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<h1 class="title toc-ignore">Mediation Analysis using Bayesian Regression Models</h1>
<div id="TOC">
<ul>
<li><a href="#mediation-analysis-in-brms-and-rstanarm">Mediation Analysis in brms and rstanarm</a></li>
<li><a href="#comparison-to-the-mediation-package">Comparison to the mediation package</a></li>
<li><a href="#comparison-to-sem-from-the-lavaan-package">Comparison to SEM from the lavaan package</a></li>
</ul>
</div>
<p>This vignettes demonstrates the <code>mediation()</code>-function. Before we start, we fit some models, including a mediation-object from the <em>mediation</em>-package and a structural equation modelling approach with the <em>lavaan</em>-package, both of which we use for comparison with <em>brms</em> and <em>rstanarm</em>.</p>
<div id="mediation-analysis-in-brms-and-rstanarm" class="section level2">
<h2>Mediation Analysis in brms and rstanarm</h2>
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(bayestestR)</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(mediation)</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(brms)</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(rstanarm)</span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a><span class="co"># load sample data</span></span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(jobs)</span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">123</span>)</span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a><span class="co"># linear models, for mediation analysis</span></span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a>b1 <span class="ot"><-</span> <span class="fu">lm</span>(job_seek <span class="sc">~</span> treat <span class="sc">+</span> econ_hard <span class="sc">+</span> sex <span class="sc">+</span> age, <span class="at">data =</span> jobs)</span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a>b2 <span class="ot"><-</span> <span class="fu">lm</span>(depress2 <span class="sc">~</span> treat <span class="sc">+</span> job_seek <span class="sc">+</span> econ_hard <span class="sc">+</span> sex <span class="sc">+</span> age, <span class="at">data =</span> jobs)</span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a><span class="co"># mediation analysis, for comparison with brms</span></span>
<span id="cb1-15"><a href="#cb1-15" aria-hidden="true" tabindex="-1"></a>m1 <span class="ot"><-</span> <span class="fu">mediate</span>(b1, b2, <span class="at">sims =</span> <span class="dv">1000</span>, <span class="at">treat =</span> <span class="st">"treat"</span>, <span class="at">mediator =</span> <span class="st">"job_seek"</span>)</span></code></pre></div>
<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Fit Bayesian mediation model in brms</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>f1 <span class="ot"><-</span> <span class="fu">bf</span>(job_seek <span class="sc">~</span> treat <span class="sc">+</span> econ_hard <span class="sc">+</span> sex <span class="sc">+</span> age)</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a>f2 <span class="ot"><-</span> <span class="fu">bf</span>(depress2 <span class="sc">~</span> treat <span class="sc">+</span> job_seek <span class="sc">+</span> econ_hard <span class="sc">+</span> sex <span class="sc">+</span> age)</span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a>m2 <span class="ot"><-</span> <span class="fu">brm</span>(f1 <span class="sc">+</span> f2 <span class="sc">+</span> <span class="fu">set_rescor</span>(<span class="cn">FALSE</span>), <span class="at">data =</span> jobs, <span class="at">cores =</span> <span class="dv">4</span>)</span></code></pre></div>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Fit Bayesian mediation model in rstanarm</span></span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>m3 <span class="ot"><-</span> <span class="fu">stan_mvmer</span>(</span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">list</span>(job_seek <span class="sc">~</span> treat <span class="sc">+</span> econ_hard <span class="sc">+</span> sex <span class="sc">+</span> age <span class="sc">+</span> (<span class="dv">1</span> <span class="sc">|</span> occp),</span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a> depress2 <span class="sc">~</span> treat <span class="sc">+</span> job_seek <span class="sc">+</span> econ_hard <span class="sc">+</span> sex <span class="sc">+</span> age <span class="sc">+</span> (<span class="dv">1</span> <span class="sc">|</span> occp)),</span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a> <span class="at">data =</span> jobs,</span>
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a> <span class="at">cores =</span> <span class="dv">4</span>,</span>
<span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a> <span class="at">refresh =</span> <span class="dv">0</span></span>
<span id="cb3-8"><a href="#cb3-8" aria-hidden="true" tabindex="-1"></a>)</span></code></pre></div>
<p><code>mediation()</code> is a summary function, especially for mediation analysis, i.e. for multivariate response models with casual mediation effects.</p>
<p>In the models <code>m2</code> and <code>m3</code>, <code>treat</code> is the treatment effect and <code>job_seek</code> is the mediator effect. For the <em>brms</em> model (<code>m2</code>), <code>f1</code> describes the mediator model and <code>f2</code> describes the outcome model. This is similar for the <em>rstanarm</em> model.</p>
<p><code>mediation()</code> returns a data frame with information on the <em>direct effect</em> (median value of posterior samples from treatment of the outcome model), <em>mediator effect</em> (median value of posterior samples from mediator of the outcome model), <em>indirect effect</em> (median value of the multiplication of the posterior samples from mediator of the outcome model and the posterior samples from treatment of the mediation model) and the <em>total effect</em> (median value of sums of posterior samples used for the direct and indirect effect). The <em>proportion mediated</em> is the indirect effect divided by the total effect.</p>
<p>The simplest call just needs the model-object.</p>
<div class="sourceCode" id="cb4"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="co"># for brms</span></span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a><span class="fu">mediation</span>(m2)</span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a><span class="co">#> # Causal Mediation Analysis for Stan Model</span></span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a><span class="co">#> Treatment: treat</span></span>
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a><span class="co">#> Mediator : job_seek</span></span>
<span id="cb4-7"><a href="#cb4-7" aria-hidden="true" tabindex="-1"></a><span class="co">#> Response : depress2</span></span>
<span id="cb4-8"><a href="#cb4-8" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb4-9"><a href="#cb4-9" aria-hidden="true" tabindex="-1"></a><span class="co">#> Effect | Estimate | 95% ETI</span></span>
<span id="cb4-10"><a href="#cb4-10" aria-hidden="true" tabindex="-1"></a><span class="co">#> ----------------------------------------------------</span></span>
<span id="cb4-11"><a href="#cb4-11" aria-hidden="true" tabindex="-1"></a><span class="co">#> Direct Effect (ADE) | -0.040 | [-0.124, 0.046]</span></span>
<span id="cb4-12"><a href="#cb4-12" aria-hidden="true" tabindex="-1"></a><span class="co">#> Indirect Effect (ACME) | -0.015 | [-0.041, 0.008]</span></span>
<span id="cb4-13"><a href="#cb4-13" aria-hidden="true" tabindex="-1"></a><span class="co">#> Mediator Effect | -0.240 | [-0.294, -0.185]</span></span>
<span id="cb4-14"><a href="#cb4-14" aria-hidden="true" tabindex="-1"></a><span class="co">#> Total Effect | -0.055 | [-0.145, 0.034]</span></span>
<span id="cb4-15"><a href="#cb4-15" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb4-16"><a href="#cb4-16" aria-hidden="true" tabindex="-1"></a><span class="co">#> Proportion mediated: 28.14% [-181.46%, 237.75%]</span></span>
<span id="cb4-17"><a href="#cb4-17" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-18"><a href="#cb4-18" aria-hidden="true" tabindex="-1"></a><span class="co"># for rstanarm</span></span>
<span id="cb4-19"><a href="#cb4-19" aria-hidden="true" tabindex="-1"></a><span class="fu">mediation</span>(m3)</span>
<span id="cb4-20"><a href="#cb4-20" aria-hidden="true" tabindex="-1"></a><span class="co">#> # Causal Mediation Analysis for Stan Model</span></span>
<span id="cb4-21"><a href="#cb4-21" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb4-22"><a href="#cb4-22" aria-hidden="true" tabindex="-1"></a><span class="co">#> Treatment: treat</span></span>
<span id="cb4-23"><a href="#cb4-23" aria-hidden="true" tabindex="-1"></a><span class="co">#> Mediator : job_seek</span></span>
<span id="cb4-24"><a href="#cb4-24" aria-hidden="true" tabindex="-1"></a><span class="co">#> Response : depress2</span></span>
<span id="cb4-25"><a href="#cb4-25" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb4-26"><a href="#cb4-26" aria-hidden="true" tabindex="-1"></a><span class="co">#> Effect | Estimate | 95% ETI</span></span>
<span id="cb4-27"><a href="#cb4-27" aria-hidden="true" tabindex="-1"></a><span class="co">#> ----------------------------------------------------</span></span>
<span id="cb4-28"><a href="#cb4-28" aria-hidden="true" tabindex="-1"></a><span class="co">#> Direct Effect (ADE) | -0.040 | [-0.129, 0.048]</span></span>
<span id="cb4-29"><a href="#cb4-29" aria-hidden="true" tabindex="-1"></a><span class="co">#> Indirect Effect (ACME) | -0.018 | [-0.042, 0.006]</span></span>
<span id="cb4-30"><a href="#cb4-30" aria-hidden="true" tabindex="-1"></a><span class="co">#> Mediator Effect | -0.241 | [-0.296, -0.187]</span></span>
<span id="cb4-31"><a href="#cb4-31" aria-hidden="true" tabindex="-1"></a><span class="co">#> Total Effect | -0.057 | [-0.151, 0.033]</span></span>
<span id="cb4-32"><a href="#cb4-32" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb4-33"><a href="#cb4-33" aria-hidden="true" tabindex="-1"></a><span class="co">#> Proportion mediated: 30.59% [-221.09%, 282.26%]</span></span></code></pre></div>
<p>Typically, <code>mediation()</code> finds the treatment and mediator variables automatically. If this does not work, use the <code>treatment</code> and <code>mediator</code> arguments to specify the related variable names. For all values, the 89% credible intervals are calculated by default. Use <code>ci</code> to calculate a different interval.</p>
</div>
<div id="comparison-to-the-mediation-package" class="section level2">
<h2>Comparison to the mediation package</h2>
<p>Here is a comparison with the <em>mediation</em> package. Note that the <code>summary()</code>-output of the <em>mediation</em> package shows the indirect effect first, followed by the direct effect.</p>
<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(m1)</span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a><span class="co">#> Causal Mediation Analysis </span></span>
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a><span class="co">#> Quasi-Bayesian Confidence Intervals</span></span>
<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a><span class="co">#> Estimate 95% CI Lower 95% CI Upper p-value</span></span>
<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a><span class="co">#> ACME -0.0157 -0.0387 0.01 0.19</span></span>
<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a><span class="co">#> ADE -0.0438 -0.1315 0.04 0.35</span></span>
<span id="cb5-10"><a href="#cb5-10" aria-hidden="true" tabindex="-1"></a><span class="co">#> Total Effect -0.0595 -0.1530 0.02 0.21</span></span>
<span id="cb5-11"><a href="#cb5-11" aria-hidden="true" tabindex="-1"></a><span class="co">#> Prop. Mediated 0.2137 -2.0277 2.70 0.32</span></span>
<span id="cb5-12"><a href="#cb5-12" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb5-13"><a href="#cb5-13" aria-hidden="true" tabindex="-1"></a><span class="co">#> Sample Size Used: 899 </span></span>
<span id="cb5-14"><a href="#cb5-14" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb5-15"><a href="#cb5-15" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb5-16"><a href="#cb5-16" aria-hidden="true" tabindex="-1"></a><span class="co">#> Simulations: 1000</span></span>
<span id="cb5-17"><a href="#cb5-17" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-18"><a href="#cb5-18" aria-hidden="true" tabindex="-1"></a><span class="fu">mediation</span>(m2, <span class="at">ci =</span> .<span class="dv">95</span>)</span>
<span id="cb5-19"><a href="#cb5-19" aria-hidden="true" tabindex="-1"></a><span class="co">#> # Causal Mediation Analysis for Stan Model</span></span>
<span id="cb5-20"><a href="#cb5-20" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb5-21"><a href="#cb5-21" aria-hidden="true" tabindex="-1"></a><span class="co">#> Treatment: treat</span></span>
<span id="cb5-22"><a href="#cb5-22" aria-hidden="true" tabindex="-1"></a><span class="co">#> Mediator : job_seek</span></span>
<span id="cb5-23"><a href="#cb5-23" aria-hidden="true" tabindex="-1"></a><span class="co">#> Response : depress2</span></span>
<span id="cb5-24"><a href="#cb5-24" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb5-25"><a href="#cb5-25" aria-hidden="true" tabindex="-1"></a><span class="co">#> Effect | Estimate | 95% ETI</span></span>
<span id="cb5-26"><a href="#cb5-26" aria-hidden="true" tabindex="-1"></a><span class="co">#> ----------------------------------------------------</span></span>
<span id="cb5-27"><a href="#cb5-27" aria-hidden="true" tabindex="-1"></a><span class="co">#> Direct Effect (ADE) | -0.040 | [-0.124, 0.046]</span></span>
<span id="cb5-28"><a href="#cb5-28" aria-hidden="true" tabindex="-1"></a><span class="co">#> Indirect Effect (ACME) | -0.015 | [-0.041, 0.008]</span></span>
<span id="cb5-29"><a href="#cb5-29" aria-hidden="true" tabindex="-1"></a><span class="co">#> Mediator Effect | -0.240 | [-0.294, -0.185]</span></span>
<span id="cb5-30"><a href="#cb5-30" aria-hidden="true" tabindex="-1"></a><span class="co">#> Total Effect | -0.055 | [-0.145, 0.034]</span></span>
<span id="cb5-31"><a href="#cb5-31" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb5-32"><a href="#cb5-32" aria-hidden="true" tabindex="-1"></a><span class="co">#> Proportion mediated: 28.14% [-181.46%, 237.75%]</span></span>
<span id="cb5-33"><a href="#cb5-33" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-34"><a href="#cb5-34" aria-hidden="true" tabindex="-1"></a><span class="fu">mediation</span>(m3, <span class="at">ci =</span> .<span class="dv">95</span>)</span>
<span id="cb5-35"><a href="#cb5-35" aria-hidden="true" tabindex="-1"></a><span class="co">#> # Causal Mediation Analysis for Stan Model</span></span>
<span id="cb5-36"><a href="#cb5-36" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb5-37"><a href="#cb5-37" aria-hidden="true" tabindex="-1"></a><span class="co">#> Treatment: treat</span></span>
<span id="cb5-38"><a href="#cb5-38" aria-hidden="true" tabindex="-1"></a><span class="co">#> Mediator : job_seek</span></span>
<span id="cb5-39"><a href="#cb5-39" aria-hidden="true" tabindex="-1"></a><span class="co">#> Response : depress2</span></span>
<span id="cb5-40"><a href="#cb5-40" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb5-41"><a href="#cb5-41" aria-hidden="true" tabindex="-1"></a><span class="co">#> Effect | Estimate | 95% ETI</span></span>
<span id="cb5-42"><a href="#cb5-42" aria-hidden="true" tabindex="-1"></a><span class="co">#> ----------------------------------------------------</span></span>
<span id="cb5-43"><a href="#cb5-43" aria-hidden="true" tabindex="-1"></a><span class="co">#> Direct Effect (ADE) | -0.040 | [-0.129, 0.048]</span></span>
<span id="cb5-44"><a href="#cb5-44" aria-hidden="true" tabindex="-1"></a><span class="co">#> Indirect Effect (ACME) | -0.018 | [-0.042, 0.006]</span></span>
<span id="cb5-45"><a href="#cb5-45" aria-hidden="true" tabindex="-1"></a><span class="co">#> Mediator Effect | -0.241 | [-0.296, -0.187]</span></span>
<span id="cb5-46"><a href="#cb5-46" aria-hidden="true" tabindex="-1"></a><span class="co">#> Total Effect | -0.057 | [-0.151, 0.033]</span></span>
<span id="cb5-47"><a href="#cb5-47" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb5-48"><a href="#cb5-48" aria-hidden="true" tabindex="-1"></a><span class="co">#> Proportion mediated: 30.59% [-221.09%, 282.26%]</span></span></code></pre></div>
<p>If you want to calculate mean instead of median values from the posterior samples, use the <code>centrality</code>-argument. Furthermore, there is a <code>print()</code>-method, which allows to print more digits.</p>
<div class="sourceCode" id="cb6"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a>m <span class="ot"><-</span> <span class="fu">mediation</span>(m2, <span class="at">centrality =</span> <span class="st">"mean"</span>, <span class="at">ci =</span> .<span class="dv">95</span>)</span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(m, <span class="at">digits =</span> <span class="dv">4</span>)</span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a><span class="co">#> # Causal Mediation Analysis for Stan Model</span></span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a><span class="co">#> Treatment: treat</span></span>
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a><span class="co">#> Mediator : job_seek</span></span>
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a><span class="co">#> Response : depress2</span></span>
<span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb6-9"><a href="#cb6-9" aria-hidden="true" tabindex="-1"></a><span class="co">#> Effect | Estimate | 95% ETI</span></span>
<span id="cb6-10"><a href="#cb6-10" aria-hidden="true" tabindex="-1"></a><span class="co">#> ------------------------------------------------------</span></span>
<span id="cb6-11"><a href="#cb6-11" aria-hidden="true" tabindex="-1"></a><span class="co">#> Direct Effect (ADE) | -0.0395 | [-0.1237, 0.0456]</span></span>
<span id="cb6-12"><a href="#cb6-12" aria-hidden="true" tabindex="-1"></a><span class="co">#> Indirect Effect (ACME) | -0.0158 | [-0.0405, 0.0083]</span></span>
<span id="cb6-13"><a href="#cb6-13" aria-hidden="true" tabindex="-1"></a><span class="co">#> Mediator Effect | -0.2401 | [-0.2944, -0.1846]</span></span>
<span id="cb6-14"><a href="#cb6-14" aria-hidden="true" tabindex="-1"></a><span class="co">#> Total Effect | -0.0553 | [-0.1454, 0.0341]</span></span>
<span id="cb6-15"><a href="#cb6-15" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb6-16"><a href="#cb6-16" aria-hidden="true" tabindex="-1"></a><span class="co">#> Proportion mediated: 28.60% [-181.01%, 238.20%]</span></span></code></pre></div>
<p>As you can see, the results are similar to what the <em>mediation</em> package produces for non-Bayesian models.</p>
</div>
<div id="comparison-to-sem-from-the-lavaan-package" class="section level2">
<h2>Comparison to SEM from the lavaan package</h2>
<p>Finally, we also compare the results to a SEM model, using <em>lavaan</em>. This example should demonstrate how to “translate” the same model in different packages or modeling approached.</p>
<div class="sourceCode" id="cb7"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(lavaan)</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(jobs)</span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1234</span>)</span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a>model <span class="ot"><-</span> <span class="st">' # direct effects</span></span>
<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a><span class="st"> depress2 ~ c1*treat + c2*econ_hard + c3*sex + c4*age + b*job_seek</span></span>
<span id="cb7-7"><a href="#cb7-7" aria-hidden="true" tabindex="-1"></a><span class="st"> </span></span>
<span id="cb7-8"><a href="#cb7-8" aria-hidden="true" tabindex="-1"></a><span class="st"> # mediation</span></span>
<span id="cb7-9"><a href="#cb7-9" aria-hidden="true" tabindex="-1"></a><span class="st"> job_seek ~ a1*treat + a2*econ_hard + a3*sex + a4*age</span></span>
<span id="cb7-10"><a href="#cb7-10" aria-hidden="true" tabindex="-1"></a><span class="st"> </span></span>
<span id="cb7-11"><a href="#cb7-11" aria-hidden="true" tabindex="-1"></a><span class="st"> # indirect effects (a*b)</span></span>
<span id="cb7-12"><a href="#cb7-12" aria-hidden="true" tabindex="-1"></a><span class="st"> indirect_treat := a1*b</span></span>
<span id="cb7-13"><a href="#cb7-13" aria-hidden="true" tabindex="-1"></a><span class="st"> indirect_econ_hard := a2*b</span></span>
<span id="cb7-14"><a href="#cb7-14" aria-hidden="true" tabindex="-1"></a><span class="st"> indirect_sex := a3*b</span></span>
<span id="cb7-15"><a href="#cb7-15" aria-hidden="true" tabindex="-1"></a><span class="st"> indirect_age := a4*b</span></span>
<span id="cb7-16"><a href="#cb7-16" aria-hidden="true" tabindex="-1"></a><span class="st"> </span></span>
<span id="cb7-17"><a href="#cb7-17" aria-hidden="true" tabindex="-1"></a><span class="st"> # total effects</span></span>
<span id="cb7-18"><a href="#cb7-18" aria-hidden="true" tabindex="-1"></a><span class="st"> total_treat := c1 + (a1*b)</span></span>
<span id="cb7-19"><a href="#cb7-19" aria-hidden="true" tabindex="-1"></a><span class="st"> total_econ_hard := c2 + (a2*b)</span></span>
<span id="cb7-20"><a href="#cb7-20" aria-hidden="true" tabindex="-1"></a><span class="st"> total_sex := c3 + (a3*b)</span></span>
<span id="cb7-21"><a href="#cb7-21" aria-hidden="true" tabindex="-1"></a><span class="st"> total_age := c4 + (a4*b)</span></span>
<span id="cb7-22"><a href="#cb7-22" aria-hidden="true" tabindex="-1"></a><span class="st"> '</span></span>
<span id="cb7-23"><a href="#cb7-23" aria-hidden="true" tabindex="-1"></a>m4 <span class="ot"><-</span> <span class="fu">sem</span>(model, <span class="at">data =</span> jobs)</span>
<span id="cb7-24"><a href="#cb7-24" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(m4)</span>
<span id="cb7-25"><a href="#cb7-25" aria-hidden="true" tabindex="-1"></a><span class="co">#> lavaan 0.6-8 ended normally after 25 iterations</span></span>
<span id="cb7-26"><a href="#cb7-26" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-27"><a href="#cb7-27" aria-hidden="true" tabindex="-1"></a><span class="co">#> Estimator ML</span></span>
<span id="cb7-28"><a href="#cb7-28" aria-hidden="true" tabindex="-1"></a><span class="co">#> Optimization method NLMINB</span></span>
<span id="cb7-29"><a href="#cb7-29" aria-hidden="true" tabindex="-1"></a><span class="co">#> Number of model parameters 11</span></span>
<span id="cb7-30"><a href="#cb7-30" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-31"><a href="#cb7-31" aria-hidden="true" tabindex="-1"></a><span class="co">#> Number of observations 899</span></span>
<span id="cb7-32"><a href="#cb7-32" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-33"><a href="#cb7-33" aria-hidden="true" tabindex="-1"></a><span class="co">#> Model Test User Model:</span></span>
<span id="cb7-34"><a href="#cb7-34" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-35"><a href="#cb7-35" aria-hidden="true" tabindex="-1"></a><span class="co">#> Test statistic 0.000</span></span>
<span id="cb7-36"><a href="#cb7-36" aria-hidden="true" tabindex="-1"></a><span class="co">#> Degrees of freedom 0</span></span>
<span id="cb7-37"><a href="#cb7-37" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-38"><a href="#cb7-38" aria-hidden="true" tabindex="-1"></a><span class="co">#> Parameter Estimates:</span></span>
<span id="cb7-39"><a href="#cb7-39" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-40"><a href="#cb7-40" aria-hidden="true" tabindex="-1"></a><span class="co">#> Standard errors Standard</span></span>
<span id="cb7-41"><a href="#cb7-41" aria-hidden="true" tabindex="-1"></a><span class="co">#> Information Expected</span></span>
<span id="cb7-42"><a href="#cb7-42" aria-hidden="true" tabindex="-1"></a><span class="co">#> Information saturated (h1) model Structured</span></span>
<span id="cb7-43"><a href="#cb7-43" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-44"><a href="#cb7-44" aria-hidden="true" tabindex="-1"></a><span class="co">#> Regressions:</span></span>
<span id="cb7-45"><a href="#cb7-45" aria-hidden="true" tabindex="-1"></a><span class="co">#> Estimate Std.Err z-value P(>|z|)</span></span>
<span id="cb7-46"><a href="#cb7-46" aria-hidden="true" tabindex="-1"></a><span class="co">#> depress2 ~ </span></span>
<span id="cb7-47"><a href="#cb7-47" aria-hidden="true" tabindex="-1"></a><span class="co">#> treat (c1) -0.040 0.043 -0.929 0.353</span></span>
<span id="cb7-48"><a href="#cb7-48" aria-hidden="true" tabindex="-1"></a><span class="co">#> econ_hard (c2) 0.149 0.021 7.156 0.000</span></span>
<span id="cb7-49"><a href="#cb7-49" aria-hidden="true" tabindex="-1"></a><span class="co">#> sex (c3) 0.107 0.041 2.604 0.009</span></span>
<span id="cb7-50"><a href="#cb7-50" aria-hidden="true" tabindex="-1"></a><span class="co">#> age (c4) 0.001 0.002 0.332 0.740</span></span>
<span id="cb7-51"><a href="#cb7-51" aria-hidden="true" tabindex="-1"></a><span class="co">#> job_seek (b) -0.240 0.028 -8.524 0.000</span></span>
<span id="cb7-52"><a href="#cb7-52" aria-hidden="true" tabindex="-1"></a><span class="co">#> job_seek ~ </span></span>
<span id="cb7-53"><a href="#cb7-53" aria-hidden="true" tabindex="-1"></a><span class="co">#> treat (a1) 0.066 0.051 1.278 0.201</span></span>
<span id="cb7-54"><a href="#cb7-54" aria-hidden="true" tabindex="-1"></a><span class="co">#> econ_hard (a2) 0.053 0.025 2.167 0.030</span></span>
<span id="cb7-55"><a href="#cb7-55" aria-hidden="true" tabindex="-1"></a><span class="co">#> sex (a3) -0.008 0.049 -0.157 0.875</span></span>
<span id="cb7-56"><a href="#cb7-56" aria-hidden="true" tabindex="-1"></a><span class="co">#> age (a4) 0.005 0.002 1.983 0.047</span></span>
<span id="cb7-57"><a href="#cb7-57" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-58"><a href="#cb7-58" aria-hidden="true" tabindex="-1"></a><span class="co">#> Variances:</span></span>
<span id="cb7-59"><a href="#cb7-59" aria-hidden="true" tabindex="-1"></a><span class="co">#> Estimate Std.Err z-value P(>|z|)</span></span>
<span id="cb7-60"><a href="#cb7-60" aria-hidden="true" tabindex="-1"></a><span class="co">#> .depress2 0.373 0.018 21.201 0.000</span></span>
<span id="cb7-61"><a href="#cb7-61" aria-hidden="true" tabindex="-1"></a><span class="co">#> .job_seek 0.524 0.025 21.201 0.000</span></span>
<span id="cb7-62"><a href="#cb7-62" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-63"><a href="#cb7-63" aria-hidden="true" tabindex="-1"></a><span class="co">#> Defined Parameters:</span></span>
<span id="cb7-64"><a href="#cb7-64" aria-hidden="true" tabindex="-1"></a><span class="co">#> Estimate Std.Err z-value P(>|z|)</span></span>
<span id="cb7-65"><a href="#cb7-65" aria-hidden="true" tabindex="-1"></a><span class="co">#> indirect_treat -0.016 0.012 -1.264 0.206</span></span>
<span id="cb7-66"><a href="#cb7-66" aria-hidden="true" tabindex="-1"></a><span class="co">#> indirct_cn_hrd -0.013 0.006 -2.100 0.036</span></span>
<span id="cb7-67"><a href="#cb7-67" aria-hidden="true" tabindex="-1"></a><span class="co">#> indirect_sex 0.002 0.012 0.157 0.875</span></span>
<span id="cb7-68"><a href="#cb7-68" aria-hidden="true" tabindex="-1"></a><span class="co">#> indirect_age -0.001 0.001 -1.932 0.053</span></span>
<span id="cb7-69"><a href="#cb7-69" aria-hidden="true" tabindex="-1"></a><span class="co">#> total_treat -0.056 0.045 -1.244 0.214</span></span>
<span id="cb7-70"><a href="#cb7-70" aria-hidden="true" tabindex="-1"></a><span class="co">#> total_econ_hrd 0.136 0.022 6.309 0.000</span></span>
<span id="cb7-71"><a href="#cb7-71" aria-hidden="true" tabindex="-1"></a><span class="co">#> total_sex 0.109 0.043 2.548 0.011</span></span>
<span id="cb7-72"><a href="#cb7-72" aria-hidden="true" tabindex="-1"></a><span class="co">#> total_age -0.000 0.002 -0.223 0.824</span></span>
<span id="cb7-73"><a href="#cb7-73" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-74"><a href="#cb7-74" aria-hidden="true" tabindex="-1"></a><span class="co"># just to have the numbers right at hand and you don't need to scroll up</span></span>
<span id="cb7-75"><a href="#cb7-75" aria-hidden="true" tabindex="-1"></a><span class="fu">mediation</span>(m2, <span class="at">ci =</span> .<span class="dv">95</span>)</span>
<span id="cb7-76"><a href="#cb7-76" aria-hidden="true" tabindex="-1"></a><span class="co">#> # Causal Mediation Analysis for Stan Model</span></span>
<span id="cb7-77"><a href="#cb7-77" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-78"><a href="#cb7-78" aria-hidden="true" tabindex="-1"></a><span class="co">#> Treatment: treat</span></span>
<span id="cb7-79"><a href="#cb7-79" aria-hidden="true" tabindex="-1"></a><span class="co">#> Mediator : job_seek</span></span>
<span id="cb7-80"><a href="#cb7-80" aria-hidden="true" tabindex="-1"></a><span class="co">#> Response : depress2</span></span>
<span id="cb7-81"><a href="#cb7-81" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-82"><a href="#cb7-82" aria-hidden="true" tabindex="-1"></a><span class="co">#> Effect | Estimate | 95% ETI</span></span>
<span id="cb7-83"><a href="#cb7-83" aria-hidden="true" tabindex="-1"></a><span class="co">#> ----------------------------------------------------</span></span>
<span id="cb7-84"><a href="#cb7-84" aria-hidden="true" tabindex="-1"></a><span class="co">#> Direct Effect (ADE) | -0.040 | [-0.124, 0.046]</span></span>
<span id="cb7-85"><a href="#cb7-85" aria-hidden="true" tabindex="-1"></a><span class="co">#> Indirect Effect (ACME) | -0.015 | [-0.041, 0.008]</span></span>
<span id="cb7-86"><a href="#cb7-86" aria-hidden="true" tabindex="-1"></a><span class="co">#> Mediator Effect | -0.240 | [-0.294, -0.185]</span></span>
<span id="cb7-87"><a href="#cb7-87" aria-hidden="true" tabindex="-1"></a><span class="co">#> Total Effect | -0.055 | [-0.145, 0.034]</span></span>
<span id="cb7-88"><a href="#cb7-88" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-89"><a href="#cb7-89" aria-hidden="true" tabindex="-1"></a><span class="co">#> Proportion mediated: 28.14% [-181.46%, 237.75%]</span></span></code></pre></div>
<p>The summary output from <em>lavaan</em> is longer, but we can find the related numbers quite easily:</p>
<ul>
<li>the <em>direct effect</em> of treatment is <code>treat (c1)</code>, which is <code>-0.040</code></li>
<li>the <em>indirect effect</em> of treatment is <code>indirect_treat</code>, which is <code>-0.016</code></li>
<li>the <em>mediator effect</em> of job_seek is <code>job_seek (b)</code>, which is <code>-0.240</code></li>
<li>the <em>total effect</em> is <code>total_treat</code>, which is <code>-0.056</code></li>
</ul>
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