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<body>

<h1 id="in-depth-1-comparison-of-point-estimates">In-Depth 1: Comparison of Point-Estimates</h1>
<ul>
<li><a href="#effect-point-estimates-in-the-bayesian-framework">Effect Point-Estimates in the Bayesian Framework</a>
<ul>
<li><a href="#introduction">Introduction</a></li>
<li><a href="#experiment-1-relationship-with-error-noise-and-sample-size">Experiment 1: Relationship with Error (Noise) and Sample Size</a>
<ul>
<li><a href="#methods">Methods</a></li>
<li><a href="#results">Results</a></li>
</ul></li>
<li><a href="#experiment-2-relationship-with-sampling-characteristics">Experiment 2: Relationship with Sampling Characteristics</a>
<ul>
<li><a href="#methods-1">Methods</a></li>
<li><a href="#results-1">Results</a></li>
</ul></li>
<li><a href="#experiment-3-relationship-with-priors-specification">Experiment 3: Relationship with Priors Specification</a></li>
<li><a href="#discussion">Discussion</a></li>
</ul></li>
</ul>
<p>This vignette can be referred to by citing the package:</p>
<ul>
<li>Makowski, D., Ben-Shachar M. S. &amp; Lüdecke, D. (2019). <em>Understand and Describe Bayesian Models and Posterior Distributions using bayestestR</em>. Available from <a href="https://github.com/easystats/bayestestR">https://github.com/easystats/bayestestR</a>. DOI: <a href="https://zenodo.org/record/2556486">10.5281/zenodo.2556486</a>.</li>
</ul>
<hr />
<h1 id="effect-point-estimates-in-the-bayesian-framework">Effect Point-Estimates in the Bayesian Framework</h1>
<h2 id="introduction">Introduction</h2>
<p>One of the main difference between the Bayesian and the frequentist frameworks is that the former returns a probability distribution of each effect (<em>i.e.</em>, parameter of interest of a model, such as a regression slope) instead of a single value. However, there is still a need and demand, for reporting or use in further analysis, for a single value (<strong>point-estimate</strong>) that best characterise the underlying posterior distribution.</p>
<p>There are three main indices used in the literature for effect estimation: the <strong>mean</strong>, the <strong>median</strong> or the <strong>MAP</strong> (Maximum A Posteriori) estimate (roughly corresponding to the mode (the “peak”) of the distribution). Unfortunately, there is no consensus about which one to use, as no systematic comparison has ever been done.</p>
<p>In the present work, we will compare these three point-estimates of effect between themselves, as well as with the widely known <strong>beta</strong>, extracted from a comparable frequentist model. With this comparison, we expect to draw bridges and relationships between the two frameworks, helping and easing the transition for the public.</p>
<h2 id="experiment-1-relationship-with-error-noise-and-sample-size">Experiment 1: Relationship with Error (Noise) and Sample Size</h2>
<h3 id="methods">Methods</h3>
<p>The simulation aimed at modulating the following characteristics:</p>
<ul>
<li><strong>Model type</strong>: linear or logistic.</li>
<li><strong>“True” effect</strong> (original regression coefficient from which data is drawn): Can be 1 or 0 (no effect).</li>
<li><strong>Sample size</strong>: From 20 to 100 by steps of 10.</li>
<li><strong>Error</strong>: Gaussian noise applied to the predictor with SD uniformly spread between 0.33 and 6.66 (with 1000 different values).</li>
</ul>
<p>We generated a dataset for each combination of these characteristics, resulting in a total of <code>2 * 2 * 9 * 1000 = 36000</code> Bayesian and frequentist models. The code used for generation is avaible <a href="https://easystats.github.io/circus/articles/bayesian_indices.html">here</a> (please note that it takes usually several days/weeks to complete).</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb1-1" title="1"><span class="kw">library</span>(ggplot2)</a>
<a class="sourceLine" id="cb1-2" title="2"><span class="kw">library</span>(dplyr)</a>
<a class="sourceLine" id="cb1-3" title="3"><span class="kw">library</span>(tidyr)</a>
<a class="sourceLine" id="cb1-4" title="4"><span class="kw">library</span>(see)</a>
<a class="sourceLine" id="cb1-5" title="5"></a>
<a class="sourceLine" id="cb1-6" title="6">df &lt;-<span class="st"> </span><span class="kw">read.csv</span>(<span class="st">&quot;https://raw.github.com/easystats/circus/master/data/bayesSim_study1.csv&quot;</span>)</a></code></pre></div>
<h3 id="results">Results</h3>
<h4 id="sensitivity-to-noise">Sensitivity to Noise</h4>
<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb2-1" title="1">df <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb2-2" title="2"><span class="st">  </span><span class="kw">select</span>(error, true_effect, outcome_type, Coefficient, Median, Mean, MAP) <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb2-3" title="3"><span class="st">  </span><span class="kw">gather</span>(estimate, value, <span class="op">-</span>error, <span class="op">-</span>true_effect, <span class="op">-</span>outcome_type) <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb2-4" title="4"><span class="st">  </span><span class="kw">mutate</span>(<span class="dt">temp =</span> <span class="kw">as.factor</span>(<span class="kw">cut</span>(error, <span class="dv">10</span>, <span class="dt">labels =</span> <span class="ot">FALSE</span>))) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb2-5" title="5"><span class="st">  </span><span class="kw">group_by</span>(temp) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb2-6" title="6"><span class="st">  </span><span class="kw">mutate</span>(<span class="dt">error_group =</span> <span class="kw">round</span>(<span class="kw">mean</span>(error), <span class="dv">1</span>)) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb2-7" title="7"><span class="st">  </span><span class="kw">ungroup</span>() <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb2-8" title="8"><span class="st">  </span><span class="kw">filter</span>(value <span class="op">&lt;</span><span class="st"> </span><span class="dv">6</span>) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb2-9" title="9"><span class="st">  </span><span class="kw">ggplot</span>(<span class="kw">aes</span>(<span class="dt">x =</span> error_group, <span class="dt">y =</span> value, <span class="dt">fill =</span> estimate, <span class="dt">group =</span> <span class="kw">interaction</span>(estimate, error_group))) <span class="op">+</span></a>
<a class="sourceLine" id="cb2-10" title="10"><span class="st">  </span><span class="co"># geom_hline(yintercept = 0) +</span></a>
<a class="sourceLine" id="cb2-11" title="11"><span class="st">  </span><span class="co"># geom_point(alpha=0.05, size=2, stroke = 0, shape=16) +</span></a>
<a class="sourceLine" id="cb2-12" title="12"><span class="st">  </span><span class="co"># geom_smooth(method=&quot;loess&quot;) +</span></a>
<a class="sourceLine" id="cb2-13" title="13"><span class="st">  </span><span class="kw">geom_boxplot</span>(<span class="dt">outlier.shape=</span><span class="ot">NA</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb2-14" title="14"><span class="st">  </span><span class="kw">theme_modern</span>() <span class="op">+</span></a>
<a class="sourceLine" id="cb2-15" title="15"><span class="st">  </span><span class="kw">scale_fill_manual</span>(<span class="dt">values =</span> <span class="kw">c</span>(<span class="st">&quot;Coefficient&quot;</span> =<span class="st"> &quot;#607D8B&quot;</span>, <span class="st">&quot;MAP&quot;</span> =<span class="st"> &quot;#795548&quot;</span>, <span class="st">&quot;Mean&quot;</span> =<span class="st"> &quot;#FF9800&quot;</span>, <span class="st">&quot;Median&quot;</span> =<span class="st"> &quot;#FFEB3B&quot;</span>),</a>
<a class="sourceLine" id="cb2-16" title="16">                    <span class="dt">name =</span> <span class="st">&quot;Index&quot;</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb2-17" title="17"><span class="st">  </span><span class="kw">ylab</span>(<span class="st">&quot;Point-estimate&quot;</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb2-18" title="18"><span class="st">  </span><span class="kw">xlab</span>(<span class="st">&quot;Noise&quot;</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb2-19" title="19"><span class="st">  </span><span class="kw">facet_wrap</span>(<span class="op">~</span><span class="st"> </span>outcome_type <span class="op">*</span><span class="st"> </span>true_effect, <span class="dt">scales=</span><span class="st">&quot;free&quot;</span>) </a></code></pre></div>
<p><img 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" /><!-- --></p>
<h4 id="sensitivity-to-sample-size">Sensitivity to Sample Size</h4>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb3-1" title="1">df <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb3-2" title="2"><span class="st">  </span><span class="kw">select</span>(sample_size, true_effect, outcome_type, Coefficient, Median, Mean, MAP) <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb3-3" title="3"><span class="st">  </span><span class="kw">gather</span>(estimate, value, <span class="op">-</span>sample_size, <span class="op">-</span>true_effect, <span class="op">-</span>outcome_type) <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb3-4" title="4"><span class="st">  </span><span class="kw">mutate</span>(<span class="dt">temp =</span> <span class="kw">as.factor</span>(<span class="kw">cut</span>(sample_size, <span class="dv">10</span>, <span class="dt">labels =</span> <span class="ot">FALSE</span>))) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb3-5" title="5"><span class="st">  </span><span class="kw">group_by</span>(temp) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb3-6" title="6"><span class="st">  </span><span class="kw">mutate</span>(<span class="dt">size_group =</span> <span class="kw">round</span>(<span class="kw">mean</span>(sample_size))) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb3-7" title="7"><span class="st">  </span><span class="kw">ungroup</span>() <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb3-8" title="8"><span class="st">  </span><span class="kw">filter</span>(value <span class="op">&lt;</span><span class="st"> </span><span class="dv">6</span>) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb3-9" title="9"><span class="st">  </span><span class="kw">ggplot</span>(<span class="kw">aes</span>(<span class="dt">x =</span> size_group, <span class="dt">y =</span> value, <span class="dt">fill =</span> estimate, <span class="dt">group =</span> <span class="kw">interaction</span>(estimate, size_group))) <span class="op">+</span></a>
<a class="sourceLine" id="cb3-10" title="10"><span class="st">  </span><span class="co"># geom_hline(yintercept = 0) +</span></a>
<a class="sourceLine" id="cb3-11" title="11"><span class="st">  </span><span class="co"># geom_point(alpha=0.05, size=2, stroke = 0, shape=16) +</span></a>
<a class="sourceLine" id="cb3-12" title="12"><span class="st">  </span><span class="co"># geom_smooth(method=&quot;loess&quot;) +</span></a>
<a class="sourceLine" id="cb3-13" title="13"><span class="st">  </span><span class="kw">geom_boxplot</span>(<span class="dt">outlier.shape=</span><span class="ot">NA</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb3-14" title="14"><span class="st">  </span><span class="kw">theme_modern</span>() <span class="op">+</span></a>
<a class="sourceLine" id="cb3-15" title="15"><span class="st">  </span><span class="kw">scale_fill_manual</span>(<span class="dt">values =</span> <span class="kw">c</span>(<span class="st">&quot;Coefficient&quot;</span> =<span class="st"> &quot;#607D8B&quot;</span>, <span class="st">&quot;MAP&quot;</span> =<span class="st"> &quot;#795548&quot;</span>, <span class="st">&quot;Mean&quot;</span> =<span class="st"> &quot;#FF9800&quot;</span>, <span class="st">&quot;Median&quot;</span> =<span class="st"> &quot;#FFEB3B&quot;</span>),</a>
<a class="sourceLine" id="cb3-16" title="16">                    <span class="dt">name =</span> <span class="st">&quot;Index&quot;</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb3-17" title="17"><span class="st">  </span><span class="kw">ylab</span>(<span class="st">&quot;Point-estimate&quot;</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb3-18" title="18"><span class="st">  </span><span class="kw">xlab</span>(<span class="st">&quot;Sample size&quot;</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb3-19" title="19"><span class="st">  </span><span class="kw">facet_wrap</span>(<span class="op">~</span><span class="st"> </span>outcome_type <span class="op">*</span><span class="st"> </span>true_effect, <span class="dt">scales=</span><span class="st">&quot;free&quot;</span>)</a></code></pre></div>
<p><img 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" /><!-- --></p>
<h4 id="statistical-modelling">Statistical Modelling</h4>
<p>We fitted a (frequentist) multiple linear regression to statistically test the the predict the presence or absence of effect with the estimates as well as their interaction with noise and sample size.</p>
<div class="sourceCode" id="cb4"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb4-1" title="1">df <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb4-2" title="2"><span class="st">  </span><span class="kw">select</span>(sample_size, error, true_effect, outcome_type, Coefficient, Median, Mean, MAP) <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb4-3" title="3"><span class="st">  </span><span class="kw">gather</span>(estimate, value, <span class="op">-</span>sample_size, <span class="op">-</span>error, <span class="op">-</span>true_effect, <span class="op">-</span>outcome_type) <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb4-4" title="4"><span class="st">  </span><span class="kw">glm</span>(true_effect <span class="op">~</span><span class="st"> </span>outcome_type <span class="op">/</span><span class="st"> </span>estimate <span class="op">/</span><span class="st"> </span>value, <span class="dt">data=</span>., <span class="dt">family=</span><span class="st">&quot;binomial&quot;</span>) <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb4-5" title="5"><span class="st">  </span>broom<span class="op">::</span><span class="kw">tidy</span>() <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb4-6" title="6"><span class="st">  </span><span class="kw">select</span>(term, estimate, <span class="dt">p=</span>p.value) <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb4-7" title="7"><span class="st">  </span><span class="kw">filter</span>(stringr<span class="op">::</span><span class="kw">str_detect</span>(term, <span class="st">&#39;outcome_type&#39;</span>),</a>
<a class="sourceLine" id="cb4-8" title="8">         stringr<span class="op">::</span><span class="kw">str_detect</span>(term, <span class="st">&#39;:value&#39;</span>)) <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb4-9" title="9"><span class="st">  </span><span class="kw">arrange</span>(<span class="kw">desc</span>(estimate)) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb4-10" title="10"><span class="st">  </span>knitr<span class="op">::</span><span class="kw">kable</span>(<span class="dt">digits=</span><span class="dv">2</span>) </a></code></pre></div>
<table>
<thead>
<tr class="header">
<th align="left">term</th>
<th align="right">estimate</th>
<th align="right">p</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left">outcome_typelinear:estimateMean:value</td>
<td align="right">10.8</td>
<td align="right">0</td>
</tr>
<tr class="even">
<td align="left">outcome_typelinear:estimateMedian:value</td>
<td align="right">10.8</td>
<td align="right">0</td>
</tr>
<tr class="odd">
<td align="left">outcome_typelinear:estimateMAP:value</td>
<td align="right">10.7</td>
<td align="right">0</td>
</tr>
<tr class="even">
<td align="left">outcome_typelinear:estimateCoefficient:value</td>
<td align="right">10.5</td>
<td align="right">0</td>
</tr>
<tr class="odd">
<td align="left">outcome_typebinary:estimateMAP:value</td>
<td align="right">4.4</td>
<td align="right">0</td>
</tr>
<tr class="even">
<td align="left">outcome_typebinary:estimateMedian:value</td>
<td align="right">4.3</td>
<td align="right">0</td>
</tr>
<tr class="odd">
<td align="left">outcome_typebinary:estimateMean:value</td>
<td align="right">4.2</td>
<td align="right">0</td>
</tr>
<tr class="even">
<td align="left">outcome_typebinary:estimateCoefficient:value</td>
<td align="right">3.9</td>
<td align="right">0</td>
</tr>
</tbody>
</table>
<p>This suggests that, in order to delineate between the presence and the absence of an effect, compared to the frequentist’s beta:</p>
<ul>
<li>For linear models, the <strong>Mean</strong> was the better predictor, closely followed by the <strong>Median</strong>, the <strong>MAP</strong> and the frequentist <strong>Coefficient</strong>.</li>
<li>For logistic models, the <strong>MAP</strong> was the better predictor, followed by the <strong>Median</strong>, the <strong>Mean</strong> and, behind, the frequentist <strong>Coefficient</strong>.</li>
</ul>
<p>Overall, the <strong>median</strong> seems to be appears as a safe and approriate choice, maintaining a a high performance accross different types of models.</p>
<!-- ```{r, message=FALSE, warning=FALSE} -->

<!-- df %>% -->

<!--   select(sample_size, error, true_effect, outcome_type, beta, Median, Mean, MAP) %>% -->

<!--   gather(estimate, value, -sample_size, -error, -true_effect, -outcome_type) %>% -->

<!--   glm(true_effect ~ outcome_type / value * estimate * sample_size * error, data=., family="binomial") %>% -->

<!--   broom::tidy() %>% -->

<!--   select(term, estimate, p=p.value) %>% -->

<!--   filter(stringr::str_detect(term, 'outcome_type'), -->

<!--          stringr::str_detect(term, ':value')) %>% -->

<!--   mutate( -->

<!--     sample_size = stringr::str_detect(term, 'sample_size'), -->

<!--     error = stringr::str_detect(term, 'error'), -->

<!--     term = stringr::str_remove(term, "estimate"), -->

<!--     term = stringr::str_remove(term, "outcome_type"), -->

<!--     p = paste0(sprintf("%.2f", p), ifelse(p < .001, "***", ifelse(p < .01, "**", ifelse(p < .05, "*", ""))))) %>% -->

<!--   arrange(sample_size, error, term) %>%  -->

<!--   select(-sample_size, -error) %>%  -->

<!--   knitr::kable(digits=2)  -->

<!-- ``` -->

<!-- This suggests that, in order to delineate between the presence and the absence of an effect, compared to the frequentist's beta: -->

<!-- - For linear Models; -->

<!--   - The **mean**, followed closely by the **median**, and the **MAP** estimate had a superior performance, altough not significantly. -->

<!--   - The **mean**, followed closely by the **median**, and the **MAP** estimate, were less affected by noise, altough not significantly. -->

<!--   - No difference for the sensitivity to sample size was found. -->

<!-- - For logistic models: -->

<!--   - The **MAP** estimate, followed by the **median** and the **mean**, estimate had a superior performance. -->

<!--   - The **MAP** estimate, followed by the **median**, and the **mean**, were less affected by noise, altough not significantly. -->

<!--   - The **MAP** estimate, followed by the **mean**, and the **median**, were less affected by sample size, altough not significantly. -->

<h2 id="experiment-2-relationship-with-sampling-characteristics">Experiment 2: Relationship with Sampling Characteristics</h2>
<h3 id="methods-1">Methods</h3>
<p>The simulation aimed at modulating the following characteristics:</p>
<ul>
<li><strong>Model type</strong>: linear or logistic.</li>
<li><strong>“True” effect</strong> (original regression coefficient from which data is drawn): Can be 1 or 0 (no effect).</li>
<li><strong>draws</strong>: from 10 to 5000 by step of 5 (1000 iterations).</li>
<li><strong>warmup</strong>: Ratio of warmup iterations. from 1/10 to 9/10 by step of 0.1 (9 iterations).</li>
</ul>
<p>We generated 3 datasets for each combination of these characteristics, resulting in a total of <code>2 * 2 * 8 * 40 * 9 * 3 = 34560</code> Bayesian and frequentist models. The code used for generation is avaible <a href="https://easystats.github.io/circus/articles/bayesian_indices.html">here</a> (please note that it takes usually several days/weeks to complete).</p>
<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb5-1" title="1">df &lt;-<span class="st"> </span><span class="kw">read.csv</span>(<span class="st">&quot;https://raw.github.com/easystats/circus/master/data/bayesSim_study2.csv&quot;</span>)</a></code></pre></div>
<h3 id="results-1">Results</h3>
<h4 id="sensitivity-to-number-of-iterations">Sensitivity to number of iterations</h4>
<div class="sourceCode" id="cb6"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb6-1" title="1">df <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb6-2" title="2"><span class="st">  </span><span class="kw">select</span>(iterations, true_effect, outcome_type, beta, Median, Mean, MAP) <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb6-3" title="3"><span class="st">  </span><span class="kw">gather</span>(estimate, value, <span class="op">-</span>iterations, <span class="op">-</span>true_effect, <span class="op">-</span>outcome_type) <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb6-4" title="4"><span class="st">  </span><span class="kw">mutate</span>(<span class="dt">temp =</span> <span class="kw">as.factor</span>(<span class="kw">cut</span>(iterations, <span class="dv">5</span>, <span class="dt">labels =</span> <span class="ot">FALSE</span>))) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb6-5" title="5"><span class="st">  </span><span class="kw">group_by</span>(temp) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb6-6" title="6"><span class="st">  </span><span class="kw">mutate</span>(<span class="dt">iterations_group =</span> <span class="kw">round</span>(<span class="kw">mean</span>(iterations), <span class="dv">1</span>)) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb6-7" title="7"><span class="st">  </span><span class="kw">ungroup</span>() <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb6-8" title="8"><span class="st">  </span><span class="kw">filter</span>(value <span class="op">&lt;</span><span class="st"> </span><span class="dv">6</span>) <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb6-9" title="9"><span class="st">  </span><span class="kw">ggplot</span>(<span class="kw">aes</span>(<span class="dt">x =</span> iterations_group, <span class="dt">y =</span> value, <span class="dt">fill =</span> estimate, <span class="dt">group =</span> <span class="kw">interaction</span>(estimate, iterations_group))) <span class="op">+</span></a>
<a class="sourceLine" id="cb6-10" title="10"><span class="st">  </span><span class="kw">geom_boxplot</span>(<span class="dt">outlier.shape=</span><span class="ot">NA</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb6-11" title="11"><span class="st">  </span><span class="kw">theme_classic</span>() <span class="op">+</span></a>
<a class="sourceLine" id="cb6-12" title="12"><span class="st">  </span><span class="kw">scale_fill_manual</span>(<span class="dt">values =</span> <span class="kw">c</span>(<span class="st">&quot;beta&quot;</span> =<span class="st"> &quot;#607D8B&quot;</span>, <span class="st">&quot;MAP&quot;</span> =<span class="st"> &quot;#795548&quot;</span>, <span class="st">&quot;Mean&quot;</span> =<span class="st"> &quot;#FF9800&quot;</span>, <span class="st">&quot;Median&quot;</span> =<span class="st"> &quot;#FFEB3B&quot;</span>),</a>
<a class="sourceLine" id="cb6-13" title="13">                    <span class="dt">name =</span> <span class="st">&quot;Index&quot;</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb6-14" title="14"><span class="st">  </span><span class="kw">ylab</span>(<span class="st">&quot;Point-estimate of the true value 0</span><span class="ch">\n</span><span class="st">&quot;</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb6-15" title="15"><span class="st">  </span><span class="kw">xlab</span>(<span class="st">&quot;</span><span class="ch">\n</span><span class="st">Number of Iterations&quot;</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb6-16" title="16"><span class="st">  </span><span class="kw">facet_wrap</span>(<span class="op">~</span><span class="st"> </span>outcome_type <span class="op">*</span><span class="st"> </span>true_effect, <span class="dt">scales=</span><span class="st">&quot;free&quot;</span>) </a></code></pre></div>
<p><img 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" /><!-- --></p>
<h4 id="sensitivity-to-warmup-ratio">Sensitivity to warmup ratio</h4>
<div class="sourceCode" id="cb7"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb7-1" title="1">df <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb7-2" title="2"><span class="st">  </span><span class="kw">mutate</span>(<span class="dt">warmup =</span> warmup <span class="op">/</span><span class="st"> </span>iterations) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb7-3" title="3"><span class="st">  </span><span class="kw">select</span>(warmup, true_effect, outcome_type, beta, Median, Mean, MAP) <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb7-4" title="4"><span class="st">  </span><span class="kw">gather</span>(estimate, value, <span class="op">-</span>warmup, <span class="op">-</span>true_effect, <span class="op">-</span>outcome_type) <span class="op">%&gt;%</span></a>
<a class="sourceLine" id="cb7-5" title="5"><span class="st">  </span><span class="kw">mutate</span>(<span class="dt">temp =</span> <span class="kw">as.factor</span>(<span class="kw">cut</span>(warmup, <span class="dv">3</span>, <span class="dt">labels =</span> <span class="ot">FALSE</span>))) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb7-6" title="6"><span class="st">  </span><span class="kw">group_by</span>(temp) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb7-7" title="7"><span class="st">  </span><span class="kw">mutate</span>(<span class="dt">warmup_group =</span> <span class="kw">round</span>(<span class="kw">mean</span>(warmup), <span class="dv">1</span>)) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb7-8" title="8"><span class="st">  </span><span class="kw">ungroup</span>() <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb7-9" title="9"><span class="st">  </span><span class="kw">filter</span>(value <span class="op">&lt;</span><span class="st"> </span><span class="dv">6</span>) <span class="op">%&gt;%</span><span class="st"> </span></a>
<a class="sourceLine" id="cb7-10" title="10"><span class="st">  </span><span class="kw">ggplot</span>(<span class="kw">aes</span>(<span class="dt">x =</span> warmup_group, <span class="dt">y =</span> value, <span class="dt">fill =</span> estimate, <span class="dt">group =</span> <span class="kw">interaction</span>(estimate, warmup_group))) <span class="op">+</span></a>
<a class="sourceLine" id="cb7-11" title="11"><span class="st">  </span><span class="kw">geom_boxplot</span>(<span class="dt">outlier.shape=</span><span class="ot">NA</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb7-12" title="12"><span class="st">  </span><span class="kw">theme_classic</span>() <span class="op">+</span></a>
<a class="sourceLine" id="cb7-13" title="13"><span class="st">  </span><span class="kw">scale_fill_manual</span>(<span class="dt">values =</span> <span class="kw">c</span>(<span class="st">&quot;beta&quot;</span> =<span class="st"> &quot;#607D8B&quot;</span>, <span class="st">&quot;MAP&quot;</span> =<span class="st"> &quot;#795548&quot;</span>, <span class="st">&quot;Mean&quot;</span> =<span class="st"> &quot;#FF9800&quot;</span>, <span class="st">&quot;Median&quot;</span> =<span class="st"> &quot;#FFEB3B&quot;</span>),</a>
<a class="sourceLine" id="cb7-14" title="14">                    <span class="dt">name =</span> <span class="st">&quot;Index&quot;</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb7-15" title="15"><span class="st">  </span><span class="kw">ylab</span>(<span class="st">&quot;Point-estimate of the true value 0</span><span class="ch">\n</span><span class="st">&quot;</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb7-16" title="16"><span class="st">  </span><span class="kw">xlab</span>(<span class="st">&quot;</span><span class="ch">\n</span><span class="st">Number of Iterations&quot;</span>) <span class="op">+</span></a>
<a class="sourceLine" id="cb7-17" title="17"><span class="st">  </span><span class="kw">facet_wrap</span>(<span class="op">~</span><span class="st"> </span>outcome_type <span class="op">*</span><span class="st"> </span>true_effect, <span class="dt">scales=</span><span class="st">&quot;free&quot;</span>) </a></code></pre></div>
<p><img 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" /><!-- --></p>
<h2 id="experiment-3-relationship-with-priors-specification">Experiment 3: Relationship with Priors Specification</h2>
<h2 id="discussion">Discussion</h2>
<p>Conclusions can be found in the <a href="https://easystats.github.io/bayestestR/articles/guidelines.html">guidelines section</a>.</p>

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