2 &lt;- c(s1[1], 6*s1[2], 21*s1[3], 337.25*s1[4], s2)</span></span>
+<span id="cb31-658"><a href="#cb31-658" aria-hidden="true" tabindex="-1"></a><span class="in">cs2 &lt;- cumsum(s2)</span></span>
+<span id="cb31-659"><a href="#cb31-659" aria-hidden="true" tabindex="-1"></a><span class="in">age2 &lt;- c(1/365.25, 7/365.25, 28/365.25, 1:110)</span></span>
+<span id="cb31-660"><a href="#cb31-660" aria-hidden="true" tabindex="-1"></a><span class="in">plot(age2,exp(-cs2),type="l",lwd=2,xlab="Age",ylab="Survival")</span></span>
+<span id="cb31-661"><a href="#cb31-661" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-662"><a href="#cb31-662" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-663"><a href="#cb31-663" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-664"><a href="#cb31-664" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-665"><a href="#cb31-665" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-666"><a href="#cb31-666" aria-hidden="true" tabindex="-1"></a><span class="fu">### Likelihood with censoring</span></span>
+<span id="cb31-667"><a href="#cb31-667" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-668"><a href="#cb31-668" aria-hidden="true" tabindex="-1"></a>If an event time $T$ is observed exactly as $T=t$, then the likelihood</span>
+<span id="cb31-669"><a href="#cb31-669" aria-hidden="true" tabindex="-1"></a>of that observation is just its probability density function:</span>
+<span id="cb31-670"><a href="#cb31-670" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-671"><a href="#cb31-671" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-672"><a href="#cb31-672" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-673"><a href="#cb31-673" aria-hidden="true" tabindex="-1"></a>\mathcal L(t) </span>
+<span id="cb31-674"><a href="#cb31-674" aria-hidden="true" tabindex="-1"></a>&amp;= p(T=t)<span class="sc">\\</span> </span>
+<span id="cb31-675"><a href="#cb31-675" aria-hidden="true" tabindex="-1"></a>&amp;\eqdef  f_T(t)<span class="sc">\\</span></span>
+<span id="cb31-676"><a href="#cb31-676" aria-hidden="true" tabindex="-1"></a>&amp;= h_T(t)S_T(t)<span class="sc">\\</span></span>
+<span id="cb31-677"><a href="#cb31-677" aria-hidden="true" tabindex="-1"></a>\ell(t) </span>
+<span id="cb31-678"><a href="#cb31-678" aria-hidden="true" tabindex="-1"></a>&amp;\eqdef \text{log}\left<span class="sc">\{</span>\mathcal L(t)\right<span class="sc">\}\\</span></span>
+<span id="cb31-679"><a href="#cb31-679" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>h_T(t)S_T(t)\right<span class="sc">\}\\</span></span>
+<span id="cb31-680"><a href="#cb31-680" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>h_T(t)\right<span class="sc">\}</span> + \text{log}\left<span class="sc">\{</span>S_T(t)\right<span class="sc">\}\\</span></span>
+<span id="cb31-681"><a href="#cb31-681" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>h_T(t)\right<span class="sc">\}</span> - H_T(t)<span class="sc">\\</span></span>
+<span id="cb31-682"><a href="#cb31-682" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-683"><a href="#cb31-683" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-684"><a href="#cb31-684" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-685"><a href="#cb31-685" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-686"><a href="#cb31-686" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-687"><a href="#cb31-687" aria-hidden="true" tabindex="-1"></a>If instead the event time $T$ is censored and only known to be after</span>
+<span id="cb31-688"><a href="#cb31-688" aria-hidden="true" tabindex="-1"></a>time $y$, then the likelihood of that censored observation is instead</span>
+<span id="cb31-689"><a href="#cb31-689" aria-hidden="true" tabindex="-1"></a>the survival function evaluated at the censoring time:</span>
+<span id="cb31-690"><a href="#cb31-690" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-691"><a href="#cb31-691" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-692"><a href="#cb31-692" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-693"><a href="#cb31-693" aria-hidden="true" tabindex="-1"></a>\mathcal L(y) </span>
+<span id="cb31-694"><a href="#cb31-694" aria-hidden="true" tabindex="-1"></a>&amp;=p_T(T&gt;y)<span class="sc">\\</span></span>
+<span id="cb31-695"><a href="#cb31-695" aria-hidden="true" tabindex="-1"></a>&amp;\eqdef  S_T(y)<span class="sc">\\</span></span>
+<span id="cb31-696"><a href="#cb31-696" aria-hidden="true" tabindex="-1"></a>\ell(y)</span>
+<span id="cb31-697"><a href="#cb31-697" aria-hidden="true" tabindex="-1"></a>&amp;\eqdef \text{log}\left<span class="sc">\{</span>\mathcal L(y)\right<span class="sc">\}\\</span></span>
+<span id="cb31-698"><a href="#cb31-698" aria-hidden="true" tabindex="-1"></a>&amp;=\text{log}\left<span class="sc">\{</span>S(y)\right<span class="sc">\}\\</span></span>
+<span id="cb31-699"><a href="#cb31-699" aria-hidden="true" tabindex="-1"></a>&amp;=-H(y)<span class="sc">\\</span></span>
+<span id="cb31-700"><a href="#cb31-700" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-701"><a href="#cb31-701" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-702"><a href="#cb31-702" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-703"><a href="#cb31-703" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-704"><a href="#cb31-704" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-705"><a href="#cb31-705" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-706"><a href="#cb31-706" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-707"><a href="#cb31-707" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-708"><a href="#cb31-708" aria-hidden="true" tabindex="-1"></a>Typically, survival analyses assume that $C$ and $T$ are mutually</span>
-<span id="cb31-709"><a href="#cb31-709" aria-hidden="true" tabindex="-1"></a>independent; this assumption is called "non-informative" censoring.</span>
-<span id="cb31-710"><a href="#cb31-710" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-711"><a href="#cb31-711" aria-hidden="true" tabindex="-1"></a>Then the joint likelihood $p(Y,D)$ factors into the product</span>
-<span id="cb31-712"><a href="#cb31-712" aria-hidden="true" tabindex="-1"></a>$p(Y), p(D)$, and the likelihood reduces to:</span>
-<span id="cb31-713"><a href="#cb31-713" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-714"><a href="#cb31-714" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-715"><a href="#cb31-715" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-716"><a href="#cb31-716" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-717"><a href="#cb31-717" aria-hidden="true" tabindex="-1"></a>\mathcal L(y,d) </span>
-<span id="cb31-718"><a href="#cb31-718" aria-hidden="true" tabindex="-1"></a>&amp;= \left<span class="co">[</span><span class="ot">p(T=y,C&gt; y)\right</span><span class="co">]</span>^d\cdot </span>
-<span id="cb31-719"><a href="#cb31-719" aria-hidden="true" tabindex="-1"></a>\left<span class="co">[</span><span class="ot">p(T&gt;y,C=y)\right</span><span class="co">]</span>^{1-d}<span class="sc">\\</span></span>
-<span id="cb31-720"><a href="#cb31-720" aria-hidden="true" tabindex="-1"></a>&amp;= \left<span class="co">[</span><span class="ot">p(T=y)p(C&gt; y)\right</span><span class="co">]</span>^d\cdot </span>
-<span id="cb31-721"><a href="#cb31-721" aria-hidden="true" tabindex="-1"></a>\left<span class="co">[</span><span class="ot">p(T&gt;y)p(C=y)\right</span><span class="co">]</span>^{1-d}<span class="sc">\\</span></span>
-<span id="cb31-722"><a href="#cb31-722" aria-hidden="true" tabindex="-1"></a>&amp;= \left<span class="co">[</span><span class="ot">f_T(y)S_C(y)\right</span><span class="co">]</span>^d\cdot </span>
-<span id="cb31-723"><a href="#cb31-723" aria-hidden="true" tabindex="-1"></a>\left<span class="co">[</span><span class="ot">S(y)f_C(y)\right</span><span class="co">]</span>^{1-d}<span class="sc">\\</span></span>
-<span id="cb31-724"><a href="#cb31-724" aria-hidden="true" tabindex="-1"></a>&amp;= \left<span class="co">[</span><span class="ot">f_T(y)^d S_C(y)^d\right</span><span class="co">]</span>\cdot </span>
-<span id="cb31-725"><a href="#cb31-725" aria-hidden="true" tabindex="-1"></a>\left<span class="co">[</span><span class="ot">S_T(y)^{1-d}f_C(y)^{1-d}\right</span><span class="co">]</span><span class="sc">\\</span></span>
-<span id="cb31-726"><a href="#cb31-726" aria-hidden="true" tabindex="-1"></a>&amp;= \left(f_T(y)^d \cdot S_T(y)^{1-d}\right)\cdot </span>
-<span id="cb31-727"><a href="#cb31-727" aria-hidden="true" tabindex="-1"></a>\left(f_C(y)^{1-d} \cdot S_C(y)^{d}\right)</span>
+<span id="cb31-705"><a href="#cb31-705" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-706"><a href="#cb31-706" aria-hidden="true" tabindex="-1"></a>What's written above is incomplete. We also observed whether or not the</span>
+<span id="cb31-707"><a href="#cb31-707" aria-hidden="true" tabindex="-1"></a>observation was censored. Let $C$ denote the time when censoring would</span>
+<span id="cb31-708"><a href="#cb31-708" aria-hidden="true" tabindex="-1"></a>occur (if the event did not occur first); let $f_C(y)$ and $S_C(y)$ be</span>
+<span id="cb31-709"><a href="#cb31-709" aria-hidden="true" tabindex="-1"></a>the corresponding density and survival functions for the censoring</span>
+<span id="cb31-710"><a href="#cb31-710" aria-hidden="true" tabindex="-1"></a>event.</span>
+<span id="cb31-711"><a href="#cb31-711" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-712"><a href="#cb31-712" aria-hidden="true" tabindex="-1"></a>Let $Y$ denote the time when observation ended (either by censoring or</span>
+<span id="cb31-713"><a href="#cb31-713" aria-hidden="true" tabindex="-1"></a>by the event of interest occurring), and let $D$ be an indicator</span>
+<span id="cb31-714"><a href="#cb31-714" aria-hidden="true" tabindex="-1"></a>variable for the event occurring at $Y$ (so $D=0$ represents a censored</span>
+<span id="cb31-715"><a href="#cb31-715" aria-hidden="true" tabindex="-1"></a>observation and $D=1$ represents an uncensored observation). In other</span>
+<span id="cb31-716"><a href="#cb31-716" aria-hidden="true" tabindex="-1"></a>words, let $Y \eqdef  \min(T,C)$ and</span>
+<span id="cb31-717"><a href="#cb31-717" aria-hidden="true" tabindex="-1"></a>$D \eqdef  \mathbb 1{<span class="sc">\{</span>T&lt;=C<span class="sc">\}</span>}$.</span>
+<span id="cb31-718"><a href="#cb31-718" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-719"><a href="#cb31-719" aria-hidden="true" tabindex="-1"></a>Then the complete likelihood of the observed data $(Y,D)$ is:</span>
+<span id="cb31-720"><a href="#cb31-720" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-721"><a href="#cb31-721" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-722"><a href="#cb31-722" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-723"><a href="#cb31-723" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-724"><a href="#cb31-724" aria-hidden="true" tabindex="-1"></a>\mathcal L(y,d) </span>
+<span id="cb31-725"><a href="#cb31-725" aria-hidden="true" tabindex="-1"></a>&amp;= p(Y=y, D=d)<span class="sc">\\</span></span>
+<span id="cb31-726"><a href="#cb31-726" aria-hidden="true" tabindex="-1"></a>&amp;= \left<span class="co">[</span><span class="ot">p(T=y,C&gt; y)\right</span><span class="co">]</span>^d \cdot </span>
+<span id="cb31-727"><a href="#cb31-727" aria-hidden="true" tabindex="-1"></a>\left<span class="co">[</span><span class="ot">p(T&gt;y,C=y)\right</span><span class="co">]</span>^{1-d}<span class="sc">\\</span></span>
 <span id="cb31-728"><a href="#cb31-728" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
 <span id="cb31-729"><a href="#cb31-729" aria-hidden="true" tabindex="-1"></a>$$</span>
 <span id="cb31-730"><a href="#cb31-730" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb31-731"><a href="#cb31-731" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-732"><a href="#cb31-732" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb31-733"><a href="#cb31-733" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-734"><a href="#cb31-734" aria-hidden="true" tabindex="-1"></a>The corresponding log-likelihood is:</span>
-<span id="cb31-735"><a href="#cb31-735" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-734"><a href="#cb31-734" aria-hidden="true" tabindex="-1"></a>Typically, survival analyses assume that $C$ and $T$ are mutually</span>
+<span id="cb31-735"><a href="#cb31-735" aria-hidden="true" tabindex="-1"></a>independent; this assumption is called "non-informative" censoring.</span>
 <span id="cb31-736"><a href="#cb31-736" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-737"><a href="#cb31-737" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-738"><a href="#cb31-738" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-739"><a href="#cb31-739" aria-hidden="true" tabindex="-1"></a>\ell(y,d) </span>
-<span id="cb31-740"><a href="#cb31-740" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>\mathcal L(y,d) \right<span class="sc">\}\\</span></span>
-<span id="cb31-741"><a href="#cb31-741" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span></span>
-<span id="cb31-742"><a href="#cb31-742" aria-hidden="true" tabindex="-1"></a>\left(f_T(y)^d \cdot S_T(y)^{1-d}\right)\cdot </span>
-<span id="cb31-743"><a href="#cb31-743" aria-hidden="true" tabindex="-1"></a>\left(f_C(y)^{1-d} \cdot S_C(y)^{d}\right)</span>
-<span id="cb31-744"><a href="#cb31-744" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}\\</span></span>
-<span id="cb31-745"><a href="#cb31-745" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span></span>
-<span id="cb31-746"><a href="#cb31-746" aria-hidden="true" tabindex="-1"></a>f_T(y)^d \cdot S_T(y)^{1-d}</span>
-<span id="cb31-747"><a href="#cb31-747" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}</span> </span>
-<span id="cb31-748"><a href="#cb31-748" aria-hidden="true" tabindex="-1"></a>+</span>
-<span id="cb31-749"><a href="#cb31-749" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
-<span id="cb31-750"><a href="#cb31-750" aria-hidden="true" tabindex="-1"></a>f_C(y)^{1-d} \cdot S_C(y)^{d}</span>
-<span id="cb31-751"><a href="#cb31-751" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}\\</span></span>
-<span id="cb31-752"><a href="#cb31-752" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-753"><a href="#cb31-753" aria-hidden="true" tabindex="-1"></a>$$ Let</span>
-<span id="cb31-754"><a href="#cb31-754" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-755"><a href="#cb31-755" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>$\theta_T$ represent the parameters of $p_T(t)$,</span>
-<span id="cb31-756"><a href="#cb31-756" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>$\theta_C$ represent the parameters of $p_C(c)$,</span>
-<span id="cb31-757"><a href="#cb31-757" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>$\theta = (\theta_T, \theta_C)$ be the combined vector of all</span>
-<span id="cb31-758"><a href="#cb31-758" aria-hidden="true" tabindex="-1"></a>    parameters.</span>
-<span id="cb31-759"><a href="#cb31-759" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-760"><a href="#cb31-760" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-761"><a href="#cb31-761" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-762"><a href="#cb31-762" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-763"><a href="#cb31-763" aria-hidden="true" tabindex="-1"></a>The corresponding score function is:</span>
-<span id="cb31-764"><a href="#cb31-764" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-765"><a href="#cb31-765" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-766"><a href="#cb31-766" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-767"><a href="#cb31-767" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-768"><a href="#cb31-768" aria-hidden="true" tabindex="-1"></a>\ell'(y,d) </span>
-<span id="cb31-769"><a href="#cb31-769" aria-hidden="true" tabindex="-1"></a>&amp;= \deriv{\theta} </span>
-<span id="cb31-770"><a href="#cb31-770" aria-hidden="true" tabindex="-1"></a>\left[</span>
-<span id="cb31-771"><a href="#cb31-771" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
+<span id="cb31-737"><a href="#cb31-737" aria-hidden="true" tabindex="-1"></a>Then the joint likelihood $p(Y,D)$ factors into the product</span>
+<span id="cb31-738"><a href="#cb31-738" aria-hidden="true" tabindex="-1"></a>$p(Y), p(D)$, and the likelihood reduces to:</span>
+<span id="cb31-739"><a href="#cb31-739" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-740"><a href="#cb31-740" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-741"><a href="#cb31-741" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-742"><a href="#cb31-742" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-743"><a href="#cb31-743" aria-hidden="true" tabindex="-1"></a>\mathcal L(y,d) </span>
+<span id="cb31-744"><a href="#cb31-744" aria-hidden="true" tabindex="-1"></a>&amp;= \left<span class="co">[</span><span class="ot">p(T=y,C&gt; y)\right</span><span class="co">]</span>^d\cdot </span>
+<span id="cb31-745"><a href="#cb31-745" aria-hidden="true" tabindex="-1"></a>\left<span class="co">[</span><span class="ot">p(T&gt;y,C=y)\right</span><span class="co">]</span>^{1-d}<span class="sc">\\</span></span>
+<span id="cb31-746"><a href="#cb31-746" aria-hidden="true" tabindex="-1"></a>&amp;= \left<span class="co">[</span><span class="ot">p(T=y)p(C&gt; y)\right</span><span class="co">]</span>^d\cdot </span>
+<span id="cb31-747"><a href="#cb31-747" aria-hidden="true" tabindex="-1"></a>\left<span class="co">[</span><span class="ot">p(T&gt;y)p(C=y)\right</span><span class="co">]</span>^{1-d}<span class="sc">\\</span></span>
+<span id="cb31-748"><a href="#cb31-748" aria-hidden="true" tabindex="-1"></a>&amp;= \left<span class="co">[</span><span class="ot">f_T(y)S_C(y)\right</span><span class="co">]</span>^d\cdot </span>
+<span id="cb31-749"><a href="#cb31-749" aria-hidden="true" tabindex="-1"></a>\left<span class="co">[</span><span class="ot">S(y)f_C(y)\right</span><span class="co">]</span>^{1-d}<span class="sc">\\</span></span>
+<span id="cb31-750"><a href="#cb31-750" aria-hidden="true" tabindex="-1"></a>&amp;= \left<span class="co">[</span><span class="ot">f_T(y)^d S_C(y)^d\right</span><span class="co">]</span>\cdot </span>
+<span id="cb31-751"><a href="#cb31-751" aria-hidden="true" tabindex="-1"></a>\left<span class="co">[</span><span class="ot">S_T(y)^{1-d}f_C(y)^{1-d}\right</span><span class="co">]</span><span class="sc">\\</span></span>
+<span id="cb31-752"><a href="#cb31-752" aria-hidden="true" tabindex="-1"></a>&amp;= \left(f_T(y)^d \cdot S_T(y)^{1-d}\right)\cdot </span>
+<span id="cb31-753"><a href="#cb31-753" aria-hidden="true" tabindex="-1"></a>\left(f_C(y)^{1-d} \cdot S_C(y)^{d}\right)</span>
+<span id="cb31-754"><a href="#cb31-754" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-755"><a href="#cb31-755" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-756"><a href="#cb31-756" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-757"><a href="#cb31-757" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-758"><a href="#cb31-758" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-759"><a href="#cb31-759" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-760"><a href="#cb31-760" aria-hidden="true" tabindex="-1"></a>The corresponding log-likelihood is:</span>
+<span id="cb31-761"><a href="#cb31-761" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-762"><a href="#cb31-762" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-763"><a href="#cb31-763" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-764"><a href="#cb31-764" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-765"><a href="#cb31-765" aria-hidden="true" tabindex="-1"></a>\ell(y,d) </span>
+<span id="cb31-766"><a href="#cb31-766" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>\mathcal L(y,d) \right<span class="sc">\}\\</span></span>
+<span id="cb31-767"><a href="#cb31-767" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span></span>
+<span id="cb31-768"><a href="#cb31-768" aria-hidden="true" tabindex="-1"></a>\left(f_T(y)^d \cdot S_T(y)^{1-d}\right)\cdot </span>
+<span id="cb31-769"><a href="#cb31-769" aria-hidden="true" tabindex="-1"></a>\left(f_C(y)^{1-d} \cdot S_C(y)^{d}\right)</span>
+<span id="cb31-770"><a href="#cb31-770" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}\\</span></span>
+<span id="cb31-771"><a href="#cb31-771" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span></span>
 <span id="cb31-772"><a href="#cb31-772" aria-hidden="true" tabindex="-1"></a>f_T(y)^d \cdot S_T(y)^{1-d}</span>
 <span id="cb31-773"><a href="#cb31-773" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}</span> </span>
 <span id="cb31-774"><a href="#cb31-774" aria-hidden="true" tabindex="-1"></a>+</span>
 <span id="cb31-775"><a href="#cb31-775" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
 <span id="cb31-776"><a href="#cb31-776" aria-hidden="true" tabindex="-1"></a>f_C(y)^{1-d} \cdot S_C(y)^{d}</span>
-<span id="cb31-777"><a href="#cb31-777" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}</span></span>
-<span id="cb31-778"><a href="#cb31-778" aria-hidden="true" tabindex="-1"></a>\right]<span class="sc">\\</span></span>
-<span id="cb31-779"><a href="#cb31-779" aria-hidden="true" tabindex="-1"></a>&amp;= </span>
-<span id="cb31-780"><a href="#cb31-780" aria-hidden="true" tabindex="-1"></a>\left(</span>
-<span id="cb31-781"><a href="#cb31-781" aria-hidden="true" tabindex="-1"></a>\deriv{\theta} </span>
-<span id="cb31-782"><a href="#cb31-782" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
-<span id="cb31-783"><a href="#cb31-783" aria-hidden="true" tabindex="-1"></a>f_T(y)^d \cdot S_T(y)^{1-d}</span>
-<span id="cb31-784"><a href="#cb31-784" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}</span></span>
-<span id="cb31-785"><a href="#cb31-785" aria-hidden="true" tabindex="-1"></a>\right)</span>
-<span id="cb31-786"><a href="#cb31-786" aria-hidden="true" tabindex="-1"></a>+</span>
-<span id="cb31-787"><a href="#cb31-787" aria-hidden="true" tabindex="-1"></a>\left(</span>
-<span id="cb31-788"><a href="#cb31-788" aria-hidden="true" tabindex="-1"></a>\deriv{\theta} </span>
-<span id="cb31-789"><a href="#cb31-789" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
-<span id="cb31-790"><a href="#cb31-790" aria-hidden="true" tabindex="-1"></a>f_C(y)^{1-d} \cdot S_C(y)^{d}</span>
-<span id="cb31-791"><a href="#cb31-791" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}</span></span>
-<span id="cb31-792"><a href="#cb31-792" aria-hidden="true" tabindex="-1"></a>\right)<span class="sc">\\</span></span>
-<span id="cb31-793"><a href="#cb31-793" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-794"><a href="#cb31-794" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-795"><a href="#cb31-795" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-796"><a href="#cb31-796" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-797"><a href="#cb31-797" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-798"><a href="#cb31-798" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-799"><a href="#cb31-799" aria-hidden="true" tabindex="-1"></a>As long as $\theta_C$ and $\theta_T$ don't share any parameters, then if</span>
-<span id="cb31-800"><a href="#cb31-800" aria-hidden="true" tabindex="-1"></a>censoring is non-informative, the partial derivative with respect to</span>
-<span id="cb31-801"><a href="#cb31-801" aria-hidden="true" tabindex="-1"></a>$\theta_T$ is:</span>
-<span id="cb31-802"><a href="#cb31-802" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-803"><a href="#cb31-803" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-804"><a href="#cb31-804" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-805"><a href="#cb31-805" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-806"><a href="#cb31-806" aria-hidden="true" tabindex="-1"></a>\ell'_{\theta_T}(y,d)</span>
-<span id="cb31-807"><a href="#cb31-807" aria-hidden="true" tabindex="-1"></a>&amp;\eqdef  \deriv{\theta_T}\ell(y,d)<span class="sc">\\</span></span>
-<span id="cb31-808"><a href="#cb31-808" aria-hidden="true" tabindex="-1"></a>&amp;= </span>
-<span id="cb31-809"><a href="#cb31-809" aria-hidden="true" tabindex="-1"></a>\left(</span>
-<span id="cb31-810"><a href="#cb31-810" aria-hidden="true" tabindex="-1"></a>\deriv{\theta_T} </span>
-<span id="cb31-811"><a href="#cb31-811" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
-<span id="cb31-812"><a href="#cb31-812" aria-hidden="true" tabindex="-1"></a>f_T(y)^d \cdot S_T(y)^{1-d}</span>
-<span id="cb31-813"><a href="#cb31-813" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}</span></span>
-<span id="cb31-814"><a href="#cb31-814" aria-hidden="true" tabindex="-1"></a>\right)</span>
-<span id="cb31-815"><a href="#cb31-815" aria-hidden="true" tabindex="-1"></a>+</span>
-<span id="cb31-816"><a href="#cb31-816" aria-hidden="true" tabindex="-1"></a>\left(</span>
-<span id="cb31-817"><a href="#cb31-817" aria-hidden="true" tabindex="-1"></a>\deriv{\theta_T} </span>
-<span id="cb31-818"><a href="#cb31-818" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
-<span id="cb31-819"><a href="#cb31-819" aria-hidden="true" tabindex="-1"></a>f_C(y)^{1-d} \cdot S_C(y)^{d}</span>
-<span id="cb31-820"><a href="#cb31-820" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}</span></span>
-<span id="cb31-821"><a href="#cb31-821" aria-hidden="true" tabindex="-1"></a>\right)<span class="sc">\\</span></span>
-<span id="cb31-822"><a href="#cb31-822" aria-hidden="true" tabindex="-1"></a>&amp;= </span>
-<span id="cb31-823"><a href="#cb31-823" aria-hidden="true" tabindex="-1"></a>\left(</span>
-<span id="cb31-824"><a href="#cb31-824" aria-hidden="true" tabindex="-1"></a>\deriv{\theta_T} </span>
-<span id="cb31-825"><a href="#cb31-825" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
-<span id="cb31-826"><a href="#cb31-826" aria-hidden="true" tabindex="-1"></a>f_T(y)^d \cdot S_T(y)^{1-d}</span>
-<span id="cb31-827"><a href="#cb31-827" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}</span></span>
-<span id="cb31-828"><a href="#cb31-828" aria-hidden="true" tabindex="-1"></a>\right) + 0<span class="sc">\\</span></span>
-<span id="cb31-829"><a href="#cb31-829" aria-hidden="true" tabindex="-1"></a>&amp;= </span>
-<span id="cb31-830"><a href="#cb31-830" aria-hidden="true" tabindex="-1"></a>\deriv{\theta_T} </span>
-<span id="cb31-831"><a href="#cb31-831" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
-<span id="cb31-832"><a href="#cb31-832" aria-hidden="true" tabindex="-1"></a>f_T(y)^d \cdot S_T(y)^{1-d}</span>
-<span id="cb31-833"><a href="#cb31-833" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}\\</span></span>
-<span id="cb31-834"><a href="#cb31-834" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-835"><a href="#cb31-835" aria-hidden="true" tabindex="-1"></a>$$ </span>
-<span id="cb31-836"><a href="#cb31-836" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-837"><a href="#cb31-837" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-838"><a href="#cb31-838" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-839"><a href="#cb31-839" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-840"><a href="#cb31-840" aria-hidden="true" tabindex="-1"></a>Thus, the MLE for $\theta_T$ won't depend on $\theta_C$, and we can</span>
-<span id="cb31-841"><a href="#cb31-841" aria-hidden="true" tabindex="-1"></a>ignore the distribution of $C$ when estimating the parameters of</span>
-<span id="cb31-842"><a href="#cb31-842" aria-hidden="true" tabindex="-1"></a>$f_T(t)=p(T=t)$.</span>
-<span id="cb31-843"><a href="#cb31-843" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-844"><a href="#cb31-844" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-845"><a href="#cb31-845" aria-hidden="true" tabindex="-1"></a>Then:</span>
-<span id="cb31-846"><a href="#cb31-846" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-847"><a href="#cb31-847" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-848"><a href="#cb31-848" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-849"><a href="#cb31-849" aria-hidden="true" tabindex="-1"></a>\mathcal L(y,d) </span>
-<span id="cb31-850"><a href="#cb31-850" aria-hidden="true" tabindex="-1"></a>&amp;= f_T(y)^d \cdot S_T(y)^{1-d}<span class="sc">\\</span></span>
-<span id="cb31-851"><a href="#cb31-851" aria-hidden="true" tabindex="-1"></a>&amp;= \left(h_T(y)^d  S_T(y)^d\right) \cdot S_T(y)^{1-d}<span class="sc">\\</span></span>
-<span id="cb31-852"><a href="#cb31-852" aria-hidden="true" tabindex="-1"></a>&amp;= h_T(y)^d  \cdot S_T(y)^d \cdot S_T(y)^{1-d}<span class="sc">\\</span></span>
-<span id="cb31-853"><a href="#cb31-853" aria-hidden="true" tabindex="-1"></a>&amp;= h_T(y)^d \cdot S_T(y)<span class="sc">\\</span></span>
-<span id="cb31-854"><a href="#cb31-854" aria-hidden="true" tabindex="-1"></a>&amp;= S_T(y) \cdot h_T(y)^d <span class="sc">\\</span></span>
-<span id="cb31-855"><a href="#cb31-855" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-856"><a href="#cb31-856" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-857"><a href="#cb31-857" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-858"><a href="#cb31-858" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-859"><a href="#cb31-859" aria-hidden="true" tabindex="-1"></a>That is, if the event occurred at time $y$ (i.e., if $d=1$), then the</span>
-<span id="cb31-860"><a href="#cb31-860" aria-hidden="true" tabindex="-1"></a>likelihood of $(Y,D) = (y,d)$ is equal to the hazard function at $y$</span>
-<span id="cb31-861"><a href="#cb31-861" aria-hidden="true" tabindex="-1"></a>times the survival function at $y$. Otherwise, the likelihood is equal</span>
-<span id="cb31-862"><a href="#cb31-862" aria-hidden="true" tabindex="-1"></a>to just the survival function at $y$.</span>
-<span id="cb31-863"><a href="#cb31-863" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-777"><a href="#cb31-777" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}\\</span></span>
+<span id="cb31-778"><a href="#cb31-778" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-779"><a href="#cb31-779" aria-hidden="true" tabindex="-1"></a>$$ Let</span>
+<span id="cb31-780"><a href="#cb31-780" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-781"><a href="#cb31-781" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>$\theta_T$ represent the parameters of $p_T(t)$,</span>
+<span id="cb31-782"><a href="#cb31-782" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>$\theta_C$ represent the parameters of $p_C(c)$,</span>
+<span id="cb31-783"><a href="#cb31-783" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>$\theta = (\theta_T, \theta_C)$ be the combined vector of all</span>
+<span id="cb31-784"><a href="#cb31-784" aria-hidden="true" tabindex="-1"></a>    parameters.</span>
+<span id="cb31-785"><a href="#cb31-785" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-786"><a href="#cb31-786" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-787"><a href="#cb31-787" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-788"><a href="#cb31-788" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-789"><a href="#cb31-789" aria-hidden="true" tabindex="-1"></a>The corresponding score function is:</span>
+<span id="cb31-790"><a href="#cb31-790" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-791"><a href="#cb31-791" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-792"><a href="#cb31-792" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-793"><a href="#cb31-793" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-794"><a href="#cb31-794" aria-hidden="true" tabindex="-1"></a>\ell'(y,d) </span>
+<span id="cb31-795"><a href="#cb31-795" aria-hidden="true" tabindex="-1"></a>&amp;= \deriv{\theta} </span>
+<span id="cb31-796"><a href="#cb31-796" aria-hidden="true" tabindex="-1"></a>\left[</span>
+<span id="cb31-797"><a href="#cb31-797" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
+<span id="cb31-798"><a href="#cb31-798" aria-hidden="true" tabindex="-1"></a>f_T(y)^d \cdot S_T(y)^{1-d}</span>
+<span id="cb31-799"><a href="#cb31-799" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}</span> </span>
+<span id="cb31-800"><a href="#cb31-800" aria-hidden="true" tabindex="-1"></a>+</span>
+<span id="cb31-801"><a href="#cb31-801" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
+<span id="cb31-802"><a href="#cb31-802" aria-hidden="true" tabindex="-1"></a>f_C(y)^{1-d} \cdot S_C(y)^{d}</span>
+<span id="cb31-803"><a href="#cb31-803" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}</span></span>
+<span id="cb31-804"><a href="#cb31-804" aria-hidden="true" tabindex="-1"></a>\right]<span class="sc">\\</span></span>
+<span id="cb31-805"><a href="#cb31-805" aria-hidden="true" tabindex="-1"></a>&amp;= </span>
+<span id="cb31-806"><a href="#cb31-806" aria-hidden="true" tabindex="-1"></a>\left(</span>
+<span id="cb31-807"><a href="#cb31-807" aria-hidden="true" tabindex="-1"></a>\deriv{\theta} </span>
+<span id="cb31-808"><a href="#cb31-808" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
+<span id="cb31-809"><a href="#cb31-809" aria-hidden="true" tabindex="-1"></a>f_T(y)^d \cdot S_T(y)^{1-d}</span>
+<span id="cb31-810"><a href="#cb31-810" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}</span></span>
+<span id="cb31-811"><a href="#cb31-811" aria-hidden="true" tabindex="-1"></a>\right)</span>
+<span id="cb31-812"><a href="#cb31-812" aria-hidden="true" tabindex="-1"></a>+</span>
+<span id="cb31-813"><a href="#cb31-813" aria-hidden="true" tabindex="-1"></a>\left(</span>
+<span id="cb31-814"><a href="#cb31-814" aria-hidden="true" tabindex="-1"></a>\deriv{\theta} </span>
+<span id="cb31-815"><a href="#cb31-815" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
+<span id="cb31-816"><a href="#cb31-816" aria-hidden="true" tabindex="-1"></a>f_C(y)^{1-d} \cdot S_C(y)^{d}</span>
+<span id="cb31-817"><a href="#cb31-817" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}</span></span>
+<span id="cb31-818"><a href="#cb31-818" aria-hidden="true" tabindex="-1"></a>\right)<span class="sc">\\</span></span>
+<span id="cb31-819"><a href="#cb31-819" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-820"><a href="#cb31-820" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-821"><a href="#cb31-821" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-822"><a href="#cb31-822" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-823"><a href="#cb31-823" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-824"><a href="#cb31-824" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-825"><a href="#cb31-825" aria-hidden="true" tabindex="-1"></a>As long as $\theta_C$ and $\theta_T$ don't share any parameters, then if</span>
+<span id="cb31-826"><a href="#cb31-826" aria-hidden="true" tabindex="-1"></a>censoring is non-informative, the partial derivative with respect to</span>
+<span id="cb31-827"><a href="#cb31-827" aria-hidden="true" tabindex="-1"></a>$\theta_T$ is:</span>
+<span id="cb31-828"><a href="#cb31-828" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-829"><a href="#cb31-829" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-830"><a href="#cb31-830" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-831"><a href="#cb31-831" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-832"><a href="#cb31-832" aria-hidden="true" tabindex="-1"></a>\ell'_{\theta_T}(y,d)</span>
+<span id="cb31-833"><a href="#cb31-833" aria-hidden="true" tabindex="-1"></a>&amp;\eqdef  \deriv{\theta_T}\ell(y,d)<span class="sc">\\</span></span>
+<span id="cb31-834"><a href="#cb31-834" aria-hidden="true" tabindex="-1"></a>&amp;= </span>
+<span id="cb31-835"><a href="#cb31-835" aria-hidden="true" tabindex="-1"></a>\left(</span>
+<span id="cb31-836"><a href="#cb31-836" aria-hidden="true" tabindex="-1"></a>\deriv{\theta_T} </span>
+<span id="cb31-837"><a href="#cb31-837" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
+<span id="cb31-838"><a href="#cb31-838" aria-hidden="true" tabindex="-1"></a>f_T(y)^d \cdot S_T(y)^{1-d}</span>
+<span id="cb31-839"><a href="#cb31-839" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}</span></span>
+<span id="cb31-840"><a href="#cb31-840" aria-hidden="true" tabindex="-1"></a>\right)</span>
+<span id="cb31-841"><a href="#cb31-841" aria-hidden="true" tabindex="-1"></a>+</span>
+<span id="cb31-842"><a href="#cb31-842" aria-hidden="true" tabindex="-1"></a>\left(</span>
+<span id="cb31-843"><a href="#cb31-843" aria-hidden="true" tabindex="-1"></a>\deriv{\theta_T} </span>
+<span id="cb31-844"><a href="#cb31-844" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
+<span id="cb31-845"><a href="#cb31-845" aria-hidden="true" tabindex="-1"></a>f_C(y)^{1-d} \cdot S_C(y)^{d}</span>
+<span id="cb31-846"><a href="#cb31-846" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}</span></span>
+<span id="cb31-847"><a href="#cb31-847" aria-hidden="true" tabindex="-1"></a>\right)<span class="sc">\\</span></span>
+<span id="cb31-848"><a href="#cb31-848" aria-hidden="true" tabindex="-1"></a>&amp;= </span>
+<span id="cb31-849"><a href="#cb31-849" aria-hidden="true" tabindex="-1"></a>\left(</span>
+<span id="cb31-850"><a href="#cb31-850" aria-hidden="true" tabindex="-1"></a>\deriv{\theta_T} </span>
+<span id="cb31-851"><a href="#cb31-851" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
+<span id="cb31-852"><a href="#cb31-852" aria-hidden="true" tabindex="-1"></a>f_T(y)^d \cdot S_T(y)^{1-d}</span>
+<span id="cb31-853"><a href="#cb31-853" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}</span></span>
+<span id="cb31-854"><a href="#cb31-854" aria-hidden="true" tabindex="-1"></a>\right) + 0<span class="sc">\\</span></span>
+<span id="cb31-855"><a href="#cb31-855" aria-hidden="true" tabindex="-1"></a>&amp;= </span>
+<span id="cb31-856"><a href="#cb31-856" aria-hidden="true" tabindex="-1"></a>\deriv{\theta_T} </span>
+<span id="cb31-857"><a href="#cb31-857" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span></span>
+<span id="cb31-858"><a href="#cb31-858" aria-hidden="true" tabindex="-1"></a>f_T(y)^d \cdot S_T(y)^{1-d}</span>
+<span id="cb31-859"><a href="#cb31-859" aria-hidden="true" tabindex="-1"></a>\right<span class="sc">\}\\</span></span>
+<span id="cb31-860"><a href="#cb31-860" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-861"><a href="#cb31-861" aria-hidden="true" tabindex="-1"></a>$$ </span>
+<span id="cb31-862"><a href="#cb31-862" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-863"><a href="#cb31-863" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-864"><a href="#cb31-864" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-865"><a href="#cb31-865" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-866"><a href="#cb31-866" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-867"><a href="#cb31-867" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-868"><a href="#cb31-868" aria-hidden="true" tabindex="-1"></a>The corresponding log-likelihood is:</span>
+<span id="cb31-865"><a href="#cb31-865" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-866"><a href="#cb31-866" aria-hidden="true" tabindex="-1"></a>Thus, the MLE for $\theta_T$ won't depend on $\theta_C$, and we can</span>
+<span id="cb31-867"><a href="#cb31-867" aria-hidden="true" tabindex="-1"></a>ignore the distribution of $C$ when estimating the parameters of</span>
+<span id="cb31-868"><a href="#cb31-868" aria-hidden="true" tabindex="-1"></a>$f_T(t)=p(T=t)$.</span>
 <span id="cb31-869"><a href="#cb31-869" aria-hidden="true" tabindex="-1"></a>:::</span>
 <span id="cb31-870"><a href="#cb31-870" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-871"><a href="#cb31-871" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-872"><a href="#cb31-872" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-873"><a href="#cb31-873" aria-hidden="true" tabindex="-1"></a>\ell(y,d)</span>
-<span id="cb31-874"><a href="#cb31-874" aria-hidden="true" tabindex="-1"></a>&amp;=\text{log}\left<span class="sc">\{</span>\mathcal L(y,d)\right<span class="sc">\}\\</span></span>
-<span id="cb31-875"><a href="#cb31-875" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>S_T(y) \cdot h_T(y)^d\right<span class="sc">\}\\</span></span>
-<span id="cb31-876"><a href="#cb31-876" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>S_T(y)\right<span class="sc">\}</span> + \text{log}\left<span class="sc">\{</span>h_T(y)^d\right<span class="sc">\}\\</span></span>
-<span id="cb31-877"><a href="#cb31-877" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>S_T(y)\right<span class="sc">\}</span> + d\cdot \text{log}\left<span class="sc">\{</span>h_T(y)\right<span class="sc">\}\\</span></span>
-<span id="cb31-878"><a href="#cb31-878" aria-hidden="true" tabindex="-1"></a>&amp;= -H_T(y) + d\cdot \text{log}\left<span class="sc">\{</span>h_T(y)\right<span class="sc">\}\\</span></span>
-<span id="cb31-879"><a href="#cb31-879" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-880"><a href="#cb31-880" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-881"><a href="#cb31-881" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-882"><a href="#cb31-882" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-883"><a href="#cb31-883" aria-hidden="true" tabindex="-1"></a>In other words, the log-likelihood contribution from a single</span>
-<span id="cb31-884"><a href="#cb31-884" aria-hidden="true" tabindex="-1"></a>observation $(Y,D) = (y,d)$ is equal to the negative cumulative hazard</span>
-<span id="cb31-885"><a href="#cb31-885" aria-hidden="true" tabindex="-1"></a>at $y$, plus the log of the hazard at $y$ if the event occurred at time</span>
-<span id="cb31-886"><a href="#cb31-886" aria-hidden="true" tabindex="-1"></a>$y$.</span>
-<span id="cb31-887"><a href="#cb31-887" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-888"><a href="#cb31-888" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-889"><a href="#cb31-889" aria-hidden="true" tabindex="-1"></a><span class="fu">## Parametric Models for Time-to-Event Outcomes</span></span>
+<span id="cb31-871"><a href="#cb31-871" aria-hidden="true" tabindex="-1"></a>Then:</span>
+<span id="cb31-872"><a href="#cb31-872" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-873"><a href="#cb31-873" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-874"><a href="#cb31-874" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-875"><a href="#cb31-875" aria-hidden="true" tabindex="-1"></a>\mathcal L(y,d) </span>
+<span id="cb31-876"><a href="#cb31-876" aria-hidden="true" tabindex="-1"></a>&amp;= f_T(y)^d \cdot S_T(y)^{1-d}<span class="sc">\\</span></span>
+<span id="cb31-877"><a href="#cb31-877" aria-hidden="true" tabindex="-1"></a>&amp;= \left(h_T(y)^d  S_T(y)^d\right) \cdot S_T(y)^{1-d}<span class="sc">\\</span></span>
+<span id="cb31-878"><a href="#cb31-878" aria-hidden="true" tabindex="-1"></a>&amp;= h_T(y)^d  \cdot S_T(y)^d \cdot S_T(y)^{1-d}<span class="sc">\\</span></span>
+<span id="cb31-879"><a href="#cb31-879" aria-hidden="true" tabindex="-1"></a>&amp;= h_T(y)^d \cdot S_T(y)<span class="sc">\\</span></span>
+<span id="cb31-880"><a href="#cb31-880" aria-hidden="true" tabindex="-1"></a>&amp;= S_T(y) \cdot h_T(y)^d <span class="sc">\\</span></span>
+<span id="cb31-881"><a href="#cb31-881" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-882"><a href="#cb31-882" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-883"><a href="#cb31-883" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-884"><a href="#cb31-884" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-885"><a href="#cb31-885" aria-hidden="true" tabindex="-1"></a>That is, if the event occurred at time $y$ (i.e., if $d=1$), then the</span>
+<span id="cb31-886"><a href="#cb31-886" aria-hidden="true" tabindex="-1"></a>likelihood of $(Y,D) = (y,d)$ is equal to the hazard function at $y$</span>
+<span id="cb31-887"><a href="#cb31-887" aria-hidden="true" tabindex="-1"></a>times the survival function at $y$. Otherwise, the likelihood is equal</span>
+<span id="cb31-888"><a href="#cb31-888" aria-hidden="true" tabindex="-1"></a>to just the survival function at $y$.</span>
+<span id="cb31-889"><a href="#cb31-889" aria-hidden="true" tabindex="-1"></a>:::</span>
 <span id="cb31-890"><a href="#cb31-890" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-891"><a href="#cb31-891" aria-hidden="true" tabindex="-1"></a><span class="fu">### Exponential Distribution</span></span>
+<span id="cb31-891"><a href="#cb31-891" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-892"><a href="#cb31-892" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-893"><a href="#cb31-893" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The exponential distribution is the base distribution for survival</span>
-<span id="cb31-894"><a href="#cb31-894" aria-hidden="true" tabindex="-1"></a>    analysis.</span>
-<span id="cb31-895"><a href="#cb31-895" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The distribution has a constant hazard $\lambda$</span>
-<span id="cb31-896"><a href="#cb31-896" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The mean survival time is $\lambda^{-1}$</span>
-<span id="cb31-897"><a href="#cb31-897" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-898"><a href="#cb31-898" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-899"><a href="#cb31-899" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-900"><a href="#cb31-900" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Mathematical details of exponential distribution</span></span>
-<span id="cb31-901"><a href="#cb31-901" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-902"><a href="#cb31-902" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-903"><a href="#cb31-903" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-904"><a href="#cb31-904" aria-hidden="true" tabindex="-1"></a>f(t) &amp;= \lambda \text{e}^{-\lambda t}<span class="sc">\\</span></span>
-<span id="cb31-905"><a href="#cb31-905" aria-hidden="true" tabindex="-1"></a>E(t) &amp;= \lambda^{-1}<span class="sc">\\</span></span>
-<span id="cb31-906"><a href="#cb31-906" aria-hidden="true" tabindex="-1"></a>Var(t) &amp;= \lambda^{-2}<span class="sc">\\</span></span>
-<span id="cb31-907"><a href="#cb31-907" aria-hidden="true" tabindex="-1"></a>F(t) &amp;= 1-\text{e}^{-\lambda x}<span class="sc">\\</span></span>
-<span id="cb31-908"><a href="#cb31-908" aria-hidden="true" tabindex="-1"></a>S(t)&amp;= \text{e}^{-\lambda x}<span class="sc">\\</span></span>
-<span id="cb31-909"><a href="#cb31-909" aria-hidden="true" tabindex="-1"></a>\ln(S(t))&amp;=-\lambda x<span class="sc">\\</span></span>
-<span id="cb31-910"><a href="#cb31-910" aria-hidden="true" tabindex="-1"></a>h(t) &amp;= -\frac{f(t)}{S(t)} = -\frac{\lambda \text{e}^{-\lambda t}}{\text{e}^{-\lambda t}}=\lambda</span>
-<span id="cb31-911"><a href="#cb31-911" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-912"><a href="#cb31-912" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-913"><a href="#cb31-913" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-914"><a href="#cb31-914" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-915"><a href="#cb31-915" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-916"><a href="#cb31-916" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Estimating $\lambda$ {.smaller}</span></span>
-<span id="cb31-917"><a href="#cb31-917" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-918"><a href="#cb31-918" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Suppose we have $m$ exponential survival times of</span>
-<span id="cb31-919"><a href="#cb31-919" aria-hidden="true" tabindex="-1"></a>    $t_1, t_2,\ldots,t_m$ and $k$ right-censored values at</span>
-<span id="cb31-920"><a href="#cb31-920" aria-hidden="true" tabindex="-1"></a>    $u_1,u_2,\ldots,u_k$.</span>
-<span id="cb31-921"><a href="#cb31-921" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-922"><a href="#cb31-922" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>A survival time of $t_i=10$ means that subject $i$ died at time 10.</span>
-<span id="cb31-923"><a href="#cb31-923" aria-hidden="true" tabindex="-1"></a>    A right-censored time $u_i=10$ means that at time 10, subject $i$</span>
-<span id="cb31-924"><a href="#cb31-924" aria-hidden="true" tabindex="-1"></a>    was still alive and that we have no further follow-up.</span>
+<span id="cb31-893"><a href="#cb31-893" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-894"><a href="#cb31-894" aria-hidden="true" tabindex="-1"></a>The corresponding log-likelihood is:</span>
+<span id="cb31-895"><a href="#cb31-895" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-896"><a href="#cb31-896" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-897"><a href="#cb31-897" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-898"><a href="#cb31-898" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-899"><a href="#cb31-899" aria-hidden="true" tabindex="-1"></a>\ell(y,d)</span>
+<span id="cb31-900"><a href="#cb31-900" aria-hidden="true" tabindex="-1"></a>&amp;=\text{log}\left<span class="sc">\{</span>\mathcal L(y,d)\right<span class="sc">\}\\</span></span>
+<span id="cb31-901"><a href="#cb31-901" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>S_T(y) \cdot h_T(y)^d\right<span class="sc">\}\\</span></span>
+<span id="cb31-902"><a href="#cb31-902" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>S_T(y)\right<span class="sc">\}</span> + \text{log}\left<span class="sc">\{</span>h_T(y)^d\right<span class="sc">\}\\</span></span>
+<span id="cb31-903"><a href="#cb31-903" aria-hidden="true" tabindex="-1"></a>&amp;= \text{log}\left<span class="sc">\{</span>S_T(y)\right<span class="sc">\}</span> + d\cdot \text{log}\left<span class="sc">\{</span>h_T(y)\right<span class="sc">\}\\</span></span>
+<span id="cb31-904"><a href="#cb31-904" aria-hidden="true" tabindex="-1"></a>&amp;= -H_T(y) + d\cdot \text{log}\left<span class="sc">\{</span>h_T(y)\right<span class="sc">\}\\</span></span>
+<span id="cb31-905"><a href="#cb31-905" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-906"><a href="#cb31-906" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-907"><a href="#cb31-907" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-908"><a href="#cb31-908" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-909"><a href="#cb31-909" aria-hidden="true" tabindex="-1"></a>In other words, the log-likelihood contribution from a single</span>
+<span id="cb31-910"><a href="#cb31-910" aria-hidden="true" tabindex="-1"></a>observation $(Y,D) = (y,d)$ is equal to the negative cumulative hazard</span>
+<span id="cb31-911"><a href="#cb31-911" aria-hidden="true" tabindex="-1"></a>at $y$, plus the log of the hazard at $y$ if the event occurred at time</span>
+<span id="cb31-912"><a href="#cb31-912" aria-hidden="true" tabindex="-1"></a>$y$.</span>
+<span id="cb31-913"><a href="#cb31-913" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-914"><a href="#cb31-914" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-915"><a href="#cb31-915" aria-hidden="true" tabindex="-1"></a><span class="fu">## Parametric Models for Time-to-Event Outcomes</span></span>
+<span id="cb31-916"><a href="#cb31-916" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-917"><a href="#cb31-917" aria-hidden="true" tabindex="-1"></a><span class="fu">### Exponential Distribution</span></span>
+<span id="cb31-918"><a href="#cb31-918" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-919"><a href="#cb31-919" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The exponential distribution is the base distribution for survival</span>
+<span id="cb31-920"><a href="#cb31-920" aria-hidden="true" tabindex="-1"></a>    analysis.</span>
+<span id="cb31-921"><a href="#cb31-921" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The distribution has a constant hazard $\lambda$</span>
+<span id="cb31-922"><a href="#cb31-922" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The mean survival time is $\lambda^{-1}$</span>
+<span id="cb31-923"><a href="#cb31-923" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-924"><a href="#cb31-924" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-925"><a href="#cb31-925" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-926"><a href="#cb31-926" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>For the moment we will assume that the survival distribution is</span>
-<span id="cb31-927"><a href="#cb31-927" aria-hidden="true" tabindex="-1"></a>    exponential and that all the subjects have the same parameter</span>
-<span id="cb31-928"><a href="#cb31-928" aria-hidden="true" tabindex="-1"></a>    $\lambda$.</span>
-<span id="cb31-929"><a href="#cb31-929" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-930"><a href="#cb31-930" aria-hidden="true" tabindex="-1"></a>We have $m$ exponential survival times of $t_1, t_2,\ldots,t_m$ and $k$</span>
-<span id="cb31-931"><a href="#cb31-931" aria-hidden="true" tabindex="-1"></a>right-censored values at $u_1,u_2,\ldots,u_k$. The log-likelihood of an</span>
-<span id="cb31-932"><a href="#cb31-932" aria-hidden="true" tabindex="-1"></a>observed survival time $t_i$ is $$</span>
-<span id="cb31-933"><a href="#cb31-933" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span>\lambda \text{e}^{-\lambda t_i}\right<span class="sc">\}</span> =</span>
-<span id="cb31-934"><a href="#cb31-934" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span>\lambda\right<span class="sc">\}</span>-\lambda t_i</span>
-<span id="cb31-935"><a href="#cb31-935" aria-hidden="true" tabindex="-1"></a>$$ and the likelihood of a censored value is the probability of that</span>
-<span id="cb31-936"><a href="#cb31-936" aria-hidden="true" tabindex="-1"></a>outcome (survival greater than $u_j$) so the log-likelihood is </span>
-<span id="cb31-937"><a href="#cb31-937" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-926"><a href="#cb31-926" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Mathematical details of exponential distribution</span></span>
+<span id="cb31-927"><a href="#cb31-927" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-928"><a href="#cb31-928" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-929"><a href="#cb31-929" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-930"><a href="#cb31-930" aria-hidden="true" tabindex="-1"></a>f(t) &amp;= \lambda \text{e}^{-\lambda t}<span class="sc">\\</span></span>
+<span id="cb31-931"><a href="#cb31-931" aria-hidden="true" tabindex="-1"></a>E(t) &amp;= \lambda^{-1}<span class="sc">\\</span></span>
+<span id="cb31-932"><a href="#cb31-932" aria-hidden="true" tabindex="-1"></a>Var(t) &amp;= \lambda^{-2}<span class="sc">\\</span></span>
+<span id="cb31-933"><a href="#cb31-933" aria-hidden="true" tabindex="-1"></a>F(t) &amp;= 1-\text{e}^{-\lambda x}<span class="sc">\\</span></span>
+<span id="cb31-934"><a href="#cb31-934" aria-hidden="true" tabindex="-1"></a>S(t)&amp;= \text{e}^{-\lambda x}<span class="sc">\\</span></span>
+<span id="cb31-935"><a href="#cb31-935" aria-hidden="true" tabindex="-1"></a>\ln(S(t))&amp;=-\lambda x<span class="sc">\\</span></span>
+<span id="cb31-936"><a href="#cb31-936" aria-hidden="true" tabindex="-1"></a>h(t) &amp;= -\frac{f(t)}{S(t)} = -\frac{\lambda \text{e}^{-\lambda t}}{\text{e}^{-\lambda t}}=\lambda</span>
+<span id="cb31-937"><a href="#cb31-937" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
 <span id="cb31-938"><a href="#cb31-938" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-939"><a href="#cb31-939" aria-hidden="true" tabindex="-1"></a>\ba</span>
-<span id="cb31-940"><a href="#cb31-940" aria-hidden="true" tabindex="-1"></a>\ell_j(\lambda) &amp;= \text{log}\left<span class="sc">\{</span>\lambda \text{e}^{u_j}\right<span class="sc">\}</span> </span>
-<span id="cb31-941"><a href="#cb31-941" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= -\lambda u_j</span>
-<span id="cb31-942"><a href="#cb31-942" aria-hidden="true" tabindex="-1"></a>\ea</span>
-<span id="cb31-943"><a href="#cb31-943" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-944"><a href="#cb31-944" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-945"><a href="#cb31-945" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-946"><a href="#cb31-946" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-947"><a href="#cb31-947" aria-hidden="true" tabindex="-1"></a>:::{#thm-mle-exp}</span>
-<span id="cb31-948"><a href="#cb31-948" aria-hidden="true" tabindex="-1"></a>Let $T=\sum t_i$ and $U=\sum u_j$. Then:</span>
-<span id="cb31-949"><a href="#cb31-949" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-950"><a href="#cb31-950" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-951"><a href="#cb31-951" aria-hidden="true" tabindex="-1"></a>\ell(\lambda) = \frac{m}{T+U}</span>
-<span id="cb31-952"><a href="#cb31-952" aria-hidden="true" tabindex="-1"></a>$$ {#eq-mle-exp}</span>
-<span id="cb31-953"><a href="#cb31-953" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-954"><a href="#cb31-954" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-955"><a href="#cb31-955" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-956"><a href="#cb31-956" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-957"><a href="#cb31-957" aria-hidden="true" tabindex="-1"></a>::: proof</span>
-<span id="cb31-958"><a href="#cb31-958" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-959"><a href="#cb31-959" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-960"><a href="#cb31-960" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-961"><a href="#cb31-961" aria-hidden="true" tabindex="-1"></a>\ell(\lambda) &amp;= \sum_{i=1}^m( \ln \lambda-\lambda t_i) + \sum_{j=1}^k (-\lambda u_j)<span class="sc">\\</span></span>
-<span id="cb31-962"><a href="#cb31-962" aria-hidden="true" tabindex="-1"></a>&amp;= m \ln \lambda -(T+U)\lambda<span class="sc">\\</span></span>
-<span id="cb31-963"><a href="#cb31-963" aria-hidden="true" tabindex="-1"></a>\ell'(\lambda) </span>
-<span id="cb31-964"><a href="#cb31-964" aria-hidden="true" tabindex="-1"></a>&amp;=m\lambda^{-1} -(T+U)<span class="sc">\\</span></span>
-<span id="cb31-965"><a href="#cb31-965" aria-hidden="true" tabindex="-1"></a>\hat{\lambda} &amp;= \frac{m}{T+U}</span>
-<span id="cb31-966"><a href="#cb31-966" aria-hidden="true" tabindex="-1"></a>\ea</span>
-<span id="cb31-967"><a href="#cb31-967" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-968"><a href="#cb31-968" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-969"><a href="#cb31-969" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-970"><a href="#cb31-970" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-971"><a href="#cb31-971" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-972"><a href="#cb31-972" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-973"><a href="#cb31-973" aria-hidden="true" tabindex="-1"></a>\ba</span>
-<span id="cb31-974"><a href="#cb31-974" aria-hidden="true" tabindex="-1"></a>\ell''&amp;=-m/\lambda^2<span class="sc">\\</span></span>
-<span id="cb31-975"><a href="#cb31-975" aria-hidden="true" tabindex="-1"></a>&amp;&lt; 0<span class="sc">\\</span></span>
-<span id="cb31-976"><a href="#cb31-976" aria-hidden="true" tabindex="-1"></a>\hat E<span class="co">[</span><span class="ot">T</span><span class="co">]</span> &amp;= \hat\lambda^{-1}<span class="sc">\\</span></span>
-<span id="cb31-977"><a href="#cb31-977" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{T+U}{m}</span>
-<span id="cb31-978"><a href="#cb31-978" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-979"><a href="#cb31-979" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-939"><a href="#cb31-939" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-940"><a href="#cb31-940" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-941"><a href="#cb31-941" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-942"><a href="#cb31-942" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Estimating $\lambda$ {.smaller}</span></span>
+<span id="cb31-943"><a href="#cb31-943" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-944"><a href="#cb31-944" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Suppose we have $m$ exponential survival times of</span>
+<span id="cb31-945"><a href="#cb31-945" aria-hidden="true" tabindex="-1"></a>    $t_1, t_2,\ldots,t_m$ and $k$ right-censored values at</span>
+<span id="cb31-946"><a href="#cb31-946" aria-hidden="true" tabindex="-1"></a>    $u_1,u_2,\ldots,u_k$.</span>
+<span id="cb31-947"><a href="#cb31-947" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-948"><a href="#cb31-948" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>A survival time of $t_i=10$ means that subject $i$ died at time 10.</span>
+<span id="cb31-949"><a href="#cb31-949" aria-hidden="true" tabindex="-1"></a>    A right-censored time $u_i=10$ means that at time 10, subject $i$</span>
+<span id="cb31-950"><a href="#cb31-950" aria-hidden="true" tabindex="-1"></a>    was still alive and that we have no further follow-up.</span>
+<span id="cb31-951"><a href="#cb31-951" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-952"><a href="#cb31-952" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>For the moment we will assume that the survival distribution is</span>
+<span id="cb31-953"><a href="#cb31-953" aria-hidden="true" tabindex="-1"></a>    exponential and that all the subjects have the same parameter</span>
+<span id="cb31-954"><a href="#cb31-954" aria-hidden="true" tabindex="-1"></a>    $\lambda$.</span>
+<span id="cb31-955"><a href="#cb31-955" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-956"><a href="#cb31-956" aria-hidden="true" tabindex="-1"></a>We have $m$ exponential survival times of $t_1, t_2,\ldots,t_m$ and $k$</span>
+<span id="cb31-957"><a href="#cb31-957" aria-hidden="true" tabindex="-1"></a>right-censored values at $u_1,u_2,\ldots,u_k$. The log-likelihood of an</span>
+<span id="cb31-958"><a href="#cb31-958" aria-hidden="true" tabindex="-1"></a>observed survival time $t_i$ is $$</span>
+<span id="cb31-959"><a href="#cb31-959" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span>\lambda \text{e}^{-\lambda t_i}\right<span class="sc">\}</span> =</span>
+<span id="cb31-960"><a href="#cb31-960" aria-hidden="true" tabindex="-1"></a>\text{log}\left<span class="sc">\{</span>\lambda\right<span class="sc">\}</span>-\lambda t_i</span>
+<span id="cb31-961"><a href="#cb31-961" aria-hidden="true" tabindex="-1"></a>$$ and the likelihood of a censored value is the probability of that</span>
+<span id="cb31-962"><a href="#cb31-962" aria-hidden="true" tabindex="-1"></a>outcome (survival greater than $u_j$) so the log-likelihood is </span>
+<span id="cb31-963"><a href="#cb31-963" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-964"><a href="#cb31-964" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-965"><a href="#cb31-965" aria-hidden="true" tabindex="-1"></a>\ba</span>
+<span id="cb31-966"><a href="#cb31-966" aria-hidden="true" tabindex="-1"></a>\ell_j(\lambda) &amp;= \text{log}\left<span class="sc">\{</span>\lambda \text{e}^{u_j}\right<span class="sc">\}</span> </span>
+<span id="cb31-967"><a href="#cb31-967" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= -\lambda u_j</span>
+<span id="cb31-968"><a href="#cb31-968" aria-hidden="true" tabindex="-1"></a>\ea</span>
+<span id="cb31-969"><a href="#cb31-969" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-970"><a href="#cb31-970" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-971"><a href="#cb31-971" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-972"><a href="#cb31-972" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-973"><a href="#cb31-973" aria-hidden="true" tabindex="-1"></a>:::{#thm-mle-exp}</span>
+<span id="cb31-974"><a href="#cb31-974" aria-hidden="true" tabindex="-1"></a>Let $T=\sum t_i$ and $U=\sum u_j$. Then:</span>
+<span id="cb31-975"><a href="#cb31-975" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-976"><a href="#cb31-976" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-977"><a href="#cb31-977" aria-hidden="true" tabindex="-1"></a>\hat{\lambda}_{ML} = \frac{m}{T+U}</span>
+<span id="cb31-978"><a href="#cb31-978" aria-hidden="true" tabindex="-1"></a>$$ {#eq-mle-exp}</span>
+<span id="cb31-979"><a href="#cb31-979" aria-hidden="true" tabindex="-1"></a>:::</span>
 <span id="cb31-980"><a href="#cb31-980" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb31-981"><a href="#cb31-981" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-982"><a href="#cb31-982" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-983"><a href="#cb31-983" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Fisher Information and Standard Error</span></span>
+<span id="cb31-983"><a href="#cb31-983" aria-hidden="true" tabindex="-1"></a>::: proof</span>
 <span id="cb31-984"><a href="#cb31-984" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb31-985"><a href="#cb31-985" aria-hidden="true" tabindex="-1"></a>$$</span>
 <span id="cb31-986"><a href="#cb31-986" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-987"><a href="#cb31-987" aria-hidden="true" tabindex="-1"></a>E<span class="co">[</span><span class="ot">-\ell''</span><span class="co">]</span></span>
-<span id="cb31-988"><a href="#cb31-988" aria-hidden="true" tabindex="-1"></a>&amp; = m/\lambda^2<span class="sc">\\</span></span>
-<span id="cb31-989"><a href="#cb31-989" aria-hidden="true" tabindex="-1"></a>\text{Var}\left(\hat\lambda\right) </span>
-<span id="cb31-990"><a href="#cb31-990" aria-hidden="true" tabindex="-1"></a>&amp;\approx \left(E<span class="co">[</span><span class="ot">-\ell''</span><span class="co">]</span>\right)^{-1}<span class="sc">\\</span></span>
-<span id="cb31-991"><a href="#cb31-991" aria-hidden="true" tabindex="-1"></a>&amp;=\lambda^2/m<span class="sc">\\</span></span>
-<span id="cb31-992"><a href="#cb31-992" aria-hidden="true" tabindex="-1"></a>\text{SE}\left(\hat\lambda\right) </span>
-<span id="cb31-993"><a href="#cb31-993" aria-hidden="true" tabindex="-1"></a>&amp;= \sqrt{\text{Var}\left(\hat\lambda\right)}<span class="sc">\\</span></span>
-<span id="cb31-994"><a href="#cb31-994" aria-hidden="true" tabindex="-1"></a>&amp;\approx \lambda/\sqrt{m}</span>
-<span id="cb31-995"><a href="#cb31-995" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-996"><a href="#cb31-996" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-987"><a href="#cb31-987" aria-hidden="true" tabindex="-1"></a>\ell(\lambda) &amp;= \sum_{i=1}^m( \ln \lambda-\lambda t_i) + \sum_{j=1}^k (-\lambda u_j)<span class="sc">\\</span></span>
+<span id="cb31-988"><a href="#cb31-988" aria-hidden="true" tabindex="-1"></a>&amp;= m \ln \lambda -(T+U)\lambda<span class="sc">\\</span></span>
+<span id="cb31-989"><a href="#cb31-989" aria-hidden="true" tabindex="-1"></a>\ell'(\lambda) </span>
+<span id="cb31-990"><a href="#cb31-990" aria-hidden="true" tabindex="-1"></a>&amp;=m\lambda^{-1} -(T+U)<span class="sc">\\</span></span>
+<span id="cb31-991"><a href="#cb31-991" aria-hidden="true" tabindex="-1"></a>\hat{\lambda} &amp;= \frac{m}{T+U}</span>
+<span id="cb31-992"><a href="#cb31-992" aria-hidden="true" tabindex="-1"></a>\ea</span>
+<span id="cb31-993"><a href="#cb31-993" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-994"><a href="#cb31-994" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-995"><a href="#cb31-995" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-996"><a href="#cb31-996" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-997"><a href="#cb31-997" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-998"><a href="#cb31-998" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-999"><a href="#cb31-999" aria-hidden="true" tabindex="-1"></a>$\hat\lambda$ depends on the censoring times of the censored</span>
-<span id="cb31-1000"><a href="#cb31-1000" aria-hidden="true" tabindex="-1"></a>observations, but $\text{Var}\left(\hat\lambda\right)$ only depends on</span>
-<span id="cb31-1001"><a href="#cb31-1001" aria-hidden="true" tabindex="-1"></a>the number of uncensored observations, $m$, and not on the number of</span>
-<span id="cb31-1002"><a href="#cb31-1002" aria-hidden="true" tabindex="-1"></a>censored observations ($k$).</span>
-<span id="cb31-1003"><a href="#cb31-1003" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-1004"><a href="#cb31-1004" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1005"><a href="#cb31-1005" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-998"><a href="#cb31-998" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-999"><a href="#cb31-999" aria-hidden="true" tabindex="-1"></a>\ba</span>
+<span id="cb31-1000"><a href="#cb31-1000" aria-hidden="true" tabindex="-1"></a>\ell''&amp;=-m/\lambda^2<span class="sc">\\</span></span>
+<span id="cb31-1001"><a href="#cb31-1001" aria-hidden="true" tabindex="-1"></a>&amp;&lt; 0<span class="sc">\\</span></span>
+<span id="cb31-1002"><a href="#cb31-1002" aria-hidden="true" tabindex="-1"></a>\hat E<span class="co">[</span><span class="ot">T</span><span class="co">]</span> &amp;= \hat\lambda^{-1}<span class="sc">\\</span></span>
+<span id="cb31-1003"><a href="#cb31-1003" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{T+U}{m}</span>
+<span id="cb31-1004"><a href="#cb31-1004" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-1005"><a href="#cb31-1005" aria-hidden="true" tabindex="-1"></a>$$</span>
 <span id="cb31-1006"><a href="#cb31-1006" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1007"><a href="#cb31-1007" aria-hidden="true" tabindex="-1"></a><span class="fu">### Other Parametric Survival Distributions</span></span>
+<span id="cb31-1007"><a href="#cb31-1007" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-1008"><a href="#cb31-1008" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1009"><a href="#cb31-1009" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Any density on $[0,\infty)$ can be a survival distribution, but the</span>
-<span id="cb31-1010"><a href="#cb31-1010" aria-hidden="true" tabindex="-1"></a>    most useful ones are all skew right.</span>
-<span id="cb31-1011"><a href="#cb31-1011" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The most frequently used generalization of the exponential is the <span class="co">[</span><span class="ot">Weibull</span><span class="co">](probability.qmd#sec-weibull)</span>.</span>
-<span id="cb31-1012"><a href="#cb31-1012" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Other common choices are the gamma, log-normal, log-logistic,</span>
-<span id="cb31-1013"><a href="#cb31-1013" aria-hidden="true" tabindex="-1"></a>    Gompertz, inverse Gaussian, and Pareto.</span>
-<span id="cb31-1014"><a href="#cb31-1014" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Most of what we do going forward is non-parametric or</span>
-<span id="cb31-1015"><a href="#cb31-1015" aria-hidden="true" tabindex="-1"></a>    semi-parametric, but sometimes these parametric distributions</span>
-<span id="cb31-1016"><a href="#cb31-1016" aria-hidden="true" tabindex="-1"></a>    provide a useful approach.</span>
-<span id="cb31-1017"><a href="#cb31-1017" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1018"><a href="#cb31-1018" aria-hidden="true" tabindex="-1"></a><span class="fu">## Nonparametric Survival Analysis</span></span>
-<span id="cb31-1019"><a href="#cb31-1019" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1020"><a href="#cb31-1020" aria-hidden="true" tabindex="-1"></a><span class="fu">### Basic ideas</span></span>
-<span id="cb31-1021"><a href="#cb31-1021" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1022"><a href="#cb31-1022" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Mostly, we work without a parametric model.</span>
+<span id="cb31-1009"><a href="#cb31-1009" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Fisher Information and Standard Error</span></span>
+<span id="cb31-1010"><a href="#cb31-1010" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1011"><a href="#cb31-1011" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1012"><a href="#cb31-1012" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-1013"><a href="#cb31-1013" aria-hidden="true" tabindex="-1"></a>E<span class="co">[</span><span class="ot">-\ell''</span><span class="co">]</span></span>
+<span id="cb31-1014"><a href="#cb31-1014" aria-hidden="true" tabindex="-1"></a>&amp; = m/\lambda^2<span class="sc">\\</span></span>
+<span id="cb31-1015"><a href="#cb31-1015" aria-hidden="true" tabindex="-1"></a>\text{Var}\left(\hat\lambda\right) </span>
+<span id="cb31-1016"><a href="#cb31-1016" aria-hidden="true" tabindex="-1"></a>&amp;\approx \left(E<span class="co">[</span><span class="ot">-\ell''</span><span class="co">]</span>\right)^{-1}<span class="sc">\\</span></span>
+<span id="cb31-1017"><a href="#cb31-1017" aria-hidden="true" tabindex="-1"></a>&amp;=\lambda^2/m<span class="sc">\\</span></span>
+<span id="cb31-1018"><a href="#cb31-1018" aria-hidden="true" tabindex="-1"></a>\text{SE}\left(\hat\lambda\right) </span>
+<span id="cb31-1019"><a href="#cb31-1019" aria-hidden="true" tabindex="-1"></a>&amp;= \sqrt{\text{Var}\left(\hat\lambda\right)}<span class="sc">\\</span></span>
+<span id="cb31-1020"><a href="#cb31-1020" aria-hidden="true" tabindex="-1"></a>&amp;\approx \lambda/\sqrt{m}</span>
+<span id="cb31-1021"><a href="#cb31-1021" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-1022"><a href="#cb31-1022" aria-hidden="true" tabindex="-1"></a>$$</span>
 <span id="cb31-1023"><a href="#cb31-1023" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1024"><a href="#cb31-1024" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The first task is to estimate a survival function from data listing</span>
-<span id="cb31-1025"><a href="#cb31-1025" aria-hidden="true" tabindex="-1"></a>    survival times, and censoring times for censored data.</span>
-<span id="cb31-1026"><a href="#cb31-1026" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1027"><a href="#cb31-1027" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>For example one patient may have relapsed at 10 months. Another</span>
-<span id="cb31-1028"><a href="#cb31-1028" aria-hidden="true" tabindex="-1"></a>    might have been followed for 32 months without a relapse having</span>
-<span id="cb31-1029"><a href="#cb31-1029" aria-hidden="true" tabindex="-1"></a>    occurred (censored).</span>
+<span id="cb31-1024"><a href="#cb31-1024" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-1025"><a href="#cb31-1025" aria-hidden="true" tabindex="-1"></a>$\hat\lambda$ depends on the censoring times of the censored</span>
+<span id="cb31-1026"><a href="#cb31-1026" aria-hidden="true" tabindex="-1"></a>observations, but $\text{Var}\left(\hat\lambda\right)$ only depends on</span>
+<span id="cb31-1027"><a href="#cb31-1027" aria-hidden="true" tabindex="-1"></a>the number of uncensored observations, $m$, and not on the number of</span>
+<span id="cb31-1028"><a href="#cb31-1028" aria-hidden="true" tabindex="-1"></a>censored observations ($k$).</span>
+<span id="cb31-1029"><a href="#cb31-1029" aria-hidden="true" tabindex="-1"></a>:::</span>
 <span id="cb31-1030"><a href="#cb31-1030" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1031"><a href="#cb31-1031" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The minimum information we need for each patient is a time and a</span>
-<span id="cb31-1032"><a href="#cb31-1032" aria-hidden="true" tabindex="-1"></a>    censoring variable which is 1 if the event occurred at the indicated</span>
-<span id="cb31-1033"><a href="#cb31-1033" aria-hidden="true" tabindex="-1"></a>    time and 0 if this is a censoring time.</span>
+<span id="cb31-1031"><a href="#cb31-1031" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1032"><a href="#cb31-1032" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1033"><a href="#cb31-1033" aria-hidden="true" tabindex="-1"></a><span class="fu">### Other Parametric Survival Distributions</span></span>
 <span id="cb31-1034"><a href="#cb31-1034" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1035"><a href="#cb31-1035" aria-hidden="true" tabindex="-1"></a><span class="fu">## Example: clinical trial for pediatric acute leukemia</span></span>
-<span id="cb31-1036"><a href="#cb31-1036" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1037"><a href="#cb31-1037" aria-hidden="true" tabindex="-1"></a><span class="fu">### Overview of study {.smaller}</span></span>
-<span id="cb31-1038"><a href="#cb31-1038" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1039"><a href="#cb31-1039" aria-hidden="true" tabindex="-1"></a>This is from a clinical trial in 1963 for 6-MP treatment vs. placebo for</span>
-<span id="cb31-1040"><a href="#cb31-1040" aria-hidden="true" tabindex="-1"></a>Acute Leukemia in 42 children.</span>
-<span id="cb31-1041"><a href="#cb31-1041" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1042"><a href="#cb31-1042" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Pairs of children:</span>
+<span id="cb31-1035"><a href="#cb31-1035" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Any density on $[0,\infty)$ can be a survival distribution, but the</span>
+<span id="cb31-1036"><a href="#cb31-1036" aria-hidden="true" tabindex="-1"></a>    most useful ones are all skew right.</span>
+<span id="cb31-1037"><a href="#cb31-1037" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The most frequently used generalization of the exponential is the <span class="co">[</span><span class="ot">Weibull</span><span class="co">](probability.qmd#sec-weibull)</span>.</span>
+<span id="cb31-1038"><a href="#cb31-1038" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Other common choices are the gamma, log-normal, log-logistic,</span>
+<span id="cb31-1039"><a href="#cb31-1039" aria-hidden="true" tabindex="-1"></a>    Gompertz, inverse Gaussian, and Pareto.</span>
+<span id="cb31-1040"><a href="#cb31-1040" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Most of what we do going forward is non-parametric or</span>
+<span id="cb31-1041"><a href="#cb31-1041" aria-hidden="true" tabindex="-1"></a>    semi-parametric, but sometimes these parametric distributions</span>
+<span id="cb31-1042"><a href="#cb31-1042" aria-hidden="true" tabindex="-1"></a>    provide a useful approach.</span>
 <span id="cb31-1043"><a href="#cb31-1043" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1044"><a href="#cb31-1044" aria-hidden="true" tabindex="-1"></a><span class="ss">    -   </span>matched by remission status at the time of treatment (<span class="in">`remstat`</span>:</span>
-<span id="cb31-1045"><a href="#cb31-1045" aria-hidden="true" tabindex="-1"></a>        <span class="in">`1`</span> = partial, <span class="in">`2`</span> = complete)</span>
-<span id="cb31-1046"><a href="#cb31-1046" aria-hidden="true" tabindex="-1"></a><span class="ss">    -   </span>randomized to 6-MP (exit times in <span class="in">`t2`</span>) or placebo (exit times</span>
-<span id="cb31-1047"><a href="#cb31-1047" aria-hidden="true" tabindex="-1"></a>        in <span class="in">`t1`</span>)</span>
-<span id="cb31-1048"><a href="#cb31-1048" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1049"><a href="#cb31-1049" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Followed until relapse or end of study.</span>
-<span id="cb31-1050"><a href="#cb31-1050" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1051"><a href="#cb31-1051" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>All of the placebo group relapsed, but some of the 6-MP group were</span>
-<span id="cb31-1052"><a href="#cb31-1052" aria-hidden="true" tabindex="-1"></a>    censored (which means they were still in remission); indicated by</span>
-<span id="cb31-1053"><a href="#cb31-1053" aria-hidden="true" tabindex="-1"></a>    <span class="in">`relapse`</span> variable (<span class="in">`0`</span> = censored, <span class="in">`1`</span> = relapse).</span>
-<span id="cb31-1054"><a href="#cb31-1054" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1055"><a href="#cb31-1055" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>6-MP = 6-Mercaptopurine (Purinethol) is an anti-cancer</span>
-<span id="cb31-1056"><a href="#cb31-1056" aria-hidden="true" tabindex="-1"></a>    ("antineoplastic" or "cytotoxic") chemotherapy drug used currently</span>
-<span id="cb31-1057"><a href="#cb31-1057" aria-hidden="true" tabindex="-1"></a>    for Acute lymphoblastic leukemia (ALL). It is classified as an</span>
-<span id="cb31-1058"><a href="#cb31-1058" aria-hidden="true" tabindex="-1"></a>    antimetabolite.</span>
-<span id="cb31-1059"><a href="#cb31-1059" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1060"><a href="#cb31-1060" aria-hidden="true" tabindex="-1"></a><span class="fu">### Study design {.smaller}</span></span>
-<span id="cb31-1061"><a href="#cb31-1061" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1062"><a href="#cb31-1062" aria-hidden="true" tabindex="-1"></a><span class="ss">- </span>Clinical trial in 1963 for 6-MP treatment vs. placebo for Acute Leukemia</span>
-<span id="cb31-1063"><a href="#cb31-1063" aria-hidden="true" tabindex="-1"></a>in 42 children. </span>
-<span id="cb31-1064"><a href="#cb31-1064" aria-hidden="true" tabindex="-1"></a><span class="ss">- </span>Pairs of children:</span>
-<span id="cb31-1065"><a href="#cb31-1065" aria-hidden="true" tabindex="-1"></a><span class="ss">  - </span>matched by remission status at the time of treatment (<span class="in">`remstat`</span>)</span>
-<span id="cb31-1066"><a href="#cb31-1066" aria-hidden="true" tabindex="-1"></a><span class="ss">     - </span><span class="in">`remstat`</span> = 1: partial </span>
-<span id="cb31-1067"><a href="#cb31-1067" aria-hidden="true" tabindex="-1"></a><span class="ss">     - </span><span class="in">`remstat`</span> = 2: complete</span>
-<span id="cb31-1068"><a href="#cb31-1068" aria-hidden="true" tabindex="-1"></a><span class="ss">  - </span>randomized to 6-MP (exit time: <span class="in">`t2`</span>) or placebo (<span class="in">`t1`</span>). </span>
-<span id="cb31-1069"><a href="#cb31-1069" aria-hidden="true" tabindex="-1"></a><span class="ss">- </span>Followed until relapse or end of study. </span>
-<span id="cb31-1070"><a href="#cb31-1070" aria-hidden="true" tabindex="-1"></a><span class="ss">  - </span>All of the placebo group relapsed, </span>
-<span id="cb31-1071"><a href="#cb31-1071" aria-hidden="true" tabindex="-1"></a><span class="ss">  - </span>Some of the 6-MP group were censored.</span>
-<span id="cb31-1072"><a href="#cb31-1072" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1073"><a href="#cb31-1073" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1044"><a href="#cb31-1044" aria-hidden="true" tabindex="-1"></a><span class="fu">## Nonparametric Survival Analysis</span></span>
+<span id="cb31-1045"><a href="#cb31-1045" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1046"><a href="#cb31-1046" aria-hidden="true" tabindex="-1"></a><span class="fu">### Basic ideas</span></span>
+<span id="cb31-1047"><a href="#cb31-1047" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1048"><a href="#cb31-1048" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Mostly, we work without a parametric model.</span>
+<span id="cb31-1049"><a href="#cb31-1049" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1050"><a href="#cb31-1050" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The first task is to estimate a survival function from data listing</span>
+<span id="cb31-1051"><a href="#cb31-1051" aria-hidden="true" tabindex="-1"></a>    survival times, and censoring times for censored data.</span>
+<span id="cb31-1052"><a href="#cb31-1052" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1053"><a href="#cb31-1053" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>For example one patient may have relapsed at 10 months. Another</span>
+<span id="cb31-1054"><a href="#cb31-1054" aria-hidden="true" tabindex="-1"></a>    might have been followed for 32 months without a relapse having</span>
+<span id="cb31-1055"><a href="#cb31-1055" aria-hidden="true" tabindex="-1"></a>    occurred (censored).</span>
+<span id="cb31-1056"><a href="#cb31-1056" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1057"><a href="#cb31-1057" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The minimum information we need for each patient is a time and a</span>
+<span id="cb31-1058"><a href="#cb31-1058" aria-hidden="true" tabindex="-1"></a>    censoring variable which is 1 if the event occurred at the indicated</span>
+<span id="cb31-1059"><a href="#cb31-1059" aria-hidden="true" tabindex="-1"></a>    time and 0 if this is a censoring time.</span>
+<span id="cb31-1060"><a href="#cb31-1060" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1061"><a href="#cb31-1061" aria-hidden="true" tabindex="-1"></a><span class="fu">## Example: clinical trial for pediatric acute leukemia</span></span>
+<span id="cb31-1062"><a href="#cb31-1062" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1063"><a href="#cb31-1063" aria-hidden="true" tabindex="-1"></a><span class="fu">### Overview of study {.smaller}</span></span>
+<span id="cb31-1064"><a href="#cb31-1064" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1065"><a href="#cb31-1065" aria-hidden="true" tabindex="-1"></a>This is from a clinical trial in 1963 for 6-MP treatment vs. placebo for</span>
+<span id="cb31-1066"><a href="#cb31-1066" aria-hidden="true" tabindex="-1"></a>Acute Leukemia in 42 children.</span>
+<span id="cb31-1067"><a href="#cb31-1067" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1068"><a href="#cb31-1068" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Pairs of children:</span>
+<span id="cb31-1069"><a href="#cb31-1069" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1070"><a href="#cb31-1070" aria-hidden="true" tabindex="-1"></a><span class="ss">    -   </span>matched by remission status at the time of treatment (<span class="in">`remstat`</span>:</span>
+<span id="cb31-1071"><a href="#cb31-1071" aria-hidden="true" tabindex="-1"></a>        <span class="in">`1`</span> = partial, <span class="in">`2`</span> = complete)</span>
+<span id="cb31-1072"><a href="#cb31-1072" aria-hidden="true" tabindex="-1"></a><span class="ss">    -   </span>randomized to 6-MP (exit times in <span class="in">`t2`</span>) or placebo (exit times</span>
+<span id="cb31-1073"><a href="#cb31-1073" aria-hidden="true" tabindex="-1"></a>        in <span class="in">`t1`</span>)</span>
 <span id="cb31-1074"><a href="#cb31-1074" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1077"><a href="#cb31-1077" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1078"><a href="#cb31-1078" aria-hidden="true" tabindex="-1"></a><span class="co">#| tbl-cap: "`drug6mp` pediatric acute leukemia data"</span></span>
-<span id="cb31-1079"><a href="#cb31-1079" aria-hidden="true" tabindex="-1"></a><span class="co">#| label: tbl-drug6mp</span></span>
-<span id="cb31-1080"><a href="#cb31-1080" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(KMsurv)</span>
-<span id="cb31-1081"><a href="#cb31-1081" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(drug6mp)</span>
-<span id="cb31-1082"><a href="#cb31-1082" aria-hidden="true" tabindex="-1"></a>drug6mp <span class="ot">=</span> drug6mp <span class="sc">|&gt;</span> <span class="fu">as_tibble</span>() <span class="sc">|&gt;</span> <span class="fu">print</span>()</span>
-<span id="cb31-1083"><a href="#cb31-1083" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1084"><a href="#cb31-1084" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1085"><a href="#cb31-1085" aria-hidden="true" tabindex="-1"></a><span class="fu">### Data documentation for `drug6mp`</span></span>
-<span id="cb31-1086"><a href="#cb31-1086" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1087"><a href="#cb31-1087" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, printr.help.sections = c("description", "format")}</span></span>
-<span id="cb31-1088"><a href="#cb31-1088" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: Data documentation for `drug6mp`</span></span>
-<span id="cb31-1089"><a href="#cb31-1089" aria-hidden="true" tabindex="-1"></a><span class="in">#| label: fig-drug6mp-helpdoc</span></span>
-<span id="cb31-1090"><a href="#cb31-1090" aria-hidden="true" tabindex="-1"></a><span class="in"># library(printr) # inserts help-file output into markdown output</span></span>
-<span id="cb31-1091"><a href="#cb31-1091" aria-hidden="true" tabindex="-1"></a><span class="in">library(KMsurv)</span></span>
-<span id="cb31-1092"><a href="#cb31-1092" aria-hidden="true" tabindex="-1"></a><span class="in">?drug6mp</span></span>
-<span id="cb31-1093"><a href="#cb31-1093" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1094"><a href="#cb31-1094" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1095"><a href="#cb31-1095" aria-hidden="true" tabindex="-1"></a><span class="fu">### Descriptive Statistics {.smaller}</span></span>
-<span id="cb31-1096"><a href="#cb31-1096" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1099"><a href="#cb31-1099" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1100"><a href="#cb31-1100" aria-hidden="true" tabindex="-1"></a><span class="co">#| tbl-cap: "Summary statistics for `drug6mp` data"</span></span>
-<span id="cb31-1101"><a href="#cb31-1101" aria-hidden="true" tabindex="-1"></a><span class="co">#| label: tbl-drug6mp-summary</span></span>
-<span id="cb31-1102"><a href="#cb31-1102" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(drug6mp)</span>
-<span id="cb31-1103"><a href="#cb31-1103" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1104"><a href="#cb31-1104" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1105"><a href="#cb31-1105" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-1106"><a href="#cb31-1106" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1107"><a href="#cb31-1107" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The average time in each group is not useful. Some of the 6-MP</span>
-<span id="cb31-1108"><a href="#cb31-1108" aria-hidden="true" tabindex="-1"></a>    patients have not relapsed at the time recorded, while all of the</span>
-<span id="cb31-1109"><a href="#cb31-1109" aria-hidden="true" tabindex="-1"></a>    placebo patients have relapsed.</span>
-<span id="cb31-1110"><a href="#cb31-1110" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The median time is not really useful either because so many of the</span>
-<span id="cb31-1111"><a href="#cb31-1111" aria-hidden="true" tabindex="-1"></a>    6-MP patients have not relapsed (12/21).</span>
-<span id="cb31-1112"><a href="#cb31-1112" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Both are biased down in the 6-MP group. Remember that lower times</span>
-<span id="cb31-1113"><a href="#cb31-1113" aria-hidden="true" tabindex="-1"></a>    are worse since they indicate sooner recurrence.</span>
-<span id="cb31-1114"><a href="#cb31-1114" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-1115"><a href="#cb31-1115" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1116"><a href="#cb31-1116" aria-hidden="true" tabindex="-1"></a><span class="fu">### Exponential model</span></span>
-<span id="cb31-1117"><a href="#cb31-1117" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1118"><a href="#cb31-1118" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-1119"><a href="#cb31-1119" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>We *can* compute the hazard rate, assuming an exponential model:</span>
-<span id="cb31-1120"><a href="#cb31-1120" aria-hidden="true" tabindex="-1"></a>number of relapses divided by the sum of the exit times (@eq-mle-exp).</span>
-<span id="cb31-1121"><a href="#cb31-1121" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-1075"><a href="#cb31-1075" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Followed until relapse or end of study.</span>
+<span id="cb31-1076"><a href="#cb31-1076" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1077"><a href="#cb31-1077" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>All of the placebo group relapsed, but some of the 6-MP group were</span>
+<span id="cb31-1078"><a href="#cb31-1078" aria-hidden="true" tabindex="-1"></a>    censored (which means they were still in remission); indicated by</span>
+<span id="cb31-1079"><a href="#cb31-1079" aria-hidden="true" tabindex="-1"></a>    <span class="in">`relapse`</span> variable (<span class="in">`0`</span> = censored, <span class="in">`1`</span> = relapse).</span>
+<span id="cb31-1080"><a href="#cb31-1080" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1081"><a href="#cb31-1081" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>6-MP = 6-Mercaptopurine (Purinethol) is an anti-cancer</span>
+<span id="cb31-1082"><a href="#cb31-1082" aria-hidden="true" tabindex="-1"></a>    ("antineoplastic" or "cytotoxic") chemotherapy drug used currently</span>
+<span id="cb31-1083"><a href="#cb31-1083" aria-hidden="true" tabindex="-1"></a>    for Acute lymphoblastic leukemia (ALL). It is classified as an</span>
+<span id="cb31-1084"><a href="#cb31-1084" aria-hidden="true" tabindex="-1"></a>    antimetabolite.</span>
+<span id="cb31-1085"><a href="#cb31-1085" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1086"><a href="#cb31-1086" aria-hidden="true" tabindex="-1"></a><span class="fu">### Study design {.smaller}</span></span>
+<span id="cb31-1087"><a href="#cb31-1087" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1088"><a href="#cb31-1088" aria-hidden="true" tabindex="-1"></a><span class="ss">- </span>Clinical trial in 1963 for 6-MP treatment vs. placebo for Acute Leukemia</span>
+<span id="cb31-1089"><a href="#cb31-1089" aria-hidden="true" tabindex="-1"></a>in 42 children. </span>
+<span id="cb31-1090"><a href="#cb31-1090" aria-hidden="true" tabindex="-1"></a><span class="ss">- </span>Pairs of children:</span>
+<span id="cb31-1091"><a href="#cb31-1091" aria-hidden="true" tabindex="-1"></a><span class="ss">  - </span>matched by remission status at the time of treatment (<span class="in">`remstat`</span>)</span>
+<span id="cb31-1092"><a href="#cb31-1092" aria-hidden="true" tabindex="-1"></a><span class="ss">     - </span><span class="in">`remstat`</span> = 1: partial </span>
+<span id="cb31-1093"><a href="#cb31-1093" aria-hidden="true" tabindex="-1"></a><span class="ss">     - </span><span class="in">`remstat`</span> = 2: complete</span>
+<span id="cb31-1094"><a href="#cb31-1094" aria-hidden="true" tabindex="-1"></a><span class="ss">  - </span>randomized to 6-MP (exit time: <span class="in">`t2`</span>) or placebo (<span class="in">`t1`</span>). </span>
+<span id="cb31-1095"><a href="#cb31-1095" aria-hidden="true" tabindex="-1"></a><span class="ss">- </span>Followed until relapse or end of study. </span>
+<span id="cb31-1096"><a href="#cb31-1096" aria-hidden="true" tabindex="-1"></a><span class="ss">  - </span>All of the placebo group relapsed, </span>
+<span id="cb31-1097"><a href="#cb31-1097" aria-hidden="true" tabindex="-1"></a><span class="ss">  - </span>Some of the 6-MP group were censored.</span>
+<span id="cb31-1098"><a href="#cb31-1098" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1099"><a href="#cb31-1099" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1100"><a href="#cb31-1100" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1103"><a href="#cb31-1103" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1104"><a href="#cb31-1104" aria-hidden="true" tabindex="-1"></a><span class="co">#| tbl-cap: "`drug6mp` pediatric acute leukemia data"</span></span>
+<span id="cb31-1105"><a href="#cb31-1105" aria-hidden="true" tabindex="-1"></a><span class="co">#| label: tbl-drug6mp</span></span>
+<span id="cb31-1106"><a href="#cb31-1106" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(KMsurv)</span>
+<span id="cb31-1107"><a href="#cb31-1107" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(drug6mp)</span>
+<span id="cb31-1108"><a href="#cb31-1108" aria-hidden="true" tabindex="-1"></a>drug6mp <span class="ot">=</span> drug6mp <span class="sc">|&gt;</span> <span class="fu">as_tibble</span>() <span class="sc">|&gt;</span> <span class="fu">print</span>()</span>
+<span id="cb31-1109"><a href="#cb31-1109" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1110"><a href="#cb31-1110" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1111"><a href="#cb31-1111" aria-hidden="true" tabindex="-1"></a><span class="fu">### Data documentation for `drug6mp`</span></span>
+<span id="cb31-1112"><a href="#cb31-1112" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1113"><a href="#cb31-1113" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, printr.help.sections = c("description", "format")}</span></span>
+<span id="cb31-1114"><a href="#cb31-1114" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: Data documentation for `drug6mp`</span></span>
+<span id="cb31-1115"><a href="#cb31-1115" aria-hidden="true" tabindex="-1"></a><span class="in">#| label: fig-drug6mp-helpdoc</span></span>
+<span id="cb31-1116"><a href="#cb31-1116" aria-hidden="true" tabindex="-1"></a><span class="in"># library(printr) # inserts help-file output into markdown output</span></span>
+<span id="cb31-1117"><a href="#cb31-1117" aria-hidden="true" tabindex="-1"></a><span class="in">library(KMsurv)</span></span>
+<span id="cb31-1118"><a href="#cb31-1118" aria-hidden="true" tabindex="-1"></a><span class="in">?drug6mp</span></span>
+<span id="cb31-1119"><a href="#cb31-1119" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1120"><a href="#cb31-1120" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1121"><a href="#cb31-1121" aria-hidden="true" tabindex="-1"></a><span class="fu">### Descriptive Statistics {.smaller}</span></span>
 <span id="cb31-1122"><a href="#cb31-1122" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1123"><a href="#cb31-1123" aria-hidden="true" tabindex="-1"></a>$$\hat\lambda = \frac{\sumin D_i}{\sumin Y_i}$$</span>
-<span id="cb31-1124"><a href="#cb31-1124" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1125"><a href="#cb31-1125" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-1126"><a href="#cb31-1126" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>For the placebo, that is just the reciprocal of the mean time:</span>
-<span id="cb31-1127"><a href="#cb31-1127" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-1128"><a href="#cb31-1128" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1129"><a href="#cb31-1129" aria-hidden="true" tabindex="-1"></a>\ba</span>
-<span id="cb31-1130"><a href="#cb31-1130" aria-hidden="true" tabindex="-1"></a>\hat \lambda_{\text{placebo}} </span>
-<span id="cb31-1131"><a href="#cb31-1131" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{\sumin D_i}{\sumin Y_i}</span>
-<span id="cb31-1132"><a href="#cb31-1132" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \frac{\sumin 1}{\sumin Y_i}</span>
-<span id="cb31-1133"><a href="#cb31-1133" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \frac{n}{\sumin Y_i}</span>
-<span id="cb31-1134"><a href="#cb31-1134" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \frac{1}{\bar{Y}}</span>
-<span id="cb31-1135"><a href="#cb31-1135" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \frac{1}{<span class="in">`r drug6mp |&gt; pull(t1) |&gt; mean()`</span>}</span>
-<span id="cb31-1136"><a href="#cb31-1136" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= <span class="in">`r 1/(drug6mp |&gt; pull(t1) |&gt; mean())`</span></span>
-<span id="cb31-1137"><a href="#cb31-1137" aria-hidden="true" tabindex="-1"></a>\ea</span>
-<span id="cb31-1138"><a href="#cb31-1138" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1139"><a href="#cb31-1139" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1140"><a href="#cb31-1140" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1125"><a href="#cb31-1125" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1126"><a href="#cb31-1126" aria-hidden="true" tabindex="-1"></a><span class="co">#| tbl-cap: "Summary statistics for `drug6mp` data"</span></span>
+<span id="cb31-1127"><a href="#cb31-1127" aria-hidden="true" tabindex="-1"></a><span class="co">#| label: tbl-drug6mp-summary</span></span>
+<span id="cb31-1128"><a href="#cb31-1128" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(drug6mp)</span>
+<span id="cb31-1129"><a href="#cb31-1129" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1130"><a href="#cb31-1130" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1131"><a href="#cb31-1131" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-1132"><a href="#cb31-1132" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1133"><a href="#cb31-1133" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The average time in each group is not useful. Some of the 6-MP</span>
+<span id="cb31-1134"><a href="#cb31-1134" aria-hidden="true" tabindex="-1"></a>    patients have not relapsed at the time recorded, while all of the</span>
+<span id="cb31-1135"><a href="#cb31-1135" aria-hidden="true" tabindex="-1"></a>    placebo patients have relapsed.</span>
+<span id="cb31-1136"><a href="#cb31-1136" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The median time is not really useful either because so many of the</span>
+<span id="cb31-1137"><a href="#cb31-1137" aria-hidden="true" tabindex="-1"></a>    6-MP patients have not relapsed (12/21).</span>
+<span id="cb31-1138"><a href="#cb31-1138" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>Both are biased down in the 6-MP group. Remember that lower times</span>
+<span id="cb31-1139"><a href="#cb31-1139" aria-hidden="true" tabindex="-1"></a>    are worse since they indicate sooner recurrence.</span>
+<span id="cb31-1140"><a href="#cb31-1140" aria-hidden="true" tabindex="-1"></a>:::</span>
 <span id="cb31-1141"><a href="#cb31-1141" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1142"><a href="#cb31-1142" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>For the 6-MP group, $\hat\lambda = 9/359 = 0.025$</span>
+<span id="cb31-1142"><a href="#cb31-1142" aria-hidden="true" tabindex="-1"></a><span class="fu">### Exponential model</span></span>
 <span id="cb31-1143"><a href="#cb31-1143" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1144"><a href="#cb31-1144" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1145"><a href="#cb31-1145" aria-hidden="true" tabindex="-1"></a>\ba</span>
-<span id="cb31-1146"><a href="#cb31-1146" aria-hidden="true" tabindex="-1"></a>\hat \lambda_{\text{6-MP}} </span>
-<span id="cb31-1147"><a href="#cb31-1147" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{\sumin D_i}{\sumin Y_i}</span>
-<span id="cb31-1148"><a href="#cb31-1148" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \frac{9}{359}</span>
-<span id="cb31-1149"><a href="#cb31-1149" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= <span class="in">`r 9/359`</span></span>
-<span id="cb31-1150"><a href="#cb31-1150" aria-hidden="true" tabindex="-1"></a>\ea</span>
-<span id="cb31-1151"><a href="#cb31-1151" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1152"><a href="#cb31-1152" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1153"><a href="#cb31-1153" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The estimated hazard in the placebo group is 4.6 times as</span>
-<span id="cb31-1154"><a href="#cb31-1154" aria-hidden="true" tabindex="-1"></a>    large as in the 6-MP group (assuming the hazard is constant over time).</span>
-<span id="cb31-1155"><a href="#cb31-1155" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1156"><a href="#cb31-1156" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1157"><a href="#cb31-1157" aria-hidden="true" tabindex="-1"></a><span class="fu">## The Kaplan-Meier Product Limit Estimator</span></span>
-<span id="cb31-1158"><a href="#cb31-1158" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1159"><a href="#cb31-1159" aria-hidden="true" tabindex="-1"></a><span class="fu">### Estimating survival in datasets without censoring</span></span>
-<span id="cb31-1160"><a href="#cb31-1160" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1161"><a href="#cb31-1161" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-1162"><a href="#cb31-1162" aria-hidden="true" tabindex="-1"></a>In the <span class="in">`drug6mp`</span> dataset, the estimated survival function for the placebo patients is easy to compute. </span>
-<span id="cb31-1163"><a href="#cb31-1163" aria-hidden="true" tabindex="-1"></a>For any time $t$ in months, $S(t)$ is the fraction of patients with times greater than $t$:</span>
-<span id="cb31-1164"><a href="#cb31-1164" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-1144"><a href="#cb31-1144" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-1145"><a href="#cb31-1145" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>We *can* compute the hazard rate, assuming an exponential model:</span>
+<span id="cb31-1146"><a href="#cb31-1146" aria-hidden="true" tabindex="-1"></a>number of relapses divided by the sum of the exit times (@eq-mle-exp).</span>
+<span id="cb31-1147"><a href="#cb31-1147" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-1148"><a href="#cb31-1148" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1149"><a href="#cb31-1149" aria-hidden="true" tabindex="-1"></a>$$\hat\lambda = \frac{\sumin D_i}{\sumin Y_i}$$</span>
+<span id="cb31-1150"><a href="#cb31-1150" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1151"><a href="#cb31-1151" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-1152"><a href="#cb31-1152" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>For the placebo, that is just the reciprocal of the mean time:</span>
+<span id="cb31-1153"><a href="#cb31-1153" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-1154"><a href="#cb31-1154" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1155"><a href="#cb31-1155" aria-hidden="true" tabindex="-1"></a>\ba</span>
+<span id="cb31-1156"><a href="#cb31-1156" aria-hidden="true" tabindex="-1"></a>\hat \lambda_{\text{placebo}} </span>
+<span id="cb31-1157"><a href="#cb31-1157" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{\sumin D_i}{\sumin Y_i}</span>
+<span id="cb31-1158"><a href="#cb31-1158" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \frac{\sumin 1}{\sumin Y_i}</span>
+<span id="cb31-1159"><a href="#cb31-1159" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \frac{n}{\sumin Y_i}</span>
+<span id="cb31-1160"><a href="#cb31-1160" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \frac{1}{\bar{Y}}</span>
+<span id="cb31-1161"><a href="#cb31-1161" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \frac{1}{<span class="in">`r drug6mp |&gt; pull(t1) |&gt; mean()`</span>}</span>
+<span id="cb31-1162"><a href="#cb31-1162" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= <span class="in">`r 1/(drug6mp |&gt; pull(t1) |&gt; mean())`</span></span>
+<span id="cb31-1163"><a href="#cb31-1163" aria-hidden="true" tabindex="-1"></a>\ea</span>
+<span id="cb31-1164"><a href="#cb31-1164" aria-hidden="true" tabindex="-1"></a>$$</span>
 <span id="cb31-1165"><a href="#cb31-1165" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1168"><a href="#cb31-1168" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1169"><a href="#cb31-1169" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1170"><a href="#cb31-1170" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1171"><a href="#cb31-1171" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1172"><a href="#cb31-1172" aria-hidden="true" tabindex="-1"></a><span class="fu">### Estimating survival in datasets with censoring</span></span>
-<span id="cb31-1173"><a href="#cb31-1173" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1174"><a href="#cb31-1174" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>For the 6-MP patients, we cannot ignore the censored data because we</span>
-<span id="cb31-1175"><a href="#cb31-1175" aria-hidden="true" tabindex="-1"></a>    know that the time to relapse is greater than the censoring time.</span>
-<span id="cb31-1176"><a href="#cb31-1176" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1177"><a href="#cb31-1177" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>For any time $t$ in months, we know that 6-MP patients with times</span>
-<span id="cb31-1178"><a href="#cb31-1178" aria-hidden="true" tabindex="-1"></a>    greater than $t$ have not relapsed, and those with relapse time less</span>
-<span id="cb31-1179"><a href="#cb31-1179" aria-hidden="true" tabindex="-1"></a>    than $t$ have relapsed, but we don't know if patients with censored</span>
-<span id="cb31-1180"><a href="#cb31-1180" aria-hidden="true" tabindex="-1"></a>    time less than $t$ have relapsed or not.</span>
+<span id="cb31-1166"><a href="#cb31-1166" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1167"><a href="#cb31-1167" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1168"><a href="#cb31-1168" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>For the 6-MP group, $\hat\lambda = 9/359 = 0.025$</span>
+<span id="cb31-1169"><a href="#cb31-1169" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1170"><a href="#cb31-1170" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1171"><a href="#cb31-1171" aria-hidden="true" tabindex="-1"></a>\ba</span>
+<span id="cb31-1172"><a href="#cb31-1172" aria-hidden="true" tabindex="-1"></a>\hat \lambda_{\text{6-MP}} </span>
+<span id="cb31-1173"><a href="#cb31-1173" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{\sumin D_i}{\sumin Y_i}</span>
+<span id="cb31-1174"><a href="#cb31-1174" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \frac{9}{359}</span>
+<span id="cb31-1175"><a href="#cb31-1175" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= <span class="in">`r 9/359`</span></span>
+<span id="cb31-1176"><a href="#cb31-1176" aria-hidden="true" tabindex="-1"></a>\ea</span>
+<span id="cb31-1177"><a href="#cb31-1177" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1178"><a href="#cb31-1178" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1179"><a href="#cb31-1179" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The estimated hazard in the placebo group is 4.6 times as</span>
+<span id="cb31-1180"><a href="#cb31-1180" aria-hidden="true" tabindex="-1"></a>    large as in the 6-MP group (assuming the hazard is constant over time).</span>
 <span id="cb31-1181"><a href="#cb31-1181" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1182"><a href="#cb31-1182" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The procedure we usually use is the Kaplan-Meier product-limit</span>
-<span id="cb31-1183"><a href="#cb31-1183" aria-hidden="true" tabindex="-1"></a>    estimator of the survival function.</span>
+<span id="cb31-1182"><a href="#cb31-1182" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1183"><a href="#cb31-1183" aria-hidden="true" tabindex="-1"></a><span class="fu">## The Kaplan-Meier Product Limit Estimator</span></span>
 <span id="cb31-1184"><a href="#cb31-1184" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1185"><a href="#cb31-1185" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The Kaplan-Meier estimator is a step function (like the empirical</span>
-<span id="cb31-1186"><a href="#cb31-1186" aria-hidden="true" tabindex="-1"></a>    cdf), which changes value only at the event times, not at the</span>
-<span id="cb31-1187"><a href="#cb31-1187" aria-hidden="true" tabindex="-1"></a>    censoring times.</span>
-<span id="cb31-1188"><a href="#cb31-1188" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1189"><a href="#cb31-1189" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>At each event time $t$, we compute the at-risk group size $Y$, which</span>
-<span id="cb31-1190"><a href="#cb31-1190" aria-hidden="true" tabindex="-1"></a>    is all those observations whose event time or censoring time is at</span>
-<span id="cb31-1191"><a href="#cb31-1191" aria-hidden="true" tabindex="-1"></a>    least $t$.</span>
-<span id="cb31-1192"><a href="#cb31-1192" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1193"><a href="#cb31-1193" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>If $d$ of the observations have an event time (not a censoring time)</span>
-<span id="cb31-1194"><a href="#cb31-1194" aria-hidden="true" tabindex="-1"></a>    of $t$, then the group of survivors immediately following time $t$</span>
-<span id="cb31-1195"><a href="#cb31-1195" aria-hidden="true" tabindex="-1"></a>    is reduced by the fraction $$\frac{Y-d}{Y}=1-\frac{d}{Y}$$</span>
+<span id="cb31-1185"><a href="#cb31-1185" aria-hidden="true" tabindex="-1"></a><span class="fu">### Estimating survival in datasets without censoring</span></span>
+<span id="cb31-1186"><a href="#cb31-1186" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1187"><a href="#cb31-1187" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-1188"><a href="#cb31-1188" aria-hidden="true" tabindex="-1"></a>In the <span class="in">`drug6mp`</span> dataset, the estimated survival function for the placebo patients is easy to compute. </span>
+<span id="cb31-1189"><a href="#cb31-1189" aria-hidden="true" tabindex="-1"></a>For any time $t$ in months, $S(t)$ is the fraction of patients with times greater than $t$:</span>
+<span id="cb31-1190"><a href="#cb31-1190" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-1191"><a href="#cb31-1191" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1194"><a href="#cb31-1194" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1195"><a href="#cb31-1195" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
 <span id="cb31-1196"><a href="#cb31-1196" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1197"><a href="#cb31-1197" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1198"><a href="#cb31-1198" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1199"><a href="#cb31-1199" aria-hidden="true" tabindex="-1"></a>:::{#def-KM-estimator}</span>
-<span id="cb31-1200"><a href="#cb31-1200" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1201"><a href="#cb31-1201" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Kaplan-Meier Product-Limit Estimator of Survival Function</span></span>
+<span id="cb31-1197"><a href="#cb31-1197" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1198"><a href="#cb31-1198" aria-hidden="true" tabindex="-1"></a><span class="fu">### Estimating survival in datasets with censoring</span></span>
+<span id="cb31-1199"><a href="#cb31-1199" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1200"><a href="#cb31-1200" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>For the 6-MP patients, we cannot ignore the censored data because we</span>
+<span id="cb31-1201"><a href="#cb31-1201" aria-hidden="true" tabindex="-1"></a>    know that the time to relapse is greater than the censoring time.</span>
 <span id="cb31-1202"><a href="#cb31-1202" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1203"><a href="#cb31-1203" aria-hidden="true" tabindex="-1"></a>If the event times are $t_i$ with events per time of $d_i$ ($1\le i \le k$), </span>
-<span id="cb31-1204"><a href="#cb31-1204" aria-hidden="true" tabindex="-1"></a>then the **Kaplan-Meier Product-Limit Estimator** </span>
-<span id="cb31-1205"><a href="#cb31-1205" aria-hidden="true" tabindex="-1"></a>of the <span class="co">[</span><span class="ot">survival function</span><span class="co">](@def-surv-fn)</span> is:</span>
-<span id="cb31-1206"><a href="#cb31-1206" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1207"><a href="#cb31-1207" aria-hidden="true" tabindex="-1"></a>$$\hat S(t) = \prod_{t_i &lt; t} \sb{\frac{1-d_i}{Y_i}}$${#eq-km-surv-est}</span>
-<span id="cb31-1208"><a href="#cb31-1208" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1209"><a href="#cb31-1209" aria-hidden="true" tabindex="-1"></a>where $Y_i$ is the set of observations whose time (event or censored) is</span>
-<span id="cb31-1210"><a href="#cb31-1210" aria-hidden="true" tabindex="-1"></a>$\ge t_i$, the group at risk at time $t_i$.</span>
-<span id="cb31-1211"><a href="#cb31-1211" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1212"><a href="#cb31-1212" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-1213"><a href="#cb31-1213" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1214"><a href="#cb31-1214" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1215"><a href="#cb31-1215" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1216"><a href="#cb31-1216" aria-hidden="true" tabindex="-1"></a>If there are no censored data, and there are $n$ data points, then just</span>
-<span id="cb31-1217"><a href="#cb31-1217" aria-hidden="true" tabindex="-1"></a>after (say) the third event time </span>
+<span id="cb31-1203"><a href="#cb31-1203" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>For any time $t$ in months, we know that 6-MP patients with times</span>
+<span id="cb31-1204"><a href="#cb31-1204" aria-hidden="true" tabindex="-1"></a>    greater than $t$ have not relapsed, and those with relapse time less</span>
+<span id="cb31-1205"><a href="#cb31-1205" aria-hidden="true" tabindex="-1"></a>    than $t$ have relapsed, but we don't know if patients with censored</span>
+<span id="cb31-1206"><a href="#cb31-1206" aria-hidden="true" tabindex="-1"></a>    time less than $t$ have relapsed or not.</span>
+<span id="cb31-1207"><a href="#cb31-1207" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1208"><a href="#cb31-1208" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The procedure we usually use is the Kaplan-Meier product-limit</span>
+<span id="cb31-1209"><a href="#cb31-1209" aria-hidden="true" tabindex="-1"></a>    estimator of the survival function.</span>
+<span id="cb31-1210"><a href="#cb31-1210" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1211"><a href="#cb31-1211" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>The Kaplan-Meier estimator is a step function (like the empirical</span>
+<span id="cb31-1212"><a href="#cb31-1212" aria-hidden="true" tabindex="-1"></a>    cdf), which changes value only at the event times, not at the</span>
+<span id="cb31-1213"><a href="#cb31-1213" aria-hidden="true" tabindex="-1"></a>    censoring times.</span>
+<span id="cb31-1214"><a href="#cb31-1214" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1215"><a href="#cb31-1215" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>At each event time $t$, we compute the at-risk group size $Y$, which</span>
+<span id="cb31-1216"><a href="#cb31-1216" aria-hidden="true" tabindex="-1"></a>    is all those observations whose event time or censoring time is at</span>
+<span id="cb31-1217"><a href="#cb31-1217" aria-hidden="true" tabindex="-1"></a>    least $t$.</span>
 <span id="cb31-1218"><a href="#cb31-1218" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1219"><a href="#cb31-1219" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1220"><a href="#cb31-1220" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-1221"><a href="#cb31-1221" aria-hidden="true" tabindex="-1"></a>\hat S(t) </span>
-<span id="cb31-1222"><a href="#cb31-1222" aria-hidden="true" tabindex="-1"></a>&amp;= \prod_{t_i &lt; t}\sb{1-\frac{d_i}{Y_i}}</span>
-<span id="cb31-1223"><a href="#cb31-1223" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \sb{\frac{n-d_1}{n}} \sb{\frac{n-d_1-d_2}{n-d_1}} \sb{\frac{n-d_1-d_2-d_3}{n-d_1-d_2}}</span>
-<span id="cb31-1224"><a href="#cb31-1224" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \frac{n-d_1-d_2-d_3}{n}</span>
-<span id="cb31-1225"><a href="#cb31-1225" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;=1-\frac{d_1+d_2+d_3}{n}</span>
-<span id="cb31-1226"><a href="#cb31-1226" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;=1-\hat F(t)</span>
-<span id="cb31-1227"><a href="#cb31-1227" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-1228"><a href="#cb31-1228" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1229"><a href="#cb31-1229" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1230"><a href="#cb31-1230" aria-hidden="true" tabindex="-1"></a>where $\hat F(t)$ is the usual empirical CDF estimate.</span>
-<span id="cb31-1231"><a href="#cb31-1231" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1232"><a href="#cb31-1232" aria-hidden="true" tabindex="-1"></a><span class="fu">### Kaplan-Meier curve for `drug6mp` data</span></span>
-<span id="cb31-1233"><a href="#cb31-1233" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1234"><a href="#cb31-1234" aria-hidden="true" tabindex="-1"></a>Here is the Kaplan-Meier estimated survival curve for the patients who</span>
-<span id="cb31-1235"><a href="#cb31-1235" aria-hidden="true" tabindex="-1"></a>received 6-MP in the <span class="in">`drug6mp`</span> dataset (we will see code to produce</span>
-<span id="cb31-1236"><a href="#cb31-1236" aria-hidden="true" tabindex="-1"></a>figures like this one shortly):</span>
+<span id="cb31-1219"><a href="#cb31-1219" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>If $d$ of the observations have an event time (not a censoring time)</span>
+<span id="cb31-1220"><a href="#cb31-1220" aria-hidden="true" tabindex="-1"></a>    of $t$, then the group of survivors immediately following time $t$</span>
+<span id="cb31-1221"><a href="#cb31-1221" aria-hidden="true" tabindex="-1"></a>    is reduced by the fraction $$\frac{Y-d}{Y}=1-\frac{d}{Y}$$</span>
+<span id="cb31-1222"><a href="#cb31-1222" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1223"><a href="#cb31-1223" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1224"><a href="#cb31-1224" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1225"><a href="#cb31-1225" aria-hidden="true" tabindex="-1"></a>:::{#def-KM-estimator}</span>
+<span id="cb31-1226"><a href="#cb31-1226" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1227"><a href="#cb31-1227" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Kaplan-Meier Product-Limit Estimator of Survival Function</span></span>
+<span id="cb31-1228"><a href="#cb31-1228" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1229"><a href="#cb31-1229" aria-hidden="true" tabindex="-1"></a>If the event times are $t_i$ with events per time of $d_i$ ($1\le i \le k$), </span>
+<span id="cb31-1230"><a href="#cb31-1230" aria-hidden="true" tabindex="-1"></a>then the **Kaplan-Meier Product-Limit Estimator** </span>
+<span id="cb31-1231"><a href="#cb31-1231" aria-hidden="true" tabindex="-1"></a>of the <span class="co">[</span><span class="ot">survival function</span><span class="co">](@def-surv-fn)</span> is:</span>
+<span id="cb31-1232"><a href="#cb31-1232" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1233"><a href="#cb31-1233" aria-hidden="true" tabindex="-1"></a>$$\hat S(t) = \prod_{t_i &lt; t} \sb{\frac{1-d_i}{Y_i}}$${#eq-km-surv-est}</span>
+<span id="cb31-1234"><a href="#cb31-1234" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1235"><a href="#cb31-1235" aria-hidden="true" tabindex="-1"></a>where $Y_i$ is the set of observations whose time (event or censored) is</span>
+<span id="cb31-1236"><a href="#cb31-1236" aria-hidden="true" tabindex="-1"></a>$\ge t_i$, the group at risk at time $t_i$.</span>
 <span id="cb31-1237"><a href="#cb31-1237" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1240"><a href="#cb31-1240" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1241"><a href="#cb31-1241" aria-hidden="true" tabindex="-1"></a><span class="co">#| fig-cap: "Kaplan-Meier Survival Curve for 6-MP Patients"</span></span>
-<span id="cb31-1242"><a href="#cb31-1242" aria-hidden="true" tabindex="-1"></a><span class="co">#| label: fig-KM-mp6</span></span>
-<span id="cb31-1243"><a href="#cb31-1243" aria-hidden="true" tabindex="-1"></a><span class="co"># | echo: false</span></span>
+<span id="cb31-1238"><a href="#cb31-1238" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-1239"><a href="#cb31-1239" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1240"><a href="#cb31-1240" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1241"><a href="#cb31-1241" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1242"><a href="#cb31-1242" aria-hidden="true" tabindex="-1"></a>:::{#thm-KM-est-no-cens}</span>
+<span id="cb31-1243"><a href="#cb31-1243" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Kaplan-Meier Estimate with No Censored Observations</span></span>
 <span id="cb31-1244"><a href="#cb31-1244" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1245"><a href="#cb31-1245" aria-hidden="true" tabindex="-1"></a><span class="fu">require</span>(KMsurv)</span>
-<span id="cb31-1246"><a href="#cb31-1246" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(drug6mp)</span>
-<span id="cb31-1247"><a href="#cb31-1247" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(dplyr)</span>
-<span id="cb31-1248"><a href="#cb31-1248" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(survival)</span>
-<span id="cb31-1249"><a href="#cb31-1249" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1250"><a href="#cb31-1250" aria-hidden="true" tabindex="-1"></a>drug6mp_km_model1 <span class="ot">=</span> </span>
-<span id="cb31-1251"><a href="#cb31-1251" aria-hidden="true" tabindex="-1"></a>  drug6mp <span class="sc">|&gt;</span> </span>
-<span id="cb31-1252"><a href="#cb31-1252" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">surv =</span> <span class="fu">Surv</span>(t2, relapse))  <span class="sc">|&gt;</span> </span>
-<span id="cb31-1253"><a href="#cb31-1253" aria-hidden="true" tabindex="-1"></a>  <span class="fu">survfit</span>(<span class="at">formula =</span> surv <span class="sc">~</span> <span class="dv">1</span>, <span class="at">data =</span> _)</span>
-<span id="cb31-1254"><a href="#cb31-1254" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1255"><a href="#cb31-1255" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(ggfortify)</span>
-<span id="cb31-1256"><a href="#cb31-1256" aria-hidden="true" tabindex="-1"></a>drug6mp_km_model1 <span class="sc">|&gt;</span>  </span>
-<span id="cb31-1257"><a href="#cb31-1257" aria-hidden="true" tabindex="-1"></a>  <span class="fu">autoplot</span>(</span>
-<span id="cb31-1258"><a href="#cb31-1258" aria-hidden="true" tabindex="-1"></a>    <span class="at">mark.time =</span> <span class="cn">TRUE</span>,</span>
-<span id="cb31-1259"><a href="#cb31-1259" aria-hidden="true" tabindex="-1"></a>    <span class="at">conf.int =</span> <span class="cn">FALSE</span>) <span class="sc">+</span></span>
-<span id="cb31-1260"><a href="#cb31-1260" aria-hidden="true" tabindex="-1"></a>  <span class="fu">expand_limits</span>(<span class="at">y =</span> <span class="dv">0</span>) <span class="sc">+</span></span>
-<span id="cb31-1261"><a href="#cb31-1261" aria-hidden="true" tabindex="-1"></a>  <span class="fu">xlab</span>(<span class="st">'Time since diagnosis (months)'</span>) <span class="sc">+</span></span>
-<span id="cb31-1262"><a href="#cb31-1262" aria-hidden="true" tabindex="-1"></a>  <span class="fu">ylab</span>(<span class="st">"KM Survival Curve"</span>)</span>
-<span id="cb31-1263"><a href="#cb31-1263" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1264"><a href="#cb31-1264" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1265"><a href="#cb31-1265" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1266"><a href="#cb31-1266" aria-hidden="true" tabindex="-1"></a><span class="fu">### Kaplan-Meier calculations {.smaller}</span></span>
-<span id="cb31-1267"><a href="#cb31-1267" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1268"><a href="#cb31-1268" aria-hidden="true" tabindex="-1"></a>Let's compute these estimates and build the chart by hand:</span>
-<span id="cb31-1269"><a href="#cb31-1269" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1272"><a href="#cb31-1272" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1273"><a href="#cb31-1273" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(KMsurv)</span>
-<span id="cb31-1274"><a href="#cb31-1274" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(dplyr)</span>
-<span id="cb31-1275"><a href="#cb31-1275" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(drug6mp)</span>
-<span id="cb31-1276"><a href="#cb31-1276" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1277"><a href="#cb31-1277" aria-hidden="true" tabindex="-1"></a>drug6mp.v2 <span class="ot">=</span> </span>
-<span id="cb31-1278"><a href="#cb31-1278" aria-hidden="true" tabindex="-1"></a>  drug6mp <span class="sc">|&gt;</span> </span>
-<span id="cb31-1279"><a href="#cb31-1279" aria-hidden="true" tabindex="-1"></a>  <span class="fu">as_tibble</span>() <span class="sc">|&gt;</span> </span>
-<span id="cb31-1280"><a href="#cb31-1280" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(</span>
-<span id="cb31-1281"><a href="#cb31-1281" aria-hidden="true" tabindex="-1"></a>    <span class="at">remstat =</span> remstat <span class="sc">|&gt;</span> </span>
-<span id="cb31-1282"><a href="#cb31-1282" aria-hidden="true" tabindex="-1"></a>      <span class="fu">case_match</span>(</span>
-<span id="cb31-1283"><a href="#cb31-1283" aria-hidden="true" tabindex="-1"></a>        <span class="dv">1</span> <span class="sc">~</span> <span class="st">"partial"</span>,</span>
-<span id="cb31-1284"><a href="#cb31-1284" aria-hidden="true" tabindex="-1"></a>        <span class="dv">2</span> <span class="sc">~</span> <span class="st">"complete"</span></span>
-<span id="cb31-1285"><a href="#cb31-1285" aria-hidden="true" tabindex="-1"></a>      ),</span>
-<span id="cb31-1286"><a href="#cb31-1286" aria-hidden="true" tabindex="-1"></a>    <span class="co"># renaming to "outcome" while relabeling is just a style choice:</span></span>
-<span id="cb31-1287"><a href="#cb31-1287" aria-hidden="true" tabindex="-1"></a>    <span class="at">outcome =</span> relapse <span class="sc">|&gt;</span> </span>
-<span id="cb31-1288"><a href="#cb31-1288" aria-hidden="true" tabindex="-1"></a>      <span class="fu">case_match</span>(</span>
-<span id="cb31-1289"><a href="#cb31-1289" aria-hidden="true" tabindex="-1"></a>        <span class="dv">0</span> <span class="sc">~</span> <span class="st">"censored"</span>,</span>
-<span id="cb31-1290"><a href="#cb31-1290" aria-hidden="true" tabindex="-1"></a>        <span class="dv">1</span> <span class="sc">~</span> <span class="st">"relapsed"</span></span>
-<span id="cb31-1291"><a href="#cb31-1291" aria-hidden="true" tabindex="-1"></a>      )</span>
-<span id="cb31-1292"><a href="#cb31-1292" aria-hidden="true" tabindex="-1"></a>  )</span>
-<span id="cb31-1293"><a href="#cb31-1293" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1294"><a href="#cb31-1294" aria-hidden="true" tabindex="-1"></a>km<span class="fl">.6</span>mp <span class="ot">=</span></span>
-<span id="cb31-1295"><a href="#cb31-1295" aria-hidden="true" tabindex="-1"></a>  drug6mp.v2 <span class="sc">|&gt;</span> </span>
-<span id="cb31-1296"><a href="#cb31-1296" aria-hidden="true" tabindex="-1"></a>  <span class="fu">summarize</span>(</span>
-<span id="cb31-1297"><a href="#cb31-1297" aria-hidden="true" tabindex="-1"></a>    <span class="at">.by =</span> t2,</span>
-<span id="cb31-1298"><a href="#cb31-1298" aria-hidden="true" tabindex="-1"></a>    <span class="at">Relapses =</span> <span class="fu">sum</span>(outcome <span class="sc">==</span> <span class="st">"relapsed"</span>),</span>
-<span id="cb31-1299"><a href="#cb31-1299" aria-hidden="true" tabindex="-1"></a>    <span class="at">Censored =</span> <span class="fu">sum</span>(outcome <span class="sc">==</span> <span class="st">"censored"</span>)) <span class="sc">|&gt;</span></span>
-<span id="cb31-1300"><a href="#cb31-1300" aria-hidden="true" tabindex="-1"></a>  <span class="co"># here we add a start time row, so the graph starts at time 0:</span></span>
-<span id="cb31-1301"><a href="#cb31-1301" aria-hidden="true" tabindex="-1"></a>  <span class="fu">bind_rows</span>(</span>
-<span id="cb31-1302"><a href="#cb31-1302" aria-hidden="true" tabindex="-1"></a>    <span class="fu">tibble</span>(</span>
-<span id="cb31-1303"><a href="#cb31-1303" aria-hidden="true" tabindex="-1"></a>      <span class="at">t2 =</span> <span class="dv">0</span>, </span>
-<span id="cb31-1304"><a href="#cb31-1304" aria-hidden="true" tabindex="-1"></a>      <span class="at">Relapses =</span> <span class="dv">0</span>, </span>
-<span id="cb31-1305"><a href="#cb31-1305" aria-hidden="true" tabindex="-1"></a>      <span class="at">Censored =</span> <span class="dv">0</span>)</span>
-<span id="cb31-1306"><a href="#cb31-1306" aria-hidden="true" tabindex="-1"></a>  ) <span class="sc">|&gt;</span> </span>
-<span id="cb31-1307"><a href="#cb31-1307" aria-hidden="true" tabindex="-1"></a>  <span class="co"># sort in time order:</span></span>
-<span id="cb31-1308"><a href="#cb31-1308" aria-hidden="true" tabindex="-1"></a>  <span class="fu">arrange</span>(t2) <span class="sc">|&gt;</span></span>
-<span id="cb31-1309"><a href="#cb31-1309" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(</span>
-<span id="cb31-1310"><a href="#cb31-1310" aria-hidden="true" tabindex="-1"></a>    <span class="at">Exiting =</span> Relapses <span class="sc">+</span> Censored,</span>
-<span id="cb31-1311"><a href="#cb31-1311" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">Study Size</span><span class="st">`</span> <span class="ot">=</span> <span class="fu">sum</span>(Exiting),</span>
-<span id="cb31-1312"><a href="#cb31-1312" aria-hidden="true" tabindex="-1"></a>    <span class="at">Exited =</span> <span class="fu">cumsum</span>(Exiting) <span class="sc">|&gt;</span> dplyr<span class="sc">::</span><span class="fu">lag</span>(<span class="at">default =</span> <span class="dv">0</span>),</span>
-<span id="cb31-1313"><a href="#cb31-1313" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">At Risk</span><span class="st">`</span> <span class="ot">=</span> <span class="st">`</span><span class="at">Study Size</span><span class="st">`</span> <span class="sc">-</span> Exited,</span>
-<span id="cb31-1314"><a href="#cb31-1314" aria-hidden="true" tabindex="-1"></a>    <span class="at">Hazard =</span> Relapses <span class="sc">/</span> <span class="st">`</span><span class="at">At Risk</span><span class="st">`</span>,</span>
-<span id="cb31-1315"><a href="#cb31-1315" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">KM Factor</span><span class="st">`</span> <span class="ot">=</span> <span class="dv">1</span> <span class="sc">-</span> Hazard,</span>
-<span id="cb31-1316"><a href="#cb31-1316" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">Cumulative Hazard</span><span class="st">`</span> <span class="ot">=</span> <span class="fu">cumsum</span>(<span class="st">`</span><span class="at">Hazard</span><span class="st">`</span>),</span>
-<span id="cb31-1317"><a href="#cb31-1317" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">KM Survival Curve</span><span class="st">`</span> <span class="ot">=</span> <span class="fu">cumprod</span>(<span class="st">`</span><span class="at">KM Factor</span><span class="st">`</span>)</span>
-<span id="cb31-1318"><a href="#cb31-1318" aria-hidden="true" tabindex="-1"></a>  )</span>
-<span id="cb31-1319"><a href="#cb31-1319" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1320"><a href="#cb31-1320" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(pander)  </span>
-<span id="cb31-1321"><a href="#cb31-1321" aria-hidden="true" tabindex="-1"></a><span class="fu">pander</span>(km<span class="fl">.6</span>mp)</span>
-<span id="cb31-1322"><a href="#cb31-1322" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1323"><a href="#cb31-1323" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1324"><a href="#cb31-1324" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1325"><a href="#cb31-1325" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1326"><a href="#cb31-1326" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Summary</span></span>
-<span id="cb31-1327"><a href="#cb31-1327" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1328"><a href="#cb31-1328" aria-hidden="true" tabindex="-1"></a>For the 6-MP patients at time 6 months, there are 21 patients at risk.</span>
-<span id="cb31-1329"><a href="#cb31-1329" aria-hidden="true" tabindex="-1"></a>At $t=6$ there are 3 relapses and 1 censored observations.</span>
-<span id="cb31-1330"><a href="#cb31-1330" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1331"><a href="#cb31-1331" aria-hidden="true" tabindex="-1"></a>The Kaplan-Meier factor is $(21-3)/21 = 0.857$. The number at risk for</span>
-<span id="cb31-1332"><a href="#cb31-1332" aria-hidden="true" tabindex="-1"></a>the next time ($t=7$) is $21-3-1=17$.</span>
-<span id="cb31-1333"><a href="#cb31-1333" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1334"><a href="#cb31-1334" aria-hidden="true" tabindex="-1"></a>At time 7 months, there are 17 patients at risk. At $t=7$ there is 1</span>
-<span id="cb31-1335"><a href="#cb31-1335" aria-hidden="true" tabindex="-1"></a>relapse and 0 censored observations. The Kaplan-Meier factor is</span>
-<span id="cb31-1336"><a href="#cb31-1336" aria-hidden="true" tabindex="-1"></a>$(17-1)/17 = 0.941$. The Kaplan Meier estimate is</span>
-<span id="cb31-1337"><a href="#cb31-1337" aria-hidden="true" tabindex="-1"></a>$0.857\times0.941=0.807$. The number at risk for the next time ($t=9$)</span>
-<span id="cb31-1338"><a href="#cb31-1338" aria-hidden="true" tabindex="-1"></a>is $17-1=16$.</span>
-<span id="cb31-1339"><a href="#cb31-1339" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1340"><a href="#cb31-1340" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1341"><a href="#cb31-1341" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1342"><a href="#cb31-1342" aria-hidden="true" tabindex="-1"></a>Now, let's graph this estimated survival curve using <span class="in">`ggplot()`</span>:</span>
-<span id="cb31-1343"><a href="#cb31-1343" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1344"><a href="#cb31-1344" aria-hidden="true" tabindex="-1"></a><span class="in">```{r "estimated survival curve"}</span></span>
-<span id="cb31-1345"><a href="#cb31-1345" aria-hidden="true" tabindex="-1"></a><span class="in">#| label: fig-km-by-hand</span></span>
-<span id="cb31-1346"><a href="#cb31-1346" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "KM curve for 6MP patients, calculated by hand"</span></span>
-<span id="cb31-1347"><a href="#cb31-1347" aria-hidden="true" tabindex="-1"></a><span class="in">library(ggplot2)</span></span>
-<span id="cb31-1348"><a href="#cb31-1348" aria-hidden="true" tabindex="-1"></a><span class="in">conflicts_prefer(dplyr::filter)</span></span>
-<span id="cb31-1349"><a href="#cb31-1349" aria-hidden="true" tabindex="-1"></a><span class="in">km.6mp |&gt; </span></span>
-<span id="cb31-1350"><a href="#cb31-1350" aria-hidden="true" tabindex="-1"></a><span class="in">  ggplot(aes(x = t2, y = `KM Survival Curve`)) +</span></span>
-<span id="cb31-1351"><a href="#cb31-1351" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_step() +</span></span>
-<span id="cb31-1352"><a href="#cb31-1352" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_point(data = km.6mp |&gt; filter(Censored &gt; 0), shape = 3) +</span></span>
-<span id="cb31-1353"><a href="#cb31-1353" aria-hidden="true" tabindex="-1"></a><span class="in">  expand_limits(y = c(0,1), x = 0) +</span></span>
-<span id="cb31-1354"><a href="#cb31-1354" aria-hidden="true" tabindex="-1"></a><span class="in">  xlab('Time since diagnosis (months)') +</span></span>
-<span id="cb31-1355"><a href="#cb31-1355" aria-hidden="true" tabindex="-1"></a><span class="in">  ylab("KM Survival Curve") +</span></span>
-<span id="cb31-1356"><a href="#cb31-1356" aria-hidden="true" tabindex="-1"></a><span class="in">  scale_y_continuous(labels = scales::percent)</span></span>
-<span id="cb31-1357"><a href="#cb31-1357" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1245"><a href="#cb31-1245" aria-hidden="true" tabindex="-1"></a>If there are no censored data, and there are $n$ data points, then just</span>
+<span id="cb31-1246"><a href="#cb31-1246" aria-hidden="true" tabindex="-1"></a>after (say) the third event time </span>
+<span id="cb31-1247"><a href="#cb31-1247" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1248"><a href="#cb31-1248" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1249"><a href="#cb31-1249" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-1250"><a href="#cb31-1250" aria-hidden="true" tabindex="-1"></a>\hat S(t) </span>
+<span id="cb31-1251"><a href="#cb31-1251" aria-hidden="true" tabindex="-1"></a>&amp;= \prod_{t_i &lt; t}\sb{1-\frac{d_i}{Y_i}}</span>
+<span id="cb31-1252"><a href="#cb31-1252" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \sb{\frac{n-d_1}{n}} \sb{\frac{n-d_1-d_2}{n-d_1}} \sb{\frac{n-d_1-d_2-d_3}{n-d_1-d_2}}</span>
+<span id="cb31-1253"><a href="#cb31-1253" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \frac{n-d_1-d_2-d_3}{n}</span>
+<span id="cb31-1254"><a href="#cb31-1254" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;=1-\frac{d_1+d_2+d_3}{n}</span>
+<span id="cb31-1255"><a href="#cb31-1255" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;=1-\hat F(t)</span>
+<span id="cb31-1256"><a href="#cb31-1256" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-1257"><a href="#cb31-1257" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1258"><a href="#cb31-1258" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1259"><a href="#cb31-1259" aria-hidden="true" tabindex="-1"></a>where $\hat F(t)$ is the usual empirical CDF estimate.</span>
+<span id="cb31-1260"><a href="#cb31-1260" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1261"><a href="#cb31-1261" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-1262"><a href="#cb31-1262" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1263"><a href="#cb31-1263" aria-hidden="true" tabindex="-1"></a><span class="fu">### Kaplan-Meier curve for `drug6mp` data</span></span>
+<span id="cb31-1264"><a href="#cb31-1264" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1265"><a href="#cb31-1265" aria-hidden="true" tabindex="-1"></a>Here is the Kaplan-Meier estimated survival curve for the patients who</span>
+<span id="cb31-1266"><a href="#cb31-1266" aria-hidden="true" tabindex="-1"></a>received 6-MP in the <span class="in">`drug6mp`</span> dataset (we will see code to produce</span>
+<span id="cb31-1267"><a href="#cb31-1267" aria-hidden="true" tabindex="-1"></a>figures like this one shortly):</span>
+<span id="cb31-1268"><a href="#cb31-1268" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1271"><a href="#cb31-1271" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1272"><a href="#cb31-1272" aria-hidden="true" tabindex="-1"></a><span class="co">#| fig-cap: "Kaplan-Meier Survival Curve for 6-MP Patients"</span></span>
+<span id="cb31-1273"><a href="#cb31-1273" aria-hidden="true" tabindex="-1"></a><span class="co">#| label: fig-KM-mp6</span></span>
+<span id="cb31-1274"><a href="#cb31-1274" aria-hidden="true" tabindex="-1"></a><span class="co"># | echo: false</span></span>
+<span id="cb31-1275"><a href="#cb31-1275" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1276"><a href="#cb31-1276" aria-hidden="true" tabindex="-1"></a><span class="fu">require</span>(KMsurv)</span>
+<span id="cb31-1277"><a href="#cb31-1277" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(drug6mp)</span>
+<span id="cb31-1278"><a href="#cb31-1278" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(dplyr)</span>
+<span id="cb31-1279"><a href="#cb31-1279" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(survival)</span>
+<span id="cb31-1280"><a href="#cb31-1280" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1281"><a href="#cb31-1281" aria-hidden="true" tabindex="-1"></a>drug6mp_km_model1 <span class="ot">=</span> </span>
+<span id="cb31-1282"><a href="#cb31-1282" aria-hidden="true" tabindex="-1"></a>  drug6mp <span class="sc">|&gt;</span> </span>
+<span id="cb31-1283"><a href="#cb31-1283" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">surv =</span> <span class="fu">Surv</span>(t2, relapse))  <span class="sc">|&gt;</span> </span>
+<span id="cb31-1284"><a href="#cb31-1284" aria-hidden="true" tabindex="-1"></a>  <span class="fu">survfit</span>(<span class="at">formula =</span> surv <span class="sc">~</span> <span class="dv">1</span>, <span class="at">data =</span> _)</span>
+<span id="cb31-1285"><a href="#cb31-1285" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1286"><a href="#cb31-1286" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(ggfortify)</span>
+<span id="cb31-1287"><a href="#cb31-1287" aria-hidden="true" tabindex="-1"></a>drug6mp_km_model1 <span class="sc">|&gt;</span>  </span>
+<span id="cb31-1288"><a href="#cb31-1288" aria-hidden="true" tabindex="-1"></a>  <span class="fu">autoplot</span>(</span>
+<span id="cb31-1289"><a href="#cb31-1289" aria-hidden="true" tabindex="-1"></a>    <span class="at">mark.time =</span> <span class="cn">TRUE</span>,</span>
+<span id="cb31-1290"><a href="#cb31-1290" aria-hidden="true" tabindex="-1"></a>    <span class="at">conf.int =</span> <span class="cn">FALSE</span>) <span class="sc">+</span></span>
+<span id="cb31-1291"><a href="#cb31-1291" aria-hidden="true" tabindex="-1"></a>  <span class="fu">expand_limits</span>(<span class="at">y =</span> <span class="dv">0</span>) <span class="sc">+</span></span>
+<span id="cb31-1292"><a href="#cb31-1292" aria-hidden="true" tabindex="-1"></a>  <span class="fu">xlab</span>(<span class="st">'Time since diagnosis (months)'</span>) <span class="sc">+</span></span>
+<span id="cb31-1293"><a href="#cb31-1293" aria-hidden="true" tabindex="-1"></a>  <span class="fu">ylab</span>(<span class="st">"KM Survival Curve"</span>)</span>
+<span id="cb31-1294"><a href="#cb31-1294" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1295"><a href="#cb31-1295" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1296"><a href="#cb31-1296" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1297"><a href="#cb31-1297" aria-hidden="true" tabindex="-1"></a><span class="fu">### Kaplan-Meier calculations {.smaller}</span></span>
+<span id="cb31-1298"><a href="#cb31-1298" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1299"><a href="#cb31-1299" aria-hidden="true" tabindex="-1"></a>Let's compute these estimates and build the chart by hand:</span>
+<span id="cb31-1300"><a href="#cb31-1300" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1303"><a href="#cb31-1303" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1304"><a href="#cb31-1304" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(KMsurv)</span>
+<span id="cb31-1305"><a href="#cb31-1305" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(dplyr)</span>
+<span id="cb31-1306"><a href="#cb31-1306" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(drug6mp)</span>
+<span id="cb31-1307"><a href="#cb31-1307" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1308"><a href="#cb31-1308" aria-hidden="true" tabindex="-1"></a>drug6mp.v2 <span class="ot">=</span> </span>
+<span id="cb31-1309"><a href="#cb31-1309" aria-hidden="true" tabindex="-1"></a>  drug6mp <span class="sc">|&gt;</span> </span>
+<span id="cb31-1310"><a href="#cb31-1310" aria-hidden="true" tabindex="-1"></a>  <span class="fu">as_tibble</span>() <span class="sc">|&gt;</span> </span>
+<span id="cb31-1311"><a href="#cb31-1311" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(</span>
+<span id="cb31-1312"><a href="#cb31-1312" aria-hidden="true" tabindex="-1"></a>    <span class="at">remstat =</span> remstat <span class="sc">|&gt;</span> </span>
+<span id="cb31-1313"><a href="#cb31-1313" aria-hidden="true" tabindex="-1"></a>      <span class="fu">case_match</span>(</span>
+<span id="cb31-1314"><a href="#cb31-1314" aria-hidden="true" tabindex="-1"></a>        <span class="dv">1</span> <span class="sc">~</span> <span class="st">"partial"</span>,</span>
+<span id="cb31-1315"><a href="#cb31-1315" aria-hidden="true" tabindex="-1"></a>        <span class="dv">2</span> <span class="sc">~</span> <span class="st">"complete"</span></span>
+<span id="cb31-1316"><a href="#cb31-1316" aria-hidden="true" tabindex="-1"></a>      ),</span>
+<span id="cb31-1317"><a href="#cb31-1317" aria-hidden="true" tabindex="-1"></a>    <span class="co"># renaming to "outcome" while relabeling is just a style choice:</span></span>
+<span id="cb31-1318"><a href="#cb31-1318" aria-hidden="true" tabindex="-1"></a>    <span class="at">outcome =</span> relapse <span class="sc">|&gt;</span> </span>
+<span id="cb31-1319"><a href="#cb31-1319" aria-hidden="true" tabindex="-1"></a>      <span class="fu">case_match</span>(</span>
+<span id="cb31-1320"><a href="#cb31-1320" aria-hidden="true" tabindex="-1"></a>        <span class="dv">0</span> <span class="sc">~</span> <span class="st">"censored"</span>,</span>
+<span id="cb31-1321"><a href="#cb31-1321" aria-hidden="true" tabindex="-1"></a>        <span class="dv">1</span> <span class="sc">~</span> <span class="st">"relapsed"</span></span>
+<span id="cb31-1322"><a href="#cb31-1322" aria-hidden="true" tabindex="-1"></a>      )</span>
+<span id="cb31-1323"><a href="#cb31-1323" aria-hidden="true" tabindex="-1"></a>  )</span>
+<span id="cb31-1324"><a href="#cb31-1324" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1325"><a href="#cb31-1325" aria-hidden="true" tabindex="-1"></a>km<span class="fl">.6</span>mp <span class="ot">=</span></span>
+<span id="cb31-1326"><a href="#cb31-1326" aria-hidden="true" tabindex="-1"></a>  drug6mp.v2 <span class="sc">|&gt;</span> </span>
+<span id="cb31-1327"><a href="#cb31-1327" aria-hidden="true" tabindex="-1"></a>  <span class="fu">summarize</span>(</span>
+<span id="cb31-1328"><a href="#cb31-1328" aria-hidden="true" tabindex="-1"></a>    <span class="at">.by =</span> t2,</span>
+<span id="cb31-1329"><a href="#cb31-1329" aria-hidden="true" tabindex="-1"></a>    <span class="at">Relapses =</span> <span class="fu">sum</span>(outcome <span class="sc">==</span> <span class="st">"relapsed"</span>),</span>
+<span id="cb31-1330"><a href="#cb31-1330" aria-hidden="true" tabindex="-1"></a>    <span class="at">Censored =</span> <span class="fu">sum</span>(outcome <span class="sc">==</span> <span class="st">"censored"</span>)) <span class="sc">|&gt;</span></span>
+<span id="cb31-1331"><a href="#cb31-1331" aria-hidden="true" tabindex="-1"></a>  <span class="co"># here we add a start time row, so the graph starts at time 0:</span></span>
+<span id="cb31-1332"><a href="#cb31-1332" aria-hidden="true" tabindex="-1"></a>  <span class="fu">bind_rows</span>(</span>
+<span id="cb31-1333"><a href="#cb31-1333" aria-hidden="true" tabindex="-1"></a>    <span class="fu">tibble</span>(</span>
+<span id="cb31-1334"><a href="#cb31-1334" aria-hidden="true" tabindex="-1"></a>      <span class="at">t2 =</span> <span class="dv">0</span>, </span>
+<span id="cb31-1335"><a href="#cb31-1335" aria-hidden="true" tabindex="-1"></a>      <span class="at">Relapses =</span> <span class="dv">0</span>, </span>
+<span id="cb31-1336"><a href="#cb31-1336" aria-hidden="true" tabindex="-1"></a>      <span class="at">Censored =</span> <span class="dv">0</span>)</span>
+<span id="cb31-1337"><a href="#cb31-1337" aria-hidden="true" tabindex="-1"></a>  ) <span class="sc">|&gt;</span> </span>
+<span id="cb31-1338"><a href="#cb31-1338" aria-hidden="true" tabindex="-1"></a>  <span class="co"># sort in time order:</span></span>
+<span id="cb31-1339"><a href="#cb31-1339" aria-hidden="true" tabindex="-1"></a>  <span class="fu">arrange</span>(t2) <span class="sc">|&gt;</span></span>
+<span id="cb31-1340"><a href="#cb31-1340" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(</span>
+<span id="cb31-1341"><a href="#cb31-1341" aria-hidden="true" tabindex="-1"></a>    <span class="at">Exiting =</span> Relapses <span class="sc">+</span> Censored,</span>
+<span id="cb31-1342"><a href="#cb31-1342" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">Study Size</span><span class="st">`</span> <span class="ot">=</span> <span class="fu">sum</span>(Exiting),</span>
+<span id="cb31-1343"><a href="#cb31-1343" aria-hidden="true" tabindex="-1"></a>    <span class="at">Exited =</span> <span class="fu">cumsum</span>(Exiting) <span class="sc">|&gt;</span> dplyr<span class="sc">::</span><span class="fu">lag</span>(<span class="at">default =</span> <span class="dv">0</span>),</span>
+<span id="cb31-1344"><a href="#cb31-1344" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">At Risk</span><span class="st">`</span> <span class="ot">=</span> <span class="st">`</span><span class="at">Study Size</span><span class="st">`</span> <span class="sc">-</span> Exited,</span>
+<span id="cb31-1345"><a href="#cb31-1345" aria-hidden="true" tabindex="-1"></a>    <span class="at">Hazard =</span> Relapses <span class="sc">/</span> <span class="st">`</span><span class="at">At Risk</span><span class="st">`</span>,</span>
+<span id="cb31-1346"><a href="#cb31-1346" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">KM Factor</span><span class="st">`</span> <span class="ot">=</span> <span class="dv">1</span> <span class="sc">-</span> Hazard,</span>
+<span id="cb31-1347"><a href="#cb31-1347" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">Cumulative Hazard</span><span class="st">`</span> <span class="ot">=</span> <span class="fu">cumsum</span>(<span class="st">`</span><span class="at">Hazard</span><span class="st">`</span>),</span>
+<span id="cb31-1348"><a href="#cb31-1348" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">KM Survival Curve</span><span class="st">`</span> <span class="ot">=</span> <span class="fu">cumprod</span>(<span class="st">`</span><span class="at">KM Factor</span><span class="st">`</span>)</span>
+<span id="cb31-1349"><a href="#cb31-1349" aria-hidden="true" tabindex="-1"></a>  )</span>
+<span id="cb31-1350"><a href="#cb31-1350" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1351"><a href="#cb31-1351" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(pander)  </span>
+<span id="cb31-1352"><a href="#cb31-1352" aria-hidden="true" tabindex="-1"></a><span class="fu">pander</span>(km<span class="fl">.6</span>mp)</span>
+<span id="cb31-1353"><a href="#cb31-1353" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1354"><a href="#cb31-1354" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1355"><a href="#cb31-1355" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1356"><a href="#cb31-1356" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1357"><a href="#cb31-1357" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Summary</span></span>
 <span id="cb31-1358"><a href="#cb31-1358" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1359"><a href="#cb31-1359" aria-hidden="true" tabindex="-1"></a><span class="fu">## Using the `survival` package in R</span></span>
-<span id="cb31-1360"><a href="#cb31-1360" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1361"><a href="#cb31-1361" aria-hidden="true" tabindex="-1"></a>We don't have to do these calculations by hand every time; the</span>
-<span id="cb31-1362"><a href="#cb31-1362" aria-hidden="true" tabindex="-1"></a><span class="in">`survival`</span> package and several others have functions available to</span>
-<span id="cb31-1363"><a href="#cb31-1363" aria-hidden="true" tabindex="-1"></a>automate many of these tasks (full list:</span>
-<span id="cb31-1364"><a href="#cb31-1364" aria-hidden="true" tabindex="-1"></a><span class="ot">&lt;https://cran.r-project.org/web/views/Survival.html&gt;</span>).</span>
-<span id="cb31-1365"><a href="#cb31-1365" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1366"><a href="#cb31-1366" aria-hidden="true" tabindex="-1"></a><span class="fu">### The `Surv` function</span></span>
-<span id="cb31-1367"><a href="#cb31-1367" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1368"><a href="#cb31-1368" aria-hidden="true" tabindex="-1"></a>To use the <span class="in">`survival`</span> package, the first step is telling R how to</span>
-<span id="cb31-1369"><a href="#cb31-1369" aria-hidden="true" tabindex="-1"></a>combine the exit time and exit reason (censoring versus event) columns.</span>
-<span id="cb31-1370"><a href="#cb31-1370" aria-hidden="true" tabindex="-1"></a>The <span class="in">`Surv()`</span> function accomplishes this task.</span>
-<span id="cb31-1371"><a href="#cb31-1371" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1372"><a href="#cb31-1372" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Example: `Surv()` with `drug6mp` data</span></span>
-<span id="cb31-1373"><a href="#cb31-1373" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1376"><a href="#cb31-1376" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1377"><a href="#cb31-1377" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-fold: show</span></span>
-<span id="cb31-1378"><a href="#cb31-1378" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-line-numbers: "5-7"</span></span>
-<span id="cb31-1379"><a href="#cb31-1379" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(survival)</span>
-<span id="cb31-1380"><a href="#cb31-1380" aria-hidden="true" tabindex="-1"></a>drug6mp.v3 <span class="ot">=</span> </span>
-<span id="cb31-1381"><a href="#cb31-1381" aria-hidden="true" tabindex="-1"></a>  drug6mp.v2 <span class="sc">|&gt;</span> </span>
-<span id="cb31-1382"><a href="#cb31-1382" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(</span>
-<span id="cb31-1383"><a href="#cb31-1383" aria-hidden="true" tabindex="-1"></a>    <span class="at">surv2 =</span> <span class="fu">Surv</span>(</span>
-<span id="cb31-1384"><a href="#cb31-1384" aria-hidden="true" tabindex="-1"></a>      <span class="at">time =</span> t2,</span>
-<span id="cb31-1385"><a href="#cb31-1385" aria-hidden="true" tabindex="-1"></a>      <span class="at">event =</span> (outcome <span class="sc">==</span> <span class="st">"relapsed"</span>)))</span>
-<span id="cb31-1386"><a href="#cb31-1386" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1387"><a href="#cb31-1387" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(drug6mp.v3)</span>
-<span id="cb31-1388"><a href="#cb31-1388" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1389"><a href="#cb31-1389" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1390"><a href="#cb31-1390" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1391"><a href="#cb31-1391" aria-hidden="true" tabindex="-1"></a>The output of <span class="in">`Surv()`</span> is a vector of objects with class <span class="in">`Surv`</span>. When we</span>
-<span id="cb31-1392"><a href="#cb31-1392" aria-hidden="true" tabindex="-1"></a>print this vector:</span>
-<span id="cb31-1393"><a href="#cb31-1393" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1394"><a href="#cb31-1394" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>observations where the event was observed are printed as the event</span>
-<span id="cb31-1395"><a href="#cb31-1395" aria-hidden="true" tabindex="-1"></a>    time (for example, <span class="in">`surv2 = 10`</span> on line 1)</span>
+<span id="cb31-1359"><a href="#cb31-1359" aria-hidden="true" tabindex="-1"></a>For the 6-MP patients at time 6 months, there are 21 patients at risk.</span>
+<span id="cb31-1360"><a href="#cb31-1360" aria-hidden="true" tabindex="-1"></a>At $t=6$ there are 3 relapses and 1 censored observations.</span>
+<span id="cb31-1361"><a href="#cb31-1361" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1362"><a href="#cb31-1362" aria-hidden="true" tabindex="-1"></a>The Kaplan-Meier factor is $(21-3)/21 = 0.857$. The number at risk for</span>
+<span id="cb31-1363"><a href="#cb31-1363" aria-hidden="true" tabindex="-1"></a>the next time ($t=7$) is $21-3-1=17$.</span>
+<span id="cb31-1364"><a href="#cb31-1364" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1365"><a href="#cb31-1365" aria-hidden="true" tabindex="-1"></a>At time 7 months, there are 17 patients at risk. At $t=7$ there is 1</span>
+<span id="cb31-1366"><a href="#cb31-1366" aria-hidden="true" tabindex="-1"></a>relapse and 0 censored observations. The Kaplan-Meier factor is</span>
+<span id="cb31-1367"><a href="#cb31-1367" aria-hidden="true" tabindex="-1"></a>$(17-1)/17 = 0.941$. The Kaplan Meier estimate is</span>
+<span id="cb31-1368"><a href="#cb31-1368" aria-hidden="true" tabindex="-1"></a>$0.857\times0.941=0.807$. The number at risk for the next time ($t=9$)</span>
+<span id="cb31-1369"><a href="#cb31-1369" aria-hidden="true" tabindex="-1"></a>is $17-1=16$.</span>
+<span id="cb31-1370"><a href="#cb31-1370" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1371"><a href="#cb31-1371" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1372"><a href="#cb31-1372" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1373"><a href="#cb31-1373" aria-hidden="true" tabindex="-1"></a>Now, let's graph this estimated survival curve using <span class="in">`ggplot()`</span>:</span>
+<span id="cb31-1374"><a href="#cb31-1374" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1375"><a href="#cb31-1375" aria-hidden="true" tabindex="-1"></a><span class="in">```{r "estimated survival curve"}</span></span>
+<span id="cb31-1376"><a href="#cb31-1376" aria-hidden="true" tabindex="-1"></a><span class="in">#| label: fig-km-by-hand</span></span>
+<span id="cb31-1377"><a href="#cb31-1377" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "KM curve for 6MP patients, calculated by hand"</span></span>
+<span id="cb31-1378"><a href="#cb31-1378" aria-hidden="true" tabindex="-1"></a><span class="in">library(ggplot2)</span></span>
+<span id="cb31-1379"><a href="#cb31-1379" aria-hidden="true" tabindex="-1"></a><span class="in">conflicts_prefer(dplyr::filter)</span></span>
+<span id="cb31-1380"><a href="#cb31-1380" aria-hidden="true" tabindex="-1"></a><span class="in">km.6mp |&gt; </span></span>
+<span id="cb31-1381"><a href="#cb31-1381" aria-hidden="true" tabindex="-1"></a><span class="in">  ggplot(aes(x = t2, y = `KM Survival Curve`)) +</span></span>
+<span id="cb31-1382"><a href="#cb31-1382" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_step() +</span></span>
+<span id="cb31-1383"><a href="#cb31-1383" aria-hidden="true" tabindex="-1"></a><span class="in">  geom_point(data = km.6mp |&gt; filter(Censored &gt; 0), shape = 3) +</span></span>
+<span id="cb31-1384"><a href="#cb31-1384" aria-hidden="true" tabindex="-1"></a><span class="in">  expand_limits(y = c(0,1), x = 0) +</span></span>
+<span id="cb31-1385"><a href="#cb31-1385" aria-hidden="true" tabindex="-1"></a><span class="in">  xlab('Time since diagnosis (months)') +</span></span>
+<span id="cb31-1386"><a href="#cb31-1386" aria-hidden="true" tabindex="-1"></a><span class="in">  ylab("KM Survival Curve") +</span></span>
+<span id="cb31-1387"><a href="#cb31-1387" aria-hidden="true" tabindex="-1"></a><span class="in">  scale_y_continuous(labels = scales::percent)</span></span>
+<span id="cb31-1388"><a href="#cb31-1388" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1389"><a href="#cb31-1389" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1390"><a href="#cb31-1390" aria-hidden="true" tabindex="-1"></a><span class="fu">## Using the `survival` package in R</span></span>
+<span id="cb31-1391"><a href="#cb31-1391" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1392"><a href="#cb31-1392" aria-hidden="true" tabindex="-1"></a>We don't have to do these calculations by hand every time; the</span>
+<span id="cb31-1393"><a href="#cb31-1393" aria-hidden="true" tabindex="-1"></a><span class="in">`survival`</span> package and several others have functions available to</span>
+<span id="cb31-1394"><a href="#cb31-1394" aria-hidden="true" tabindex="-1"></a>automate many of these tasks (full list:</span>
+<span id="cb31-1395"><a href="#cb31-1395" aria-hidden="true" tabindex="-1"></a><span class="ot">&lt;https://cran.r-project.org/web/views/Survival.html&gt;</span>).</span>
 <span id="cb31-1396"><a href="#cb31-1396" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1397"><a href="#cb31-1397" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>observations where the event was right-censored are printed as the</span>
-<span id="cb31-1398"><a href="#cb31-1398" aria-hidden="true" tabindex="-1"></a>    censoring time with a plus sign (<span class="in">`+`</span>; for example, <span class="in">`surv2 = 32+`</span> on</span>
-<span id="cb31-1399"><a href="#cb31-1399" aria-hidden="true" tabindex="-1"></a>    line 3).</span>
-<span id="cb31-1400"><a href="#cb31-1400" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1401"><a href="#cb31-1401" aria-hidden="true" tabindex="-1"></a><span class="fu">### The `survfit` function</span></span>
+<span id="cb31-1397"><a href="#cb31-1397" aria-hidden="true" tabindex="-1"></a><span class="fu">### The `Surv` function</span></span>
+<span id="cb31-1398"><a href="#cb31-1398" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1399"><a href="#cb31-1399" aria-hidden="true" tabindex="-1"></a>To use the <span class="in">`survival`</span> package, the first step is telling R how to</span>
+<span id="cb31-1400"><a href="#cb31-1400" aria-hidden="true" tabindex="-1"></a>combine the exit time and exit reason (censoring versus event) columns.</span>
+<span id="cb31-1401"><a href="#cb31-1401" aria-hidden="true" tabindex="-1"></a>The <span class="in">`Surv()`</span> function accomplishes this task.</span>
 <span id="cb31-1402"><a href="#cb31-1402" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1403"><a href="#cb31-1403" aria-hidden="true" tabindex="-1"></a>Once we have constructed our <span class="in">`Surv`</span> variable, we can calculate the</span>
-<span id="cb31-1404"><a href="#cb31-1404" aria-hidden="true" tabindex="-1"></a>Kaplan-Meier estimate of the survival curve using the <span class="in">`survfit()`</span></span>
-<span id="cb31-1405"><a href="#cb31-1405" aria-hidden="true" tabindex="-1"></a>function.</span>
-<span id="cb31-1406"><a href="#cb31-1406" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1407"><a href="#cb31-1407" aria-hidden="true" tabindex="-1"></a>::: callout-note</span>
-<span id="cb31-1408"><a href="#cb31-1408" aria-hidden="true" tabindex="-1"></a>The documentation for <span class="in">`?survfit`</span> isn't too helpful; the</span>
-<span id="cb31-1409"><a href="#cb31-1409" aria-hidden="true" tabindex="-1"></a><span class="in">`survfit.formula`</span> documentation is better.</span>
-<span id="cb31-1410"><a href="#cb31-1410" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1411"><a href="#cb31-1411" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, printr.help.sections = c("description", "usage")}</span></span>
-<span id="cb31-1412"><a href="#cb31-1412" aria-hidden="true" tabindex="-1"></a><span class="in">#| include: false</span></span>
-<span id="cb31-1413"><a href="#cb31-1413" aria-hidden="true" tabindex="-1"></a><span class="in">?survfit.formula</span></span>
-<span id="cb31-1414"><a href="#cb31-1414" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1415"><a href="#cb31-1415" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-1416"><a href="#cb31-1416" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1417"><a href="#cb31-1417" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1418"><a href="#cb31-1418" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1419"><a href="#cb31-1419" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Example: `survfit()` with `drug6mp` data</span></span>
-<span id="cb31-1420"><a href="#cb31-1420" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1421"><a href="#cb31-1421" aria-hidden="true" tabindex="-1"></a>Here we use <span class="in">`survfit()`</span> to create a <span class="in">`survfit`</span> object, which contains the</span>
-<span id="cb31-1422"><a href="#cb31-1422" aria-hidden="true" tabindex="-1"></a>Kaplan-Meier estimate:</span>
-<span id="cb31-1423"><a href="#cb31-1423" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1426"><a href="#cb31-1426" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1427"><a href="#cb31-1427" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-fold: show</span></span>
-<span id="cb31-1428"><a href="#cb31-1428" aria-hidden="true" tabindex="-1"></a>drug6mp.km_model <span class="ot">=</span> <span class="fu">survfit</span>(</span>
-<span id="cb31-1429"><a href="#cb31-1429" aria-hidden="true" tabindex="-1"></a>  <span class="at">formula =</span> surv2 <span class="sc">~</span> <span class="dv">1</span>, </span>
-<span id="cb31-1430"><a href="#cb31-1430" aria-hidden="true" tabindex="-1"></a>  <span class="at">data =</span> drug6mp.v3)</span>
-<span id="cb31-1431"><a href="#cb31-1431" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1432"><a href="#cb31-1432" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1433"><a href="#cb31-1433" aria-hidden="true" tabindex="-1"></a><span class="in">`print.survfit()`</span> just gives some summary statistics:</span>
-<span id="cb31-1434"><a href="#cb31-1434" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1437"><a href="#cb31-1437" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1438"><a href="#cb31-1438" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-fold: show</span></span>
-<span id="cb31-1439"><a href="#cb31-1439" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(drug6mp.km_model)</span>
-<span id="cb31-1440"><a href="#cb31-1440" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1403"><a href="#cb31-1403" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Example: `Surv()` with `drug6mp` data</span></span>
+<span id="cb31-1404"><a href="#cb31-1404" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1407"><a href="#cb31-1407" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1408"><a href="#cb31-1408" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-fold: show</span></span>
+<span id="cb31-1409"><a href="#cb31-1409" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-line-numbers: "5-7"</span></span>
+<span id="cb31-1410"><a href="#cb31-1410" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(survival)</span>
+<span id="cb31-1411"><a href="#cb31-1411" aria-hidden="true" tabindex="-1"></a>drug6mp.v3 <span class="ot">=</span> </span>
+<span id="cb31-1412"><a href="#cb31-1412" aria-hidden="true" tabindex="-1"></a>  drug6mp.v2 <span class="sc">|&gt;</span> </span>
+<span id="cb31-1413"><a href="#cb31-1413" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(</span>
+<span id="cb31-1414"><a href="#cb31-1414" aria-hidden="true" tabindex="-1"></a>    <span class="at">surv2 =</span> <span class="fu">Surv</span>(</span>
+<span id="cb31-1415"><a href="#cb31-1415" aria-hidden="true" tabindex="-1"></a>      <span class="at">time =</span> t2,</span>
+<span id="cb31-1416"><a href="#cb31-1416" aria-hidden="true" tabindex="-1"></a>      <span class="at">event =</span> (outcome <span class="sc">==</span> <span class="st">"relapsed"</span>)))</span>
+<span id="cb31-1417"><a href="#cb31-1417" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1418"><a href="#cb31-1418" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(drug6mp.v3)</span>
+<span id="cb31-1419"><a href="#cb31-1419" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1420"><a href="#cb31-1420" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1421"><a href="#cb31-1421" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1422"><a href="#cb31-1422" aria-hidden="true" tabindex="-1"></a>The output of <span class="in">`Surv()`</span> is a vector of objects with class <span class="in">`Surv`</span>. When we</span>
+<span id="cb31-1423"><a href="#cb31-1423" aria-hidden="true" tabindex="-1"></a>print this vector:</span>
+<span id="cb31-1424"><a href="#cb31-1424" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1425"><a href="#cb31-1425" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>observations where the event was observed are printed as the event</span>
+<span id="cb31-1426"><a href="#cb31-1426" aria-hidden="true" tabindex="-1"></a>    time (for example, <span class="in">`surv2 = 10`</span> on line 1)</span>
+<span id="cb31-1427"><a href="#cb31-1427" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1428"><a href="#cb31-1428" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>observations where the event was right-censored are printed as the</span>
+<span id="cb31-1429"><a href="#cb31-1429" aria-hidden="true" tabindex="-1"></a>    censoring time with a plus sign (<span class="in">`+`</span>; for example, <span class="in">`surv2 = 32+`</span> on</span>
+<span id="cb31-1430"><a href="#cb31-1430" aria-hidden="true" tabindex="-1"></a>    line 3).</span>
+<span id="cb31-1431"><a href="#cb31-1431" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1432"><a href="#cb31-1432" aria-hidden="true" tabindex="-1"></a><span class="fu">### The `survfit` function</span></span>
+<span id="cb31-1433"><a href="#cb31-1433" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1434"><a href="#cb31-1434" aria-hidden="true" tabindex="-1"></a>Once we have constructed our <span class="in">`Surv`</span> variable, we can calculate the</span>
+<span id="cb31-1435"><a href="#cb31-1435" aria-hidden="true" tabindex="-1"></a>Kaplan-Meier estimate of the survival curve using the <span class="in">`survfit()`</span></span>
+<span id="cb31-1436"><a href="#cb31-1436" aria-hidden="true" tabindex="-1"></a>function.</span>
+<span id="cb31-1437"><a href="#cb31-1437" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1438"><a href="#cb31-1438" aria-hidden="true" tabindex="-1"></a>::: callout-note</span>
+<span id="cb31-1439"><a href="#cb31-1439" aria-hidden="true" tabindex="-1"></a>The documentation for <span class="in">`?survfit`</span> isn't too helpful; the</span>
+<span id="cb31-1440"><a href="#cb31-1440" aria-hidden="true" tabindex="-1"></a><span class="in">`survfit.formula`</span> documentation is better.</span>
 <span id="cb31-1441"><a href="#cb31-1441" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1442"><a href="#cb31-1442" aria-hidden="true" tabindex="-1"></a><span class="in">`summary.survfit()`</span> shows us the underlying Kaplan-Meier table:</span>
-<span id="cb31-1443"><a href="#cb31-1443" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1446"><a href="#cb31-1446" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1447"><a href="#cb31-1447" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-fold: show</span></span>
-<span id="cb31-1448"><a href="#cb31-1448" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(drug6mp.km_model)</span>
-<span id="cb31-1449"><a href="#cb31-1449" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1450"><a href="#cb31-1450" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1451"><a href="#cb31-1451" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1452"><a href="#cb31-1452" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1453"><a href="#cb31-1453" aria-hidden="true" tabindex="-1"></a>We can specify which time points we want using the <span class="in">`times`</span> argument:</span>
+<span id="cb31-1442"><a href="#cb31-1442" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, printr.help.sections = c("description", "usage")}</span></span>
+<span id="cb31-1443"><a href="#cb31-1443" aria-hidden="true" tabindex="-1"></a><span class="in">#| include: false</span></span>
+<span id="cb31-1444"><a href="#cb31-1444" aria-hidden="true" tabindex="-1"></a><span class="in">?survfit.formula</span></span>
+<span id="cb31-1445"><a href="#cb31-1445" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1446"><a href="#cb31-1446" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-1447"><a href="#cb31-1447" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1448"><a href="#cb31-1448" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1449"><a href="#cb31-1449" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1450"><a href="#cb31-1450" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Example: `survfit()` with `drug6mp` data</span></span>
+<span id="cb31-1451"><a href="#cb31-1451" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1452"><a href="#cb31-1452" aria-hidden="true" tabindex="-1"></a>Here we use <span class="in">`survfit()`</span> to create a <span class="in">`survfit`</span> object, which contains the</span>
+<span id="cb31-1453"><a href="#cb31-1453" aria-hidden="true" tabindex="-1"></a>Kaplan-Meier estimate:</span>
 <span id="cb31-1454"><a href="#cb31-1454" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb31-1457"><a href="#cb31-1457" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
 <span id="cb31-1458"><a href="#cb31-1458" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-fold: show</span></span>
-<span id="cb31-1459"><a href="#cb31-1459" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(</span>
-<span id="cb31-1460"><a href="#cb31-1460" aria-hidden="true" tabindex="-1"></a>  drug6mp.km_model, </span>
-<span id="cb31-1461"><a href="#cb31-1461" aria-hidden="true" tabindex="-1"></a>  <span class="at">times =</span> <span class="fu">c</span>(<span class="dv">0</span>, drug6mp.v3<span class="sc">$</span>t2))</span>
-<span id="cb31-1462"><a href="#cb31-1462" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1463"><a href="#cb31-1463" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1464"><a href="#cb31-1464" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1465"><a href="#cb31-1465" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1466"><a href="#cb31-1466" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1467"><a href="#cb31-1467" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, printr.help.sections = c("description", "usage", "arguments")}</span></span>
-<span id="cb31-1468"><a href="#cb31-1468" aria-hidden="true" tabindex="-1"></a><span class="in">#| code-fold: show</span></span>
-<span id="cb31-1469"><a href="#cb31-1469" aria-hidden="true" tabindex="-1"></a><span class="in">?summary.survfit</span></span>
-<span id="cb31-1470"><a href="#cb31-1470" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1471"><a href="#cb31-1471" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1472"><a href="#cb31-1472" aria-hidden="true" tabindex="-1"></a><span class="fu">### Plotting estimated survival functions</span></span>
-<span id="cb31-1473"><a href="#cb31-1473" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1474"><a href="#cb31-1474" aria-hidden="true" tabindex="-1"></a>We can plot <span class="in">`survfit`</span> objects with <span class="in">`plot()`</span>, <span class="in">`autoplot()`</span>, or</span>
-<span id="cb31-1475"><a href="#cb31-1475" aria-hidden="true" tabindex="-1"></a><span class="in">`ggsurvplot()`</span>:</span>
-<span id="cb31-1476"><a href="#cb31-1476" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1479"><a href="#cb31-1479" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1480"><a href="#cb31-1480" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-fold: show</span></span>
-<span id="cb31-1481"><a href="#cb31-1481" aria-hidden="true" tabindex="-1"></a><span class="co">#| fig-cap: "Kaplan-Meier Survival Curve for 6-MP Patients"</span></span>
-<span id="cb31-1482"><a href="#cb31-1482" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1483"><a href="#cb31-1483" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(ggfortify)</span>
-<span id="cb31-1484"><a href="#cb31-1484" aria-hidden="true" tabindex="-1"></a><span class="fu">autoplot</span>(drug6mp.km_model)</span>
+<span id="cb31-1459"><a href="#cb31-1459" aria-hidden="true" tabindex="-1"></a>drug6mp.km_model <span class="ot">=</span> <span class="fu">survfit</span>(</span>
+<span id="cb31-1460"><a href="#cb31-1460" aria-hidden="true" tabindex="-1"></a>  <span class="at">formula =</span> surv2 <span class="sc">~</span> <span class="dv">1</span>, </span>
+<span id="cb31-1461"><a href="#cb31-1461" aria-hidden="true" tabindex="-1"></a>  <span class="at">data =</span> drug6mp.v3)</span>
+<span id="cb31-1462"><a href="#cb31-1462" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1463"><a href="#cb31-1463" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1464"><a href="#cb31-1464" aria-hidden="true" tabindex="-1"></a><span class="in">`print.survfit()`</span> just gives some summary statistics:</span>
+<span id="cb31-1465"><a href="#cb31-1465" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1468"><a href="#cb31-1468" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1469"><a href="#cb31-1469" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-fold: show</span></span>
+<span id="cb31-1470"><a href="#cb31-1470" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(drug6mp.km_model)</span>
+<span id="cb31-1471"><a href="#cb31-1471" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1472"><a href="#cb31-1472" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1473"><a href="#cb31-1473" aria-hidden="true" tabindex="-1"></a><span class="in">`summary.survfit()`</span> shows us the underlying Kaplan-Meier table:</span>
+<span id="cb31-1474"><a href="#cb31-1474" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1477"><a href="#cb31-1477" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1478"><a href="#cb31-1478" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-fold: show</span></span>
+<span id="cb31-1479"><a href="#cb31-1479" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(drug6mp.km_model)</span>
+<span id="cb31-1480"><a href="#cb31-1480" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1481"><a href="#cb31-1481" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1482"><a href="#cb31-1482" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1483"><a href="#cb31-1483" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1484"><a href="#cb31-1484" aria-hidden="true" tabindex="-1"></a>We can specify which time points we want using the <span class="in">`times`</span> argument:</span>
 <span id="cb31-1485"><a href="#cb31-1485" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1486"><a href="#cb31-1486" aria-hidden="true" tabindex="-1"></a><span class="co"># not shown:</span></span>
-<span id="cb31-1487"><a href="#cb31-1487" aria-hidden="true" tabindex="-1"></a><span class="co"># plot(drug6mp.km_model)</span></span>
-<span id="cb31-1488"><a href="#cb31-1488" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1489"><a href="#cb31-1489" aria-hidden="true" tabindex="-1"></a><span class="co"># library(survminer)</span></span>
-<span id="cb31-1490"><a href="#cb31-1490" aria-hidden="true" tabindex="-1"></a><span class="co"># ggsurvplot(drug6mp.km_model)</span></span>
-<span id="cb31-1491"><a href="#cb31-1491" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1492"><a href="#cb31-1492" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1488"><a href="#cb31-1488" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1489"><a href="#cb31-1489" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-fold: show</span></span>
+<span id="cb31-1490"><a href="#cb31-1490" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(</span>
+<span id="cb31-1491"><a href="#cb31-1491" aria-hidden="true" tabindex="-1"></a>  drug6mp.km_model, </span>
+<span id="cb31-1492"><a href="#cb31-1492" aria-hidden="true" tabindex="-1"></a>  <span class="at">times =</span> <span class="fu">c</span>(<span class="dv">0</span>, drug6mp.v3<span class="sc">$</span>t2))</span>
 <span id="cb31-1493"><a href="#cb31-1493" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1494"><a href="#cb31-1494" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1494"><a href="#cb31-1494" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
 <span id="cb31-1495"><a href="#cb31-1495" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1496"><a href="#cb31-1496" aria-hidden="true" tabindex="-1"></a><span class="fu">#### quantiles of survival curve</span></span>
+<span id="cb31-1496"><a href="#cb31-1496" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-1497"><a href="#cb31-1497" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1498"><a href="#cb31-1498" aria-hidden="true" tabindex="-1"></a>We can extract quantiles with <span class="in">`quantile()`</span>:</span>
-<span id="cb31-1499"><a href="#cb31-1499" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1502"><a href="#cb31-1502" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1503"><a href="#cb31-1503" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-line-numbers: "2"</span></span>
-<span id="cb31-1504"><a href="#cb31-1504" aria-hidden="true" tabindex="-1"></a>drug6mp.km_model <span class="sc">|&gt;</span> </span>
-<span id="cb31-1505"><a href="#cb31-1505" aria-hidden="true" tabindex="-1"></a>  <span class="fu">quantile</span>(<span class="at">p =</span> <span class="fu">c</span>(.<span class="dv">25</span>, .<span class="dv">5</span>)) <span class="sc">|&gt;</span> </span>
-<span id="cb31-1506"><a href="#cb31-1506" aria-hidden="true" tabindex="-1"></a>  <span class="fu">as_tibble</span>() <span class="sc">|&gt;</span> </span>
-<span id="cb31-1507"><a href="#cb31-1507" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">p =</span> <span class="fu">c</span>(.<span class="dv">25</span>, .<span class="dv">5</span>)) <span class="sc">|&gt;</span> </span>
-<span id="cb31-1508"><a href="#cb31-1508" aria-hidden="true" tabindex="-1"></a>  <span class="fu">relocate</span>(p, <span class="at">.before =</span> <span class="fu">everything</span>())</span>
-<span id="cb31-1509"><a href="#cb31-1509" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1510"><a href="#cb31-1510" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1511"><a href="#cb31-1511" aria-hidden="true" tabindex="-1"></a><span class="fu">### Two-sample tests</span></span>
-<span id="cb31-1512"><a href="#cb31-1512" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1513"><a href="#cb31-1513" aria-hidden="true" tabindex="-1"></a><span class="fu">#### The `survdiff` function</span></span>
-<span id="cb31-1514"><a href="#cb31-1514" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1515"><a href="#cb31-1515" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, printr.help.sections = c("description", "usage")}</span></span>
-<span id="cb31-1516"><a href="#cb31-1516" aria-hidden="true" tabindex="-1"></a><span class="in">?survdiff</span></span>
-<span id="cb31-1517"><a href="#cb31-1517" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1518"><a href="#cb31-1518" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1519"><a href="#cb31-1519" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Example: `survdiff()` with `drug6mp` data</span></span>
-<span id="cb31-1520"><a href="#cb31-1520" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1521"><a href="#cb31-1521" aria-hidden="true" tabindex="-1"></a>Now we are going to compare the placebo and 6-MP data. We need to</span>
-<span id="cb31-1522"><a href="#cb31-1522" aria-hidden="true" tabindex="-1"></a>reshape the data to make it usable with the standard <span class="in">`survival`</span></span>
-<span id="cb31-1523"><a href="#cb31-1523" aria-hidden="true" tabindex="-1"></a>workflow:</span>
+<span id="cb31-1498"><a href="#cb31-1498" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, printr.help.sections = c("description", "usage", "arguments")}</span></span>
+<span id="cb31-1499"><a href="#cb31-1499" aria-hidden="true" tabindex="-1"></a><span class="in">#| code-fold: show</span></span>
+<span id="cb31-1500"><a href="#cb31-1500" aria-hidden="true" tabindex="-1"></a><span class="in">?summary.survfit</span></span>
+<span id="cb31-1501"><a href="#cb31-1501" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1502"><a href="#cb31-1502" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1503"><a href="#cb31-1503" aria-hidden="true" tabindex="-1"></a><span class="fu">### Plotting estimated survival functions</span></span>
+<span id="cb31-1504"><a href="#cb31-1504" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1505"><a href="#cb31-1505" aria-hidden="true" tabindex="-1"></a>We can plot <span class="in">`survfit`</span> objects with <span class="in">`plot()`</span>, <span class="in">`autoplot()`</span>, or</span>
+<span id="cb31-1506"><a href="#cb31-1506" aria-hidden="true" tabindex="-1"></a><span class="in">`ggsurvplot()`</span>:</span>
+<span id="cb31-1507"><a href="#cb31-1507" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1510"><a href="#cb31-1510" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1511"><a href="#cb31-1511" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-fold: show</span></span>
+<span id="cb31-1512"><a href="#cb31-1512" aria-hidden="true" tabindex="-1"></a><span class="co">#| fig-cap: "Kaplan-Meier Survival Curve for 6-MP Patients"</span></span>
+<span id="cb31-1513"><a href="#cb31-1513" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1514"><a href="#cb31-1514" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(ggfortify)</span>
+<span id="cb31-1515"><a href="#cb31-1515" aria-hidden="true" tabindex="-1"></a><span class="fu">autoplot</span>(drug6mp.km_model)</span>
+<span id="cb31-1516"><a href="#cb31-1516" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1517"><a href="#cb31-1517" aria-hidden="true" tabindex="-1"></a><span class="co"># not shown:</span></span>
+<span id="cb31-1518"><a href="#cb31-1518" aria-hidden="true" tabindex="-1"></a><span class="co"># plot(drug6mp.km_model)</span></span>
+<span id="cb31-1519"><a href="#cb31-1519" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1520"><a href="#cb31-1520" aria-hidden="true" tabindex="-1"></a><span class="co"># library(survminer)</span></span>
+<span id="cb31-1521"><a href="#cb31-1521" aria-hidden="true" tabindex="-1"></a><span class="co"># ggsurvplot(drug6mp.km_model)</span></span>
+<span id="cb31-1522"><a href="#cb31-1522" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1523"><a href="#cb31-1523" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
 <span id="cb31-1524"><a href="#cb31-1524" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1527"><a href="#cb31-1527" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1528"><a href="#cb31-1528" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(survival)</span>
-<span id="cb31-1529"><a href="#cb31-1529" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyr)</span>
-<span id="cb31-1530"><a href="#cb31-1530" aria-hidden="true" tabindex="-1"></a>drug6mp.v4 <span class="ot">=</span> </span>
-<span id="cb31-1531"><a href="#cb31-1531" aria-hidden="true" tabindex="-1"></a>  drug6mp.v3 <span class="sc">|&gt;</span></span>
-<span id="cb31-1532"><a href="#cb31-1532" aria-hidden="true" tabindex="-1"></a>  <span class="fu">select</span>(pair, remstat, t1, t2, outcome) <span class="sc">|&gt;</span> </span>
-<span id="cb31-1533"><a href="#cb31-1533" aria-hidden="true" tabindex="-1"></a>  <span class="co"># here we are going to change the data from a wide format to long:</span></span>
-<span id="cb31-1534"><a href="#cb31-1534" aria-hidden="true" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(</span>
-<span id="cb31-1535"><a href="#cb31-1535" aria-hidden="true" tabindex="-1"></a>    <span class="at">cols =</span> <span class="fu">c</span>(t1, t2),</span>
-<span id="cb31-1536"><a href="#cb31-1536" aria-hidden="true" tabindex="-1"></a>    <span class="at">names_to =</span> <span class="st">"treatment"</span>,</span>
-<span id="cb31-1537"><a href="#cb31-1537" aria-hidden="true" tabindex="-1"></a>    <span class="at">values_to =</span> <span class="st">"exit_time"</span>) <span class="sc">|&gt;</span> </span>
-<span id="cb31-1538"><a href="#cb31-1538" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(</span>
-<span id="cb31-1539"><a href="#cb31-1539" aria-hidden="true" tabindex="-1"></a>    <span class="at">treatment =</span> treatment <span class="sc">|&gt;</span> </span>
-<span id="cb31-1540"><a href="#cb31-1540" aria-hidden="true" tabindex="-1"></a>      <span class="fu">case_match</span>(</span>
-<span id="cb31-1541"><a href="#cb31-1541" aria-hidden="true" tabindex="-1"></a>        <span class="st">"t1"</span> <span class="sc">~</span> <span class="st">"placebo"</span>,</span>
-<span id="cb31-1542"><a href="#cb31-1542" aria-hidden="true" tabindex="-1"></a>        <span class="st">"t2"</span> <span class="sc">~</span> <span class="st">"6-MP"</span></span>
-<span id="cb31-1543"><a href="#cb31-1543" aria-hidden="true" tabindex="-1"></a>      ),</span>
-<span id="cb31-1544"><a href="#cb31-1544" aria-hidden="true" tabindex="-1"></a>    <span class="at">outcome =</span> <span class="fu">if_else</span>(</span>
-<span id="cb31-1545"><a href="#cb31-1545" aria-hidden="true" tabindex="-1"></a>      treatment <span class="sc">==</span> <span class="st">"placebo"</span>,</span>
-<span id="cb31-1546"><a href="#cb31-1546" aria-hidden="true" tabindex="-1"></a>      <span class="st">"relapsed"</span>,</span>
-<span id="cb31-1547"><a href="#cb31-1547" aria-hidden="true" tabindex="-1"></a>      outcome</span>
-<span id="cb31-1548"><a href="#cb31-1548" aria-hidden="true" tabindex="-1"></a>    ),</span>
-<span id="cb31-1549"><a href="#cb31-1549" aria-hidden="true" tabindex="-1"></a>    <span class="at">surv =</span> <span class="fu">Surv</span>(</span>
-<span id="cb31-1550"><a href="#cb31-1550" aria-hidden="true" tabindex="-1"></a>      <span class="at">time =</span> exit_time,</span>
-<span id="cb31-1551"><a href="#cb31-1551" aria-hidden="true" tabindex="-1"></a>      <span class="at">event =</span> (outcome <span class="sc">==</span> <span class="st">"relapsed"</span>))</span>
-<span id="cb31-1552"><a href="#cb31-1552" aria-hidden="true" tabindex="-1"></a>  )</span>
-<span id="cb31-1553"><a href="#cb31-1553" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1554"><a href="#cb31-1554" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1555"><a href="#cb31-1555" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1556"><a href="#cb31-1556" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1557"><a href="#cb31-1557" aria-hidden="true" tabindex="-1"></a>Using this long data format, we can fit a Kaplan-Meier curve for each</span>
-<span id="cb31-1558"><a href="#cb31-1558" aria-hidden="true" tabindex="-1"></a>treatment group simultaneously:</span>
-<span id="cb31-1559"><a href="#cb31-1559" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1562"><a href="#cb31-1562" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1563"><a href="#cb31-1563" aria-hidden="true" tabindex="-1"></a>drug6mp.km_model2 <span class="ot">=</span> </span>
-<span id="cb31-1564"><a href="#cb31-1564" aria-hidden="true" tabindex="-1"></a>   <span class="fu">survfit</span>(</span>
-<span id="cb31-1565"><a href="#cb31-1565" aria-hidden="true" tabindex="-1"></a>  <span class="at">formula =</span> surv <span class="sc">~</span> treatment, </span>
-<span id="cb31-1566"><a href="#cb31-1566" aria-hidden="true" tabindex="-1"></a>  <span class="at">data =</span> drug6mp.v4)</span>
-<span id="cb31-1567"><a href="#cb31-1567" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1568"><a href="#cb31-1568" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1569"><a href="#cb31-1569" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1570"><a href="#cb31-1570" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1571"><a href="#cb31-1571" aria-hidden="true" tabindex="-1"></a>We can plot the curves in the same graph:</span>
-<span id="cb31-1572"><a href="#cb31-1572" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1575"><a href="#cb31-1575" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1576"><a href="#cb31-1576" aria-hidden="true" tabindex="-1"></a>drug6mp.km_model2 <span class="sc">|&gt;</span> <span class="fu">autoplot</span>()</span>
-<span id="cb31-1577"><a href="#cb31-1577" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1578"><a href="#cb31-1578" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1579"><a href="#cb31-1579" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1580"><a href="#cb31-1580" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1581"><a href="#cb31-1581" aria-hidden="true" tabindex="-1"></a>We can also perform something like a t-test, where the null hypothesis</span>
-<span id="cb31-1582"><a href="#cb31-1582" aria-hidden="true" tabindex="-1"></a>is that the curves are the same:</span>
-<span id="cb31-1583"><a href="#cb31-1583" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1586"><a href="#cb31-1586" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1587"><a href="#cb31-1587" aria-hidden="true" tabindex="-1"></a><span class="fu">survdiff</span>(</span>
-<span id="cb31-1588"><a href="#cb31-1588" aria-hidden="true" tabindex="-1"></a>  <span class="at">formula =</span> surv <span class="sc">~</span> treatment,</span>
-<span id="cb31-1589"><a href="#cb31-1589" aria-hidden="true" tabindex="-1"></a>  <span class="at">data =</span> drug6mp.v4)</span>
-<span id="cb31-1590"><a href="#cb31-1590" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1591"><a href="#cb31-1591" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1592"><a href="#cb31-1592" aria-hidden="true" tabindex="-1"></a>By default, <span class="in">`survdiff()`</span> ignores any pairing, </span>
-<span id="cb31-1593"><a href="#cb31-1593" aria-hidden="true" tabindex="-1"></a>but we can use <span class="in">`strata()`</span> to perform something similar to a paired t-test:</span>
-<span id="cb31-1594"><a href="#cb31-1594" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1597"><a href="#cb31-1597" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1598"><a href="#cb31-1598" aria-hidden="true" tabindex="-1"></a><span class="fu">survdiff</span>(</span>
-<span id="cb31-1599"><a href="#cb31-1599" aria-hidden="true" tabindex="-1"></a>  <span class="at">formula =</span> surv <span class="sc">~</span> treatment <span class="sc">+</span> <span class="fu">strata</span>(pair),</span>
-<span id="cb31-1600"><a href="#cb31-1600" aria-hidden="true" tabindex="-1"></a>  <span class="at">data =</span> drug6mp.v4)</span>
+<span id="cb31-1525"><a href="#cb31-1525" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1526"><a href="#cb31-1526" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1527"><a href="#cb31-1527" aria-hidden="true" tabindex="-1"></a><span class="fu">#### quantiles of survival curve</span></span>
+<span id="cb31-1528"><a href="#cb31-1528" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1529"><a href="#cb31-1529" aria-hidden="true" tabindex="-1"></a>We can extract quantiles with <span class="in">`quantile()`</span>:</span>
+<span id="cb31-1530"><a href="#cb31-1530" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1533"><a href="#cb31-1533" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1534"><a href="#cb31-1534" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-line-numbers: "2"</span></span>
+<span id="cb31-1535"><a href="#cb31-1535" aria-hidden="true" tabindex="-1"></a>drug6mp.km_model <span class="sc">|&gt;</span> </span>
+<span id="cb31-1536"><a href="#cb31-1536" aria-hidden="true" tabindex="-1"></a>  <span class="fu">quantile</span>(<span class="at">p =</span> <span class="fu">c</span>(.<span class="dv">25</span>, .<span class="dv">5</span>)) <span class="sc">|&gt;</span> </span>
+<span id="cb31-1537"><a href="#cb31-1537" aria-hidden="true" tabindex="-1"></a>  <span class="fu">as_tibble</span>() <span class="sc">|&gt;</span> </span>
+<span id="cb31-1538"><a href="#cb31-1538" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">p =</span> <span class="fu">c</span>(.<span class="dv">25</span>, .<span class="dv">5</span>)) <span class="sc">|&gt;</span> </span>
+<span id="cb31-1539"><a href="#cb31-1539" aria-hidden="true" tabindex="-1"></a>  <span class="fu">relocate</span>(p, <span class="at">.before =</span> <span class="fu">everything</span>())</span>
+<span id="cb31-1540"><a href="#cb31-1540" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1541"><a href="#cb31-1541" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1542"><a href="#cb31-1542" aria-hidden="true" tabindex="-1"></a><span class="fu">### Two-sample tests</span></span>
+<span id="cb31-1543"><a href="#cb31-1543" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1544"><a href="#cb31-1544" aria-hidden="true" tabindex="-1"></a><span class="fu">#### The `survdiff` function</span></span>
+<span id="cb31-1545"><a href="#cb31-1545" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1546"><a href="#cb31-1546" aria-hidden="true" tabindex="-1"></a><span class="in">```{r, printr.help.sections = c("description", "usage")}</span></span>
+<span id="cb31-1547"><a href="#cb31-1547" aria-hidden="true" tabindex="-1"></a><span class="in">?survdiff</span></span>
+<span id="cb31-1548"><a href="#cb31-1548" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1549"><a href="#cb31-1549" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1550"><a href="#cb31-1550" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Example: `survdiff()` with `drug6mp` data</span></span>
+<span id="cb31-1551"><a href="#cb31-1551" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1552"><a href="#cb31-1552" aria-hidden="true" tabindex="-1"></a>Now we are going to compare the placebo and 6-MP data. We need to</span>
+<span id="cb31-1553"><a href="#cb31-1553" aria-hidden="true" tabindex="-1"></a>reshape the data to make it usable with the standard <span class="in">`survival`</span></span>
+<span id="cb31-1554"><a href="#cb31-1554" aria-hidden="true" tabindex="-1"></a>workflow:</span>
+<span id="cb31-1555"><a href="#cb31-1555" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1558"><a href="#cb31-1558" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1559"><a href="#cb31-1559" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(survival)</span>
+<span id="cb31-1560"><a href="#cb31-1560" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyr)</span>
+<span id="cb31-1561"><a href="#cb31-1561" aria-hidden="true" tabindex="-1"></a>drug6mp.v4 <span class="ot">=</span> </span>
+<span id="cb31-1562"><a href="#cb31-1562" aria-hidden="true" tabindex="-1"></a>  drug6mp.v3 <span class="sc">|&gt;</span></span>
+<span id="cb31-1563"><a href="#cb31-1563" aria-hidden="true" tabindex="-1"></a>  <span class="fu">select</span>(pair, remstat, t1, t2, outcome) <span class="sc">|&gt;</span> </span>
+<span id="cb31-1564"><a href="#cb31-1564" aria-hidden="true" tabindex="-1"></a>  <span class="co"># here we are going to change the data from a wide format to long:</span></span>
+<span id="cb31-1565"><a href="#cb31-1565" aria-hidden="true" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(</span>
+<span id="cb31-1566"><a href="#cb31-1566" aria-hidden="true" tabindex="-1"></a>    <span class="at">cols =</span> <span class="fu">c</span>(t1, t2),</span>
+<span id="cb31-1567"><a href="#cb31-1567" aria-hidden="true" tabindex="-1"></a>    <span class="at">names_to =</span> <span class="st">"treatment"</span>,</span>
+<span id="cb31-1568"><a href="#cb31-1568" aria-hidden="true" tabindex="-1"></a>    <span class="at">values_to =</span> <span class="st">"exit_time"</span>) <span class="sc">|&gt;</span> </span>
+<span id="cb31-1569"><a href="#cb31-1569" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(</span>
+<span id="cb31-1570"><a href="#cb31-1570" aria-hidden="true" tabindex="-1"></a>    <span class="at">treatment =</span> treatment <span class="sc">|&gt;</span> </span>
+<span id="cb31-1571"><a href="#cb31-1571" aria-hidden="true" tabindex="-1"></a>      <span class="fu">case_match</span>(</span>
+<span id="cb31-1572"><a href="#cb31-1572" aria-hidden="true" tabindex="-1"></a>        <span class="st">"t1"</span> <span class="sc">~</span> <span class="st">"placebo"</span>,</span>
+<span id="cb31-1573"><a href="#cb31-1573" aria-hidden="true" tabindex="-1"></a>        <span class="st">"t2"</span> <span class="sc">~</span> <span class="st">"6-MP"</span></span>
+<span id="cb31-1574"><a href="#cb31-1574" aria-hidden="true" tabindex="-1"></a>      ),</span>
+<span id="cb31-1575"><a href="#cb31-1575" aria-hidden="true" tabindex="-1"></a>    <span class="at">outcome =</span> <span class="fu">if_else</span>(</span>
+<span id="cb31-1576"><a href="#cb31-1576" aria-hidden="true" tabindex="-1"></a>      treatment <span class="sc">==</span> <span class="st">"placebo"</span>,</span>
+<span id="cb31-1577"><a href="#cb31-1577" aria-hidden="true" tabindex="-1"></a>      <span class="st">"relapsed"</span>,</span>
+<span id="cb31-1578"><a href="#cb31-1578" aria-hidden="true" tabindex="-1"></a>      outcome</span>
+<span id="cb31-1579"><a href="#cb31-1579" aria-hidden="true" tabindex="-1"></a>    ),</span>
+<span id="cb31-1580"><a href="#cb31-1580" aria-hidden="true" tabindex="-1"></a>    <span class="at">surv =</span> <span class="fu">Surv</span>(</span>
+<span id="cb31-1581"><a href="#cb31-1581" aria-hidden="true" tabindex="-1"></a>      <span class="at">time =</span> exit_time,</span>
+<span id="cb31-1582"><a href="#cb31-1582" aria-hidden="true" tabindex="-1"></a>      <span class="at">event =</span> (outcome <span class="sc">==</span> <span class="st">"relapsed"</span>))</span>
+<span id="cb31-1583"><a href="#cb31-1583" aria-hidden="true" tabindex="-1"></a>  )</span>
+<span id="cb31-1584"><a href="#cb31-1584" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1585"><a href="#cb31-1585" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1586"><a href="#cb31-1586" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1587"><a href="#cb31-1587" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1588"><a href="#cb31-1588" aria-hidden="true" tabindex="-1"></a>Using this long data format, we can fit a Kaplan-Meier curve for each</span>
+<span id="cb31-1589"><a href="#cb31-1589" aria-hidden="true" tabindex="-1"></a>treatment group simultaneously:</span>
+<span id="cb31-1590"><a href="#cb31-1590" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1593"><a href="#cb31-1593" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1594"><a href="#cb31-1594" aria-hidden="true" tabindex="-1"></a>drug6mp.km_model2 <span class="ot">=</span> </span>
+<span id="cb31-1595"><a href="#cb31-1595" aria-hidden="true" tabindex="-1"></a>   <span class="fu">survfit</span>(</span>
+<span id="cb31-1596"><a href="#cb31-1596" aria-hidden="true" tabindex="-1"></a>  <span class="at">formula =</span> surv <span class="sc">~</span> treatment, </span>
+<span id="cb31-1597"><a href="#cb31-1597" aria-hidden="true" tabindex="-1"></a>  <span class="at">data =</span> drug6mp.v4)</span>
+<span id="cb31-1598"><a href="#cb31-1598" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1599"><a href="#cb31-1599" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1600"><a href="#cb31-1600" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-1601"><a href="#cb31-1601" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1602"><a href="#cb31-1602" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1602"><a href="#cb31-1602" aria-hidden="true" tabindex="-1"></a>We can plot the curves in the same graph:</span>
 <span id="cb31-1603"><a href="#cb31-1603" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1604"><a href="#cb31-1604" aria-hidden="true" tabindex="-1"></a>Interestingly, accounting for pairing reduces the significant of the</span>
-<span id="cb31-1605"><a href="#cb31-1605" aria-hidden="true" tabindex="-1"></a>difference.</span>
-<span id="cb31-1606"><a href="#cb31-1606" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1607"><a href="#cb31-1607" aria-hidden="true" tabindex="-1"></a><span class="fu">## Example: Bone Marrow Transplant Data</span></span>
-<span id="cb31-1608"><a href="#cb31-1608" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1609"><a href="#cb31-1609" aria-hidden="true" tabindex="-1"></a>@copelan1991treatment</span>
+<span id="cb31-1606"><a href="#cb31-1606" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1607"><a href="#cb31-1607" aria-hidden="true" tabindex="-1"></a>drug6mp.km_model2 <span class="sc">|&gt;</span> <span class="fu">autoplot</span>()</span>
+<span id="cb31-1608"><a href="#cb31-1608" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1609"><a href="#cb31-1609" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb31-1610"><a href="#cb31-1610" aria-hidden="true" tabindex="-1"></a>---</span>
 <span id="cb31-1611"><a href="#cb31-1611" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1612"><a href="#cb31-1612" aria-hidden="true" tabindex="-1"></a><span class="al">![Recovery process from a bone marrow transplant (Fig. 1.1 from @klein2003survival)](images/bone marrow multi-stage model.png)</span>{#fig-bmt-mst}</span>
-<span id="cb31-1613"><a href="#cb31-1613" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1614"><a href="#cb31-1614" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1615"><a href="#cb31-1615" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1616"><a href="#cb31-1616" aria-hidden="true" tabindex="-1"></a><span class="fu">### Study design</span></span>
-<span id="cb31-1617"><a href="#cb31-1617" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1618"><a href="#cb31-1618" aria-hidden="true" tabindex="-1"></a><span class="fu">##### Treatment {.unnumbered}</span></span>
-<span id="cb31-1619"><a href="#cb31-1619" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1620"><a href="#cb31-1620" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>**allogeneic** (from a donor) **bone marrow transplant therapy**</span>
-<span id="cb31-1621"><a href="#cb31-1621" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1622"><a href="#cb31-1622" aria-hidden="true" tabindex="-1"></a><span class="fu">##### Inclusion criteria {.unnumbered}</span></span>
-<span id="cb31-1623"><a href="#cb31-1623" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1624"><a href="#cb31-1624" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>**acute myeloid leukemia (AML)**</span>
-<span id="cb31-1625"><a href="#cb31-1625" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>**acute lymphoblastic leukemia (ALL).**</span>
-<span id="cb31-1626"><a href="#cb31-1626" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1627"><a href="#cb31-1627" aria-hidden="true" tabindex="-1"></a><span class="fu">##### Possible intermediate events {.unnumbered}</span></span>
-<span id="cb31-1628"><a href="#cb31-1628" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1629"><a href="#cb31-1629" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>**graft vs. host disease (GVHD)**: an immunological rejection</span>
-<span id="cb31-1630"><a href="#cb31-1630" aria-hidden="true" tabindex="-1"></a>    response to the transplant</span>
-<span id="cb31-1631"><a href="#cb31-1631" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>**platelet recovery**: a return of platelet count to normal levels.</span>
+<span id="cb31-1612"><a href="#cb31-1612" aria-hidden="true" tabindex="-1"></a>We can also perform something like a t-test, where the null hypothesis</span>
+<span id="cb31-1613"><a href="#cb31-1613" aria-hidden="true" tabindex="-1"></a>is that the curves are the same:</span>
+<span id="cb31-1614"><a href="#cb31-1614" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1617"><a href="#cb31-1617" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1618"><a href="#cb31-1618" aria-hidden="true" tabindex="-1"></a><span class="fu">survdiff</span>(</span>
+<span id="cb31-1619"><a href="#cb31-1619" aria-hidden="true" tabindex="-1"></a>  <span class="at">formula =</span> surv <span class="sc">~</span> treatment,</span>
+<span id="cb31-1620"><a href="#cb31-1620" aria-hidden="true" tabindex="-1"></a>  <span class="at">data =</span> drug6mp.v4)</span>
+<span id="cb31-1621"><a href="#cb31-1621" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1622"><a href="#cb31-1622" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1623"><a href="#cb31-1623" aria-hidden="true" tabindex="-1"></a>By default, <span class="in">`survdiff()`</span> ignores any pairing, </span>
+<span id="cb31-1624"><a href="#cb31-1624" aria-hidden="true" tabindex="-1"></a>but we can use <span class="in">`strata()`</span> to perform something similar to a paired t-test:</span>
+<span id="cb31-1625"><a href="#cb31-1625" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1628"><a href="#cb31-1628" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1629"><a href="#cb31-1629" aria-hidden="true" tabindex="-1"></a><span class="fu">survdiff</span>(</span>
+<span id="cb31-1630"><a href="#cb31-1630" aria-hidden="true" tabindex="-1"></a>  <span class="at">formula =</span> surv <span class="sc">~</span> treatment <span class="sc">+</span> <span class="fu">strata</span>(pair),</span>
+<span id="cb31-1631"><a href="#cb31-1631" aria-hidden="true" tabindex="-1"></a>  <span class="at">data =</span> drug6mp.v4)</span>
 <span id="cb31-1632"><a href="#cb31-1632" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1633"><a href="#cb31-1633" aria-hidden="true" tabindex="-1"></a>One or the other, both in either order, or neither may occur.</span>
+<span id="cb31-1633"><a href="#cb31-1633" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
 <span id="cb31-1634"><a href="#cb31-1634" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1635"><a href="#cb31-1635" aria-hidden="true" tabindex="-1"></a><span class="fu">##### End point events</span></span>
-<span id="cb31-1636"><a href="#cb31-1636" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1637"><a href="#cb31-1637" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>relapse of the disease</span>
-<span id="cb31-1638"><a href="#cb31-1638" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>death</span>
+<span id="cb31-1635"><a href="#cb31-1635" aria-hidden="true" tabindex="-1"></a>Interestingly, accounting for pairing reduces the significant of the</span>
+<span id="cb31-1636"><a href="#cb31-1636" aria-hidden="true" tabindex="-1"></a>difference.</span>
+<span id="cb31-1637"><a href="#cb31-1637" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1638"><a href="#cb31-1638" aria-hidden="true" tabindex="-1"></a><span class="fu">## Example: Bone Marrow Transplant Data</span></span>
 <span id="cb31-1639"><a href="#cb31-1639" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1640"><a href="#cb31-1640" aria-hidden="true" tabindex="-1"></a>Any or all of these events may be censored.</span>
-<span id="cb31-1641"><a href="#cb31-1641" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1642"><a href="#cb31-1642" aria-hidden="true" tabindex="-1"></a><span class="fu">### `KMsurv::bmt` data in R</span></span>
-<span id="cb31-1643"><a href="#cb31-1643" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1646"><a href="#cb31-1646" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1647"><a href="#cb31-1647" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(KMsurv)</span>
-<span id="cb31-1648"><a href="#cb31-1648" aria-hidden="true" tabindex="-1"></a>?bmt</span>
-<span id="cb31-1649"><a href="#cb31-1649" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1640"><a href="#cb31-1640" aria-hidden="true" tabindex="-1"></a>@copelan1991treatment</span>
+<span id="cb31-1641"><a href="#cb31-1641" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1642"><a href="#cb31-1642" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1643"><a href="#cb31-1643" aria-hidden="true" tabindex="-1"></a><span class="al">![Recovery process from a bone marrow transplant (Fig. 1.1 from @klein2003survival)](images/bone marrow multi-stage model.png)</span>{#fig-bmt-mst}</span>
+<span id="cb31-1644"><a href="#cb31-1644" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1645"><a href="#cb31-1645" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1646"><a href="#cb31-1646" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1647"><a href="#cb31-1647" aria-hidden="true" tabindex="-1"></a><span class="fu">### Study design</span></span>
+<span id="cb31-1648"><a href="#cb31-1648" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1649"><a href="#cb31-1649" aria-hidden="true" tabindex="-1"></a><span class="fu">##### Treatment {.unnumbered}</span></span>
 <span id="cb31-1650"><a href="#cb31-1650" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1651"><a href="#cb31-1651" aria-hidden="true" tabindex="-1"></a><span class="fu">### Analysis plan</span></span>
+<span id="cb31-1651"><a href="#cb31-1651" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>**allogeneic** (from a donor) **bone marrow transplant therapy**</span>
 <span id="cb31-1652"><a href="#cb31-1652" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1653"><a href="#cb31-1653" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>We concentrate for now on disease-free survival (<span class="in">`t2`</span> and <span class="in">`d3`</span>) for</span>
-<span id="cb31-1654"><a href="#cb31-1654" aria-hidden="true" tabindex="-1"></a>    the three risk groups, ALL, AML Low Risk, and AML High Risk.</span>
-<span id="cb31-1655"><a href="#cb31-1655" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>We will construct the Kaplan-Meier survival curves, compare them,</span>
-<span id="cb31-1656"><a href="#cb31-1656" aria-hidden="true" tabindex="-1"></a>    and test for differences.</span>
-<span id="cb31-1657"><a href="#cb31-1657" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>We will construct the cumulative hazard curves and compare them.</span>
-<span id="cb31-1658"><a href="#cb31-1658" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>We will estimate the hazard functions, interpret, and compare them.</span>
+<span id="cb31-1653"><a href="#cb31-1653" aria-hidden="true" tabindex="-1"></a><span class="fu">##### Inclusion criteria {.unnumbered}</span></span>
+<span id="cb31-1654"><a href="#cb31-1654" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1655"><a href="#cb31-1655" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>**acute myeloid leukemia (AML)**</span>
+<span id="cb31-1656"><a href="#cb31-1656" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>**acute lymphoblastic leukemia (ALL).**</span>
+<span id="cb31-1657"><a href="#cb31-1657" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1658"><a href="#cb31-1658" aria-hidden="true" tabindex="-1"></a><span class="fu">##### Possible intermediate events {.unnumbered}</span></span>
 <span id="cb31-1659"><a href="#cb31-1659" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1660"><a href="#cb31-1660" aria-hidden="true" tabindex="-1"></a><span class="fu">### Survival Function Estimate and Variance</span></span>
-<span id="cb31-1661"><a href="#cb31-1661" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1662"><a href="#cb31-1662" aria-hidden="true" tabindex="-1"></a>$$\hat S(t) = \prod_{t_i &lt; t}\left<span class="co">[</span><span class="ot">1-\frac{d_i}{Y_i}\right</span><span class="co">]</span>$$ where</span>
-<span id="cb31-1663"><a href="#cb31-1663" aria-hidden="true" tabindex="-1"></a>$Y_i$ is the group at risk at time $t_i$.</span>
-<span id="cb31-1664"><a href="#cb31-1664" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1665"><a href="#cb31-1665" aria-hidden="true" tabindex="-1"></a>The estimated variance of $\hat S(t)$ is:</span>
-<span id="cb31-1666"><a href="#cb31-1666" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1667"><a href="#cb31-1667" aria-hidden="true" tabindex="-1"></a>:::{#thm-greenwood}</span>
-<span id="cb31-1668"><a href="#cb31-1668" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Greenwood's estimator for variance of Kaplan-Meier survival estimator</span></span>
-<span id="cb31-1669"><a href="#cb31-1669" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1670"><a href="#cb31-1670" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1671"><a href="#cb31-1671" aria-hidden="true" tabindex="-1"></a>\varhf{\hat S(t)} = \hat S(t)^2\sum_{t_i &lt;t}\frac{d_i}{Y_i(Y_i-d_i)}</span>
-<span id="cb31-1672"><a href="#cb31-1672" aria-hidden="true" tabindex="-1"></a>$${#eq-var-est-surv}</span>
-<span id="cb31-1673"><a href="#cb31-1673" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-1660"><a href="#cb31-1660" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>**graft vs. host disease (GVHD)**: an immunological rejection</span>
+<span id="cb31-1661"><a href="#cb31-1661" aria-hidden="true" tabindex="-1"></a>    response to the transplant</span>
+<span id="cb31-1662"><a href="#cb31-1662" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>**platelet recovery**: a return of platelet count to normal levels.</span>
+<span id="cb31-1663"><a href="#cb31-1663" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1664"><a href="#cb31-1664" aria-hidden="true" tabindex="-1"></a>One or the other, both in either order, or neither may occur.</span>
+<span id="cb31-1665"><a href="#cb31-1665" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1666"><a href="#cb31-1666" aria-hidden="true" tabindex="-1"></a><span class="fu">##### End point events</span></span>
+<span id="cb31-1667"><a href="#cb31-1667" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1668"><a href="#cb31-1668" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>relapse of the disease</span>
+<span id="cb31-1669"><a href="#cb31-1669" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>death</span>
+<span id="cb31-1670"><a href="#cb31-1670" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1671"><a href="#cb31-1671" aria-hidden="true" tabindex="-1"></a>Any or all of these events may be censored.</span>
+<span id="cb31-1672"><a href="#cb31-1672" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1673"><a href="#cb31-1673" aria-hidden="true" tabindex="-1"></a><span class="fu">### `KMsurv::bmt` data in R</span></span>
 <span id="cb31-1674"><a href="#cb31-1674" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1675"><a href="#cb31-1675" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1676"><a href="#cb31-1676" aria-hidden="true" tabindex="-1"></a>We can use @eq-var-est-surv for confidence intervals for a survival function or a</span>
-<span id="cb31-1677"><a href="#cb31-1677" aria-hidden="true" tabindex="-1"></a>difference of survival functions.</span>
-<span id="cb31-1678"><a href="#cb31-1678" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1679"><a href="#cb31-1679" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1680"><a href="#cb31-1680" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1681"><a href="#cb31-1681" aria-hidden="true" tabindex="-1"></a><span class="fu">##### Kaplan-Meier survival curves</span></span>
-<span id="cb31-1682"><a href="#cb31-1682" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1685"><a href="#cb31-1685" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1686"><a href="#cb31-1686" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-summary: "code to preprocess and model `bmt` data"</span></span>
-<span id="cb31-1687"><a href="#cb31-1687" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(KMsurv)</span>
-<span id="cb31-1688"><a href="#cb31-1688" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(survival)</span>
-<span id="cb31-1689"><a href="#cb31-1689" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(bmt)</span>
+<span id="cb31-1677"><a href="#cb31-1677" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1678"><a href="#cb31-1678" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(KMsurv)</span>
+<span id="cb31-1679"><a href="#cb31-1679" aria-hidden="true" tabindex="-1"></a>?bmt</span>
+<span id="cb31-1680"><a href="#cb31-1680" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1681"><a href="#cb31-1681" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1682"><a href="#cb31-1682" aria-hidden="true" tabindex="-1"></a><span class="fu">### Analysis plan</span></span>
+<span id="cb31-1683"><a href="#cb31-1683" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1684"><a href="#cb31-1684" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>We concentrate for now on disease-free survival (<span class="in">`t2`</span> and <span class="in">`d3`</span>) for</span>
+<span id="cb31-1685"><a href="#cb31-1685" aria-hidden="true" tabindex="-1"></a>    the three risk groups, ALL, AML Low Risk, and AML High Risk.</span>
+<span id="cb31-1686"><a href="#cb31-1686" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>We will construct the Kaplan-Meier survival curves, compare them,</span>
+<span id="cb31-1687"><a href="#cb31-1687" aria-hidden="true" tabindex="-1"></a>    and test for differences.</span>
+<span id="cb31-1688"><a href="#cb31-1688" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>We will construct the cumulative hazard curves and compare them.</span>
+<span id="cb31-1689"><a href="#cb31-1689" aria-hidden="true" tabindex="-1"></a><span class="ss">-   </span>We will estimate the hazard functions, interpret, and compare them.</span>
 <span id="cb31-1690"><a href="#cb31-1690" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1691"><a href="#cb31-1691" aria-hidden="true" tabindex="-1"></a>bmt <span class="ot">=</span> </span>
-<span id="cb31-1692"><a href="#cb31-1692" aria-hidden="true" tabindex="-1"></a>  bmt <span class="sc">|&gt;</span> </span>
-<span id="cb31-1693"><a href="#cb31-1693" aria-hidden="true" tabindex="-1"></a>  <span class="fu">as_tibble</span>() <span class="sc">|&gt;</span> </span>
-<span id="cb31-1694"><a href="#cb31-1694" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(</span>
-<span id="cb31-1695"><a href="#cb31-1695" aria-hidden="true" tabindex="-1"></a>    <span class="at">group =</span> </span>
-<span id="cb31-1696"><a href="#cb31-1696" aria-hidden="true" tabindex="-1"></a>      group <span class="sc">|&gt;</span> </span>
-<span id="cb31-1697"><a href="#cb31-1697" aria-hidden="true" tabindex="-1"></a>      <span class="fu">factor</span>(</span>
-<span id="cb31-1698"><a href="#cb31-1698" aria-hidden="true" tabindex="-1"></a>        <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">"ALL"</span>,<span class="st">"Low Risk AML"</span>,<span class="st">"High Risk AML"</span>)),</span>
-<span id="cb31-1699"><a href="#cb31-1699" aria-hidden="true" tabindex="-1"></a>    <span class="at">surv =</span> <span class="fu">Surv</span>(t2,d3))</span>
+<span id="cb31-1691"><a href="#cb31-1691" aria-hidden="true" tabindex="-1"></a><span class="fu">### Survival Function Estimate and Variance</span></span>
+<span id="cb31-1692"><a href="#cb31-1692" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1693"><a href="#cb31-1693" aria-hidden="true" tabindex="-1"></a>$$\hat S(t) = \prod_{t_i &lt; t}\left<span class="co">[</span><span class="ot">1-\frac{d_i}{Y_i}\right</span><span class="co">]</span>$$ where</span>
+<span id="cb31-1694"><a href="#cb31-1694" aria-hidden="true" tabindex="-1"></a>$Y_i$ is the group at risk at time $t_i$.</span>
+<span id="cb31-1695"><a href="#cb31-1695" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1696"><a href="#cb31-1696" aria-hidden="true" tabindex="-1"></a>The estimated variance of $\hat S(t)$ is:</span>
+<span id="cb31-1697"><a href="#cb31-1697" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1698"><a href="#cb31-1698" aria-hidden="true" tabindex="-1"></a>:::{#thm-greenwood}</span>
+<span id="cb31-1699"><a href="#cb31-1699" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Greenwood's estimator for variance of Kaplan-Meier survival estimator</span></span>
 <span id="cb31-1700"><a href="#cb31-1700" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1701"><a href="#cb31-1701" aria-hidden="true" tabindex="-1"></a>km_model1 <span class="ot">=</span> <span class="fu">survfit</span>(</span>
-<span id="cb31-1702"><a href="#cb31-1702" aria-hidden="true" tabindex="-1"></a>  <span class="at">formula =</span> surv <span class="sc">~</span> group, </span>
-<span id="cb31-1703"><a href="#cb31-1703" aria-hidden="true" tabindex="-1"></a>  <span class="at">data =</span> bmt)</span>
-<span id="cb31-1704"><a href="#cb31-1704" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1701"><a href="#cb31-1701" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1702"><a href="#cb31-1702" aria-hidden="true" tabindex="-1"></a>\varhf{\hat S(t)} = \hat S(t)^2\sum_{t_i &lt;t}\frac{d_i}{Y_i(Y_i-d_i)}</span>
+<span id="cb31-1703"><a href="#cb31-1703" aria-hidden="true" tabindex="-1"></a>$${#eq-var-est-surv}</span>
+<span id="cb31-1704"><a href="#cb31-1704" aria-hidden="true" tabindex="-1"></a>:::</span>
 <span id="cb31-1705"><a href="#cb31-1705" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1706"><a href="#cb31-1706" aria-hidden="true" tabindex="-1"></a><span class="in">```{r "KM survival curves for bmt data"}</span></span>
-<span id="cb31-1707"><a href="#cb31-1707" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "Disease-Free Survival by Disease Group"</span></span>
-<span id="cb31-1708"><a href="#cb31-1708" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1709"><a href="#cb31-1709" aria-hidden="true" tabindex="-1"></a><span class="in">library(ggfortify)</span></span>
-<span id="cb31-1710"><a href="#cb31-1710" aria-hidden="true" tabindex="-1"></a><span class="in">autoplot(</span></span>
-<span id="cb31-1711"><a href="#cb31-1711" aria-hidden="true" tabindex="-1"></a><span class="in">  km_model1, </span></span>
-<span id="cb31-1712"><a href="#cb31-1712" aria-hidden="true" tabindex="-1"></a><span class="in">  conf.int = TRUE,</span></span>
-<span id="cb31-1713"><a href="#cb31-1713" aria-hidden="true" tabindex="-1"></a><span class="in">  ylab = "Pr(disease-free survival)",</span></span>
-<span id="cb31-1714"><a href="#cb31-1714" aria-hidden="true" tabindex="-1"></a><span class="in">  xlab = "Time since transplant (days)") + </span></span>
-<span id="cb31-1715"><a href="#cb31-1715" aria-hidden="true" tabindex="-1"></a><span class="in">  theme_bw() +</span></span>
-<span id="cb31-1716"><a href="#cb31-1716" aria-hidden="true" tabindex="-1"></a><span class="in">  theme(legend.position="bottom")</span></span>
-<span id="cb31-1717"><a href="#cb31-1717" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1718"><a href="#cb31-1718" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1719"><a href="#cb31-1719" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1720"><a href="#cb31-1720" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1706"><a href="#cb31-1706" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1707"><a href="#cb31-1707" aria-hidden="true" tabindex="-1"></a>We can use @eq-var-est-surv for confidence intervals for a survival function or a</span>
+<span id="cb31-1708"><a href="#cb31-1708" aria-hidden="true" tabindex="-1"></a>difference of survival functions.</span>
+<span id="cb31-1709"><a href="#cb31-1709" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1710"><a href="#cb31-1710" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1711"><a href="#cb31-1711" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1712"><a href="#cb31-1712" aria-hidden="true" tabindex="-1"></a><span class="fu">##### Kaplan-Meier survival curves</span></span>
+<span id="cb31-1713"><a href="#cb31-1713" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1716"><a href="#cb31-1716" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1717"><a href="#cb31-1717" aria-hidden="true" tabindex="-1"></a><span class="co">#| code-summary: "code to preprocess and model `bmt` data"</span></span>
+<span id="cb31-1718"><a href="#cb31-1718" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(KMsurv)</span>
+<span id="cb31-1719"><a href="#cb31-1719" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(survival)</span>
+<span id="cb31-1720"><a href="#cb31-1720" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(bmt)</span>
 <span id="cb31-1721"><a href="#cb31-1721" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1722"><a href="#cb31-1722" aria-hidden="true" tabindex="-1"></a><span class="fu">### Understanding Greenwood's formula (optional)</span></span>
-<span id="cb31-1723"><a href="#cb31-1723" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1724"><a href="#cb31-1724" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-1725"><a href="#cb31-1725" aria-hidden="true" tabindex="-1"></a>To see where Greenwood's formula comes from, let $x_i = Y_i - d_i$. We</span>
-<span id="cb31-1726"><a href="#cb31-1726" aria-hidden="true" tabindex="-1"></a>approximate the solution treating each time as independent, with $Y_i$</span>
-<span id="cb31-1727"><a href="#cb31-1727" aria-hidden="true" tabindex="-1"></a>fixed and ignore randomness in times of failure and we treat $x_i$ as</span>
-<span id="cb31-1728"><a href="#cb31-1728" aria-hidden="true" tabindex="-1"></a>independent binomials $\text{Bin}(Y_i,p_i)$. Letting $S(t)$ be the</span>
-<span id="cb31-1729"><a href="#cb31-1729" aria-hidden="true" tabindex="-1"></a>"true" survival function</span>
-<span id="cb31-1730"><a href="#cb31-1730" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-1722"><a href="#cb31-1722" aria-hidden="true" tabindex="-1"></a>bmt <span class="ot">=</span> </span>
+<span id="cb31-1723"><a href="#cb31-1723" aria-hidden="true" tabindex="-1"></a>  bmt <span class="sc">|&gt;</span> </span>
+<span id="cb31-1724"><a href="#cb31-1724" aria-hidden="true" tabindex="-1"></a>  <span class="fu">as_tibble</span>() <span class="sc">|&gt;</span> </span>
+<span id="cb31-1725"><a href="#cb31-1725" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(</span>
+<span id="cb31-1726"><a href="#cb31-1726" aria-hidden="true" tabindex="-1"></a>    <span class="at">group =</span> </span>
+<span id="cb31-1727"><a href="#cb31-1727" aria-hidden="true" tabindex="-1"></a>      group <span class="sc">|&gt;</span> </span>
+<span id="cb31-1728"><a href="#cb31-1728" aria-hidden="true" tabindex="-1"></a>      <span class="fu">factor</span>(</span>
+<span id="cb31-1729"><a href="#cb31-1729" aria-hidden="true" tabindex="-1"></a>        <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">"ALL"</span>,<span class="st">"Low Risk AML"</span>,<span class="st">"High Risk AML"</span>)),</span>
+<span id="cb31-1730"><a href="#cb31-1730" aria-hidden="true" tabindex="-1"></a>    <span class="at">surv =</span> <span class="fu">Surv</span>(t2,d3))</span>
 <span id="cb31-1731"><a href="#cb31-1731" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1732"><a href="#cb31-1732" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1733"><a href="#cb31-1733" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-1734"><a href="#cb31-1734" aria-hidden="true" tabindex="-1"></a>\hat S(t) &amp;=\prod_{t_i&lt;t}x_i/Y_i<span class="sc">\\</span></span>
-<span id="cb31-1735"><a href="#cb31-1735" aria-hidden="true" tabindex="-1"></a>S(t)&amp;=\prod_{t_i&lt;t}p_i</span>
-<span id="cb31-1736"><a href="#cb31-1736" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-1737"><a href="#cb31-1737" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1738"><a href="#cb31-1738" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1739"><a href="#cb31-1739" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1740"><a href="#cb31-1740" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-1741"><a href="#cb31-1741" aria-hidden="true" tabindex="-1"></a>\frac{\hat S(t)}{S(t)} </span>
-<span id="cb31-1742"><a href="#cb31-1742" aria-hidden="true" tabindex="-1"></a>   &amp;= \prod_{t_i&lt;t} \frac{x_i}.    {p_iY_i}</span>
-<span id="cb31-1743"><a href="#cb31-1743" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \prod_{t_i&lt;t} \frac{\hat p_i}{p_i}</span>
-<span id="cb31-1744"><a href="#cb31-1744" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \prod_{t_i&lt;t} \paren{1+\frac{\hat p_i-p_i}{p_i}}</span>
-<span id="cb31-1745"><a href="#cb31-1745" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;\approx 1+\sum_{t_i&lt;t} \frac{\hat p_i-p_i}{p_i}</span>
-<span id="cb31-1746"><a href="#cb31-1746" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-1747"><a href="#cb31-1747" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1732"><a href="#cb31-1732" aria-hidden="true" tabindex="-1"></a>km_model1 <span class="ot">=</span> <span class="fu">survfit</span>(</span>
+<span id="cb31-1733"><a href="#cb31-1733" aria-hidden="true" tabindex="-1"></a>  <span class="at">formula =</span> surv <span class="sc">~</span> group, </span>
+<span id="cb31-1734"><a href="#cb31-1734" aria-hidden="true" tabindex="-1"></a>  <span class="at">data =</span> bmt)</span>
+<span id="cb31-1735"><a href="#cb31-1735" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1736"><a href="#cb31-1736" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1737"><a href="#cb31-1737" aria-hidden="true" tabindex="-1"></a><span class="in">```{r "KM survival curves for bmt data"}</span></span>
+<span id="cb31-1738"><a href="#cb31-1738" aria-hidden="true" tabindex="-1"></a><span class="in">#| fig-cap: "Disease-Free Survival by Disease Group"</span></span>
+<span id="cb31-1739"><a href="#cb31-1739" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1740"><a href="#cb31-1740" aria-hidden="true" tabindex="-1"></a><span class="in">library(ggfortify)</span></span>
+<span id="cb31-1741"><a href="#cb31-1741" aria-hidden="true" tabindex="-1"></a><span class="in">autoplot(</span></span>
+<span id="cb31-1742"><a href="#cb31-1742" aria-hidden="true" tabindex="-1"></a><span class="in">  km_model1, </span></span>
+<span id="cb31-1743"><a href="#cb31-1743" aria-hidden="true" tabindex="-1"></a><span class="in">  conf.int = TRUE,</span></span>
+<span id="cb31-1744"><a href="#cb31-1744" aria-hidden="true" tabindex="-1"></a><span class="in">  ylab = "Pr(disease-free survival)",</span></span>
+<span id="cb31-1745"><a href="#cb31-1745" aria-hidden="true" tabindex="-1"></a><span class="in">  xlab = "Time since transplant (days)") + </span></span>
+<span id="cb31-1746"><a href="#cb31-1746" aria-hidden="true" tabindex="-1"></a><span class="in">  theme_bw() +</span></span>
+<span id="cb31-1747"><a href="#cb31-1747" aria-hidden="true" tabindex="-1"></a><span class="in">  theme(legend.position="bottom")</span></span>
 <span id="cb31-1748"><a href="#cb31-1748" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1749"><a href="#cb31-1749" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1749"><a href="#cb31-1749" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
 <span id="cb31-1750"><a href="#cb31-1750" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1751"><a href="#cb31-1751" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1752"><a href="#cb31-1752" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-1753"><a href="#cb31-1753" aria-hidden="true" tabindex="-1"></a>\text{Var}\left(\frac{\hat S(t)}{S(t)}\right) </span>
-<span id="cb31-1754"><a href="#cb31-1754" aria-hidden="true" tabindex="-1"></a>&amp;\approx \text{Var}\left(1+\sum_{t_i&lt;t} \frac{\hat p_i-p_i}{p_i}\right) </span>
-<span id="cb31-1755"><a href="#cb31-1755" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;=\sum_{t_i&lt;t} \frac{1}{p_i^2}\frac{p_i(1-p_i)}{Y_i} </span>
-<span id="cb31-1756"><a href="#cb31-1756" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \sum_{t_i&lt;t} \frac{(1-p_i)}{p_iY_i}</span>
-<span id="cb31-1757"><a href="#cb31-1757" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;\approx\sum_{t_i&lt;t} \frac{(1-x_i/Y_i)}{x_i}</span>
-<span id="cb31-1758"><a href="#cb31-1758" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;=\sum_{t_i&lt;t} \frac{Y_i-x_i}{x_iY_i}</span>
-<span id="cb31-1759"><a href="#cb31-1759" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;=\sum_{t_i&lt;t} \frac{d_i}{Y_i(Y_i-d_i)}</span>
-<span id="cb31-1760"><a href="#cb31-1760" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> \tf \text{Var}\left(\hat S(t)\right)</span>
-<span id="cb31-1761"><a href="#cb31-1761" aria-hidden="true" tabindex="-1"></a>&amp;\approx \hat S(t)^2\sum_{t_i&lt;t} \frac{d_i}{Y_i(Y_i-d_i)}</span>
-<span id="cb31-1762"><a href="#cb31-1762" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-1751"><a href="#cb31-1751" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1752"><a href="#cb31-1752" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1753"><a href="#cb31-1753" aria-hidden="true" tabindex="-1"></a><span class="fu">### Understanding Greenwood's formula (optional)</span></span>
+<span id="cb31-1754"><a href="#cb31-1754" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1755"><a href="#cb31-1755" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-1756"><a href="#cb31-1756" aria-hidden="true" tabindex="-1"></a>To see where Greenwood's formula comes from, let $x_i = Y_i - d_i$. We</span>
+<span id="cb31-1757"><a href="#cb31-1757" aria-hidden="true" tabindex="-1"></a>approximate the solution treating each time as independent, with $Y_i$</span>
+<span id="cb31-1758"><a href="#cb31-1758" aria-hidden="true" tabindex="-1"></a>fixed and ignore randomness in times of failure and we treat $x_i$ as</span>
+<span id="cb31-1759"><a href="#cb31-1759" aria-hidden="true" tabindex="-1"></a>independent binomials $\text{Bin}(Y_i,p_i)$. Letting $S(t)$ be the</span>
+<span id="cb31-1760"><a href="#cb31-1760" aria-hidden="true" tabindex="-1"></a>"true" survival function</span>
+<span id="cb31-1761"><a href="#cb31-1761" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-1762"><a href="#cb31-1762" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb31-1763"><a href="#cb31-1763" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1764"><a href="#cb31-1764" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1765"><a href="#cb31-1765" aria-hidden="true" tabindex="-1"></a><span class="fu">### Test for differences among the disease groups</span></span>
-<span id="cb31-1766"><a href="#cb31-1766" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1767"><a href="#cb31-1767" aria-hidden="true" tabindex="-1"></a>Here we compute a chi-square test for assocation between disease group</span>
-<span id="cb31-1768"><a href="#cb31-1768" aria-hidden="true" tabindex="-1"></a>(<span class="in">`group`</span>) and disease-free survival:</span>
+<span id="cb31-1764"><a href="#cb31-1764" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-1765"><a href="#cb31-1765" aria-hidden="true" tabindex="-1"></a>\hat S(t) &amp;=\prod_{t_i&lt;t}x_i/Y_i<span class="sc">\\</span></span>
+<span id="cb31-1766"><a href="#cb31-1766" aria-hidden="true" tabindex="-1"></a>S(t)&amp;=\prod_{t_i&lt;t}p_i</span>
+<span id="cb31-1767"><a href="#cb31-1767" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-1768"><a href="#cb31-1768" aria-hidden="true" tabindex="-1"></a>$$</span>
 <span id="cb31-1769"><a href="#cb31-1769" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1772"><a href="#cb31-1772" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1773"><a href="#cb31-1773" aria-hidden="true" tabindex="-1"></a><span class="fu">survdiff</span>(surv <span class="sc">~</span> group, <span class="at">data =</span> bmt)</span>
-<span id="cb31-1774"><a href="#cb31-1774" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1775"><a href="#cb31-1775" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1776"><a href="#cb31-1776" aria-hidden="true" tabindex="-1"></a><span class="fu">### Cumulative Hazard</span></span>
-<span id="cb31-1777"><a href="#cb31-1777" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1770"><a href="#cb31-1770" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1771"><a href="#cb31-1771" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-1772"><a href="#cb31-1772" aria-hidden="true" tabindex="-1"></a>\frac{\hat S(t)}{S(t)} </span>
+<span id="cb31-1773"><a href="#cb31-1773" aria-hidden="true" tabindex="-1"></a>   &amp;= \prod_{t_i&lt;t} \frac{x_i}.    {p_iY_i}</span>
+<span id="cb31-1774"><a href="#cb31-1774" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \prod_{t_i&lt;t} \frac{\hat p_i}{p_i}</span>
+<span id="cb31-1775"><a href="#cb31-1775" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \prod_{t_i&lt;t} \paren{1+\frac{\hat p_i-p_i}{p_i}}</span>
+<span id="cb31-1776"><a href="#cb31-1776" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;\approx 1+\sum_{t_i&lt;t} \frac{\hat p_i-p_i}{p_i}</span>
+<span id="cb31-1777"><a href="#cb31-1777" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
 <span id="cb31-1778"><a href="#cb31-1778" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1779"><a href="#cb31-1779" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-1780"><a href="#cb31-1780" aria-hidden="true" tabindex="-1"></a>h(t) </span>
-<span id="cb31-1781"><a href="#cb31-1781" aria-hidden="true" tabindex="-1"></a>&amp;\eqdef   P(T=t|T\ge t)<span class="sc">\\</span></span>
-<span id="cb31-1782"><a href="#cb31-1782" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{p(T=t)}{P(T\ge t)}<span class="sc">\\</span></span>
-<span id="cb31-1783"><a href="#cb31-1783" aria-hidden="true" tabindex="-1"></a>&amp;= -\deriv{t}\text{log}\left<span class="sc">\{</span>S(t)\right<span class="sc">\}</span></span>
-<span id="cb31-1784"><a href="#cb31-1784" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-1785"><a href="#cb31-1785" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1786"><a href="#cb31-1786" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1787"><a href="#cb31-1787" aria-hidden="true" tabindex="-1"></a>The **cumulative hazard** (or **integrated hazard**) function is</span>
-<span id="cb31-1788"><a href="#cb31-1788" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1789"><a href="#cb31-1789" aria-hidden="true" tabindex="-1"></a>$$H(t)\eqdef  \int_0^t h(t) dt$$ Since</span>
-<span id="cb31-1790"><a href="#cb31-1790" aria-hidden="true" tabindex="-1"></a>$h(t) = -\deriv{t}\text{log}\left<span class="sc">\{</span>S(t)\right<span class="sc">\}</span>$ as shown above, we</span>
-<span id="cb31-1791"><a href="#cb31-1791" aria-hidden="true" tabindex="-1"></a>have:</span>
-<span id="cb31-1792"><a href="#cb31-1792" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1793"><a href="#cb31-1793" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1794"><a href="#cb31-1794" aria-hidden="true" tabindex="-1"></a>H(t)=-\text{log}\left<span class="sc">\{</span>S\right<span class="sc">\}</span>(t)</span>
-<span id="cb31-1795"><a href="#cb31-1795" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1796"><a href="#cb31-1796" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1797"><a href="#cb31-1797" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1798"><a href="#cb31-1798" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1799"><a href="#cb31-1799" aria-hidden="true" tabindex="-1"></a>So we can estimate $H(t)$ as:</span>
+<span id="cb31-1779"><a href="#cb31-1779" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1780"><a href="#cb31-1780" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1781"><a href="#cb31-1781" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1782"><a href="#cb31-1782" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1783"><a href="#cb31-1783" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-1784"><a href="#cb31-1784" aria-hidden="true" tabindex="-1"></a>\text{Var}\left(\frac{\hat S(t)}{S(t)}\right) </span>
+<span id="cb31-1785"><a href="#cb31-1785" aria-hidden="true" tabindex="-1"></a>&amp;\approx \text{Var}\left(1+\sum_{t_i&lt;t} \frac{\hat p_i-p_i}{p_i}\right) </span>
+<span id="cb31-1786"><a href="#cb31-1786" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;=\sum_{t_i&lt;t} \frac{1}{p_i^2}\frac{p_i(1-p_i)}{Y_i} </span>
+<span id="cb31-1787"><a href="#cb31-1787" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;= \sum_{t_i&lt;t} \frac{(1-p_i)}{p_iY_i}</span>
+<span id="cb31-1788"><a href="#cb31-1788" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;\approx\sum_{t_i&lt;t} \frac{(1-x_i/Y_i)}{x_i}</span>
+<span id="cb31-1789"><a href="#cb31-1789" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;=\sum_{t_i&lt;t} \frac{Y_i-x_i}{x_iY_i}</span>
+<span id="cb31-1790"><a href="#cb31-1790" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> &amp;=\sum_{t_i&lt;t} \frac{d_i}{Y_i(Y_i-d_i)}</span>
+<span id="cb31-1791"><a href="#cb31-1791" aria-hidden="true" tabindex="-1"></a><span class="sc">\\</span> \tf \text{Var}\left(\hat S(t)\right)</span>
+<span id="cb31-1792"><a href="#cb31-1792" aria-hidden="true" tabindex="-1"></a>&amp;\approx \hat S(t)^2\sum_{t_i&lt;t} \frac{d_i}{Y_i(Y_i-d_i)}</span>
+<span id="cb31-1793"><a href="#cb31-1793" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-1794"><a href="#cb31-1794" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1795"><a href="#cb31-1795" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1796"><a href="#cb31-1796" aria-hidden="true" tabindex="-1"></a><span class="fu">### Test for differences among the disease groups</span></span>
+<span id="cb31-1797"><a href="#cb31-1797" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1798"><a href="#cb31-1798" aria-hidden="true" tabindex="-1"></a>Here we compute a chi-square test for assocation between disease group</span>
+<span id="cb31-1799"><a href="#cb31-1799" aria-hidden="true" tabindex="-1"></a>(<span class="in">`group`</span>) and disease-free survival:</span>
 <span id="cb31-1800"><a href="#cb31-1800" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1801"><a href="#cb31-1801" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1802"><a href="#cb31-1802" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-1803"><a href="#cb31-1803" aria-hidden="true" tabindex="-1"></a>\hat H(t) </span>
-<span id="cb31-1804"><a href="#cb31-1804" aria-hidden="true" tabindex="-1"></a>&amp;= -\text{log}\left<span class="sc">\{</span>\hat S(t)\right<span class="sc">\}\\</span></span>
-<span id="cb31-1805"><a href="#cb31-1805" aria-hidden="true" tabindex="-1"></a>&amp;= -\text{log}\left<span class="sc">\{</span>\prod_{t_i &lt; t}\left<span class="co">[</span><span class="ot">1-\frac{d_i}{Y_i}\right</span><span class="co">]</span>\right<span class="sc">\}\\</span></span>
-<span id="cb31-1806"><a href="#cb31-1806" aria-hidden="true" tabindex="-1"></a>&amp;= -\sum_{t_i &lt; t}\text{log}\left<span class="sc">\{</span>1-\frac{d_i}{Y_i}\right<span class="sc">\}\\</span></span>
-<span id="cb31-1807"><a href="#cb31-1807" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-1808"><a href="#cb31-1808" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1809"><a href="#cb31-1809" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1810"><a href="#cb31-1810" aria-hidden="true" tabindex="-1"></a>This is the **Kaplan-Meier (product-limit) estimate of cumulative</span>
-<span id="cb31-1811"><a href="#cb31-1811" aria-hidden="true" tabindex="-1"></a>hazard**.</span>
-<span id="cb31-1812"><a href="#cb31-1812" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1813"><a href="#cb31-1813" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1814"><a href="#cb31-1814" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1815"><a href="#cb31-1815" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Example: Cumulative Hazard Curves for Bone-Marrow Transplant (`bmt`) data</span></span>
-<span id="cb31-1816"><a href="#cb31-1816" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1819"><a href="#cb31-1819" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1820"><a href="#cb31-1820" aria-hidden="true" tabindex="-1"></a><span class="co">#| fig-cap: "Disease-Free Cumulative Hazard by Disease Group"</span></span>
-<span id="cb31-1821"><a href="#cb31-1821" aria-hidden="true" tabindex="-1"></a><span class="co">#| label: fig-cuhaz-bmt</span></span>
-<span id="cb31-1822"><a href="#cb31-1822" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1823"><a href="#cb31-1823" aria-hidden="true" tabindex="-1"></a><span class="fu">autoplot</span>(</span>
-<span id="cb31-1824"><a href="#cb31-1824" aria-hidden="true" tabindex="-1"></a>  <span class="at">fun =</span> <span class="st">"cumhaz"</span>,</span>
-<span id="cb31-1825"><a href="#cb31-1825" aria-hidden="true" tabindex="-1"></a>  km_model1, </span>
-<span id="cb31-1826"><a href="#cb31-1826" aria-hidden="true" tabindex="-1"></a>  <span class="at">conf.int =</span> <span class="cn">FALSE</span>,</span>
-<span id="cb31-1827"><a href="#cb31-1827" aria-hidden="true" tabindex="-1"></a>  <span class="at">ylab =</span> <span class="st">"Cumulative hazard (disease-free survival)"</span>,</span>
-<span id="cb31-1828"><a href="#cb31-1828" aria-hidden="true" tabindex="-1"></a>  <span class="at">xlab =</span> <span class="st">"Time since transplant (days)"</span>) <span class="sc">+</span> </span>
-<span id="cb31-1829"><a href="#cb31-1829" aria-hidden="true" tabindex="-1"></a>  <span class="fu">theme_bw</span>() <span class="sc">+</span></span>
-<span id="cb31-1830"><a href="#cb31-1830" aria-hidden="true" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position=</span><span class="st">"bottom"</span>)</span>
-<span id="cb31-1831"><a href="#cb31-1831" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1832"><a href="#cb31-1832" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1833"><a href="#cb31-1833" aria-hidden="true" tabindex="-1"></a><span class="fu">## Nelson-Aalen Estimates of Cumulative Hazard and Survival</span></span>
-<span id="cb31-1834"><a href="#cb31-1834" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1835"><a href="#cb31-1835" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1836"><a href="#cb31-1836" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1837"><a href="#cb31-1837" aria-hidden="true" tabindex="-1"></a>:::{#def-na-cuhaz-est}</span>
-<span id="cb31-1838"><a href="#cb31-1838" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Nelson-Aalen Cumulative Hazard Estimator</span></span>
-<span id="cb31-1839"><a href="#cb31-1839" aria-hidden="true" tabindex="-1"></a>:::: notes</span>
-<span id="cb31-1840"><a href="#cb31-1840" aria-hidden="true" tabindex="-1"></a>The point hazard at time $t_i$ can be estimated by $d_i/Y_i$, which</span>
-<span id="cb31-1841"><a href="#cb31-1841" aria-hidden="true" tabindex="-1"></a>leads to the **Nelson-Aalen estimator of the cumulative hazard**:</span>
-<span id="cb31-1842"><a href="#cb31-1842" aria-hidden="true" tabindex="-1"></a>::::</span>
-<span id="cb31-1843"><a href="#cb31-1843" aria-hidden="true" tabindex="-1"></a>$$\hat H_{NA}(t) \eqdef \sum_{t_i &lt; t}\frac{d_i}{Y_i}$${#eq-NA-cuhaz-est}</span>
-<span id="cb31-1844"><a href="#cb31-1844" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1845"><a href="#cb31-1845" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-1846"><a href="#cb31-1846" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1847"><a href="#cb31-1847" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1848"><a href="#cb31-1848" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1849"><a href="#cb31-1849" aria-hidden="true" tabindex="-1"></a>:::{#thm-var-NA-est}</span>
-<span id="cb31-1850"><a href="#cb31-1850" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1851"><a href="#cb31-1851" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Variance of Nelson-Aalen estimator</span></span>
-<span id="cb31-1852"><a href="#cb31-1852" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1853"><a href="#cb31-1853" aria-hidden="true" tabindex="-1"></a>:::: notes</span>
-<span id="cb31-1854"><a href="#cb31-1854" aria-hidden="true" tabindex="-1"></a>The variance of this estimator is approximately: </span>
-<span id="cb31-1855"><a href="#cb31-1855" aria-hidden="true" tabindex="-1"></a>::::</span>
-<span id="cb31-1856"><a href="#cb31-1856" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1857"><a href="#cb31-1857" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1858"><a href="#cb31-1858" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-1859"><a href="#cb31-1859" aria-hidden="true" tabindex="-1"></a>\hat{\text{Var}}\left(\hat H_{NA} (t)\right) </span>
-<span id="cb31-1860"><a href="#cb31-1860" aria-hidden="true" tabindex="-1"></a>&amp;= \sum_{t_i &lt;t}\frac{(d_i/Y_i)(1-d_i/Y_i)}{Y_i}<span class="sc">\\</span></span>
-<span id="cb31-1861"><a href="#cb31-1861" aria-hidden="true" tabindex="-1"></a>&amp;\approx \sum_{t_i &lt;t}\frac{d_i}{Y_i^2}</span>
-<span id="cb31-1862"><a href="#cb31-1862" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-1863"><a href="#cb31-1863" aria-hidden="true" tabindex="-1"></a>$${#eq-var-NA-cuhaz-est}</span>
-<span id="cb31-1864"><a href="#cb31-1864" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1865"><a href="#cb31-1865" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-1866"><a href="#cb31-1866" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1867"><a href="#cb31-1867" aria-hidden="true" tabindex="-1"></a>---</span>
-<span id="cb31-1868"><a href="#cb31-1868" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1869"><a href="#cb31-1869" aria-hidden="true" tabindex="-1"></a>Since $S(t)=\text{exp}\left<span class="sc">\{</span>-H(t)\right<span class="sc">\}</span>$, the Nelson-Aalen cumulative</span>
-<span id="cb31-1870"><a href="#cb31-1870" aria-hidden="true" tabindex="-1"></a>hazard estimate can be converted into an alternate estimate of the</span>
-<span id="cb31-1871"><a href="#cb31-1871" aria-hidden="true" tabindex="-1"></a>survival function:</span>
-<span id="cb31-1872"><a href="#cb31-1872" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1873"><a href="#cb31-1873" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1874"><a href="#cb31-1874" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-1875"><a href="#cb31-1875" aria-hidden="true" tabindex="-1"></a>\hat S_{NA}(t)</span>
-<span id="cb31-1876"><a href="#cb31-1876" aria-hidden="true" tabindex="-1"></a>&amp;= \text{exp}\left<span class="sc">\{</span>-\hat H_{NA}(t)\right<span class="sc">\}\\</span></span>
-<span id="cb31-1877"><a href="#cb31-1877" aria-hidden="true" tabindex="-1"></a>&amp;= \text{exp}\left<span class="sc">\{</span>-\sum_{t_i &lt; t}\frac{d_i}{Y_i}\right<span class="sc">\}\\</span></span>
-<span id="cb31-1878"><a href="#cb31-1878" aria-hidden="true" tabindex="-1"></a>&amp;= \prod_{t_i &lt; t}\text{exp}\left<span class="sc">\{</span>-\frac{d_i}{Y_i}\right<span class="sc">\}\\</span></span>
-<span id="cb31-1879"><a href="#cb31-1879" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-1880"><a href="#cb31-1880" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1803"><a href="#cb31-1803" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1804"><a href="#cb31-1804" aria-hidden="true" tabindex="-1"></a><span class="fu">survdiff</span>(surv <span class="sc">~</span> group, <span class="at">data =</span> bmt)</span>
+<span id="cb31-1805"><a href="#cb31-1805" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1806"><a href="#cb31-1806" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1807"><a href="#cb31-1807" aria-hidden="true" tabindex="-1"></a><span class="fu">### Cumulative Hazard</span></span>
+<span id="cb31-1808"><a href="#cb31-1808" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1809"><a href="#cb31-1809" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1810"><a href="#cb31-1810" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-1811"><a href="#cb31-1811" aria-hidden="true" tabindex="-1"></a>h(t) </span>
+<span id="cb31-1812"><a href="#cb31-1812" aria-hidden="true" tabindex="-1"></a>&amp;\eqdef   P(T=t|T\ge t)<span class="sc">\\</span></span>
+<span id="cb31-1813"><a href="#cb31-1813" aria-hidden="true" tabindex="-1"></a>&amp;= \frac{p(T=t)}{P(T\ge t)}<span class="sc">\\</span></span>
+<span id="cb31-1814"><a href="#cb31-1814" aria-hidden="true" tabindex="-1"></a>&amp;= -\deriv{t}\text{log}\left<span class="sc">\{</span>S(t)\right<span class="sc">\}</span></span>
+<span id="cb31-1815"><a href="#cb31-1815" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-1816"><a href="#cb31-1816" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1817"><a href="#cb31-1817" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1818"><a href="#cb31-1818" aria-hidden="true" tabindex="-1"></a>The **cumulative hazard** (or **integrated hazard**) function is</span>
+<span id="cb31-1819"><a href="#cb31-1819" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1820"><a href="#cb31-1820" aria-hidden="true" tabindex="-1"></a>$$H(t)\eqdef  \int_0^t h(t) dt$$ Since</span>
+<span id="cb31-1821"><a href="#cb31-1821" aria-hidden="true" tabindex="-1"></a>$h(t) = -\deriv{t}\text{log}\left<span class="sc">\{</span>S(t)\right<span class="sc">\}</span>$ as shown above, we</span>
+<span id="cb31-1822"><a href="#cb31-1822" aria-hidden="true" tabindex="-1"></a>have:</span>
+<span id="cb31-1823"><a href="#cb31-1823" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1824"><a href="#cb31-1824" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1825"><a href="#cb31-1825" aria-hidden="true" tabindex="-1"></a>H(t)=-\text{log}\left<span class="sc">\{</span>S\right<span class="sc">\}</span>(t)</span>
+<span id="cb31-1826"><a href="#cb31-1826" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1827"><a href="#cb31-1827" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1828"><a href="#cb31-1828" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1829"><a href="#cb31-1829" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1830"><a href="#cb31-1830" aria-hidden="true" tabindex="-1"></a>So we can estimate $H(t)$ as:</span>
+<span id="cb31-1831"><a href="#cb31-1831" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1832"><a href="#cb31-1832" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1833"><a href="#cb31-1833" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-1834"><a href="#cb31-1834" aria-hidden="true" tabindex="-1"></a>\hat H(t) </span>
+<span id="cb31-1835"><a href="#cb31-1835" aria-hidden="true" tabindex="-1"></a>&amp;= -\text{log}\left<span class="sc">\{</span>\hat S(t)\right<span class="sc">\}\\</span></span>
+<span id="cb31-1836"><a href="#cb31-1836" aria-hidden="true" tabindex="-1"></a>&amp;= -\text{log}\left<span class="sc">\{</span>\prod_{t_i &lt; t}\left<span class="co">[</span><span class="ot">1-\frac{d_i}{Y_i}\right</span><span class="co">]</span>\right<span class="sc">\}\\</span></span>
+<span id="cb31-1837"><a href="#cb31-1837" aria-hidden="true" tabindex="-1"></a>&amp;= -\sum_{t_i &lt; t}\text{log}\left<span class="sc">\{</span>1-\frac{d_i}{Y_i}\right<span class="sc">\}\\</span></span>
+<span id="cb31-1838"><a href="#cb31-1838" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-1839"><a href="#cb31-1839" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1840"><a href="#cb31-1840" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1841"><a href="#cb31-1841" aria-hidden="true" tabindex="-1"></a>This is the **Kaplan-Meier (product-limit) estimate of cumulative</span>
+<span id="cb31-1842"><a href="#cb31-1842" aria-hidden="true" tabindex="-1"></a>hazard**.</span>
+<span id="cb31-1843"><a href="#cb31-1843" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1844"><a href="#cb31-1844" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1845"><a href="#cb31-1845" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1846"><a href="#cb31-1846" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Example: Cumulative Hazard Curves for Bone-Marrow Transplant (`bmt`) data</span></span>
+<span id="cb31-1847"><a href="#cb31-1847" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1850"><a href="#cb31-1850" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1851"><a href="#cb31-1851" aria-hidden="true" tabindex="-1"></a><span class="co">#| fig-cap: "Disease-Free Cumulative Hazard by Disease Group"</span></span>
+<span id="cb31-1852"><a href="#cb31-1852" aria-hidden="true" tabindex="-1"></a><span class="co">#| label: fig-cuhaz-bmt</span></span>
+<span id="cb31-1853"><a href="#cb31-1853" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1854"><a href="#cb31-1854" aria-hidden="true" tabindex="-1"></a><span class="fu">autoplot</span>(</span>
+<span id="cb31-1855"><a href="#cb31-1855" aria-hidden="true" tabindex="-1"></a>  <span class="at">fun =</span> <span class="st">"cumhaz"</span>,</span>
+<span id="cb31-1856"><a href="#cb31-1856" aria-hidden="true" tabindex="-1"></a>  km_model1, </span>
+<span id="cb31-1857"><a href="#cb31-1857" aria-hidden="true" tabindex="-1"></a>  <span class="at">conf.int =</span> <span class="cn">FALSE</span>,</span>
+<span id="cb31-1858"><a href="#cb31-1858" aria-hidden="true" tabindex="-1"></a>  <span class="at">ylab =</span> <span class="st">"Cumulative hazard (disease-free survival)"</span>,</span>
+<span id="cb31-1859"><a href="#cb31-1859" aria-hidden="true" tabindex="-1"></a>  <span class="at">xlab =</span> <span class="st">"Time since transplant (days)"</span>) <span class="sc">+</span> </span>
+<span id="cb31-1860"><a href="#cb31-1860" aria-hidden="true" tabindex="-1"></a>  <span class="fu">theme_bw</span>() <span class="sc">+</span></span>
+<span id="cb31-1861"><a href="#cb31-1861" aria-hidden="true" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position=</span><span class="st">"bottom"</span>)</span>
+<span id="cb31-1862"><a href="#cb31-1862" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1863"><a href="#cb31-1863" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1864"><a href="#cb31-1864" aria-hidden="true" tabindex="-1"></a><span class="fu">## Nelson-Aalen Estimates of Cumulative Hazard and Survival</span></span>
+<span id="cb31-1865"><a href="#cb31-1865" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1866"><a href="#cb31-1866" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1867"><a href="#cb31-1867" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1868"><a href="#cb31-1868" aria-hidden="true" tabindex="-1"></a>:::{#def-na-cuhaz-est}</span>
+<span id="cb31-1869"><a href="#cb31-1869" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Nelson-Aalen Cumulative Hazard Estimator</span></span>
+<span id="cb31-1870"><a href="#cb31-1870" aria-hidden="true" tabindex="-1"></a>:::: notes</span>
+<span id="cb31-1871"><a href="#cb31-1871" aria-hidden="true" tabindex="-1"></a>The point hazard at time $t_i$ can be estimated by $d_i/Y_i$, which</span>
+<span id="cb31-1872"><a href="#cb31-1872" aria-hidden="true" tabindex="-1"></a>leads to the **Nelson-Aalen estimator of the cumulative hazard**:</span>
+<span id="cb31-1873"><a href="#cb31-1873" aria-hidden="true" tabindex="-1"></a>::::</span>
+<span id="cb31-1874"><a href="#cb31-1874" aria-hidden="true" tabindex="-1"></a>$$\hat H_{NA}(t) \eqdef \sum_{t_i &lt; t}\frac{d_i}{Y_i}$${#eq-NA-cuhaz-est}</span>
+<span id="cb31-1875"><a href="#cb31-1875" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1876"><a href="#cb31-1876" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-1877"><a href="#cb31-1877" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1878"><a href="#cb31-1878" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1879"><a href="#cb31-1879" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1880"><a href="#cb31-1880" aria-hidden="true" tabindex="-1"></a>:::{#thm-var-NA-est}</span>
 <span id="cb31-1881"><a href="#cb31-1881" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1882"><a href="#cb31-1882" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1882"><a href="#cb31-1882" aria-hidden="true" tabindex="-1"></a><span class="fu">#### Variance of Nelson-Aalen estimator</span></span>
 <span id="cb31-1883"><a href="#cb31-1883" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1884"><a href="#cb31-1884" aria-hidden="true" tabindex="-1"></a>Compare these with the corresponding Kaplan-Meier estimates:</span>
-<span id="cb31-1885"><a href="#cb31-1885" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1886"><a href="#cb31-1886" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1887"><a href="#cb31-1887" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
-<span id="cb31-1888"><a href="#cb31-1888" aria-hidden="true" tabindex="-1"></a>\hat H_{KM}(t) &amp;= -\sum_{t_i &lt; t}\text{log}\left<span class="sc">\{</span>1-\frac{d_i}{Y_i}\right<span class="sc">\}\\</span></span>
-<span id="cb31-1889"><a href="#cb31-1889" aria-hidden="true" tabindex="-1"></a>\hat S_{KM}(t) &amp;= \prod_{t_i &lt; t}\left<span class="co">[</span><span class="ot">1-\frac{d_i}{Y_i}\right</span><span class="co">]</span></span>
-<span id="cb31-1890"><a href="#cb31-1890" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
-<span id="cb31-1891"><a href="#cb31-1891" aria-hidden="true" tabindex="-1"></a>$$</span>
-<span id="cb31-1892"><a href="#cb31-1892" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1893"><a href="#cb31-1893" aria-hidden="true" tabindex="-1"></a>::: notes</span>
-<span id="cb31-1894"><a href="#cb31-1894" aria-hidden="true" tabindex="-1"></a>The product limit estimate and the Nelson-Aalen estimate often do not</span>
-<span id="cb31-1895"><a href="#cb31-1895" aria-hidden="true" tabindex="-1"></a>differ by much. The latter is considered more accurate in small samples</span>
-<span id="cb31-1896"><a href="#cb31-1896" aria-hidden="true" tabindex="-1"></a>and also directly estimates the cumulative hazard. </span>
-<span id="cb31-1897"><a href="#cb31-1897" aria-hidden="true" tabindex="-1"></a>The <span class="in">`"fleming-harrington"`</span> method for <span class="in">`survfit()`</span> reduces to Nelson-Aalen</span>
-<span id="cb31-1898"><a href="#cb31-1898" aria-hidden="true" tabindex="-1"></a>when the data are unweighted. </span>
-<span id="cb31-1899"><a href="#cb31-1899" aria-hidden="true" tabindex="-1"></a>We can also estimate the cumulative hazard</span>
-<span id="cb31-1900"><a href="#cb31-1900" aria-hidden="true" tabindex="-1"></a>as the negative log of the KM survival function estimate.</span>
-<span id="cb31-1901"><a href="#cb31-1901" aria-hidden="true" tabindex="-1"></a>:::</span>
-<span id="cb31-1902"><a href="#cb31-1902" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1903"><a href="#cb31-1903" aria-hidden="true" tabindex="-1"></a><span class="fu">### Application to `bmt` dataset</span></span>
-<span id="cb31-1904"><a href="#cb31-1904" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1907"><a href="#cb31-1907" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1908"><a href="#cb31-1908" aria-hidden="true" tabindex="-1"></a>na_fit <span class="ot">=</span> <span class="fu">survfit</span>(</span>
-<span id="cb31-1909"><a href="#cb31-1909" aria-hidden="true" tabindex="-1"></a>  <span class="at">formula =</span> surv <span class="sc">~</span> group,</span>
-<span id="cb31-1910"><a href="#cb31-1910" aria-hidden="true" tabindex="-1"></a>  <span class="at">type =</span> <span class="st">"fleming-harrington"</span>,</span>
-<span id="cb31-1911"><a href="#cb31-1911" aria-hidden="true" tabindex="-1"></a>  <span class="at">data =</span> bmt)</span>
+<span id="cb31-1884"><a href="#cb31-1884" aria-hidden="true" tabindex="-1"></a>:::: notes</span>
+<span id="cb31-1885"><a href="#cb31-1885" aria-hidden="true" tabindex="-1"></a>The variance of this estimator is approximately: </span>
+<span id="cb31-1886"><a href="#cb31-1886" aria-hidden="true" tabindex="-1"></a>::::</span>
+<span id="cb31-1887"><a href="#cb31-1887" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1888"><a href="#cb31-1888" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1889"><a href="#cb31-1889" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-1890"><a href="#cb31-1890" aria-hidden="true" tabindex="-1"></a>\hat{\text{Var}}\left(\hat H_{NA} (t)\right) </span>
+<span id="cb31-1891"><a href="#cb31-1891" aria-hidden="true" tabindex="-1"></a>&amp;= \sum_{t_i &lt;t}\frac{(d_i/Y_i)(1-d_i/Y_i)}{Y_i}<span class="sc">\\</span></span>
+<span id="cb31-1892"><a href="#cb31-1892" aria-hidden="true" tabindex="-1"></a>&amp;\approx \sum_{t_i &lt;t}\frac{d_i}{Y_i^2}</span>
+<span id="cb31-1893"><a href="#cb31-1893" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-1894"><a href="#cb31-1894" aria-hidden="true" tabindex="-1"></a>$${#eq-var-NA-cuhaz-est}</span>
+<span id="cb31-1895"><a href="#cb31-1895" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1896"><a href="#cb31-1896" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-1897"><a href="#cb31-1897" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1898"><a href="#cb31-1898" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1899"><a href="#cb31-1899" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1900"><a href="#cb31-1900" aria-hidden="true" tabindex="-1"></a>Since $S(t)=\text{exp}\left<span class="sc">\{</span>-H(t)\right<span class="sc">\}</span>$, the Nelson-Aalen cumulative</span>
+<span id="cb31-1901"><a href="#cb31-1901" aria-hidden="true" tabindex="-1"></a>hazard estimate can be converted into an alternate estimate of the</span>
+<span id="cb31-1902"><a href="#cb31-1902" aria-hidden="true" tabindex="-1"></a>survival function:</span>
+<span id="cb31-1903"><a href="#cb31-1903" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1904"><a href="#cb31-1904" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1905"><a href="#cb31-1905" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-1906"><a href="#cb31-1906" aria-hidden="true" tabindex="-1"></a>\hat S_{NA}(t)</span>
+<span id="cb31-1907"><a href="#cb31-1907" aria-hidden="true" tabindex="-1"></a>&amp;= \text{exp}\left<span class="sc">\{</span>-\hat H_{NA}(t)\right<span class="sc">\}\\</span></span>
+<span id="cb31-1908"><a href="#cb31-1908" aria-hidden="true" tabindex="-1"></a>&amp;= \text{exp}\left<span class="sc">\{</span>-\sum_{t_i &lt; t}\frac{d_i}{Y_i}\right<span class="sc">\}\\</span></span>
+<span id="cb31-1909"><a href="#cb31-1909" aria-hidden="true" tabindex="-1"></a>&amp;= \prod_{t_i &lt; t}\text{exp}\left<span class="sc">\{</span>-\frac{d_i}{Y_i}\right<span class="sc">\}\\</span></span>
+<span id="cb31-1910"><a href="#cb31-1910" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-1911"><a href="#cb31-1911" aria-hidden="true" tabindex="-1"></a>$$</span>
 <span id="cb31-1912"><a href="#cb31-1912" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1913"><a href="#cb31-1913" aria-hidden="true" tabindex="-1"></a>km_fit <span class="ot">=</span> <span class="fu">survfit</span>(</span>
-<span id="cb31-1914"><a href="#cb31-1914" aria-hidden="true" tabindex="-1"></a>  <span class="at">formula =</span> surv <span class="sc">~</span> group,</span>
-<span id="cb31-1915"><a href="#cb31-1915" aria-hidden="true" tabindex="-1"></a>  <span class="at">type =</span> <span class="st">"kaplan-meier"</span>,</span>
-<span id="cb31-1916"><a href="#cb31-1916" aria-hidden="true" tabindex="-1"></a>  <span class="at">data =</span> bmt)</span>
-<span id="cb31-1917"><a href="#cb31-1917" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1918"><a href="#cb31-1918" aria-hidden="true" tabindex="-1"></a>km_and_na <span class="ot">=</span> </span>
-<span id="cb31-1919"><a href="#cb31-1919" aria-hidden="true" tabindex="-1"></a>  <span class="fu">bind_rows</span>(</span>
-<span id="cb31-1920"><a href="#cb31-1920" aria-hidden="true" tabindex="-1"></a>    <span class="at">.id =</span> <span class="st">"model"</span>,</span>
-<span id="cb31-1921"><a href="#cb31-1921" aria-hidden="true" tabindex="-1"></a>    <span class="st">"Kaplan-Meier"</span> <span class="ot">=</span> km_fit <span class="sc">|&gt;</span> <span class="fu">fortify</span>(<span class="at">surv.connect =</span> <span class="cn">TRUE</span>),</span>
-<span id="cb31-1922"><a href="#cb31-1922" aria-hidden="true" tabindex="-1"></a>    <span class="st">"Nelson-Aalen"</span> <span class="ot">=</span> na_fit <span class="sc">|&gt;</span> <span class="fu">fortify</span>(<span class="at">surv.connect =</span> <span class="cn">TRUE</span>)</span>
-<span id="cb31-1923"><a href="#cb31-1923" aria-hidden="true" tabindex="-1"></a>  ) <span class="sc">|&gt;</span> </span>
-<span id="cb31-1924"><a href="#cb31-1924" aria-hidden="true" tabindex="-1"></a>  <span class="fu">as_tibble</span>()</span>
-<span id="cb31-1925"><a href="#cb31-1925" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1926"><a href="#cb31-1926" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
-<span id="cb31-1927"><a href="#cb31-1927" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1930"><a href="#cb31-1930" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
-<span id="cb31-1931"><a href="#cb31-1931" aria-hidden="true" tabindex="-1"></a><span class="co">#| fig-cap: "Kaplan-Meier and Nelson-Aalen Survival Function Estimates, stratified by disease group"</span></span>
-<span id="cb31-1932"><a href="#cb31-1932" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1933"><a href="#cb31-1933" aria-hidden="true" tabindex="-1"></a>km_and_na <span class="sc">|&gt;</span> </span>
-<span id="cb31-1934"><a href="#cb31-1934" aria-hidden="true" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> time, <span class="at">y =</span> surv, <span class="at">col =</span> model)) <span class="sc">+</span></span>
-<span id="cb31-1935"><a href="#cb31-1935" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_step</span>() <span class="sc">+</span></span>
-<span id="cb31-1936"><a href="#cb31-1936" aria-hidden="true" tabindex="-1"></a>  <span class="fu">facet_grid</span>(. <span class="sc">~</span> strata) <span class="sc">+</span></span>
-<span id="cb31-1937"><a href="#cb31-1937" aria-hidden="true" tabindex="-1"></a>  <span class="fu">theme_bw</span>() <span class="sc">+</span> </span>
-<span id="cb31-1938"><a href="#cb31-1938" aria-hidden="true" tabindex="-1"></a>  <span class="fu">ylab</span>(<span class="st">"S(t) = P(T&gt;=t)"</span>) <span class="sc">+</span></span>
-<span id="cb31-1939"><a href="#cb31-1939" aria-hidden="true" tabindex="-1"></a>  <span class="fu">xlab</span>(<span class="st">"Survival time (t, days)"</span>) <span class="sc">+</span></span>
-<span id="cb31-1940"><a href="#cb31-1940" aria-hidden="true" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"bottom"</span>)</span>
-<span id="cb31-1941"><a href="#cb31-1941" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1942"><a href="#cb31-1942" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1913"><a href="#cb31-1913" aria-hidden="true" tabindex="-1"></a>---</span>
+<span id="cb31-1914"><a href="#cb31-1914" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1915"><a href="#cb31-1915" aria-hidden="true" tabindex="-1"></a>Compare these with the corresponding Kaplan-Meier estimates:</span>
+<span id="cb31-1916"><a href="#cb31-1916" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1917"><a href="#cb31-1917" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1918"><a href="#cb31-1918" aria-hidden="true" tabindex="-1"></a>\begin{aligned}</span>
+<span id="cb31-1919"><a href="#cb31-1919" aria-hidden="true" tabindex="-1"></a>\hat H_{KM}(t) &amp;= -\sum_{t_i &lt; t}\text{log}\left<span class="sc">\{</span>1-\frac{d_i}{Y_i}\right<span class="sc">\}\\</span></span>
+<span id="cb31-1920"><a href="#cb31-1920" aria-hidden="true" tabindex="-1"></a>\hat S_{KM}(t) &amp;= \prod_{t_i &lt; t}\left<span class="co">[</span><span class="ot">1-\frac{d_i}{Y_i}\right</span><span class="co">]</span></span>
+<span id="cb31-1921"><a href="#cb31-1921" aria-hidden="true" tabindex="-1"></a>\end{aligned}</span>
+<span id="cb31-1922"><a href="#cb31-1922" aria-hidden="true" tabindex="-1"></a>$$</span>
+<span id="cb31-1923"><a href="#cb31-1923" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1924"><a href="#cb31-1924" aria-hidden="true" tabindex="-1"></a>::: notes</span>
+<span id="cb31-1925"><a href="#cb31-1925" aria-hidden="true" tabindex="-1"></a>The product limit estimate and the Nelson-Aalen estimate often do not</span>
+<span id="cb31-1926"><a href="#cb31-1926" aria-hidden="true" tabindex="-1"></a>differ by much. The latter is considered more accurate in small samples</span>
+<span id="cb31-1927"><a href="#cb31-1927" aria-hidden="true" tabindex="-1"></a>and also directly estimates the cumulative hazard. </span>
+<span id="cb31-1928"><a href="#cb31-1928" aria-hidden="true" tabindex="-1"></a>The <span class="in">`"fleming-harrington"`</span> method for <span class="in">`survfit()`</span> reduces to Nelson-Aalen</span>
+<span id="cb31-1929"><a href="#cb31-1929" aria-hidden="true" tabindex="-1"></a>when the data are unweighted. </span>
+<span id="cb31-1930"><a href="#cb31-1930" aria-hidden="true" tabindex="-1"></a>We can also estimate the cumulative hazard</span>
+<span id="cb31-1931"><a href="#cb31-1931" aria-hidden="true" tabindex="-1"></a>as the negative log of the KM survival function estimate.</span>
+<span id="cb31-1932"><a href="#cb31-1932" aria-hidden="true" tabindex="-1"></a>:::</span>
+<span id="cb31-1933"><a href="#cb31-1933" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1934"><a href="#cb31-1934" aria-hidden="true" tabindex="-1"></a><span class="fu">### Application to `bmt` dataset</span></span>
+<span id="cb31-1935"><a href="#cb31-1935" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1938"><a href="#cb31-1938" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1939"><a href="#cb31-1939" aria-hidden="true" tabindex="-1"></a>na_fit <span class="ot">=</span> <span class="fu">survfit</span>(</span>
+<span id="cb31-1940"><a href="#cb31-1940" aria-hidden="true" tabindex="-1"></a>  <span class="at">formula =</span> surv <span class="sc">~</span> group,</span>
+<span id="cb31-1941"><a href="#cb31-1941" aria-hidden="true" tabindex="-1"></a>  <span class="at">type =</span> <span class="st">"fleming-harrington"</span>,</span>
+<span id="cb31-1942"><a href="#cb31-1942" aria-hidden="true" tabindex="-1"></a>  <span class="at">data =</span> bmt)</span>
 <span id="cb31-1943"><a href="#cb31-1943" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb31-1944"><a href="#cb31-1944" aria-hidden="true" tabindex="-1"></a>The Kaplan-Meier and Nelson-Aalen survival estimates are very similar</span>
-<span id="cb31-1945"><a href="#cb31-1945" aria-hidden="true" tabindex="-1"></a>for this dataset.</span>
+<span id="cb31-1944"><a href="#cb31-1944" aria-hidden="true" tabindex="-1"></a>km_fit <span class="ot">=</span> <span class="fu">survfit</span>(</span>
+<span id="cb31-1945"><a href="#cb31-1945" aria-hidden="true" tabindex="-1"></a>  <span class="at">formula =</span> surv <span class="sc">~</span> group,</span>
+<span id="cb31-1946"><a href="#cb31-1946" aria-hidden="true" tabindex="-1"></a>  <span class="at">type =</span> <span class="st">"kaplan-meier"</span>,</span>
+<span id="cb31-1947"><a href="#cb31-1947" aria-hidden="true" tabindex="-1"></a>  <span class="at">data =</span> bmt)</span>
+<span id="cb31-1948"><a href="#cb31-1948" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1949"><a href="#cb31-1949" aria-hidden="true" tabindex="-1"></a>km_and_na <span class="ot">=</span> </span>
+<span id="cb31-1950"><a href="#cb31-1950" aria-hidden="true" tabindex="-1"></a>  <span class="fu">bind_rows</span>(</span>
+<span id="cb31-1951"><a href="#cb31-1951" aria-hidden="true" tabindex="-1"></a>    <span class="at">.id =</span> <span class="st">"model"</span>,</span>
+<span id="cb31-1952"><a href="#cb31-1952" aria-hidden="true" tabindex="-1"></a>    <span class="st">"Kaplan-Meier"</span> <span class="ot">=</span> km_fit <span class="sc">|&gt;</span> <span class="fu">fortify</span>(<span class="at">surv.connect =</span> <span class="cn">TRUE</span>),</span>
+<span id="cb31-1953"><a href="#cb31-1953" aria-hidden="true" tabindex="-1"></a>    <span class="st">"Nelson-Aalen"</span> <span class="ot">=</span> na_fit <span class="sc">|&gt;</span> <span class="fu">fortify</span>(<span class="at">surv.connect =</span> <span class="cn">TRUE</span>)</span>
+<span id="cb31-1954"><a href="#cb31-1954" aria-hidden="true" tabindex="-1"></a>  ) <span class="sc">|&gt;</span> </span>
+<span id="cb31-1955"><a href="#cb31-1955" aria-hidden="true" tabindex="-1"></a>  <span class="fu">as_tibble</span>()</span>
+<span id="cb31-1956"><a href="#cb31-1956" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1957"><a href="#cb31-1957" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1958"><a href="#cb31-1958" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1961"><a href="#cb31-1961" aria-hidden="true" tabindex="-1"></a><span class="in">```{r}</span></span>
+<span id="cb31-1962"><a href="#cb31-1962" aria-hidden="true" tabindex="-1"></a><span class="co">#| fig-cap: "Kaplan-Meier and Nelson-Aalen Survival Function Estimates, stratified by disease group"</span></span>
+<span id="cb31-1963"><a href="#cb31-1963" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1964"><a href="#cb31-1964" aria-hidden="true" tabindex="-1"></a>km_and_na <span class="sc">|&gt;</span> </span>
+<span id="cb31-1965"><a href="#cb31-1965" aria-hidden="true" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> time, <span class="at">y =</span> surv, <span class="at">col =</span> model)) <span class="sc">+</span></span>
+<span id="cb31-1966"><a href="#cb31-1966" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_step</span>() <span class="sc">+</span></span>
+<span id="cb31-1967"><a href="#cb31-1967" aria-hidden="true" tabindex="-1"></a>  <span class="fu">facet_grid</span>(. <span class="sc">~</span> strata) <span class="sc">+</span></span>
+<span id="cb31-1968"><a href="#cb31-1968" aria-hidden="true" tabindex="-1"></a>  <span class="fu">theme_bw</span>() <span class="sc">+</span> </span>
+<span id="cb31-1969"><a href="#cb31-1969" aria-hidden="true" tabindex="-1"></a>  <span class="fu">ylab</span>(<span class="st">"S(t) = P(T&gt;=t)"</span>) <span class="sc">+</span></span>
+<span id="cb31-1970"><a href="#cb31-1970" aria-hidden="true" tabindex="-1"></a>  <span class="fu">xlab</span>(<span class="st">"Survival time (t, days)"</span>) <span class="sc">+</span></span>
+<span id="cb31-1971"><a href="#cb31-1971" aria-hidden="true" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"bottom"</span>)</span>
+<span id="cb31-1972"><a href="#cb31-1972" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1973"><a href="#cb31-1973" aria-hidden="true" tabindex="-1"></a><span class="in">```</span></span>
+<span id="cb31-1974"><a href="#cb31-1974" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb31-1975"><a href="#cb31-1975" aria-hidden="true" tabindex="-1"></a>The Kaplan-Meier and Nelson-Aalen survival estimates are very similar</span>
+<span id="cb31-1976"><a href="#cb31-1976" aria-hidden="true" tabindex="-1"></a>for this dataset.</span>
 </code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </div></div></div></div></div>
 </div> <!-- /content -->
diff --git a/logistic-regression.html b/logistic-regression.html
index 15cfaa2be..8f50aaaf7 100644
--- a/logistic-regression.html
+++ b/logistic-regression.html
@@ -3703,8 +3703,8 @@ <h3 data-number="3.9.2" class="anchored" data-anchor-id="hosmer-lemeshow-test"><
 <div class="sourceCode cell-code" id="cb42"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb42-1"><a href="#cb42-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplotly</span>(HL_plot)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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@@ -3855,8 +3855,8 @@ <h3 data-number="3.10.2" class="anchored" data-anchor-id="response-residuals"><s
 <div class="sourceCode cell-code" id="cb44"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb44-1"><a href="#cb44-1" aria-hidden="true" tabindex="-1"></a>wcgs_response_resid_plot <span class="sc">|&gt;</span> <span class="fu">ggplotly</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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 <p>We can see a slight fan-shape here: observations on the right have larger variance (as expected since <span class="math inline">\(var(\bar y) = \pi(1-\pi)/n\)</span> is maximized when <span class="math inline">\(\pi = 0.5\)</span>).</p>
@@ -3992,8 +3992,8 @@ <h4 class="anchored" data-anchor-id="residuals-plot-by-hand-optional-section">Re
 <div class="sourceCode cell-code" id="cb51"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb51-1"><a href="#cb51-1" aria-hidden="true" tabindex="-1"></a>wcgs_resid_plot1 <span class="sc">|&gt;</span> <span class="fu">ggplotly</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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diff --git a/search.json b/search.json
index 1b0abc1f5..d6a7c9f5e 100644
--- a/search.json
+++ b/search.json
@@ -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 &lt;- lm(carbohydrate ~ ., data = carbohydrate)\ncv_full = \n  full_model |&gt; cvFit(\n    data = carbohydrate, K = 5, R = 10,\n    y = carbohydrate$carbohydrate)\n\nreduced_model = update(full_model, \n                       formula = ~ . - age)\n\ncv_reduced = \n  reduced_model |&gt; cvFit(\n    data = carbohydrate, K = 5, R = 10,\n    y = carbohydrate$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 |&gt; \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)$adj.r.squared, sigma(full_model), logLik(full_model) |&gt; as.numeric(),\n  \"reduced\", cv_reduced$cv, summary(reduced_model)$r.squared, summary(reduced_model)$adj.r.squared, sigma(reduced_model), logLik(reduced_model) |&gt; as.numeric())\n\ncompare_results\n\n\n\nmodel\ncvRMSE\nr.squared\nadj.r.squared\ntrainRMSE\nloglik\n\n\n\nfull\n6.900\n0.4805\n0.3831\n5.956\n-61.84\n\n\nreduced\n6.581\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(&gt;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#&gt; \n#&gt; \n#&gt;                              Stepwise Summary                              \n#&gt; -------------------------------------------------------------------------\n#&gt; Step    Variable         AIC        SBC       SBIC       R2       Adj. R2 \n#&gt; -------------------------------------------------------------------------\n#&gt;  0      Base Model     140.773    142.764    83.068    0.00000    0.00000 \n#&gt;  1      protein (+)    137.950    140.937    80.438    0.21427    0.17061 \n#&gt;  2      weight (+)     132.981    136.964    77.191    0.44544    0.38020 \n#&gt; -------------------------------------------------------------------------\n#&gt; \n#&gt; Final Model Output \n#&gt; ------------------\n#&gt; \n#&gt;                          Model Summary                          \n#&gt; ---------------------------------------------------------------\n#&gt; R                       0.667       RMSE                 5.505 \n#&gt; R-Squared               0.445       MSE                 35.648 \n#&gt; Adj. R-Squared          0.380       Coef. Var           15.879 \n#&gt; Pred R-Squared          0.236       AIC                132.981 \n#&gt; MAE                     4.593       SBC                136.964 \n#&gt; ---------------------------------------------------------------\n#&gt;  RMSE: Root Mean Square Error \n#&gt;  MSE: Mean Square Error \n#&gt;  MAE: Mean Absolute Error \n#&gt;  AIC: Akaike Information Criteria \n#&gt;  SBC: Schwarz Bayesian Criteria \n#&gt; \n#&gt;                                ANOVA                                \n#&gt; -------------------------------------------------------------------\n#&gt;                 Sum of                                             \n#&gt;                Squares        DF    Mean Square      F        Sig. \n#&gt; -------------------------------------------------------------------\n#&gt; Regression     486.778         2        243.389    6.827    0.0067 \n#&gt; Residual       606.022        17         35.648                    \n#&gt; Total         1092.800        19                                   \n#&gt; -------------------------------------------------------------------\n#&gt; \n#&gt;                                   Parameter Estimates                                    \n#&gt; ----------------------------------------------------------------------------------------\n#&gt;       model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper \n#&gt; ----------------------------------------------------------------------------------------\n#&gt; (Intercept)    33.130        12.572                  2.635    0.017     6.607    59.654 \n#&gt;     protein     1.824         0.623        0.534     2.927    0.009     0.509     3.139 \n#&gt;      weight    -0.222         0.083       -0.486    -2.662    0.016    -0.397    -0.046 \n#&gt; ----------------------------------------------------------------------------------------\n\n\nLasso\n\\[\\arg min_{\\theta} \\ell(\\theta) + \\lambda \\sum_{j=1}^p|\\beta_j|\\]\n\nShow R codelibrary(glmnet)\ny = carbohydrate$carbohydrate\nx = carbohydrate |&gt; \n  select(age, weight, protein) |&gt; \n  as.matrix()\nfit = glmnet(x,y)\n\n\n\n\nShow R codeautoplot(fit, xvar = 'lambda')\n\n\n\nFigure 2.19: Lasso selection\n\n\n\n\n\n\n\n\n\nShow R codecvfit = cv.glmnet(x,y)\nplot(cvfit)\n\n\n\n\n\n\n\n\n\nShow R codecoef(cvfit, s = \"lambda.1se\")\n#&gt; 4 x 1 sparse Matrix of class \"dgCMatrix\"\n#&gt;                  s1\n#&gt; (Intercept) 34.2044\n#&gt; age          .     \n#&gt; weight      -0.0926\n#&gt; protein      0.8582",
+    "text": "2.9 Model selection\n(adapted from Dobson and Barnett (2018) §6.3.3; for more information on prediction, see James et al. (2013) and Harrell (2015)).\n\nIf we have a lot of covariates in our dataset, we might want to choose a small subset to use in our model.\nThere are a few possible metrics to consider for choosing a “best” model.\n\n\n2.9.1 Mean squared error\nWe might want to minimize the mean squared error, \\(\\text E[(y-\\hat y)^2]\\), for new observations that weren’t in our data set when we fit the model.\nUnfortunately, \\[\\frac{1}{n}\\sum_{i=1}^n (y_i-\\hat y_i)^2\\] gives a biased estimate of \\(\\text E[(y-\\hat y)^2]\\) for new data. If we want an unbiased estimate, we will have to be clever.\n\nCross-validation\n\nShow R codedata(\"carbohydrate\", package = \"dobson\")\nlibrary(cvTools)\nfull_model &lt;- lm(carbohydrate ~ ., data = carbohydrate)\ncv_full = \n  full_model |&gt; cvFit(\n    data = carbohydrate, K = 5, R = 10,\n    y = carbohydrate$carbohydrate)\n\nreduced_model = update(full_model, \n                       formula = ~ . - age)\n\ncv_reduced = \n  reduced_model |&gt; 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$reps[])\n\ncv_results = \n  bind_rows(results_reduced, results_full)\n\ncv_results |&gt; \n  ggplot(aes(y = model, x = errs)) +\n  geom_boxplot()\n\n\n\n\n\n\n\n\ncomparing metrics\n\nShow R code\ncompare_results = tribble(\n  ~ model, ~ cvRMSE, ~ r.squared, ~adj.r.squared, ~ trainRMSE, ~loglik,\n  \"full\", cv_full$cv, summary(full_model)$r.squared,  summary(full_model)$adj.r.squared, sigma(full_model), logLik(full_model) |&gt; as.numeric(),\n  \"reduced\", cv_reduced$cv, summary(reduced_model)$r.squared, summary(reduced_model)$adj.r.squared, sigma(reduced_model), logLik(reduced_model) |&gt; as.numeric())\n\ncompare_results\n\n\n\nmodel\ncvRMSE\nr.squared\nadj.r.squared\ntrainRMSE\nloglik\n\n\n\nfull\n7.071\n0.4805\n0.3831\n5.956\n-61.84\n\n\nreduced\n6.767\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(&gt;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#&gt; \n#&gt; \n#&gt;                              Stepwise Summary                              \n#&gt; -------------------------------------------------------------------------\n#&gt; Step    Variable         AIC        SBC       SBIC       R2       Adj. R2 \n#&gt; -------------------------------------------------------------------------\n#&gt;  0      Base Model     140.773    142.764    83.068    0.00000    0.00000 \n#&gt;  1      protein (+)    137.950    140.937    80.438    0.21427    0.17061 \n#&gt;  2      weight (+)     132.981    136.964    77.191    0.44544    0.38020 \n#&gt; -------------------------------------------------------------------------\n#&gt; \n#&gt; Final Model Output \n#&gt; ------------------\n#&gt; \n#&gt;                          Model Summary                          \n#&gt; ---------------------------------------------------------------\n#&gt; R                       0.667       RMSE                 5.505 \n#&gt; R-Squared               0.445       MSE                 35.648 \n#&gt; Adj. R-Squared          0.380       Coef. Var           15.879 \n#&gt; Pred R-Squared          0.236       AIC                132.981 \n#&gt; MAE                     4.593       SBC                136.964 \n#&gt; ---------------------------------------------------------------\n#&gt;  RMSE: Root Mean Square Error \n#&gt;  MSE: Mean Square Error \n#&gt;  MAE: Mean Absolute Error \n#&gt;  AIC: Akaike Information Criteria \n#&gt;  SBC: Schwarz Bayesian Criteria \n#&gt; \n#&gt;                                ANOVA                                \n#&gt; -------------------------------------------------------------------\n#&gt;                 Sum of                                             \n#&gt;                Squares        DF    Mean Square      F        Sig. \n#&gt; -------------------------------------------------------------------\n#&gt; Regression     486.778         2        243.389    6.827    0.0067 \n#&gt; Residual       606.022        17         35.648                    \n#&gt; Total         1092.800        19                                   \n#&gt; -------------------------------------------------------------------\n#&gt; \n#&gt;                                   Parameter Estimates                                    \n#&gt; ----------------------------------------------------------------------------------------\n#&gt;       model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper \n#&gt; ----------------------------------------------------------------------------------------\n#&gt; (Intercept)    33.130        12.572                  2.635    0.017     6.607    59.654 \n#&gt;     protein     1.824         0.623        0.534     2.927    0.009     0.509     3.139 \n#&gt;      weight    -0.222         0.083       -0.486    -2.662    0.016    -0.397    -0.046 \n#&gt; ----------------------------------------------------------------------------------------\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 |&gt; \n  select(age, weight, protein) |&gt; \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#&gt; 4 x 1 sparse Matrix of class \"dgCMatrix\"\n#&gt;                  s1\n#&gt; (Intercept) 34.2044\n#&gt; age          .     \n#&gt; weight      -0.0926\n#&gt; protein      0.8582",
     "crumbs": [
       "Generalized Linear Models",
       "<span class='chapter-number'>2</span>  <span class='chapter-title'>Linear (Gaussian) Models</span>"
@@ -533,7 +533,7 @@
     "href": "intro-to-survival-analysis.html#distribution-functions-for-time-to-event-variables",
     "title": "\n5  Introduction to Survival Analysis\n",
     "section": "\n5.3 Distribution functions for time-to-event variables",
-    "text": "5.3 Distribution functions for time-to-event variables\n\n5.3.1 The Probability Density Function (PDF)\nFor a time-to-event variable \\(T\\) with a continuous distribution, the probability density function is defined as usual (see Section B.4.1).\n\nIn most time-to-event models, this density is assumed to be 0 for all \\(t&lt;0\\); that is, \\(f(t) = 0, \\forall t&lt;0\\). In other words, the support of \\(T\\) is typically \\([0,\\infty)\\).\n\n\n\nExample 5.1 (exponential distribution) Recall from Epi 202: the pdf of the exponential distribution family of models is:\n\\[p(T=t) = \\mathbb{1}_{t \\ge 0} \\cdot \\lambda \\text{e}^{-\\lambda t}\\]\nwhere \\(\\lambda &gt; 0\\).\n\nHere are some examples of exponential pdfs:\n\n\n\n\n\n\n\n\n\n\n5.3.2 The Cumulative Distribution Function (CDF)\nThe cumulative distribution function is defined as:\n\\[\n\\begin{aligned}\nF(t) &\\stackrel{\\text{def}}{=}\\Pr(T \\le t)\\\\\n&=\\int_{u=-\\infty}^t f(u) du\n\\end{aligned}\n\\]\n\nExample 5.2 (exponential distribution) Recall from Epi 202: the cdf of the exponential distribution family of models is:\n\\[\nP(T\\le t) = \\mathbb{1}_{t \\ge 0} \\cdot (1- \\text{e}^{-\\lambda t})\n\\] where \\(\\lambda &gt; 0\\).\n\nHere are some examples of exponential cdfs:\n\n\n\n\n\n\n\n\n\n5.3.3 The Survival Function\nFor survival data, a more important quantity is the survival function:\n\\[\n\\begin{aligned}\nS(t) &\\stackrel{\\text{def}}{=}\\Pr(T &gt; t)\\\\\n&=\\int_{u=t}^\\infty p(u) du\\\\\n&=1-F(t)\\\\\n\\end{aligned}\n\\]\n\n\nDefinition 5.1 (Survival function)  \n\nThe survival function \\(S(t)\\) is the probability that the event time is later than \\(t\\). If the event in a clinical trial is death, then \\(S(t)\\) is the expected fraction of the original population at time 0 who have survived up to time \\(t\\) and are still alive at time \\(t\\); that is:\n\nIf \\(X_t\\) represents survival status at time \\(t\\), with \\(X_t = 1\\) denoting alive at time \\(t\\) and \\(X_t = 0\\) denoting deceased at time \\(t\\), then:\n\\[S(t) = \\Pr(X_t=1) = \\mathbb{E}[X_t]\\]\n\n\n\nExample 5.3 (exponential distribution) Since \\(S(t) = 1 - F(t)\\), the survival function of the exponential distribution family of models is:\n\\[\nP(T&gt; t) = \\left\\{ {{\\text{e}^{-\\lambda t}, t\\ge0} \\atop {1, t \\le 0}}\\right.\n\\] where \\(\\lambda &gt; 0\\).\nFigure 5.1 shows some examples of exponential survival functions.\n\n\n\n\n\n\nFigure 5.1: Exponential Survival Functions\n\n\n\n\n\n\n\n\n5.3.4 The Hazard Function\nAnother important quantity is the hazard function:\n\nDefinition 5.2 (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\n\nThe hazard function has an important relationship to the density and survival functions, which we can use to derive the hazard function for a given probability distribution (Theorem 5.1).\n\n\nLemma 5.1 (Joint probability of a variable with itself) \\[p(T=t, T\\ge t) = p(T=t)\\]\n\nProof. Recall from Epi 202: if \\(A\\) and \\(B\\) are statistical events and \\(A\\subseteq B\\), then \\(p(A, B) = p(A)\\). In particular, \\(\\{T=t\\} \\subseteq \\{T\\geq t\\}\\), so \\(p(T=t, T\\ge t) = p(T=t)\\).\n\n\n\n\nTheorem 5.1 \\[h(t)=\\frac{f(t)}{S(t)}\\]\n\n\n\nProof. \\[\n\\begin{aligned}\nh(t) &=p(T=t|T\\ge t)\\\\\n&=\\frac{p(T=t, T\\ge t)}{p(T \\ge t)}\\\\\n&=\\frac{p(T=t)}{p(T \\ge t)}\\\\\n&=\\frac{f(t)}{S(t)}\n\\end{aligned}\n\\]\n\n\n\nExample 5.4 (exponential distribution) The hazard function of the exponential distribution family of models is:\n\\[\n\\begin{aligned}\nP(T=t|T \\ge t)\n&= \\frac{f(t)}{S(t)}\\\\\n&= \\frac{\\mathbb{1}_{t \\ge 0}\\cdot \\lambda  \\text{e}^{-\\lambda t}}{\\text{e}^{-\\lambda t}}\\\\\n&=\\mathbb{1}_{t \\ge 0}\\cdot \\lambda\n\\end{aligned}\n\\] Figure 5.2 shows some examples of exponential hazard functions.\n\n\n\n\n\n\nFigure 5.2: Examples of hazard functions for exponential distributions\n\n\n\n\n\n\n\n\nWe can also view the hazard function as the derivative of the negative of the logarithm of the survival function:\n\nTheorem 5.2 (transform survival to hazard) \\[h(t) = \\frac{\\partial}{\\partial t}\\left\\{-\\text{log}\\left\\{S(t)\\right\\}\\right\\}\\]\n\n\n\nProof. \\[\n\\begin{aligned}\nh(t)\n&= \\frac{f(t)}{S(t)}\\\\\n&= \\frac{-S'(t)}{S(t)}\\\\\n&= -\\frac{S'(t)}{S(t)}\\\\\n&=-\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\}\\\\\n&=\\frac{\\partial}{\\partial t}\\left\\{-\\text{log}\\left\\{S(t)\\right\\}\\right\\}\n\\end{aligned}\n\\]\n\n\n5.3.5 The Cumulative Hazard Function\nSince \\(h(t) = \\frac{\\partial}{\\partial t}\\left\\{-\\text{log}\\left\\{S(t)\\right\\}\\right\\}\\) (see Theorem 5.2), we also have:\n\nCorollary 5.1 \\[S(t) = \\text{exp}\\left\\{-\\int_{u=0}^t h(u)du\\right\\} \\tag{5.1}\\]\n\n\n\nThe integral in Equation 5.1 is important enough to have its own name: cumulative hazard.\n\n\nDefinition 5.3 (cumulative hazard) The cumulative hazard function \\(H(t)\\) is defined as:\n\\[H(t) \\stackrel{\\text{def}}{=}\\int_{u=0}^t h(u) du\\]\n\nAs we will see below, \\(H(t)\\) is tractable to estimate, and we can then derive an estimate of the hazard function using an approximate derivative of the estimated cumulative hazard.\n\n\nExample 5.5 The cumulative hazard function of the exponential distribution family of models is:\n\\[\nH(t) = \\mathbb{1}_{t \\ge 0}\\cdot \\lambda t\n\\]\nFigure 5.3 shows some examples of exponential cumulative hazard functions.\n\n\n\n\n\n\nFigure 5.3: Examples of exponential cumulative hazard functions\n\n\n\n\n\n\n\n\n5.3.6 Some Key Mathematical Relationships among Survival Concepts\nDiagram:\n\\[\nh(t) \\xrightarrow[]{\\int_{u=0}^t h(u)du} H(t)\n\\xrightarrow[]{\\text{exp}\\left\\{-H(t)\\right\\}} S(t)\n\\xrightarrow[]{1-S(t)} F(t)\n\\]\n\\[\nh(t) \\xleftarrow[\\frac{\\partial}{\\partial t}H(t)]{} H(t)\n\\xleftarrow[-\\text{log}\\left\\{S(t)\\right\\}]{} S(t)\n\\xleftarrow[1-F(t)]{} F(t)\n\\]\nIdentities:\n\\[\n\\begin{aligned}\nS(t) &= 1 - F(t)\\\\\n&= \\text{exp}\\left\\{-H(t)\\right\\}\\\\\nS'(t) &= -f(t)\\\\\nH(t) &= -\\text{log}\\left\\{S(t)\\right\\}\\\\\nH'(t) &= h(t)\\\\\nh(t) &= \\frac{f(t)}{S(t)}\\\\\n&= -\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\} \\\\\nf(t) &= h(t)\\cdot S(t)\\\\\n\\end{aligned}\n\\]\n\nSome proofs (others left as exercises):\n\\[\n\\begin{aligned}\nS'(t) &= \\frac{\\partial}{\\partial t}(1-F(t))\\\\\n&= -F'(t)\\\\\n&= -f(t)\\\\\n\\end{aligned}\n\\]\n\n\\[\n\\begin{aligned}\n\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\}\n&= \\frac{S'(t)}{S(t)}\\\\\n&= -\\frac{f(t)}{S(t)}\\\\\n&= -h(t)\\\\\n\\end{aligned}\n\\]\n\n\\[\n\\begin{aligned}\nH(t)\n&\\stackrel{\\text{def}}{=}\\int_{u=0}^t h(u) du\\\\\n&= \\int_0^t -\\frac{\\partial}{\\partial u}\\text{log}\\left\\{S(u)\\right\\} du\\\\\n&= \\left[-\\text{log}\\left\\{S(u)\\right\\}\\right]_{u=0}^{u=t}\\\\\n&= \\left[\\text{log}\\left\\{S(u)\\right\\}\\right]_{u=t}^{u=0}\\\\\n&= \\text{log}\\left\\{S(0)\\right\\} - \\text{log}\\left\\{S(t)\\right\\}\\\\\n&= \\text{log}\\left\\{1\\right\\} - \\text{log}\\left\\{S(t)\\right\\}\\\\\n&= 0 - \\text{log}\\left\\{S(t)\\right\\}\\\\\n&=-\\text{log}\\left\\{S(t)\\right\\}\n\\end{aligned}\n\\]\n\nCorollary:\n\\[S(t) = \\text{exp}\\left\\{-H(t)\\right\\}\\]\nExample: Time to death the US in 2004\nThe first day is the most dangerous:\n\n\n\nDaily Hazard Rates in 2004 for US Females\n\n\n\n\n\nExercise: hypothesize why these curves differ where they do?\n\n\n\nDaily Hazard Rates in 2004 for US Males and Females 1-40\n\n\n\n\n\nExercise: compare and contrast this curve with the corresponding hazard curve.\n\n\n\nSurvival Curve in 2004 for US Females\n\n\n\n\n\n5.3.7 Likelihood with censoring\nIf an event time \\(T\\) is observed exactly as \\(T=t\\), then the likelihood of that observation is just its probability density function:\n\\[\n\\begin{aligned}\n\\mathcal L(t)\n&= p(T=t)\\\\\n&\\stackrel{\\text{def}}{=}f_T(t)\\\\\n&= h_T(t)S_T(t)\\\\\n\\ell(t)\n&\\stackrel{\\text{def}}{=}\\text{log}\\left\\{\\mathcal L(t)\\right\\}\\\\\n&= \\text{log}\\left\\{h_T(t)S_T(t)\\right\\}\\\\\n&= \\text{log}\\left\\{h_T(t)\\right\\} + \\text{log}\\left\\{S_T(t)\\right\\}\\\\\n&= \\text{log}\\left\\{h_T(t)\\right\\} - H_T(t)\\\\\n\\end{aligned}\n\\]\n\nIf instead the event time \\(T\\) is censored and only known to be after time \\(y\\), then the likelihood of that censored observation is instead the survival function evaluated at the censoring time:\n\\[\n\\begin{aligned}\n\\mathcal L(y)\n&=p_T(T&gt;y)\\\\\n&\\stackrel{\\text{def}}{=}S_T(y)\\\\\n\\ell(y)\n&\\stackrel{\\text{def}}{=}\\text{log}\\left\\{\\mathcal L(y)\\right\\}\\\\\n&=\\text{log}\\left\\{S(y)\\right\\}\\\\\n&=-H(y)\\\\\n\\end{aligned}\n\\]\n\n\nWhat’s written above is incomplete. We also observed whether or not the observation was censored. Let \\(C\\) denote the time when censoring would occur (if the event did not occur first); let \\(f_C(y)\\) and \\(S_C(y)\\) be the corresponding density and survival functions for the censoring event.\nLet \\(Y\\) denote the time when observation ended (either by censoring or by the event of interest occurring), and let \\(D\\) be an indicator variable for the event occurring at \\(Y\\) (so \\(D=0\\) represents a censored observation and \\(D=1\\) represents an uncensored observation). In other words, let \\(Y \\stackrel{\\text{def}}{=}\\min(T,C)\\) and \\(D \\stackrel{\\text{def}}{=}\\mathbb 1{\\{T&lt;=C\\}}\\).\nThen the complete likelihood of the observed data \\((Y,D)\\) is:\n\n\\[\n\\begin{aligned}\n\\mathcal L(y,d)\n&= p(Y=y, D=d)\\\\\n&= \\left[p(T=y,C&gt; y)\\right]^d \\cdot\n\\left[p(T&gt;y,C=y)\\right]^{1-d}\\\\\n\\end{aligned}\n\\]\n\n\nTypically, survival analyses assume that \\(C\\) and \\(T\\) are mutually independent; this assumption is called “non-informative” censoring.\nThen the joint likelihood \\(p(Y,D)\\) factors into the product \\(p(Y), p(D)\\), and the likelihood reduces to:\n\n\\[\n\\begin{aligned}\n\\mathcal L(y,d)\n&= \\left[p(T=y,C&gt; y)\\right]^d\\cdot\n\\left[p(T&gt;y,C=y)\\right]^{1-d}\\\\\n&= \\left[p(T=y)p(C&gt; y)\\right]^d\\cdot\n\\left[p(T&gt;y)p(C=y)\\right]^{1-d}\\\\\n&= \\left[f_T(y)S_C(y)\\right]^d\\cdot\n\\left[S(y)f_C(y)\\right]^{1-d}\\\\\n&= \\left[f_T(y)^d S_C(y)^d\\right]\\cdot\n\\left[S_T(y)^{1-d}f_C(y)^{1-d}\\right]\\\\\n&= \\left(f_T(y)^d \\cdot S_T(y)^{1-d}\\right)\\cdot\n\\left(f_C(y)^{1-d} \\cdot S_C(y)^{d}\\right)\n\\end{aligned}\n\\]\n\n\nThe corresponding log-likelihood is:\n\n\\[\n\\begin{aligned}\n\\ell(y,d)\n&= \\text{log}\\left\\{\\mathcal L(y,d) \\right\\}\\\\\n&= \\text{log}\\left\\{\n\\left(f_T(y)^d \\cdot S_T(y)^{1-d}\\right)\\cdot\n\\left(f_C(y)^{1-d} \\cdot S_C(y)^{d}\\right)\n\\right\\}\\\\\n&= \\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n+\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\\\\\n\\end{aligned}\n\\] Let\n\n\n\\(\\theta_T\\) represent the parameters of \\(p_T(t)\\),\n\n\\(\\theta_C\\) represent the parameters of \\(p_C(c)\\),\n\n\\(\\theta = (\\theta_T, \\theta_C)\\) be the combined vector of all parameters.\n\n\n\nThe corresponding score function is:\n\n\\[\n\\begin{aligned}\n\\ell'(y,d)\n&= \\frac{\\partial}{\\partial \\theta}\n\\left[\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n+\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\n\\right]\\\\\n&=\n\\left(\n\\frac{\\partial}{\\partial \\theta}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n\\right)\n+\n\\left(\n\\frac{\\partial}{\\partial \\theta}\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\n\\right)\\\\\n\\end{aligned}\n\\]\n\n\nAs long as \\(\\theta_C\\) and \\(\\theta_T\\) don’t share any parameters, then if censoring is non-informative, the partial derivative with respect to \\(\\theta_T\\) is:\n\n\\[\n\\begin{aligned}\n\\ell'_{\\theta_T}(y,d)\n&\\stackrel{\\text{def}}{=}\\frac{\\partial}{\\partial \\theta_T}\\ell(y,d)\\\\\n&=\n\\left(\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n\\right)\n+\n\\left(\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\n\\right)\\\\\n&=\n\\left(\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n\\right) + 0\\\\\n&=\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\\\\\n\\end{aligned}\n\\]\n\n\nThus, the MLE for \\(\\theta_T\\) won’t depend on \\(\\theta_C\\), and we can ignore the distribution of \\(C\\) when estimating the parameters of \\(f_T(t)=p(T=t)\\).\n\nThen:\n\\[\n\\begin{aligned}\n\\mathcal L(y,d)\n&= f_T(y)^d \\cdot S_T(y)^{1-d}\\\\\n&= \\left(h_T(y)^d  S_T(y)^d\\right) \\cdot S_T(y)^{1-d}\\\\\n&= h_T(y)^d  \\cdot S_T(y)^d \\cdot S_T(y)^{1-d}\\\\\n&= h_T(y)^d \\cdot S_T(y)\\\\\n&= S_T(y) \\cdot h_T(y)^d \\\\\n\\end{aligned}\n\\]\n\nThat is, if the event occurred at time \\(y\\) (i.e., if \\(d=1\\)), then the likelihood of \\((Y,D) = (y,d)\\) is equal to the hazard function at \\(y\\) times the survival function at \\(y\\). Otherwise, the likelihood is equal to just the survival function at \\(y\\).\n\n\n\nThe corresponding log-likelihood is:\n\n\\[\n\\begin{aligned}\n\\ell(y,d)\n&=\\text{log}\\left\\{\\mathcal L(y,d)\\right\\}\\\\\n&= \\text{log}\\left\\{S_T(y) \\cdot h_T(y)^d\\right\\}\\\\\n&= \\text{log}\\left\\{S_T(y)\\right\\} + \\text{log}\\left\\{h_T(y)^d\\right\\}\\\\\n&= \\text{log}\\left\\{S_T(y)\\right\\} + d\\cdot \\text{log}\\left\\{h_T(y)\\right\\}\\\\\n&= -H_T(y) + d\\cdot \\text{log}\\left\\{h_T(y)\\right\\}\\\\\n\\end{aligned}\n\\]\n\nIn other words, the log-likelihood contribution from a single observation \\((Y,D) = (y,d)\\) is equal to the negative cumulative hazard at \\(y\\), plus the log of the hazard at \\(y\\) if the event occurred at time \\(y\\).",
+    "text": "5.3 Distribution functions for time-to-event variables\n\n5.3.1 The Probability Density Function (PDF)\nFor a time-to-event variable \\(T\\) with a continuous distribution, the probability density function is defined as usual (see Section B.4.1).\n\nIn most time-to-event models, this density is assumed to be 0 for all \\(t&lt;0\\); that is, \\(f(t) = 0, \\forall t&lt;0\\). In other words, the support of \\(T\\) is typically \\([0,\\infty)\\).\n\n\n\nExample 5.1 (exponential distribution) Recall from Epi 202: the pdf of the exponential distribution family of models is:\n\\[p(T=t) = \\mathbb{1}_{t \\ge 0} \\cdot \\lambda \\text{e}^{-\\lambda t}\\]\nwhere \\(\\lambda &gt; 0\\).\n\nHere are some examples of exponential pdfs:\n\n\n\n\n\n\n\n\n\n\n5.3.2 The Cumulative Distribution Function (CDF)\nThe cumulative distribution function is defined as:\n\\[\n\\begin{aligned}\nF(t) &\\stackrel{\\text{def}}{=}\\Pr(T \\le t)\\\\\n&=\\int_{u=-\\infty}^t f(u) du\n\\end{aligned}\n\\]\n\nExample 5.2 (exponential distribution) Recall from Epi 202: the cdf of the exponential distribution family of models is:\n\\[\nP(T\\le t) = \\mathbb{1}_{t \\ge 0} \\cdot (1- \\text{e}^{-\\lambda t})\n\\] where \\(\\lambda &gt; 0\\).\n\nHere are some examples of exponential cdfs:\n\n\n\n\n\n\n\n\n\n5.3.3 The Survival Function\nFor survival data, a more important quantity is the survival function:\n\\[\n\\begin{aligned}\nS(t) &\\stackrel{\\text{def}}{=}\\Pr(T &gt; t)\\\\\n&=\\int_{u=t}^\\infty p(u) du\\\\\n&=1-F(t)\\\\\n\\end{aligned}\n\\]\n\n\nDefinition 5.1 (Survival function)  \n\nThe survival function \\(S(t)\\) is the probability that the event time is later than \\(t\\). If the event in a clinical trial is death, then \\(S(t)\\) is the expected fraction of the original population at time 0 who have survived up to time \\(t\\) and are still alive at time \\(t\\); that is:\n\n\\[S(t) \\stackrel{\\text{def}}{=}\\Pr(T &gt; t) \\tag{5.1}\\]\n\n\n\nExample 5.3 (exponential distribution) Since \\(S(t) = 1 - F(t)\\), the survival function of the exponential distribution family of models is:\n\\[\nP(T&gt; t) = \\left\\{ {{\\text{e}^{-\\lambda t}, t\\ge0} \\atop {1, t \\le 0}}\\right.\n\\] where \\(\\lambda &gt; 0\\).\nFigure 5.1 shows some examples of exponential survival functions.\n\n\n\n\n\n\nFigure 5.1: Exponential Survival Functions\n\n\n\n\n\n\n\n\n\nTheorem 5.1 If \\(A_t\\) represents survival status at time \\(t\\), with \\(A_t = 1\\) denoting alive at time \\(t\\) and \\(A_t = 0\\) denoting deceased at time \\(t\\), then:\n\\[S(t) = \\Pr(A_t=1) = \\mathbb{E}[A_t]\\]\n\n\n\nTheorem 5.2 If \\(T\\) is a nonnegative random variable, then:\n\\[\\mathbb{E}[T] = \\int_{t=0}^{\\infty} S(t)dt\\]\n\n\n\nProof. See https://statproofbook.github.io/P/mean-nnrvar.html or\n\n\n5.3.4 The Hazard Function\nAnother important quantity is the hazard function:\n\nDefinition 5.2 (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\n\nThe hazard function has an important relationship to the density and survival functions, which we can use to derive the hazard function for a given probability distribution (Theorem 5.3).\n\n\nLemma 5.1 (Joint probability of a variable with itself) \\[p(T=t, T\\ge t) = p(T=t)\\]\n\nProof. Recall from Epi 202: if \\(A\\) and \\(B\\) are statistical events and \\(A\\subseteq B\\), then \\(p(A, B) = p(A)\\). In particular, \\(\\{T=t\\} \\subseteq \\{T\\geq t\\}\\), so \\(p(T=t, T\\ge t) = p(T=t)\\).\n\n\n\n\nTheorem 5.3 \\[h(t)=\\frac{f(t)}{S(t)}\\]\n\n\n\nProof. \\[\n\\begin{aligned}\nh(t) &=p(T=t|T\\ge t)\\\\\n&=\\frac{p(T=t, T\\ge t)}{p(T \\ge t)}\\\\\n&=\\frac{p(T=t)}{p(T \\ge t)}\\\\\n&=\\frac{f(t)}{S(t)}\n\\end{aligned}\n\\]\n\n\n\nExample 5.4 (exponential distribution) The hazard function of the exponential distribution family of models is:\n\\[\n\\begin{aligned}\nP(T=t|T \\ge t)\n&= \\frac{f(t)}{S(t)}\\\\\n&= \\frac{\\mathbb{1}_{t \\ge 0}\\cdot \\lambda  \\text{e}^{-\\lambda t}}{\\text{e}^{-\\lambda t}}\\\\\n&=\\mathbb{1}_{t \\ge 0}\\cdot \\lambda\n\\end{aligned}\n\\] Figure 5.2 shows some examples of exponential hazard functions.\n\n\n\n\n\n\nFigure 5.2: Examples of hazard functions for exponential distributions\n\n\n\n\n\n\n\n\nWe can also view the hazard function as the derivative of the negative of the logarithm of the survival function:\n\nTheorem 5.4 (transform survival to hazard) \\[h(t) = \\frac{\\partial}{\\partial t}\\left\\{-\\text{log}\\left\\{S(t)\\right\\}\\right\\}\\]\n\n\n\nProof. \\[\n\\begin{aligned}\nh(t)\n&= \\frac{f(t)}{S(t)}\\\\\n&= \\frac{-S'(t)}{S(t)}\\\\\n&= -\\frac{S'(t)}{S(t)}\\\\\n&=-\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\}\\\\\n&=\\frac{\\partial}{\\partial t}\\left\\{-\\text{log}\\left\\{S(t)\\right\\}\\right\\}\n\\end{aligned}\n\\]\n\n\n5.3.5 The Cumulative Hazard Function\nSince \\(h(t) = \\frac{\\partial}{\\partial t}\\left\\{-\\text{log}\\left\\{S(t)\\right\\}\\right\\}\\) (see Theorem 5.4), we also have:\n\nCorollary 5.1 \\[S(t) = \\text{exp}\\left\\{-\\int_{u=0}^t h(u)du\\right\\} \\tag{5.2}\\]\n\n\n\nThe integral in Equation 5.2 is important enough to have its own name: cumulative hazard.\n\n\nDefinition 5.3 (cumulative hazard) The cumulative hazard function \\(H(t)\\) is defined as:\n\\[H(t) \\stackrel{\\text{def}}{=}\\int_{u=0}^t h(u) du\\]\n\nAs we will see below, \\(H(t)\\) is tractable to estimate, and we can then derive an estimate of the hazard function using an approximate derivative of the estimated cumulative hazard.\n\n\nExample 5.5 The cumulative hazard function of the exponential distribution family of models is:\n\\[\nH(t) = \\mathbb{1}_{t \\ge 0}\\cdot \\lambda t\n\\]\nFigure 5.3 shows some examples of exponential cumulative hazard functions.\n\n\n\n\n\n\nFigure 5.3: Examples of exponential cumulative hazard functions\n\n\n\n\n\n\n\n\n5.3.6 Some Key Mathematical Relationships among Survival Concepts\nDiagram:\n\\[\nh(t) \\xrightarrow[]{\\int_{u=0}^t h(u)du} H(t)\n\\xrightarrow[]{\\text{exp}\\left\\{-H(t)\\right\\}} S(t)\n\\xrightarrow[]{1-S(t)} F(t)\n\\]\n\\[\nh(t) \\xleftarrow[\\frac{\\partial}{\\partial t}H(t)]{} H(t)\n\\xleftarrow[-\\text{log}\\left\\{S(t)\\right\\}]{} S(t)\n\\xleftarrow[1-F(t)]{} F(t)\n\\]\nIdentities:\n\\[\n\\begin{aligned}\nS(t) &= 1 - F(t)\\\\\n&= \\text{exp}\\left\\{-H(t)\\right\\}\\\\\nS'(t) &= -f(t)\\\\\nH(t) &= -\\text{log}\\left\\{S(t)\\right\\}\\\\\nH'(t) &= h(t)\\\\\nh(t) &= \\frac{f(t)}{S(t)}\\\\\n&= -\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\} \\\\\nf(t) &= h(t)\\cdot S(t)\\\\\n\\end{aligned}\n\\]\n\nSome proofs (others left as exercises):\n\\[\n\\begin{aligned}\nS'(t) &= \\frac{\\partial}{\\partial t}(1-F(t))\\\\\n&= -F'(t)\\\\\n&= -f(t)\\\\\n\\end{aligned}\n\\]\n\n\\[\n\\begin{aligned}\n\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\}\n&= \\frac{S'(t)}{S(t)}\\\\\n&= -\\frac{f(t)}{S(t)}\\\\\n&= -h(t)\\\\\n\\end{aligned}\n\\]\n\n\\[\n\\begin{aligned}\nH(t)\n&\\stackrel{\\text{def}}{=}\\int_{u=0}^t h(u) du\\\\\n&= \\int_0^t -\\frac{\\partial}{\\partial u}\\text{log}\\left\\{S(u)\\right\\} du\\\\\n&= \\left[-\\text{log}\\left\\{S(u)\\right\\}\\right]_{u=0}^{u=t}\\\\\n&= \\left[\\text{log}\\left\\{S(u)\\right\\}\\right]_{u=t}^{u=0}\\\\\n&= \\text{log}\\left\\{S(0)\\right\\} - \\text{log}\\left\\{S(t)\\right\\}\\\\\n&= \\text{log}\\left\\{1\\right\\} - \\text{log}\\left\\{S(t)\\right\\}\\\\\n&= 0 - \\text{log}\\left\\{S(t)\\right\\}\\\\\n&=-\\text{log}\\left\\{S(t)\\right\\}\n\\end{aligned}\n\\]\n\nCorollary:\n\\[S(t) = \\text{exp}\\left\\{-H(t)\\right\\}\\]\nExample: Time to death the US in 2004\nThe first day is the most dangerous:\n\n\n\nDaily Hazard Rates in 2004 for US Females\n\n\n\n\n\nExercise: hypothesize why these curves differ where they do?\n\n\n\nDaily Hazard Rates in 2004 for US Males and Females 1-40\n\n\n\n\n\nExercise: compare and contrast this curve with the corresponding hazard curve.\n\n\n\nSurvival Curve in 2004 for US Females\n\n\n\n\n\n5.3.7 Likelihood with censoring\nIf an event time \\(T\\) is observed exactly as \\(T=t\\), then the likelihood of that observation is just its probability density function:\n\\[\n\\begin{aligned}\n\\mathcal L(t)\n&= p(T=t)\\\\\n&\\stackrel{\\text{def}}{=}f_T(t)\\\\\n&= h_T(t)S_T(t)\\\\\n\\ell(t)\n&\\stackrel{\\text{def}}{=}\\text{log}\\left\\{\\mathcal L(t)\\right\\}\\\\\n&= \\text{log}\\left\\{h_T(t)S_T(t)\\right\\}\\\\\n&= \\text{log}\\left\\{h_T(t)\\right\\} + \\text{log}\\left\\{S_T(t)\\right\\}\\\\\n&= \\text{log}\\left\\{h_T(t)\\right\\} - H_T(t)\\\\\n\\end{aligned}\n\\]\n\nIf instead the event time \\(T\\) is censored and only known to be after time \\(y\\), then the likelihood of that censored observation is instead the survival function evaluated at the censoring time:\n\\[\n\\begin{aligned}\n\\mathcal L(y)\n&=p_T(T&gt;y)\\\\\n&\\stackrel{\\text{def}}{=}S_T(y)\\\\\n\\ell(y)\n&\\stackrel{\\text{def}}{=}\\text{log}\\left\\{\\mathcal L(y)\\right\\}\\\\\n&=\\text{log}\\left\\{S(y)\\right\\}\\\\\n&=-H(y)\\\\\n\\end{aligned}\n\\]\n\n\nWhat’s written above is incomplete. We also observed whether or not the observation was censored. Let \\(C\\) denote the time when censoring would occur (if the event did not occur first); let \\(f_C(y)\\) and \\(S_C(y)\\) be the corresponding density and survival functions for the censoring event.\nLet \\(Y\\) denote the time when observation ended (either by censoring or by the event of interest occurring), and let \\(D\\) be an indicator variable for the event occurring at \\(Y\\) (so \\(D=0\\) represents a censored observation and \\(D=1\\) represents an uncensored observation). In other words, let \\(Y \\stackrel{\\text{def}}{=}\\min(T,C)\\) and \\(D \\stackrel{\\text{def}}{=}\\mathbb 1{\\{T&lt;=C\\}}\\).\nThen the complete likelihood of the observed data \\((Y,D)\\) is:\n\n\\[\n\\begin{aligned}\n\\mathcal L(y,d)\n&= p(Y=y, D=d)\\\\\n&= \\left[p(T=y,C&gt; y)\\right]^d \\cdot\n\\left[p(T&gt;y,C=y)\\right]^{1-d}\\\\\n\\end{aligned}\n\\]\n\n\nTypically, survival analyses assume that \\(C\\) and \\(T\\) are mutually independent; this assumption is called “non-informative” censoring.\nThen the joint likelihood \\(p(Y,D)\\) factors into the product \\(p(Y), p(D)\\), and the likelihood reduces to:\n\n\\[\n\\begin{aligned}\n\\mathcal L(y,d)\n&= \\left[p(T=y,C&gt; y)\\right]^d\\cdot\n\\left[p(T&gt;y,C=y)\\right]^{1-d}\\\\\n&= \\left[p(T=y)p(C&gt; y)\\right]^d\\cdot\n\\left[p(T&gt;y)p(C=y)\\right]^{1-d}\\\\\n&= \\left[f_T(y)S_C(y)\\right]^d\\cdot\n\\left[S(y)f_C(y)\\right]^{1-d}\\\\\n&= \\left[f_T(y)^d S_C(y)^d\\right]\\cdot\n\\left[S_T(y)^{1-d}f_C(y)^{1-d}\\right]\\\\\n&= \\left(f_T(y)^d \\cdot S_T(y)^{1-d}\\right)\\cdot\n\\left(f_C(y)^{1-d} \\cdot S_C(y)^{d}\\right)\n\\end{aligned}\n\\]\n\n\nThe corresponding log-likelihood is:\n\n\\[\n\\begin{aligned}\n\\ell(y,d)\n&= \\text{log}\\left\\{\\mathcal L(y,d) \\right\\}\\\\\n&= \\text{log}\\left\\{\n\\left(f_T(y)^d \\cdot S_T(y)^{1-d}\\right)\\cdot\n\\left(f_C(y)^{1-d} \\cdot S_C(y)^{d}\\right)\n\\right\\}\\\\\n&= \\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n+\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\\\\\n\\end{aligned}\n\\] Let\n\n\n\\(\\theta_T\\) represent the parameters of \\(p_T(t)\\),\n\n\\(\\theta_C\\) represent the parameters of \\(p_C(c)\\),\n\n\\(\\theta = (\\theta_T, \\theta_C)\\) be the combined vector of all parameters.\n\n\n\nThe corresponding score function is:\n\n\\[\n\\begin{aligned}\n\\ell'(y,d)\n&= \\frac{\\partial}{\\partial \\theta}\n\\left[\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n+\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\n\\right]\\\\\n&=\n\\left(\n\\frac{\\partial}{\\partial \\theta}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n\\right)\n+\n\\left(\n\\frac{\\partial}{\\partial \\theta}\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\n\\right)\\\\\n\\end{aligned}\n\\]\n\n\nAs long as \\(\\theta_C\\) and \\(\\theta_T\\) don’t share any parameters, then if censoring is non-informative, the partial derivative with respect to \\(\\theta_T\\) is:\n\n\\[\n\\begin{aligned}\n\\ell'_{\\theta_T}(y,d)\n&\\stackrel{\\text{def}}{=}\\frac{\\partial}{\\partial \\theta_T}\\ell(y,d)\\\\\n&=\n\\left(\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n\\right)\n+\n\\left(\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_C(y)^{1-d} \\cdot S_C(y)^{d}\n\\right\\}\n\\right)\\\\\n&=\n\\left(\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\n\\right) + 0\\\\\n&=\n\\frac{\\partial}{\\partial \\theta_T}\n\\text{log}\\left\\{\nf_T(y)^d \\cdot S_T(y)^{1-d}\n\\right\\}\\\\\n\\end{aligned}\n\\]\n\n\nThus, the MLE for \\(\\theta_T\\) won’t depend on \\(\\theta_C\\), and we can ignore the distribution of \\(C\\) when estimating the parameters of \\(f_T(t)=p(T=t)\\).\n\nThen:\n\\[\n\\begin{aligned}\n\\mathcal L(y,d)\n&= f_T(y)^d \\cdot S_T(y)^{1-d}\\\\\n&= \\left(h_T(y)^d  S_T(y)^d\\right) \\cdot S_T(y)^{1-d}\\\\\n&= h_T(y)^d  \\cdot S_T(y)^d \\cdot S_T(y)^{1-d}\\\\\n&= h_T(y)^d \\cdot S_T(y)\\\\\n&= S_T(y) \\cdot h_T(y)^d \\\\\n\\end{aligned}\n\\]\n\nThat is, if the event occurred at time \\(y\\) (i.e., if \\(d=1\\)), then the likelihood of \\((Y,D) = (y,d)\\) is equal to the hazard function at \\(y\\) times the survival function at \\(y\\). Otherwise, the likelihood is equal to just the survival function at \\(y\\).\n\n\n\nThe corresponding log-likelihood is:\n\n\\[\n\\begin{aligned}\n\\ell(y,d)\n&=\\text{log}\\left\\{\\mathcal L(y,d)\\right\\}\\\\\n&= \\text{log}\\left\\{S_T(y) \\cdot h_T(y)^d\\right\\}\\\\\n&= \\text{log}\\left\\{S_T(y)\\right\\} + \\text{log}\\left\\{h_T(y)^d\\right\\}\\\\\n&= \\text{log}\\left\\{S_T(y)\\right\\} + d\\cdot \\text{log}\\left\\{h_T(y)\\right\\}\\\\\n&= -H_T(y) + d\\cdot \\text{log}\\left\\{h_T(y)\\right\\}\\\\\n\\end{aligned}\n\\]\n\nIn other words, the log-likelihood contribution from a single observation \\((Y,D) = (y,d)\\) is equal to the negative cumulative hazard at \\(y\\), plus the log of the hazard at \\(y\\) if the event occurred at time \\(y\\).",
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       "Time to Event Models",
       "<span class='chapter-number'>5</span>  <span class='chapter-title'>Introduction to Survival Analysis</span>"
@@ -544,7 +544,7 @@
     "href": "intro-to-survival-analysis.html#parametric-models-for-time-to-event-outcomes",
     "title": "\n5  Introduction to Survival Analysis\n",
     "section": "\n5.4 Parametric Models for Time-to-Event Outcomes",
-    "text": "5.4 Parametric Models for Time-to-Event Outcomes\n\n5.4.1 Exponential Distribution\n\nThe exponential distribution is the base distribution for survival analysis.\nThe distribution has a constant hazard \\(\\lambda\\)\n\nThe mean survival time is \\(\\lambda^{-1}\\)\n\n\n\nMathematical details of exponential distribution\n\\[\n\\begin{aligned}\nf(t) &= \\lambda \\text{e}^{-\\lambda t}\\\\\nE(t) &= \\lambda^{-1}\\\\\nVar(t) &= \\lambda^{-2}\\\\\nF(t) &= 1-\\text{e}^{-\\lambda x}\\\\\nS(t)&= \\text{e}^{-\\lambda x}\\\\\n\\ln(S(t))&=-\\lambda x\\\\\nh(t) &= -\\frac{f(t)}{S(t)} = -\\frac{\\lambda \\text{e}^{-\\lambda t}}{\\text{e}^{-\\lambda t}}=\\lambda\n\\end{aligned}\n\\]\nEstimating \\(\\lambda\\)\n\n\nSuppose we have \\(m\\) exponential survival times of \\(t_1, t_2,\\ldots,t_m\\) and \\(k\\) right-censored values at \\(u_1,u_2,\\ldots,u_k\\).\nA survival time of \\(t_i=10\\) means that subject \\(i\\) died at time 10. A right-censored time \\(u_i=10\\) means that at time 10, subject \\(i\\) was still alive and that we have no further follow-up.\nFor the moment we will assume that the survival distribution is exponential and that all the subjects have the same parameter \\(\\lambda\\).\n\nWe have \\(m\\) exponential survival times of \\(t_1, t_2,\\ldots,t_m\\) and \\(k\\) right-censored values at \\(u_1,u_2,\\ldots,u_k\\). The log-likelihood of an observed survival time \\(t_i\\) is \\[\n\\text{log}\\left\\{\\lambda \\text{e}^{-\\lambda t_i}\\right\\} =\n\\text{log}\\left\\{\\lambda\\right\\}-\\lambda t_i\n\\] and the likelihood of a censored value is the probability of that outcome (survival greater than \\(u_j\\)) so the log-likelihood is\n\\[\n\\begin{aligned}\n\\ell_j(\\lambda) &= \\text{log}\\left\\{\\lambda \\text{e}^{u_j}\\right\\}\n\\\\ &= -\\lambda u_j\n\\end{aligned}\n\\]\n\n\nTheorem 5.3 Let \\(T=\\sum t_i\\) and \\(U=\\sum u_j\\). Then:\n\\[\n\\ell(\\lambda) = \\frac{m}{T+U}\n\\tag{5.2}\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n\\ell(\\lambda) &= \\sum_{i=1}^m( \\ln \\lambda-\\lambda t_i) + \\sum_{j=1}^k (-\\lambda u_j)\\\\\n&= m \\ln \\lambda -(T+U)\\lambda\\\\\n\\ell'(\\lambda)\n&=m\\lambda^{-1} -(T+U)\\\\\n\\hat{\\lambda} &= \\frac{m}{T+U}\n\\end{aligned}\n\\]\n\n\n\\[\n\\begin{aligned}\n\\ell''&=-m/\\lambda^2\\\\\n&&lt; 0\\\\\n\\hat E[T] &= \\hat\\lambda^{-1}\\\\\n&= \\frac{T+U}{m}\n\\end{aligned}\n\\]\nFisher Information and Standard Error\n\\[\n\\begin{aligned}\nE[-\\ell'']\n& = m/\\lambda^2\\\\\n\\text{Var}\\left(\\hat\\lambda\\right)\n&\\approx \\left(E[-\\ell'']\\right)^{-1}\\\\\n&=\\lambda^2/m\\\\\n\\text{SE}\\left(\\hat\\lambda\\right)\n&= \\sqrt{\\text{Var}\\left(\\hat\\lambda\\right)}\\\\\n&\\approx \\lambda/\\sqrt{m}\n\\end{aligned}\n\\]\n\n\\(\\hat\\lambda\\) depends on the censoring times of the censored observations, but \\(\\text{Var}\\left(\\hat\\lambda\\right)\\) only depends on the number of uncensored observations, \\(m\\), and not on the number of censored observations (\\(k\\)).\n\n\n5.4.2 Other Parametric Survival Distributions\n\nAny density on \\([0,\\infty)\\) can be a survival distribution, but the most useful ones are all skew right.\nThe most frequently used generalization of the exponential is the Weibull.\nOther common choices are the gamma, log-normal, log-logistic, Gompertz, inverse Gaussian, and Pareto.\nMost of what we do going forward is non-parametric or semi-parametric, but sometimes these parametric distributions provide a useful approach.",
+    "text": "5.4 Parametric Models for Time-to-Event Outcomes\n\n5.4.1 Exponential Distribution\n\nThe exponential distribution is the base distribution for survival analysis.\nThe distribution has a constant hazard \\(\\lambda\\)\n\nThe mean survival time is \\(\\lambda^{-1}\\)\n\n\n\nMathematical details of exponential distribution\n\\[\n\\begin{aligned}\nf(t) &= \\lambda \\text{e}^{-\\lambda t}\\\\\nE(t) &= \\lambda^{-1}\\\\\nVar(t) &= \\lambda^{-2}\\\\\nF(t) &= 1-\\text{e}^{-\\lambda x}\\\\\nS(t)&= \\text{e}^{-\\lambda x}\\\\\n\\ln(S(t))&=-\\lambda x\\\\\nh(t) &= -\\frac{f(t)}{S(t)} = -\\frac{\\lambda \\text{e}^{-\\lambda t}}{\\text{e}^{-\\lambda t}}=\\lambda\n\\end{aligned}\n\\]\nEstimating \\(\\lambda\\)\n\n\nSuppose we have \\(m\\) exponential survival times of \\(t_1, t_2,\\ldots,t_m\\) and \\(k\\) right-censored values at \\(u_1,u_2,\\ldots,u_k\\).\nA survival time of \\(t_i=10\\) means that subject \\(i\\) died at time 10. A right-censored time \\(u_i=10\\) means that at time 10, subject \\(i\\) was still alive and that we have no further follow-up.\nFor the moment we will assume that the survival distribution is exponential and that all the subjects have the same parameter \\(\\lambda\\).\n\nWe have \\(m\\) exponential survival times of \\(t_1, t_2,\\ldots,t_m\\) and \\(k\\) right-censored values at \\(u_1,u_2,\\ldots,u_k\\). The log-likelihood of an observed survival time \\(t_i\\) is \\[\n\\text{log}\\left\\{\\lambda \\text{e}^{-\\lambda t_i}\\right\\} =\n\\text{log}\\left\\{\\lambda\\right\\}-\\lambda t_i\n\\] and the likelihood of a censored value is the probability of that outcome (survival greater than \\(u_j\\)) so the log-likelihood is\n\\[\n\\begin{aligned}\n\\ell_j(\\lambda) &= \\text{log}\\left\\{\\lambda \\text{e}^{u_j}\\right\\}\n\\\\ &= -\\lambda u_j\n\\end{aligned}\n\\]\n\n\nTheorem 5.5 Let \\(T=\\sum t_i\\) and \\(U=\\sum u_j\\). Then:\n\\[\n\\hat{\\lambda}_{ML} = \\frac{m}{T+U}\n\\tag{5.3}\\]\n\n\n\nProof. \\[\n\\begin{aligned}\n\\ell(\\lambda) &= \\sum_{i=1}^m( \\ln \\lambda-\\lambda t_i) + \\sum_{j=1}^k (-\\lambda u_j)\\\\\n&= m \\ln \\lambda -(T+U)\\lambda\\\\\n\\ell'(\\lambda)\n&=m\\lambda^{-1} -(T+U)\\\\\n\\hat{\\lambda} &= \\frac{m}{T+U}\n\\end{aligned}\n\\]\n\n\n\\[\n\\begin{aligned}\n\\ell''&=-m/\\lambda^2\\\\\n&&lt; 0\\\\\n\\hat E[T] &= \\hat\\lambda^{-1}\\\\\n&= \\frac{T+U}{m}\n\\end{aligned}\n\\]\nFisher Information and Standard Error\n\\[\n\\begin{aligned}\nE[-\\ell'']\n& = m/\\lambda^2\\\\\n\\text{Var}\\left(\\hat\\lambda\\right)\n&\\approx \\left(E[-\\ell'']\\right)^{-1}\\\\\n&=\\lambda^2/m\\\\\n\\text{SE}\\left(\\hat\\lambda\\right)\n&= \\sqrt{\\text{Var}\\left(\\hat\\lambda\\right)}\\\\\n&\\approx \\lambda/\\sqrt{m}\n\\end{aligned}\n\\]\n\n\\(\\hat\\lambda\\) depends on the censoring times of the censored observations, but \\(\\text{Var}\\left(\\hat\\lambda\\right)\\) only depends on the number of uncensored observations, \\(m\\), and not on the number of censored observations (\\(k\\)).\n\n\n5.4.2 Other Parametric Survival Distributions\n\nAny density on \\([0,\\infty)\\) can be a survival distribution, but the most useful ones are all skew right.\nThe most frequently used generalization of the exponential is the Weibull.\nOther common choices are the gamma, log-normal, log-logistic, Gompertz, inverse Gaussian, and Pareto.\nMost of what we do going forward is non-parametric or semi-parametric, but sometimes these parametric distributions provide a useful approach.",
     "crumbs": [
       "Time to Event Models",
       "<span class='chapter-number'>5</span>  <span class='chapter-title'>Introduction to Survival Analysis</span>"
@@ -566,7 +566,7 @@
     "href": "intro-to-survival-analysis.html#example-clinical-trial-for-pediatric-acute-leukemia",
     "title": "\n5  Introduction to Survival Analysis\n",
     "section": "\n5.6 Example: clinical trial for pediatric acute leukemia",
-    "text": "5.6 Example: clinical trial for pediatric acute leukemia\n\n5.6.1 Overview of study\nThis is from a clinical trial in 1963 for 6-MP treatment vs. placebo for Acute Leukemia in 42 children.\n\n\nPairs of children:\n\nmatched by remission status at the time of treatment (remstat: 1 = partial, 2 = complete)\nrandomized to 6-MP (exit times in t2) or placebo (exit times in t1)\n\n\nFollowed until relapse or end of study.\nAll of the placebo group relapsed, but some of the 6-MP group were censored (which means they were still in remission); indicated by relapse variable (0 = censored, 1 = relapse).\n6-MP = 6-Mercaptopurine (Purinethol) is an anti-cancer (“antineoplastic” or “cytotoxic”) chemotherapy drug used currently for Acute lymphoblastic leukemia (ALL). It is classified as an antimetabolite.\n\n5.6.2 Study design\n\nClinical trial in 1963 for 6-MP treatment vs. placebo for Acute Leukemia in 42 children.\nPairs of children:\n\nmatched by remission status at the time of treatment (remstat)\n\n\nremstat = 1: partial\n\nremstat = 2: complete\n\n\nrandomized to 6-MP (exit time: t2) or placebo (t1).\n\n\nFollowed until relapse or end of study.\n\nAll of the placebo group relapsed,\nSome of the 6-MP group were censored.\n\n\n\n\n\n\nTable 5.1: drug6mp pediatric acute leukemia data\n\nShow R codelibrary(KMsurv)\ndata(drug6mp)\ndrug6mp = drug6mp |&gt; as_tibble() |&gt; print()\n#&gt; # A tibble: 21 × 5\n#&gt;     pair remstat    t1    t2 relapse\n#&gt;    &lt;int&gt;   &lt;int&gt; &lt;int&gt; &lt;int&gt;   &lt;int&gt;\n#&gt;  1     1       1     1    10       1\n#&gt;  2     2       2    22     7       1\n#&gt;  3     3       2     3    32       0\n#&gt;  4     4       2    12    23       1\n#&gt;  5     5       2     8    22       1\n#&gt;  6     6       1    17     6       1\n#&gt;  7     7       2     2    16       1\n#&gt;  8     8       2    11    34       0\n#&gt;  9     9       2     8    32       0\n#&gt; 10    10       2    12    25       0\n#&gt; # ℹ 11 more rows\n\n\n\n\n\n5.6.3 Data documentation for drug6mp\n\n\nShow R code# library(printr) # inserts help-file output into markdown output\nlibrary(KMsurv)\n?drug6mp\n\n\n\n5.6.4 Descriptive Statistics\n\n\nTable 5.2: Summary statistics for drug6mp data\n\nShow R codesummary(drug6mp)\n#&gt;       pair       remstat           t1              t2          relapse     \n#&gt;  Min.   : 1   Min.   :1.00   Min.   : 1.00   Min.   : 6.0   Min.   :0.000  \n#&gt;  1st Qu.: 6   1st Qu.:2.00   1st Qu.: 4.00   1st Qu.: 9.0   1st Qu.:0.000  \n#&gt;  Median :11   Median :2.00   Median : 8.00   Median :16.0   Median :0.000  \n#&gt;  Mean   :11   Mean   :1.76   Mean   : 8.67   Mean   :17.1   Mean   :0.429  \n#&gt;  3rd Qu.:16   3rd Qu.:2.00   3rd Qu.:12.00   3rd Qu.:23.0   3rd Qu.:1.000  \n#&gt;  Max.   :21   Max.   :2.00   Max.   :23.00   Max.   :35.0   Max.   :1.000\n\n\n\n\n\n\nThe average time in each group is not useful. Some of the 6-MP patients have not relapsed at the time recorded, while all of the placebo patients have relapsed.\nThe median time is not really useful either because so many of the 6-MP patients have not relapsed (12/21).\nBoth are biased down in the 6-MP group. Remember that lower times are worse since they indicate sooner recurrence.\n\n\n\n5.6.5 Exponential model\n\n\nWe can compute the hazard rate, assuming an exponential model: number of relapses divided by the sum of the exit times (Equation 5.2).\n\n\n\\[\\hat\\lambda = \\frac{\\sum_{i=1}^nD_i}{\\sum_{i=1}^nY_i}\\]\n\n\nFor the placebo, that is just the reciprocal of the mean time:\n\n\n\\[\n\\begin{aligned}\n\\hat \\lambda_{\\text{placebo}}\n&= \\frac{\\sum_{i=1}^nD_i}{\\sum_{i=1}^nY_i}\n\\\\ &= \\frac{\\sum_{i=1}^n1}{\\sum_{i=1}^nY_i}\n\\\\ &= \\frac{n}{\\sum_{i=1}^nY_i}\n\\\\ &= \\frac{1}{\\bar{Y}}\n\\\\ &= \\frac{1}{8.6667}\n\\\\ &= 0.1154\n\\end{aligned}\n\\]\n\n\nFor the 6-MP group, \\(\\hat\\lambda = 9/359 = 0.025\\)\n\n\n\\[\n\\begin{aligned}\n\\hat \\lambda_{\\text{6-MP}}\n&= \\frac{\\sum_{i=1}^nD_i}{\\sum_{i=1}^nY_i}\n\\\\ &= \\frac{9}{359}\n\\\\ &= 0.0251\n\\end{aligned}\n\\]\n\nThe estimated hazard in the placebo group is 4.6 times as large as in the 6-MP group (assuming the hazard is constant over time).",
+    "text": "5.6 Example: clinical trial for pediatric acute leukemia\n\n5.6.1 Overview of study\nThis is from a clinical trial in 1963 for 6-MP treatment vs. placebo for Acute Leukemia in 42 children.\n\n\nPairs of children:\n\nmatched by remission status at the time of treatment (remstat: 1 = partial, 2 = complete)\nrandomized to 6-MP (exit times in t2) or placebo (exit times in t1)\n\n\nFollowed until relapse or end of study.\nAll of the placebo group relapsed, but some of the 6-MP group were censored (which means they were still in remission); indicated by relapse variable (0 = censored, 1 = relapse).\n6-MP = 6-Mercaptopurine (Purinethol) is an anti-cancer (“antineoplastic” or “cytotoxic”) chemotherapy drug used currently for Acute lymphoblastic leukemia (ALL). It is classified as an antimetabolite.\n\n5.6.2 Study design\n\nClinical trial in 1963 for 6-MP treatment vs. placebo for Acute Leukemia in 42 children.\nPairs of children:\n\nmatched by remission status at the time of treatment (remstat)\n\n\nremstat = 1: partial\n\nremstat = 2: complete\n\n\nrandomized to 6-MP (exit time: t2) or placebo (t1).\n\n\nFollowed until relapse or end of study.\n\nAll of the placebo group relapsed,\nSome of the 6-MP group were censored.\n\n\n\n\n\n\nTable 5.1: drug6mp pediatric acute leukemia data\n\nShow R codelibrary(KMsurv)\ndata(drug6mp)\ndrug6mp = drug6mp |&gt; as_tibble() |&gt; print()\n#&gt; # A tibble: 21 × 5\n#&gt;     pair remstat    t1    t2 relapse\n#&gt;    &lt;int&gt;   &lt;int&gt; &lt;int&gt; &lt;int&gt;   &lt;int&gt;\n#&gt;  1     1       1     1    10       1\n#&gt;  2     2       2    22     7       1\n#&gt;  3     3       2     3    32       0\n#&gt;  4     4       2    12    23       1\n#&gt;  5     5       2     8    22       1\n#&gt;  6     6       1    17     6       1\n#&gt;  7     7       2     2    16       1\n#&gt;  8     8       2    11    34       0\n#&gt;  9     9       2     8    32       0\n#&gt; 10    10       2    12    25       0\n#&gt; # ℹ 11 more rows\n\n\n\n\n\n5.6.3 Data documentation for drug6mp\n\n\nShow R code# library(printr) # inserts help-file output into markdown output\nlibrary(KMsurv)\n?drug6mp\n\n\n\n5.6.4 Descriptive Statistics\n\n\nTable 5.2: Summary statistics for drug6mp data\n\nShow R codesummary(drug6mp)\n#&gt;       pair       remstat           t1              t2          relapse     \n#&gt;  Min.   : 1   Min.   :1.00   Min.   : 1.00   Min.   : 6.0   Min.   :0.000  \n#&gt;  1st Qu.: 6   1st Qu.:2.00   1st Qu.: 4.00   1st Qu.: 9.0   1st Qu.:0.000  \n#&gt;  Median :11   Median :2.00   Median : 8.00   Median :16.0   Median :0.000  \n#&gt;  Mean   :11   Mean   :1.76   Mean   : 8.67   Mean   :17.1   Mean   :0.429  \n#&gt;  3rd Qu.:16   3rd Qu.:2.00   3rd Qu.:12.00   3rd Qu.:23.0   3rd Qu.:1.000  \n#&gt;  Max.   :21   Max.   :2.00   Max.   :23.00   Max.   :35.0   Max.   :1.000\n\n\n\n\n\n\nThe average time in each group is not useful. Some of the 6-MP patients have not relapsed at the time recorded, while all of the placebo patients have relapsed.\nThe median time is not really useful either because so many of the 6-MP patients have not relapsed (12/21).\nBoth are biased down in the 6-MP group. Remember that lower times are worse since they indicate sooner recurrence.\n\n\n\n5.6.5 Exponential model\n\n\nWe can compute the hazard rate, assuming an exponential model: number of relapses divided by the sum of the exit times (Equation 5.3).\n\n\n\\[\\hat\\lambda = \\frac{\\sum_{i=1}^nD_i}{\\sum_{i=1}^nY_i}\\]\n\n\nFor the placebo, that is just the reciprocal of the mean time:\n\n\n\\[\n\\begin{aligned}\n\\hat \\lambda_{\\text{placebo}}\n&= \\frac{\\sum_{i=1}^nD_i}{\\sum_{i=1}^nY_i}\n\\\\ &= \\frac{\\sum_{i=1}^n1}{\\sum_{i=1}^nY_i}\n\\\\ &= \\frac{n}{\\sum_{i=1}^nY_i}\n\\\\ &= \\frac{1}{\\bar{Y}}\n\\\\ &= \\frac{1}{8.6667}\n\\\\ &= 0.1154\n\\end{aligned}\n\\]\n\n\nFor the 6-MP group, \\(\\hat\\lambda = 9/359 = 0.025\\)\n\n\n\\[\n\\begin{aligned}\n\\hat \\lambda_{\\text{6-MP}}\n&= \\frac{\\sum_{i=1}^nD_i}{\\sum_{i=1}^nY_i}\n\\\\ &= \\frac{9}{359}\n\\\\ &= 0.0251\n\\end{aligned}\n\\]\n\nThe estimated hazard in the placebo group is 4.6 times as large as in the 6-MP group (assuming the hazard is constant over time).",
     "crumbs": [
       "Time to Event Models",
       "<span class='chapter-number'>5</span>  <span class='chapter-title'>Introduction to Survival Analysis</span>"
@@ -577,7 +577,7 @@
     "href": "intro-to-survival-analysis.html#the-kaplan-meier-product-limit-estimator",
     "title": "\n5  Introduction to Survival Analysis\n",
     "section": "\n5.7 The Kaplan-Meier Product Limit Estimator",
-    "text": "5.7 The Kaplan-Meier Product Limit Estimator\n\n5.7.1 Estimating survival in datasets without censoring\n\nIn the drug6mp dataset, the estimated survival function for the placebo patients is easy to compute. For any time \\(t\\) in months, \\(S(t)\\) is the fraction of patients with times greater than \\(t\\):\n\n\n5.7.2 Estimating survival in datasets with censoring\n\nFor the 6-MP patients, we cannot ignore the censored data because we know that the time to relapse is greater than the censoring time.\nFor any time \\(t\\) in months, we know that 6-MP patients with times greater than \\(t\\) have not relapsed, and those with relapse time less than \\(t\\) have relapsed, but we don’t know if patients with censored time less than \\(t\\) have relapsed or not.\nThe procedure we usually use is the Kaplan-Meier product-limit estimator of the survival function.\nThe Kaplan-Meier estimator is a step function (like the empirical cdf), which changes value only at the event times, not at the censoring times.\nAt each event time \\(t\\), we compute the at-risk group size \\(Y\\), which is all those observations whose event time or censoring time is at least \\(t\\).\nIf \\(d\\) of the observations have an event time (not a censoring time) of \\(t\\), then the group of survivors immediately following time \\(t\\) is reduced by the fraction \\[\\frac{Y-d}{Y}=1-\\frac{d}{Y}\\]\n\n\n\nDefinition 5.4 (Kaplan-Meier Product-Limit Estimator of Survival Function) If the event times are \\(t_i\\) with events per time of \\(d_i\\) (\\(1\\le i \\le k\\)), then the Kaplan-Meier Product-Limit Estimator of the survival function is:\n\\[\\hat S(t) = \\prod_{t_i &lt; t} \\left[\\frac{1-d_i}{Y_i}\\right] \\tag{5.3}\\]\nwhere \\(Y_i\\) is the set of observations whose time (event or censored) is \\(\\ge t_i\\), the group at risk at time \\(t_i\\).\n\n\nIf there are no censored data, and there are \\(n\\) data points, then just after (say) the third event time\n\\[\n\\begin{aligned}\n\\hat S(t)\n&= \\prod_{t_i &lt; t}\\left[1-\\frac{d_i}{Y_i}\\right]\n\\\\ &= \\left[\\frac{n-d_1}{n}\\right] \\left[\\frac{n-d_1-d_2}{n-d_1}\\right] \\left[\\frac{n-d_1-d_2-d_3}{n-d_1-d_2}\\right]\n\\\\ &= \\frac{n-d_1-d_2-d_3}{n}\n\\\\ &=1-\\frac{d_1+d_2+d_3}{n}\n\\\\ &=1-\\hat F(t)\n\\end{aligned}\n\\]\nwhere \\(\\hat F(t)\\) is the usual empirical CDF estimate.\n\n5.7.3 Kaplan-Meier curve for drug6mp data\nHere is the Kaplan-Meier estimated survival curve for the patients who received 6-MP in the drug6mp dataset (we will see code to produce figures like this one shortly):\n\nShow R code# | echo: false\n\nrequire(KMsurv)\ndata(drug6mp)\nlibrary(dplyr)\nlibrary(survival)\n\ndrug6mp_km_model1 = \n  drug6mp |&gt; \n  mutate(surv = Surv(t2, relapse))  |&gt; \n  survfit(formula = surv ~ 1, data = _)\n\nlibrary(ggfortify)\ndrug6mp_km_model1 |&gt;  \n  autoplot(\n    mark.time = TRUE,\n    conf.int = FALSE) +\n  expand_limits(y = 0) +\n  xlab('Time since diagnosis (months)') +\n  ylab(\"KM Survival Curve\")\n\n\n\nFigure 5.4: Kaplan-Meier Survival Curve for 6-MP Patients\n\n\n\n\n\n\n\n\n5.7.4 Kaplan-Meier calculations\nLet’s compute these estimates and build the chart by hand:\n\nShow R codelibrary(KMsurv)\nlibrary(dplyr)\ndata(drug6mp)\n\ndrug6mp.v2 = \n  drug6mp |&gt; \n  as_tibble() |&gt; \n  mutate(\n    remstat = remstat |&gt; \n      case_match(\n        1 ~ \"partial\",\n        2 ~ \"complete\"\n      ),\n    # renaming to \"outcome\" while relabeling is just a style choice:\n    outcome = relapse |&gt; \n      case_match(\n        0 ~ \"censored\",\n        1 ~ \"relapsed\"\n      )\n  )\n\nkm.6mp =\n  drug6mp.v2 |&gt; \n  summarize(\n    .by = t2,\n    Relapses = sum(outcome == \"relapsed\"),\n    Censored = sum(outcome == \"censored\")) |&gt;\n  # here we add a start time row, so the graph starts at time 0:\n  bind_rows(\n    tibble(\n      t2 = 0, \n      Relapses = 0, \n      Censored = 0)\n  ) |&gt; \n  # sort in time order:\n  arrange(t2) |&gt;\n  mutate(\n    Exiting = Relapses + Censored,\n    `Study Size` = sum(Exiting),\n    Exited = cumsum(Exiting) |&gt; dplyr::lag(default = 0),\n    `At Risk` = `Study Size` - Exited,\n    Hazard = Relapses / `At Risk`,\n    `KM Factor` = 1 - Hazard,\n    `Cumulative Hazard` = cumsum(`Hazard`),\n    `KM Survival Curve` = cumprod(`KM Factor`)\n  )\n\nlibrary(pander)  \npander(km.6mp)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nt2\nRelapses\nCensored\nExiting\nStudy Size\nExited\nAt Risk\nHazard\nKM Factor\nCumulative Hazard\nKM Survival Curve\n\n\n\n0\n0\n0\n0\n21\n0\n21\n0\n1\n0\n1\n\n\n6\n3\n1\n4\n21\n0\n21\n0.1429\n0.8571\n0.1429\n0.8571\n\n\n7\n1\n0\n1\n21\n4\n17\n0.05882\n0.9412\n0.2017\n0.8067\n\n\n9\n0\n1\n1\n21\n5\n16\n0\n1\n0.2017\n0.8067\n\n\n10\n1\n1\n2\n21\n6\n15\n0.06667\n0.9333\n0.2683\n0.7529\n\n\n11\n0\n1\n1\n21\n8\n13\n0\n1\n0.2683\n0.7529\n\n\n13\n1\n0\n1\n21\n9\n12\n0.08333\n0.9167\n0.3517\n0.6902\n\n\n16\n1\n0\n1\n21\n10\n11\n0.09091\n0.9091\n0.4426\n0.6275\n\n\n17\n0\n1\n1\n21\n11\n10\n0\n1\n0.4426\n0.6275\n\n\n19\n0\n1\n1\n21\n12\n9\n0\n1\n0.4426\n0.6275\n\n\n20\n0\n1\n1\n21\n13\n8\n0\n1\n0.4426\n0.6275\n\n\n22\n1\n0\n1\n21\n14\n7\n0.1429\n0.8571\n0.5854\n0.5378\n\n\n23\n1\n0\n1\n21\n15\n6\n0.1667\n0.8333\n0.7521\n0.4482\n\n\n25\n0\n1\n1\n21\n16\n5\n0\n1\n0.7521\n0.4482\n\n\n32\n0\n2\n2\n21\n17\n4\n0\n1\n0.7521\n0.4482\n\n\n34\n0\n1\n1\n21\n19\n2\n0\n1\n0.7521\n0.4482\n\n\n35\n0\n1\n1\n21\n20\n1\n0\n1\n0.7521\n0.4482\n\n\n\n\n\n\nSummary\nFor the 6-MP patients at time 6 months, there are 21 patients at risk. At \\(t=6\\) there are 3 relapses and 1 censored observations.\nThe Kaplan-Meier factor is \\((21-3)/21 = 0.857\\). The number at risk for the next time (\\(t=7\\)) is \\(21-3-1=17\\).\nAt time 7 months, there are 17 patients at risk. At \\(t=7\\) there is 1 relapse and 0 censored observations. The Kaplan-Meier factor is \\((17-1)/17 = 0.941\\). The Kaplan Meier estimate is \\(0.857\\times0.941=0.807\\). The number at risk for the next time (\\(t=9\\)) is \\(17-1=16\\).\n\nNow, let’s graph this estimated survival curve using ggplot():\n\nShow R codelibrary(ggplot2)\nconflicts_prefer(dplyr::filter)\nkm.6mp |&gt; \n  ggplot(aes(x = t2, y = `KM Survival Curve`)) +\n  geom_step() +\n  geom_point(data = km.6mp |&gt; filter(Censored &gt; 0), shape = 3) +\n  expand_limits(y = c(0,1), x = 0) +\n  xlab('Time since diagnosis (months)') +\n  ylab(\"KM Survival Curve\") +\n  scale_y_continuous(labels = scales::percent)\n\n\n\nFigure 5.5: KM curve for 6MP patients, calculated by hand",
+    "text": "5.7 The Kaplan-Meier Product Limit Estimator\n\n5.7.1 Estimating survival in datasets without censoring\n\nIn the drug6mp dataset, the estimated survival function for the placebo patients is easy to compute. For any time \\(t\\) in months, \\(S(t)\\) is the fraction of patients with times greater than \\(t\\):\n\n\n5.7.2 Estimating survival in datasets with censoring\n\nFor the 6-MP patients, we cannot ignore the censored data because we know that the time to relapse is greater than the censoring time.\nFor any time \\(t\\) in months, we know that 6-MP patients with times greater than \\(t\\) have not relapsed, and those with relapse time less than \\(t\\) have relapsed, but we don’t know if patients with censored time less than \\(t\\) have relapsed or not.\nThe procedure we usually use is the Kaplan-Meier product-limit estimator of the survival function.\nThe Kaplan-Meier estimator is a step function (like the empirical cdf), which changes value only at the event times, not at the censoring times.\nAt each event time \\(t\\), we compute the at-risk group size \\(Y\\), which is all those observations whose event time or censoring time is at least \\(t\\).\nIf \\(d\\) of the observations have an event time (not a censoring time) of \\(t\\), then the group of survivors immediately following time \\(t\\) is reduced by the fraction \\[\\frac{Y-d}{Y}=1-\\frac{d}{Y}\\]\n\n\n\nDefinition 5.4 (Kaplan-Meier Product-Limit Estimator of Survival Function) If the event times are \\(t_i\\) with events per time of \\(d_i\\) (\\(1\\le i \\le k\\)), then the Kaplan-Meier Product-Limit Estimator of the survival function is:\n\\[\\hat S(t) = \\prod_{t_i &lt; t} \\left[\\frac{1-d_i}{Y_i}\\right] \\tag{5.4}\\]\nwhere \\(Y_i\\) is the set of observations whose time (event or censored) is \\(\\ge t_i\\), the group at risk at time \\(t_i\\).\n\n\n\nTheorem 5.6 (Kaplan-Meier Estimate with No Censored Observations) If there are no censored data, and there are \\(n\\) data points, then just after (say) the third event time\n\\[\n\\begin{aligned}\n\\hat S(t)\n&= \\prod_{t_i &lt; t}\\left[1-\\frac{d_i}{Y_i}\\right]\n\\\\ &= \\left[\\frac{n-d_1}{n}\\right] \\left[\\frac{n-d_1-d_2}{n-d_1}\\right] \\left[\\frac{n-d_1-d_2-d_3}{n-d_1-d_2}\\right]\n\\\\ &= \\frac{n-d_1-d_2-d_3}{n}\n\\\\ &=1-\\frac{d_1+d_2+d_3}{n}\n\\\\ &=1-\\hat F(t)\n\\end{aligned}\n\\]\nwhere \\(\\hat F(t)\\) is the usual empirical CDF estimate.\n\n\n5.7.3 Kaplan-Meier curve for drug6mp data\nHere is the Kaplan-Meier estimated survival curve for the patients who received 6-MP in the drug6mp dataset (we will see code to produce figures like this one shortly):\n\nShow R code# | echo: false\n\nrequire(KMsurv)\ndata(drug6mp)\nlibrary(dplyr)\nlibrary(survival)\n\ndrug6mp_km_model1 = \n  drug6mp |&gt; \n  mutate(surv = Surv(t2, relapse))  |&gt; \n  survfit(formula = surv ~ 1, data = _)\n\nlibrary(ggfortify)\ndrug6mp_km_model1 |&gt;  \n  autoplot(\n    mark.time = TRUE,\n    conf.int = FALSE) +\n  expand_limits(y = 0) +\n  xlab('Time since diagnosis (months)') +\n  ylab(\"KM Survival Curve\")\n\n\n\nFigure 5.4: Kaplan-Meier Survival Curve for 6-MP Patients\n\n\n\n\n\n\n\n\n5.7.4 Kaplan-Meier calculations\nLet’s compute these estimates and build the chart by hand:\n\nShow R codelibrary(KMsurv)\nlibrary(dplyr)\ndata(drug6mp)\n\ndrug6mp.v2 = \n  drug6mp |&gt; \n  as_tibble() |&gt; \n  mutate(\n    remstat = remstat |&gt; \n      case_match(\n        1 ~ \"partial\",\n        2 ~ \"complete\"\n      ),\n    # renaming to \"outcome\" while relabeling is just a style choice:\n    outcome = relapse |&gt; \n      case_match(\n        0 ~ \"censored\",\n        1 ~ \"relapsed\"\n      )\n  )\n\nkm.6mp =\n  drug6mp.v2 |&gt; \n  summarize(\n    .by = t2,\n    Relapses = sum(outcome == \"relapsed\"),\n    Censored = sum(outcome == \"censored\")) |&gt;\n  # here we add a start time row, so the graph starts at time 0:\n  bind_rows(\n    tibble(\n      t2 = 0, \n      Relapses = 0, \n      Censored = 0)\n  ) |&gt; \n  # sort in time order:\n  arrange(t2) |&gt;\n  mutate(\n    Exiting = Relapses + Censored,\n    `Study Size` = sum(Exiting),\n    Exited = cumsum(Exiting) |&gt; dplyr::lag(default = 0),\n    `At Risk` = `Study Size` - Exited,\n    Hazard = Relapses / `At Risk`,\n    `KM Factor` = 1 - Hazard,\n    `Cumulative Hazard` = cumsum(`Hazard`),\n    `KM Survival Curve` = cumprod(`KM Factor`)\n  )\n\nlibrary(pander)  \npander(km.6mp)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nt2\nRelapses\nCensored\nExiting\nStudy Size\nExited\nAt Risk\nHazard\nKM Factor\nCumulative Hazard\nKM Survival Curve\n\n\n\n0\n0\n0\n0\n21\n0\n21\n0\n1\n0\n1\n\n\n6\n3\n1\n4\n21\n0\n21\n0.1429\n0.8571\n0.1429\n0.8571\n\n\n7\n1\n0\n1\n21\n4\n17\n0.05882\n0.9412\n0.2017\n0.8067\n\n\n9\n0\n1\n1\n21\n5\n16\n0\n1\n0.2017\n0.8067\n\n\n10\n1\n1\n2\n21\n6\n15\n0.06667\n0.9333\n0.2683\n0.7529\n\n\n11\n0\n1\n1\n21\n8\n13\n0\n1\n0.2683\n0.7529\n\n\n13\n1\n0\n1\n21\n9\n12\n0.08333\n0.9167\n0.3517\n0.6902\n\n\n16\n1\n0\n1\n21\n10\n11\n0.09091\n0.9091\n0.4426\n0.6275\n\n\n17\n0\n1\n1\n21\n11\n10\n0\n1\n0.4426\n0.6275\n\n\n19\n0\n1\n1\n21\n12\n9\n0\n1\n0.4426\n0.6275\n\n\n20\n0\n1\n1\n21\n13\n8\n0\n1\n0.4426\n0.6275\n\n\n22\n1\n0\n1\n21\n14\n7\n0.1429\n0.8571\n0.5854\n0.5378\n\n\n23\n1\n0\n1\n21\n15\n6\n0.1667\n0.8333\n0.7521\n0.4482\n\n\n25\n0\n1\n1\n21\n16\n5\n0\n1\n0.7521\n0.4482\n\n\n32\n0\n2\n2\n21\n17\n4\n0\n1\n0.7521\n0.4482\n\n\n34\n0\n1\n1\n21\n19\n2\n0\n1\n0.7521\n0.4482\n\n\n35\n0\n1\n1\n21\n20\n1\n0\n1\n0.7521\n0.4482\n\n\n\n\n\n\nSummary\nFor the 6-MP patients at time 6 months, there are 21 patients at risk. At \\(t=6\\) there are 3 relapses and 1 censored observations.\nThe Kaplan-Meier factor is \\((21-3)/21 = 0.857\\). The number at risk for the next time (\\(t=7\\)) is \\(21-3-1=17\\).\nAt time 7 months, there are 17 patients at risk. At \\(t=7\\) there is 1 relapse and 0 censored observations. The Kaplan-Meier factor is \\((17-1)/17 = 0.941\\). The Kaplan Meier estimate is \\(0.857\\times0.941=0.807\\). The number at risk for the next time (\\(t=9\\)) is \\(17-1=16\\).\n\nNow, let’s graph this estimated survival curve using ggplot():\n\nShow R codelibrary(ggplot2)\nconflicts_prefer(dplyr::filter)\nkm.6mp |&gt; \n  ggplot(aes(x = t2, y = `KM Survival Curve`)) +\n  geom_step() +\n  geom_point(data = km.6mp |&gt; filter(Censored &gt; 0), shape = 3) +\n  expand_limits(y = c(0,1), x = 0) +\n  xlab('Time since diagnosis (months)') +\n  ylab(\"KM Survival Curve\") +\n  scale_y_continuous(labels = scales::percent)\n\n\n\nFigure 5.5: KM curve for 6MP patients, calculated by hand",
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       "Time to Event Models",
       "<span class='chapter-number'>5</span>  <span class='chapter-title'>Introduction to Survival Analysis</span>"
@@ -599,7 +599,7 @@
     "href": "intro-to-survival-analysis.html#example-bone-marrow-transplant-data",
     "title": "\n5  Introduction to Survival Analysis\n",
     "section": "\n5.9 Example: Bone Marrow Transplant Data",
-    "text": "5.9 Example: Bone Marrow Transplant Data\nCopelan et al. (1991)\n\n\n\nFigure 5.6: Recovery process from a bone marrow transplant (Fig. 1.1 from Klein, Moeschberger, et al. (2003))\n\n\n\n\n\n\n\n5.9.1 Study design\nTreatment\n\n\nallogeneic (from a donor) bone marrow transplant therapy\n\nInclusion criteria\n\nacute myeloid leukemia (AML)\nacute lymphoblastic leukemia (ALL).\nPossible intermediate events\n\n\ngraft vs. host disease (GVHD): an immunological rejection response to the transplant\n\nplatelet recovery: a return of platelet count to normal levels.\n\nOne or the other, both in either order, or neither may occur.\nEnd point events\n\nrelapse of the disease\ndeath\n\nAny or all of these events may be censored.\n\n5.9.2 KMsurv::bmt data in R\n\nShow R codelibrary(KMsurv)\n?bmt\n\n\n\n5.9.3 Analysis plan\n\nWe concentrate for now on disease-free survival (t2 and d3) for the three risk groups, ALL, AML Low Risk, and AML High Risk.\nWe will construct the Kaplan-Meier survival curves, compare them, and test for differences.\nWe will construct the cumulative hazard curves and compare them.\nWe will estimate the hazard functions, interpret, and compare them.\n\n5.9.4 Survival Function Estimate and Variance\n\\[\\hat S(t) = \\prod_{t_i &lt; t}\\left[1-\\frac{d_i}{Y_i}\\right]\\] where \\(Y_i\\) is the group at risk at time \\(t_i\\).\nThe estimated variance of \\(\\hat S(t)\\) is:\n\nTheorem 5.4 (Greenwood’s estimator for variance of Kaplan-Meier survival estimator) \\[\n\\widehat{\\text{Var}}\\left(\\hat S(t)\\right) = \\hat S(t)^2\\sum_{t_i &lt;t}\\frac{d_i}{Y_i(Y_i-d_i)}\n\\tag{5.4}\\]\n\nWe can use Equation 5.4 for confidence intervals for a survival function or a difference of survival functions.\n\nKaplan-Meier survival curves\n\ncode to preprocess and model bmt datalibrary(KMsurv)\nlibrary(survival)\ndata(bmt)\n\nbmt = \n  bmt |&gt; \n  as_tibble() |&gt; \n  mutate(\n    group = \n      group |&gt; \n      factor(\n        labels = c(\"ALL\",\"Low Risk AML\",\"High Risk AML\")),\n    surv = Surv(t2,d3))\n\nkm_model1 = survfit(\n  formula = surv ~ group, \n  data = bmt)\n\n\n\nShow R codelibrary(ggfortify)\nautoplot(\n  km_model1, \n  conf.int = TRUE,\n  ylab = \"Pr(disease-free survival)\",\n  xlab = \"Time since transplant (days)\") + \n  theme_bw() +\n  theme(legend.position=\"bottom\")\n\n\nDisease-Free Survival by Disease Group\n\n\n\n\n\n5.9.5 Understanding Greenwood’s formula (optional)\n\nTo see where Greenwood’s formula comes from, let \\(x_i = Y_i - d_i\\). We approximate the solution treating each time as independent, with \\(Y_i\\) fixed and ignore randomness in times of failure and we treat \\(x_i\\) as independent binomials \\(\\text{Bin}(Y_i,p_i)\\). Letting \\(S(t)\\) be the “true” survival function\n\n\\[\n\\begin{aligned}\n\\hat S(t) &=\\prod_{t_i&lt;t}x_i/Y_i\\\\\nS(t)&=\\prod_{t_i&lt;t}p_i\n\\end{aligned}\n\\]\n\\[\n\\begin{aligned}\n\\frac{\\hat S(t)}{S(t)}\n   &= \\prod_{t_i&lt;t} \\frac{x_i}.    {p_iY_i}\n\\\\ &= \\prod_{t_i&lt;t} \\frac{\\hat p_i}{p_i}\n\\\\ &= \\prod_{t_i&lt;t} \\left(1+\\frac{\\hat p_i-p_i}{p_i}\\right)\n\\\\ &\\approx 1+\\sum_{t_i&lt;t} \\frac{\\hat p_i-p_i}{p_i}\n\\end{aligned}\n\\]\n\n\\[\n\\begin{aligned}\n\\text{Var}\\left(\\frac{\\hat S(t)}{S(t)}\\right)\n&\\approx \\text{Var}\\left(1+\\sum_{t_i&lt;t} \\frac{\\hat p_i-p_i}{p_i}\\right)\n\\\\ &=\\sum_{t_i&lt;t} \\frac{1}{p_i^2}\\frac{p_i(1-p_i)}{Y_i}\n\\\\ &= \\sum_{t_i&lt;t} \\frac{(1-p_i)}{p_iY_i}\n\\\\ &\\approx\\sum_{t_i&lt;t} \\frac{(1-x_i/Y_i)}{x_i}\n\\\\ &=\\sum_{t_i&lt;t} \\frac{Y_i-x_i}{x_iY_i}\n\\\\ &=\\sum_{t_i&lt;t} \\frac{d_i}{Y_i(Y_i-d_i)}\n\\\\ \\therefore\\text{Var}\\left(\\hat S(t)\\right)\n&\\approx \\hat S(t)^2\\sum_{t_i&lt;t} \\frac{d_i}{Y_i(Y_i-d_i)}\n\\end{aligned}\n\\]\n\n5.9.6 Test for differences among the disease groups\nHere we compute a chi-square test for assocation between disease group (group) and disease-free survival:\n\nShow R codesurvdiff(surv ~ group, data = bmt)\n#&gt; Call:\n#&gt; survdiff(formula = surv ~ group, data = bmt)\n#&gt; \n#&gt;                      N Observed Expected (O-E)^2/E (O-E)^2/V\n#&gt; group=ALL           38       24     21.9     0.211     0.289\n#&gt; group=Low Risk AML  54       25     40.0     5.604    11.012\n#&gt; group=High Risk AML 45       34     21.2     7.756    10.529\n#&gt; \n#&gt;  Chisq= 13.8  on 2 degrees of freedom, p= 0.001\n\n\n\n5.9.7 Cumulative Hazard\n\\[\n\\begin{aligned}\nh(t)\n&\\stackrel{\\text{def}}{=}P(T=t|T\\ge t)\\\\\n&= \\frac{p(T=t)}{P(T\\ge t)}\\\\\n&= -\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\}\n\\end{aligned}\n\\]\nThe cumulative hazard (or integrated hazard) function is\n\\[H(t)\\stackrel{\\text{def}}{=}\\int_0^t h(t) dt\\] Since \\(h(t) = -\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\}\\) as shown above, we have:\n\\[\nH(t)=-\\text{log}\\left\\{S\\right\\}(t)\n\\]\n\nSo we can estimate \\(H(t)\\) as:\n\\[\n\\begin{aligned}\n\\hat H(t)\n&= -\\text{log}\\left\\{\\hat S(t)\\right\\}\\\\\n&= -\\text{log}\\left\\{\\prod_{t_i &lt; t}\\left[1-\\frac{d_i}{Y_i}\\right]\\right\\}\\\\\n&= -\\sum_{t_i &lt; t}\\text{log}\\left\\{1-\\frac{d_i}{Y_i}\\right\\}\\\\\n\\end{aligned}\n\\]\nThis is the Kaplan-Meier (product-limit) estimate of cumulative hazard.\n\nExample: Cumulative Hazard Curves for Bone-Marrow Transplant (bmt) data\n\nShow R codeautoplot(\n  fun = \"cumhaz\",\n  km_model1, \n  conf.int = FALSE,\n  ylab = \"Cumulative hazard (disease-free survival)\",\n  xlab = \"Time since transplant (days)\") + \n  theme_bw() +\n  theme(legend.position=\"bottom\")\n\n\n\nFigure 5.7: Disease-Free Cumulative Hazard by Disease Group",
+    "text": "5.9 Example: Bone Marrow Transplant Data\nCopelan et al. (1991)\n\n\n\nFigure 5.6: Recovery process from a bone marrow transplant (Fig. 1.1 from Klein, Moeschberger, et al. (2003))\n\n\n\n\n\n\n\n5.9.1 Study design\nTreatment\n\n\nallogeneic (from a donor) bone marrow transplant therapy\n\nInclusion criteria\n\nacute myeloid leukemia (AML)\nacute lymphoblastic leukemia (ALL).\nPossible intermediate events\n\n\ngraft vs. host disease (GVHD): an immunological rejection response to the transplant\n\nplatelet recovery: a return of platelet count to normal levels.\n\nOne or the other, both in either order, or neither may occur.\nEnd point events\n\nrelapse of the disease\ndeath\n\nAny or all of these events may be censored.\n\n5.9.2 KMsurv::bmt data in R\n\nShow R codelibrary(KMsurv)\n?bmt\n\n\n\n5.9.3 Analysis plan\n\nWe concentrate for now on disease-free survival (t2 and d3) for the three risk groups, ALL, AML Low Risk, and AML High Risk.\nWe will construct the Kaplan-Meier survival curves, compare them, and test for differences.\nWe will construct the cumulative hazard curves and compare them.\nWe will estimate the hazard functions, interpret, and compare them.\n\n5.9.4 Survival Function Estimate and Variance\n\\[\\hat S(t) = \\prod_{t_i &lt; t}\\left[1-\\frac{d_i}{Y_i}\\right]\\] where \\(Y_i\\) is the group at risk at time \\(t_i\\).\nThe estimated variance of \\(\\hat S(t)\\) is:\n\nTheorem 5.7 (Greenwood’s estimator for variance of Kaplan-Meier survival estimator) \\[\n\\widehat{\\text{Var}}\\left(\\hat S(t)\\right) = \\hat S(t)^2\\sum_{t_i &lt;t}\\frac{d_i}{Y_i(Y_i-d_i)}\n\\tag{5.5}\\]\n\nWe can use Equation 5.5 for confidence intervals for a survival function or a difference of survival functions.\n\nKaplan-Meier survival curves\n\ncode to preprocess and model bmt datalibrary(KMsurv)\nlibrary(survival)\ndata(bmt)\n\nbmt = \n  bmt |&gt; \n  as_tibble() |&gt; \n  mutate(\n    group = \n      group |&gt; \n      factor(\n        labels = c(\"ALL\",\"Low Risk AML\",\"High Risk AML\")),\n    surv = Surv(t2,d3))\n\nkm_model1 = survfit(\n  formula = surv ~ group, \n  data = bmt)\n\n\n\nShow R codelibrary(ggfortify)\nautoplot(\n  km_model1, \n  conf.int = TRUE,\n  ylab = \"Pr(disease-free survival)\",\n  xlab = \"Time since transplant (days)\") + \n  theme_bw() +\n  theme(legend.position=\"bottom\")\n\n\nDisease-Free Survival by Disease Group\n\n\n\n\n\n5.9.5 Understanding Greenwood’s formula (optional)\n\nTo see where Greenwood’s formula comes from, let \\(x_i = Y_i - d_i\\). We approximate the solution treating each time as independent, with \\(Y_i\\) fixed and ignore randomness in times of failure and we treat \\(x_i\\) as independent binomials \\(\\text{Bin}(Y_i,p_i)\\). Letting \\(S(t)\\) be the “true” survival function\n\n\\[\n\\begin{aligned}\n\\hat S(t) &=\\prod_{t_i&lt;t}x_i/Y_i\\\\\nS(t)&=\\prod_{t_i&lt;t}p_i\n\\end{aligned}\n\\]\n\\[\n\\begin{aligned}\n\\frac{\\hat S(t)}{S(t)}\n   &= \\prod_{t_i&lt;t} \\frac{x_i}.    {p_iY_i}\n\\\\ &= \\prod_{t_i&lt;t} \\frac{\\hat p_i}{p_i}\n\\\\ &= \\prod_{t_i&lt;t} \\left(1+\\frac{\\hat p_i-p_i}{p_i}\\right)\n\\\\ &\\approx 1+\\sum_{t_i&lt;t} \\frac{\\hat p_i-p_i}{p_i}\n\\end{aligned}\n\\]\n\n\\[\n\\begin{aligned}\n\\text{Var}\\left(\\frac{\\hat S(t)}{S(t)}\\right)\n&\\approx \\text{Var}\\left(1+\\sum_{t_i&lt;t} \\frac{\\hat p_i-p_i}{p_i}\\right)\n\\\\ &=\\sum_{t_i&lt;t} \\frac{1}{p_i^2}\\frac{p_i(1-p_i)}{Y_i}\n\\\\ &= \\sum_{t_i&lt;t} \\frac{(1-p_i)}{p_iY_i}\n\\\\ &\\approx\\sum_{t_i&lt;t} \\frac{(1-x_i/Y_i)}{x_i}\n\\\\ &=\\sum_{t_i&lt;t} \\frac{Y_i-x_i}{x_iY_i}\n\\\\ &=\\sum_{t_i&lt;t} \\frac{d_i}{Y_i(Y_i-d_i)}\n\\\\ \\therefore\\text{Var}\\left(\\hat S(t)\\right)\n&\\approx \\hat S(t)^2\\sum_{t_i&lt;t} \\frac{d_i}{Y_i(Y_i-d_i)}\n\\end{aligned}\n\\]\n\n5.9.6 Test for differences among the disease groups\nHere we compute a chi-square test for assocation between disease group (group) and disease-free survival:\n\nShow R codesurvdiff(surv ~ group, data = bmt)\n#&gt; Call:\n#&gt; survdiff(formula = surv ~ group, data = bmt)\n#&gt; \n#&gt;                      N Observed Expected (O-E)^2/E (O-E)^2/V\n#&gt; group=ALL           38       24     21.9     0.211     0.289\n#&gt; group=Low Risk AML  54       25     40.0     5.604    11.012\n#&gt; group=High Risk AML 45       34     21.2     7.756    10.529\n#&gt; \n#&gt;  Chisq= 13.8  on 2 degrees of freedom, p= 0.001\n\n\n\n5.9.7 Cumulative Hazard\n\\[\n\\begin{aligned}\nh(t)\n&\\stackrel{\\text{def}}{=}P(T=t|T\\ge t)\\\\\n&= \\frac{p(T=t)}{P(T\\ge t)}\\\\\n&= -\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\}\n\\end{aligned}\n\\]\nThe cumulative hazard (or integrated hazard) function is\n\\[H(t)\\stackrel{\\text{def}}{=}\\int_0^t h(t) dt\\] Since \\(h(t) = -\\frac{\\partial}{\\partial t}\\text{log}\\left\\{S(t)\\right\\}\\) as shown above, we have:\n\\[\nH(t)=-\\text{log}\\left\\{S\\right\\}(t)\n\\]\n\nSo we can estimate \\(H(t)\\) as:\n\\[\n\\begin{aligned}\n\\hat H(t)\n&= -\\text{log}\\left\\{\\hat S(t)\\right\\}\\\\\n&= -\\text{log}\\left\\{\\prod_{t_i &lt; t}\\left[1-\\frac{d_i}{Y_i}\\right]\\right\\}\\\\\n&= -\\sum_{t_i &lt; t}\\text{log}\\left\\{1-\\frac{d_i}{Y_i}\\right\\}\\\\\n\\end{aligned}\n\\]\nThis is the Kaplan-Meier (product-limit) estimate of cumulative hazard.\n\nExample: Cumulative Hazard Curves for Bone-Marrow Transplant (bmt) data\n\nShow R codeautoplot(\n  fun = \"cumhaz\",\n  km_model1, \n  conf.int = FALSE,\n  ylab = \"Cumulative hazard (disease-free survival)\",\n  xlab = \"Time since transplant (days)\") + \n  theme_bw() +\n  theme(legend.position=\"bottom\")\n\n\n\nFigure 5.7: Disease-Free Cumulative Hazard by Disease Group",
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     "href": "intro-to-survival-analysis.html#nelson-aalen-estimates-of-cumulative-hazard-and-survival",
     "title": "\n5  Introduction to Survival Analysis\n",
     "section": "\n5.10 Nelson-Aalen Estimates of Cumulative Hazard and Survival",
-    "text": "5.10 Nelson-Aalen Estimates of Cumulative Hazard and Survival\n\n\nDefinition 5.5 (Nelson-Aalen Cumulative Hazard Estimator)  \n\nThe point hazard at time \\(t_i\\) can be estimated by \\(d_i/Y_i\\), which leads to the Nelson-Aalen estimator of the cumulative hazard:\n\n\\[\\hat H_{NA}(t) \\stackrel{\\text{def}}{=}\\sum_{t_i &lt; t}\\frac{d_i}{Y_i} \\tag{5.5}\\]\n\n\n\nTheorem 5.5 (Variance of Nelson-Aalen estimator)  \n\nThe variance of this estimator is approximately:\n\n\\[\n\\begin{aligned}\n\\hat{\\text{Var}}\\left(\\hat H_{NA} (t)\\right)\n&= \\sum_{t_i &lt;t}\\frac{(d_i/Y_i)(1-d_i/Y_i)}{Y_i}\\\\\n&\\approx \\sum_{t_i &lt;t}\\frac{d_i}{Y_i^2}\n\\end{aligned}\n\\tag{5.6}\\]\n\n\nSince \\(S(t)=\\text{exp}\\left\\{-H(t)\\right\\}\\), the Nelson-Aalen cumulative hazard estimate can be converted into an alternate estimate of the survival function:\n\\[\n\\begin{aligned}\n\\hat S_{NA}(t)\n&= \\text{exp}\\left\\{-\\hat H_{NA}(t)\\right\\}\\\\\n&= \\text{exp}\\left\\{-\\sum_{t_i &lt; t}\\frac{d_i}{Y_i}\\right\\}\\\\\n&= \\prod_{t_i &lt; t}\\text{exp}\\left\\{-\\frac{d_i}{Y_i}\\right\\}\\\\\n\\end{aligned}\n\\]\n\nCompare these with the corresponding Kaplan-Meier estimates:\n\\[\n\\begin{aligned}\n\\hat H_{KM}(t) &= -\\sum_{t_i &lt; t}\\text{log}\\left\\{1-\\frac{d_i}{Y_i}\\right\\}\\\\\n\\hat S_{KM}(t) &= \\prod_{t_i &lt; t}\\left[1-\\frac{d_i}{Y_i}\\right]\n\\end{aligned}\n\\]\n\nThe product limit estimate and the Nelson-Aalen estimate often do not differ by much. The latter is considered more accurate in small samples and also directly estimates the cumulative hazard. The \"fleming-harrington\" method for survfit() reduces to Nelson-Aalen when the data are unweighted. We can also estimate the cumulative hazard as the negative log of the KM survival function estimate.\n\n\n5.10.1 Application to bmt dataset\n\nShow R code\nna_fit = survfit(\n  formula = surv ~ group,\n  type = \"fleming-harrington\",\n  data = bmt)\n\nkm_fit = survfit(\n  formula = surv ~ group,\n  type = \"kaplan-meier\",\n  data = bmt)\n\nkm_and_na = \n  bind_rows(\n    .id = \"model\",\n    \"Kaplan-Meier\" = km_fit |&gt; fortify(surv.connect = TRUE),\n    \"Nelson-Aalen\" = na_fit |&gt; fortify(surv.connect = TRUE)\n  ) |&gt; \n  as_tibble()\n\n\n\nShow R codekm_and_na |&gt; \n  ggplot(aes(x = time, y = surv, col = model)) +\n  geom_step() +\n  facet_grid(. ~ strata) +\n  theme_bw() + \n  ylab(\"S(t) = P(T&gt;=t)\") +\n  xlab(\"Survival time (t, days)\") +\n  theme(legend.position = \"bottom\")\n\n\nKaplan-Meier and Nelson-Aalen Survival Function Estimates, stratified by disease group\n\n\n\n\nThe Kaplan-Meier and Nelson-Aalen survival estimates are very similar for this dataset.\n\n\n\n\n\n\nCopelan, Edward A, James C Biggs, James M Thompson, Pamela Crilley, Jeff Szer, John P Klein, Neena Kapoor, Belinda R Avalos, Isabel Cunningham, and Kerry Atkinson. 1991. “Treatment for Acute Myelocytic Leukemia with Allogeneic Bone Marrow Transplantation Following Preparation with BuCy2.” https://doi.org/10.1182/blood.V78.3.838.838 .\n\n\nKlein, John P, Melvin L Moeschberger, et al. 2003. Survival Analysis: Techniques for Censored and Truncated Data. Vol. 1230. Springer. https://link.springer.com/book/10.1007/b97377.",
+    "text": "5.10 Nelson-Aalen Estimates of Cumulative Hazard and Survival\n\n\nDefinition 5.5 (Nelson-Aalen Cumulative Hazard Estimator)  \n\nThe point hazard at time \\(t_i\\) can be estimated by \\(d_i/Y_i\\), which leads to the Nelson-Aalen estimator of the cumulative hazard:\n\n\\[\\hat H_{NA}(t) \\stackrel{\\text{def}}{=}\\sum_{t_i &lt; t}\\frac{d_i}{Y_i} \\tag{5.6}\\]\n\n\n\nTheorem 5.8 (Variance of Nelson-Aalen estimator)  \n\nThe variance of this estimator is approximately:\n\n\\[\n\\begin{aligned}\n\\hat{\\text{Var}}\\left(\\hat H_{NA} (t)\\right)\n&= \\sum_{t_i &lt;t}\\frac{(d_i/Y_i)(1-d_i/Y_i)}{Y_i}\\\\\n&\\approx \\sum_{t_i &lt;t}\\frac{d_i}{Y_i^2}\n\\end{aligned}\n\\tag{5.7}\\]\n\n\nSince \\(S(t)=\\text{exp}\\left\\{-H(t)\\right\\}\\), the Nelson-Aalen cumulative hazard estimate can be converted into an alternate estimate of the survival function:\n\\[\n\\begin{aligned}\n\\hat S_{NA}(t)\n&= \\text{exp}\\left\\{-\\hat H_{NA}(t)\\right\\}\\\\\n&= \\text{exp}\\left\\{-\\sum_{t_i &lt; t}\\frac{d_i}{Y_i}\\right\\}\\\\\n&= \\prod_{t_i &lt; t}\\text{exp}\\left\\{-\\frac{d_i}{Y_i}\\right\\}\\\\\n\\end{aligned}\n\\]\n\nCompare these with the corresponding Kaplan-Meier estimates:\n\\[\n\\begin{aligned}\n\\hat H_{KM}(t) &= -\\sum_{t_i &lt; t}\\text{log}\\left\\{1-\\frac{d_i}{Y_i}\\right\\}\\\\\n\\hat S_{KM}(t) &= \\prod_{t_i &lt; t}\\left[1-\\frac{d_i}{Y_i}\\right]\n\\end{aligned}\n\\]\n\nThe product limit estimate and the Nelson-Aalen estimate often do not differ by much. The latter is considered more accurate in small samples and also directly estimates the cumulative hazard. The \"fleming-harrington\" method for survfit() reduces to Nelson-Aalen when the data are unweighted. We can also estimate the cumulative hazard as the negative log of the KM survival function estimate.\n\n\n5.10.1 Application to bmt dataset\n\nShow R code\nna_fit = survfit(\n  formula = surv ~ group,\n  type = \"fleming-harrington\",\n  data = bmt)\n\nkm_fit = survfit(\n  formula = surv ~ group,\n  type = \"kaplan-meier\",\n  data = bmt)\n\nkm_and_na = \n  bind_rows(\n    .id = \"model\",\n    \"Kaplan-Meier\" = km_fit |&gt; fortify(surv.connect = TRUE),\n    \"Nelson-Aalen\" = na_fit |&gt; fortify(surv.connect = TRUE)\n  ) |&gt; \n  as_tibble()\n\n\n\nShow R codekm_and_na |&gt; \n  ggplot(aes(x = time, y = surv, col = model)) +\n  geom_step() +\n  facet_grid(. ~ strata) +\n  theme_bw() + \n  ylab(\"S(t) = P(T&gt;=t)\") +\n  xlab(\"Survival time (t, days)\") +\n  theme(legend.position = \"bottom\")\n\n\nKaplan-Meier and Nelson-Aalen Survival Function Estimates, stratified by disease group\n\n\n\n\nThe Kaplan-Meier and Nelson-Aalen survival estimates are very similar for this dataset.\n\n\n\n\n\n\nCopelan, Edward A, James C Biggs, James M Thompson, Pamela Crilley, Jeff Szer, John P Klein, Neena Kapoor, Belinda R Avalos, Isabel Cunningham, and Kerry Atkinson. 1991. “Treatment for Acute Myelocytic Leukemia with Allogeneic Bone Marrow Transplantation Following Preparation with BuCy2.” https://doi.org/10.1182/blood.V78.3.838.838 .\n\n\nKlein, John P, Melvin L Moeschberger, et al. 2003. Survival Analysis: Techniques for Censored and Truncated Data. Vol. 1230. Springer. https://link.springer.com/book/10.1007/b97377.",
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       "Time to Event Models",
       "<span class='chapter-number'>5</span>  <span class='chapter-title'>Introduction to Survival Analysis</span>"