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Linear-models-overview.html

@@ -465,7 +465,7 @@
     <div>
     <div class="quarto-title-meta-heading">Published</div>
     <div class="quarto-title-meta-contents">
-      <p class="date">Last modified: 2024-05-16: 13:44:16 (PM)</p>
+      <p class="date">Last modified: 2024-05-18: 7:52:50 (AM)</p>
     </div>
   </div>
 
@@ -906,7 +906,7 @@
 </figcaption><div aria-describedby="tbl-coef-model1-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
 <details class="code-fold"><summary>Show R code</summary><div class="sourceCode" id="cb8"><pre class="downlit sourceCode r code-with-copy"><code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/coef.html">coef</a></span><span class="op">(</span><span class="va">bw_lm1</span><span class="op">)</span></span>
 <span><span class="co">#&gt; (Intercept)     sexmale         age </span></span>
-<span><span class="co">#&gt; -1773.32184   163.03930   120.89433</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<span><span class="co">#&gt;     -1773.3       163.0       120.9</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </details>
 </div>
 </figure>
@@ -929,7 +929,7 @@
 \begin{aligned}
 E[Y|A = 0, A = 36]
 &amp;= \beta_0 + \beta_M \cdot 0+ \beta_A \cdot 36 \\
-&amp;=  2578.873934
+&amp;=  2578.8739
 \end{aligned}
 \]</span></p>
 </div>
@@ -960,7 +960,7 @@
 \begin{aligned}
 E[Y|M = 1, A = 36]
 &amp;= \beta_0 + \beta_M \cdot 1+ \beta_A \cdot 36 \\
-&amp;=  2741.91323693
+&amp;=  2741.9132
 \end{aligned}
 \]</span></p>
 </div>
@@ -980,9 +980,9 @@
 \begin{aligned}
 &amp; E[Y|M = 1, A = 36] - E[Y|M = 0, A = 36]\\
 &amp;=
-2741.91323693 - 2578.873934\\
+2741.9132 - 2578.8739\\
 &amp;=
-163.03930293
+163.0393
 \end{aligned}
 \]</span></p>
 <p>Shortcut:</p>
@@ -992,7 +992,7 @@
 &amp;= (\beta_0 + \beta_M \cdot 1+ \beta_A \cdot 36) -
 (\beta_0 + \beta_M \cdot 0+ \beta_A \cdot 36) \\
 &amp;= \beta_M \\
-&amp;=  163.03930293
+&amp;=  163.0393
 \end{aligned}
 \]</span></p>
 </div>
@@ -1131,7 +1131,7 @@
 <p><span class="math display">\[
 E[Y|A = 0, X_2 = 36] =
 \beta_0 + \beta_M \cdot 0+ \beta_A \cdot 36 + \beta_{AM} \cdot (0 * 36)
-=  2552.73333333
+=  2552.7333
 \]</span></p>
 </div>
 <hr>
@@ -1153,7 +1153,7 @@
 \begin{aligned}
 E[Y|A = 0, X_2 = 36]
 &amp;= \beta_0 + \beta_M \cdot 1+ \beta_A \cdot 36 + \beta_{AM} \cdot 1 \cdot 36\\
-&amp;= 2762.70689655
+&amp;= 2762.7069
 \end{aligned}
 \]</span></p>
 </div>
@@ -1169,7 +1169,7 @@
 &amp;= (\beta_0 + \beta_M \cdot 1+ \beta_A \cdot 36 + \beta_{AM} \cdot 1 \cdot 36)\\
 &amp;\ \ \ \ \  -(\beta_0 + \beta_M \cdot 0+ \beta_A \cdot 36 + \beta_{AM} \cdot 0 \cdot 36) \\
 &amp;= \beta_{S} + \beta_{AM}\cdot 36\\
-&amp;=  209.97356322
+&amp;=  209.9736
 \end{aligned}
 \]</span></p>
 </div>
@@ -1602,7 +1602,7 @@ <h3 class="anchored">Note</h3>
 <p>Research question: is there really an interaction between sex and age?</p>
 <p><span class="math inline">\(H_0: \beta_{AM} = 0\)</span></p>
 <p><span class="math inline">\(H_A: \beta_{AM} \neq 0\)</span></p>
-<p><span class="math inline">\(P(|\hat\beta_{AM}| &gt; |-18.41724138| \mid H_0)\)</span> = ?</p>
+<p><span class="math inline">\(P(|\hat\beta_{AM}| &gt; |-18.4172| \mid H_0)\)</span> = ?</p>
 </section><section id="wald-tests-and-cis" class="level3 smaller" data-number="2.4.2"><h3 class="smaller anchored" data-number="2.4.2" data-anchor-id="wald-tests-and-cis">
 <span class="header-section-number">2.4.2</span> Wald tests and CIs</h3>
 <p>R can give you Wald tests for single coefficients and corresponding CIs:</p>
@@ -1726,17 +1726,17 @@ <h3 class="anchored">Note</h3>
 \begin{aligned}
 &amp;P\left(
 | \hat \beta_{AM}  | &gt;
-| -18.41724138| \middle| H_0
+| -18.4172| \middle| H_0
 \right)
 \\
 &amp;= \Pr \left(
 \left| \frac{\hat\beta_{AM}}{\hat{SE}(\hat\beta_{AM})} \right| &gt;
-\left| \frac{-18.41724138}{41.75581745} \right| \middle| H_0
+\left| \frac{-18.4172}{41.7558} \right| \middle| H_0
 \right)\\
 &amp;= \Pr \left(
-\left| T_{20} \right|  &gt;  0.44107007 | H_0
+\left| T_{20} \right|  &gt;  0.4411 | H_0
 \right)\\
-&amp;= 0.66389342
+&amp;= 0.6639
 \end{aligned}
 \]</span></p>
 <div class="notes">
@@ -2329,12 +2329,12 @@ <h3 class="anchored">Note</h3>
 </table>
 </div>
 </div>
-<p>The likelihood ratio statistic for this test is <span class="math display">\[\lambda = 2 * (\ell_{\text{full}} - \ell) = 10.35537421\]</span> where:</p>
+<p>The likelihood ratio statistic for this test is <span class="math display">\[\lambda = 2 * (\ell_{\text{full}} - \ell) = 10.3554\]</span> where:</p>
 <ul>
 <li>
-<span class="math inline">\(\ell_{\text{max}}\)</span> is the log-likelihood of the full model: -151.40160056</li>
+<span class="math inline">\(\ell_{\text{max}}\)</span> is the log-likelihood of the full model: -151.4016</li>
 <li>
-<span class="math inline">\(\ell\)</span> is the log-likelihood of our comparison model (two slopes, two intercepts): -156.57928767</li>
+<span class="math inline">\(\ell\)</span> is the log-likelihood of our comparison model (two slopes, two intercepts): -156.5793</li>
 </ul>
 <p>This statistic is called the <strong>deviance</strong> or <strong>residual deviance</strong> for our two-slopes and two-intercepts model; it tells us how much the likelihood of that model deviates from the likelihood of the maximal model.</p>
 <p>The corresponding p-value tells us whether there we have enough evidence to detect that our two-slopes, two-intercepts model is a worse fit for the data than the maximal model; in other words, it tells us if there’s evidence that we missed any important patterns. (Remember, a nonsignificant p-value could mean that we didn’t miss anything and a more complicated model is unnecessary, or it could mean we just don’t have enough data to tell the difference between these models.)</p>
@@ -2413,12 +2413,12 @@ <h3 class="anchored">Note</h3>
 </table>
 </div>
 </div>
-<p>The likelihood ratio statistic for the test comparing the null model to the maximal model is <span class="math display">\[\lambda = 2 * (\ell_{\text{full}} - \ell_{0}) = 35.10673188\]</span> where:</p>
+<p>The likelihood ratio statistic for the test comparing the null model to the maximal model is <span class="math display">\[\lambda = 2 * (\ell_{\text{full}} - \ell_{0}) = 35.1067\]</span> where:</p>
 <ul>
 <li>
-<span class="math inline">\(\ell_{\text{0}}\)</span> is the log-likelihood of the null model: -168.9549665</li>
+<span class="math inline">\(\ell_{\text{0}}\)</span> is the log-likelihood of the null model: -168.955</li>
 <li>
-<span class="math inline">\(\ell_{\text{full}}\)</span> is the log-likelihood of the maximal model: -151.40160056</li>
+<span class="math inline">\(\ell_{\text{full}}\)</span> is the log-likelihood of the maximal model: -151.4016</li>
 </ul>
 <p>In R, this test is:</p>
 <div class="cell">
@@ -3926,15 +3926,15 @@ <h3 class="anchored">Note</h3>
 <tbody>
 <tr class="odd">
 <td style="text-align: left;">full</td>
-<td style="text-align: right;">6.946</td>
+<td style="text-align: right;">6.958</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>
 <td style="text-align: right;">-61.84</td>
 </tr>
 <tr class="even">
 <td style="text-align: left;">reduced</td>
-<td style="text-align: right;">6.627</td>
+<td style="text-align: right;">6.778</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>
@@ -4071,10 +4071,10 @@ <h3 class="anchored">Note</h3>
 <details class="code-fold"><summary>Show R code</summary><div class="sourceCode" id="cb105"><pre class="downlit sourceCode r code-with-copy"><code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/coef.html">coef</a></span><span class="op">(</span><span class="va">cvfit</span>, s <span class="op">=</span> <span class="st">"lambda.1se"</span><span class="op">)</span></span>
 <span><span class="co">#&gt; 4 x 1 sparse Matrix of class "dgCMatrix"</span></span>
 <span><span class="co">#&gt;                  s1</span></span>
-<span><span class="co">#&gt; (Intercept) 34.4241</span></span>
+<span><span class="co">#&gt; (Intercept) 34.1090</span></span>
 <span><span class="co">#&gt; age          .     </span></span>
-<span><span class="co">#&gt; weight      -0.0662</span></span>
-<span><span class="co">#&gt; protein      0.6607</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<span><span class="co">#&gt; weight      -0.1041</span></span>
+<span><span class="co">#&gt; protein      0.9441</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </details>
 </div>
 </section></section></section><section id="categorical-covariates-with-more-than-two-levels" class="level2" data-number="2.10"><h2 data-number="2.10" class="anchored" data-anchor-id="categorical-covariates-with-more-than-two-levels">

coxph-model-building.html

@@ -872,8 +872,8 @@
 <p>The nature of time-to-event data results in very wide prediction intervals:</p>
 <ul>
 <li>Suppose a cancer patient is predicted to have a mean lifetime of 5 years after diagnosis and suppose the distribution is exponential.</li>
-<li>If we want a 95% interval for survival, the lower end is at the 0.025 percentage point of the exponential which is <code>qexp(.025, rate = 1/5)</code> = 0.12658904 years, or 1/40 of the mean lifetime.</li>
-<li>The upper end is at the 0.975 point which is <code>qexp(.975, rate = 1/5)</code> = 18.44439727 years, or 3.7 times the mean lifetime.</li>
+<li>If we want a 95% interval for survival, the lower end is at the 0.025 percentage point of the exponential which is <code>qexp(.025, rate = 1/5)</code> = 0.1266 years, or 1/40 of the mean lifetime.</li>
+<li>The upper end is at the 0.975 point which is <code>qexp(.975, rate = 1/5)</code> = 18.4444 years, or 3.7 times the mean lifetime.</li>
 <li>Saying that the survival time is somewhere between 6 weeks and 18 years does not seem very useful, but it may be the best we can do.</li>
 <li>For survival analysis, something is like a residual if it is small when the model is accurate or if the accumulation of them is in some way minimized by the estimation algorithm, but there is no exact equivalence to linear regression residuals.</li>
 <li>And if there is, they are mostly quite large!</li>

index.html

@@ -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: 2024-05-16: 19:54:50 (PM)</p>
+      <p class="date">Last modified: 2024-05-21: 12:59:54 (PM)</p>
     </div>
   </div>
 

intro-MLEs.html

@@ -1071,10 +1071,10 @@
 <div class="proof solution">
 <p><span class="proof-title"><em>Solution</em>. </span><span class="math display">\[
 \begin{aligned}
-\mathcal{L}(\lambda; \tilde x)
-&amp;= P(\tilde X = \tilde x) \\
-&amp;= P(X_1 = x_1, X_2 = x_2, ..., X_{13} = x_{13}) \\
-&amp;= \prod_{i=1}^{13} P(X_i = x_i) \\
+\mathcal{L}(\lambda; \tilde{x})
+&amp;= \text{P}(\tilde{X}= \tilde{x}) \\
+&amp;= \text{P}(X_1 = x_1, X_2 = x_2, ..., X_{13} = x_{13}) \\
+&amp;= \prod_{i=1}^{13} \text{P}(X_i = x_i) \\
 &amp;= \prod_{i=1}^{13} \frac{\lambda^{x_i} e^{-\lambda}}{x_i!}
 \end{aligned}
 \]</span></p>
@@ -1299,7 +1299,7 @@
 <details class="code-fold"><summary>Show R code</summary><div class="sourceCode" id="cb11"><pre class="downlit sourceCode r code-with-copy"><code class="sourceCode R"><span><span class="va">mle</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/mean.html">mean</a></span><span class="op">(</span><span class="va">cyclones</span><span class="op">$</span><span class="va">number</span><span class="op">)</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </details>
 </div>
-<p><span class="math display">\[\hat{\lambda}_{\text{ML}} = \bar x = 5.53846154\]</span></p>
+<p><span class="math display">\[\hat{\lambda}_{\text{ML}} = \bar x = 5.5385\]</span></p>
 </div>
 </div>
 <div id="exr-graph-mle" class="theorem exercise">
@@ -2082,7 +2082,7 @@
 </details><div class="cell-output-display">
 <div data-pagedtable="false">
   <script data-pagedtable-source="" type="application/json">
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