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‎.nojekyll

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\]</span></p>
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<span class="math inline">\(\hat y_i\)</span> estimates the conditional mean of <span class="math inline">\(y_i\)</span> given the covariate values <span class="math inline">\(\tilde x_i\)</span>. This together with the prediction error says that we are predicting the distribution of values of <span class="math inline">\(y\)</span>.</li>
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<span class="math inline">\(\hat y_i\)</span> estimates the conditional mean of <span class="math inline">\(y_i\)</span> given the covariate values <span class="math inline">\(\tilde{x}_i\)</span>. This together with the prediction error says that we are predicting the distribution of values of <span class="math inline">\(y\)</span>.</li>
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</ul></section><section id="review-residuals-in-linear-regression" class="level3" data-number="7.3.2"><h3 data-number="7.3.2" class="anchored" data-anchor-id="review-residuals-in-linear-regression">
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<p>should have an exponential distribution with constant hazard <span class="math inline">\(\lambda=1\)</span> if the estimate <span class="math inline">\(\hat S_i\)</span> is accurate, which means that these values should look like a censored sample from this exponential distribution. These values are called <strong>generalized residuals</strong> or <strong>Cox-Snell residuals</strong>.</p>
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<span id="cb69-412"><a href="#cb69-412" aria-hidden="true" tabindex="-1"></a><span class="fu">### Review: Residuals in Linear Regression</span></span>
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</section><section id="three-parts-to-regression-models" class="level3" data-number="1.2.10"><h3 data-number="1.2.10" class="anchored" data-anchor-id="three-parts-to-regression-models">
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<li><p>What distribution does the outcome have for a specific sub-population defined by covariates? (<strong>outcome model</strong>)</p></li>
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<li><p>How do the covariates combine? (<strong>linear predictor/linear component</strong>) <span class="math display">\[\eta \stackrel{\text{def}}{=}\tilde{x}^{\top} \tilde{\beta}= \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ...\]</span></p></li>
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<span id="cb10-277"><a href="#cb10-277" aria-hidden="true" tabindex="-1"></a><span class="ss">- </span>What distribution does the outcome have for a specific sub-population defined by covariates? (**outcome model**)</span>
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<span id="cb10-282"><a href="#cb10-282" aria-hidden="true" tabindex="-1"></a>$$\eta \eqdef \vx \' \vb = \b_0 + \b_1 x_1 + \b_2 x_2 + ...$$</span>
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