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index.html

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 <meta name="author" content="Phillip Alday">
 <meta name="author" content="Reinhold Kliegl">
 <meta name="author" content="Douglas Bates">
-<meta name="dcterms.date" content="2024-06-28">
+<meta name="dcterms.date" content="2024-06-30">
 
 <title>Embrace Uncertainty</title>
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@@ -252,7 +252,7 @@ <h1 class="title">Embrace Uncertainty</h1>
     <div>
     <div class="quarto-title-meta-heading">Published</div>
     <div class="quarto-title-meta-contents">
-      <p class="date">June 28, 2024</p>
+      <p class="date">June 30, 2024</p>
     </div>
   </div>
 
@@ -270,8 +270,8 @@ <h1 class="unnumbered">Preface</h1>
 <p>The practical aspects of fitting such models to experimental or observational data and in evaluating such models are introduced through the <a href="https://julialang.org">Julia programming language</a> and its <a href="https://github.com/JuliaStats/MixedModels.jl">MixedModels</a> package.</p>
 <p>This book is intended for researchers in applications areas, such as Psychology or Linguistics, and for researchers in Statistical methods. Some of the technical discussions, especially in the appendices, may be rather daunting for applications-area researchers, who should feel free to skim those sections. The purpose of those technical discussions is to establish a firm mathematical and theoretical basis for the computational methods.</p>
 <p>This book has been produced with Quarto. To learn more about Quarto books visit <a href="https://quarto.org/docs/books">https://quarto.org/docs/books</a>.</p>
-<p><em>This page was rendered from git revision <a href="https://github.com/JuliaMixedModels/EmbraceUncertainty/tree/31e9b61
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intro.html

@@ -1281,21 +1281,21 @@ <h2 data-number="1.5" class="anchored" data-anchor-id="sec-furtherassess"><span
 <span id="cb44-4"><a href="#cb44-4" aria-hidden="true" tabindex="-1"></a><span class="fu">first</span>(dsm01pars, <span class="fl">7</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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 <pre><code>┌ Warning: NLopt was roundoff limited
-└ @ MixedModels ~/.julia/packages/MixedModels/peUlR/src/optsummary.jl:160
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-└ @ MixedModels ~/.julia/packages/MixedModels/peUlR/src/optsummary.jl:160
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-└ @ MixedModels ~/.julia/packages/MixedModels/peUlR/src/optsummary.jl:160
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-└ @ MixedModels ~/.julia/packages/MixedModels/peUlR/src/optsummary.jl:160
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-└ @ MixedModels ~/.julia/packages/MixedModels/peUlR/src/optsummary.jl:160</code></pre>
+└ @ MixedModels ~/.julia/packages/MixedModels/gKejD/src/optsummary.jl:160</code></pre>
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-<p>The distribution of the estimates of <code>β₁</code> is more-or-less a Gaussian (or “normal”) shape, with a mean value of 1528.02 which is close to the estimated <code>β₁</code> of 1527.5.</p>
-<p>Similarly the standard deviation of the simulated β values, 17.7114 is close to the standard error of the parameter, 17.6946.</p>
+<p>The distribution of the estimates of <code>β₁</code> is more-or-less a Gaussian (or “normal”) shape, with a mean value of “1528.02” which is close to the estimated <code>β₁</code> of “1527.5”.</p>
+<p>Similarly the standard deviation of the simulated β values, “17.7114” is close to the standard error of the parameter, “17.6946”.</p>
 <p>In other words, the estimator of the fixed-effects parameter in this case behaves as we would expect. The estimates are approximately normally distributed centered about the “true” parameter value with a standard deviation given by the standard error of the parameter.</p>
 <p>The situation is different for the estimates of the standard deviation parameters, <span class="math inline">\(\sigma\)</span> and <span class="math inline">\(\sigma_1\)</span>, as shown in <a href="#fig-dsm01_bs_sigma_density" class="quarto-xref">Figure&nbsp;<span>1.4</span></a>.</p>
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-<p>Here the algorithm converges after 18 function evaluations to a profiled deviance of 327.327 at θ = 0.752581.</p>
+<p>Here the algorithm converges after 18 function evaluations to a profiled deviance of “327.327” at θ = “0.752581”.</p>
 </section>
 </section>
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 <p>The maximum likelihood estimates of the parameters are obtained by minimizing the deviance. For linear mixed models we can minimize the profiled deviance, which is a function of <span class="math inline">\({\boldsymbol{\theta}}\)</span> only, thereby considerably simplifying the optimization problem.</p>
 <p>To assess the precision of the parameter estimates we use a parametric bootstrap sample, when feasible. Re-fitting a simple model, such as those shown in this chapter, to a randomly generated response vector is quite fast and it is reasonable to work with bootstrap samples of tens of thousands of replicates. With larger data sets and more complex models, large bootstrap samples of parameter estimates could take much longer.</p>
 <p>Prediction intervals from the conditional distribution of the random effects, given the observed data, allow us to assess the precision of the random effects.</p>
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