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-<p>The definition of <code>MixedModels.GHnorm</code> is similar to the <code>gausshermitenorm</code> function with some extra provisions for ensuring symmetry of the abscissae and the weights and for caching values once they have been calculated.</p><article class="docstring"><header><a class="docstring-binding" id="MixedModels.GHnorm" href="#MixedModels.GHnorm"><code>MixedModels.GHnorm</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">GHnorm(k::Int)</code></pre><p>Return the (unique) GaussHermiteNormalized{k} object.</p><p>The function values are stored (memoized) when first evaluated.  Subsequent evaluations for the same <code>k</code> have very low overhead.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaStats/MixedModels.jl/blob/7e951b94e30fe6b527b3f456844da8adc9b3f9c0/src/gausshermite.jl#L72-L79">source</a></section></article><pre><code class="language-julia hljs">using MixedModels
+<p>The definition of <code>MixedModels.GHnorm</code> is similar to the <code>gausshermitenorm</code> function with some extra provisions for ensuring symmetry of the abscissae and the weights and for caching values once they have been calculated.</p><article class="docstring"><header><a class="docstring-binding" id="MixedModels.GHnorm" href="#MixedModels.GHnorm"><code>MixedModels.GHnorm</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">GHnorm(k::Int)</code></pre><p>Return the (unique) GaussHermiteNormalized{k} object.</p><p>The function values are stored (memoized) when first evaluated.  Subsequent evaluations for the same <code>k</code> have very low overhead.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaStats/MixedModels.jl/blob/08a0164b88672795eedd6e09089b455b999a9fab/src/gausshermite.jl#L72-L79">source</a></section></article><pre><code class="language-julia hljs">using MixedModels
 GHnorm(3)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">MixedModels.GaussHermiteNormalized{3}([-1.7320508075688772, 0.0, 1.7320508075688772], [0.16666666666666666, 0.6666666666666666, 0.16666666666666666])</code></pre><p>By the properties of the normal distribution, when <span>$\mathcal{X}\sim\mathscr{N}(\mu, \sigma^2)$</span></p><p class="math-container">\[\mathbb{E}[g(x)] \approx \sum_{i=1}^k g(\mu + \sigma z_i)\,w_i\]</p><p>For example, <span>$\mathbb{E}[\mathcal{X}^2]$</span> where <span>$\mathcal{X}\sim\mathcal{N}(2, 3^2)$</span> is</p><pre><code class="language-julia hljs">μ = 2; σ = 3; ghn3 = GHnorm(3);
 sum(@. ghn3.w * abs2(μ + σ * ghn3.z))  # should be μ² + σ² = 13</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">13.0</code></pre><p>(In general a dot, &#39;<code>.</code>&#39;, after the function name in a function call, as in <code>abs2.(...)</code>, or before an operator creates a <a href="https://docs.julialang.org/en/stable/manual/performance-tips/#More-dots:-Fuse-vectorized-operations-1"><em>fused vectorized</em></a> evaluation in Julia. The macro <code>@.</code> has the effect of vectorizing all operations in the subsequent expression.)</p><h2 id="Application-to-a-model-for-contraception-use"><a class="docs-heading-anchor" href="#Application-to-a-model-for-contraception-use">Application to a model for contraception use</a><a id="Application-to-a-model-for-contraception-use-1"></a><a class="docs-heading-anchor-permalink" href="#Application-to-a-model-for-contraception-use" title="Permalink"></a></h2><p>A <em>binary response</em> is a &quot;Yes&quot;/&quot;No&quot; type of answer. For example, in a 1989 fertility survey of women in Bangladesh (reported in <a href="https://www.popline.org/node/371841">Huq, N. M. and Cleland, J., 1990</a>) one response of interest was whether the woman used artificial contraception. Several covariates were recorded including the woman&#39;s age (centered at the mean), the number of live children the woman has had (in 4 categories: 0, 1, 2, and 3 or more), whether she lived in an urban setting, and the district in which she lived. The version of the data used here is that used in review of multilevel modeling software conducted by the Center for Multilevel Modelling, currently at University of Bristol (http://www.bristol.ac.uk/cmm/learning/mmsoftware/data-rev.html). These data are available as the <code>:contra</code> dataset.</p><pre><code class="language-julia hljs">contra = DataFrame(MixedModels.dataset(:contra))
 describe(contra)</code></pre><div><div style = "float: left;"><span>5×7 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">variable</th><th style = "text-align: left;">mean</th><th style = "text-align: left;">min</th><th style = "text-align: left;">median</th><th style = "text-align: left;">max</th><th style = "text-align: left;">nmissing</th><th style = "text-align: left;">eltype</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Symbol" style = "text-align: left;">Symbol</th><th title = "Union{Nothing, Float64}" style = "text-align: left;">Union…</th><th title = "Any" style = "text-align: left;">Any</th><th title = "Union{Nothing, Float64}" style = "text-align: left;">Union…</th><th title = "Any" style = "text-align: left;">Any</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "DataType" style = "text-align: left;">DataType</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">dist</td><td style = "font-style: italic; text-align: left;"></td><td style = "text-align: left;">D01</td><td style = "font-style: italic; text-align: left;"></td><td style = "text-align: left;">D61</td><td style = "text-align: right;">0</td><td style = "text-align: left;">String</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">urban</td><td style = "font-style: italic; text-align: left;"></td><td style = "text-align: left;">N</td><td style = "font-style: italic; text-align: left;"></td><td style = "text-align: left;">Y</td><td style = "text-align: right;">0</td><td style = "text-align: left;">String</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">livch</td><td style = "font-style: italic; text-align: left;"></td><td style = "text-align: left;">0</td><td style = "font-style: italic; text-align: left;"></td><td style = "text-align: left;">3+</td><td style = "text-align: right;">0</td><td style = "text-align: left;">String</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">age</td><td style = "text-align: left;">0.00204757</td><td style = "text-align: left;">-13.56</td><td style = "text-align: left;">-1.56</td><td style = "text-align: left;">19.44</td><td style = "text-align: right;">0</td><td style = "text-align: left;">Float64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">use</td><td style = "font-style: italic; text-align: left;"></td><td style = "text-align: left;">N</td><td style = "font-style: italic; text-align: left;"></td><td style = "text-align: left;">Y</td><td style = "text-align: right;">0</td><td style = "text-align: left;">String</td></tr></tbody></table></div><p>A smoothed scatterplot of contraception use versus age</p><pre><code class="language-julia hljs">plot(contra, x=:age, y=:use, Geom.smooth, Guide.xlabel(&quot;Centered age (yr)&quot;),
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-<section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>https://en.wikipedia.org/wiki/Gaussian_quadrature</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../optimization/">« Details of the parameter estimation</a><a class="docs-footer-nextpage" href="../prediction/">Prediction and simulation in Mixed-Effects Models »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Friday 4 August 2023 19:09">Friday 4 August 2023</span>. Using Julia version 1.8.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+<section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>https://en.wikipedia.org/wiki/Gaussian_quadrature</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../optimization/">« Details of the parameter estimation</a><a class="docs-footer-nextpage" href="../prediction/">Prediction and simulation in Mixed-Effects Models »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Friday 4 August 2023 22:26">Friday 4 August 2023</span>. Using Julia version 1.8.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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-  LLVM: libLLVM-6.0.0 (ORCJIT, ivybridge)</code></pre></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Friday 4 August 2023 19:09">Friday 4 August 2023</span>. Using Julia version 1.8.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+  LLVM: libLLVM-6.0.0 (ORCJIT, ivybridge)</code></pre></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Friday 4 August 2023 22:26">Friday 4 August 2023</span>. Using Julia version 1.8.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>