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<!DOCTYPE html>
<html lang="en"><head>
<script src="slides_files/libs/clipboard/clipboard.min.js"></script>
<script src="slides_files/libs/quarto-html/tabby.min.js"></script>
<script src="slides_files/libs/quarto-html/popper.min.js"></script>
<script src="slides_files/libs/quarto-html/tippy.umd.min.js"></script>
<link href="slides_files/libs/quarto-html/tippy.css" rel="stylesheet">
<link href="slides_files/libs/quarto-html/light-border.css" rel="stylesheet">
<link href="slides_files/libs/quarto-html/quarto-html.min.css" rel="stylesheet" data-mode="light">
<link href="slides_files/libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles"><meta charset="utf-8">
<meta name="generator" content="quarto-1.4.555">
<meta name="author" content="Daniel Witte, Jie Zhang">
<title>Mediation Analyses</title>
<meta name="apple-mobile-web-app-capable" content="yes">
<meta name="apple-mobile-web-app-status-bar-style" content="black-translucent">
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no, minimal-ui">
<link rel="stylesheet" href="slides_files/libs/revealjs/dist/reset.css">
<link rel="stylesheet" href="slides_files/libs/revealjs/dist/reveal.css">
<style>
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
div.columns{display: flex; gap: min(4vw, 1.5em);}
div.column{flex: auto; overflow-x: auto;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
ul.task-list li input[type="checkbox"] {
width: 0.8em;
margin: 0 0.8em 0.2em -1em; /* quarto-specific, see https://github.com/quarto-dev/quarto-cli/issues/4556 */
vertical-align: middle;
}
</style>
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<link href="slides_files/libs/revealjs/plugin/quarto-support/footer.css" rel="stylesheet">
<style type="text/css">
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margin-top: 1em;
margin-bottom: 1em;
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padding: 0em 0.5em;
border-left: solid #acacac .3rem;
border-right: solid 1px silver;
border-top: solid 1px silver;
border-bottom: solid 1px silver;
display: flex;
}
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border-left: solid #acacac .3rem;
border-right: solid 1px silver;
border-top: solid 1px silver;
border-bottom: solid 1px silver;
}
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margin-bottom: 0.5em;
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font-size: .9rem;
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background-size: 0.9rem 0.9rem;
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div.callout-important {
border-left-color: #d9534f !important;
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div.callout-important .callout-icon::before {
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background-color: #f7dddc
}
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</head>
<body class="quarto-light">
<div class="reveal">
<div class="slides">
<section id="title-slide" class="quarto-title-block center">
<h1 class="title">Mediation Analyses</h1>
<div class="quarto-title-authors">
<div class="quarto-title-author">
<div class="quarto-title-author-name">
Daniel Witte, Jie Zhang
</div>
</div>
</div>
</section>
<section id="welcome-to-the-workshop-on-mediation-analysis" class="slide level2">
<h2>Welcome to the workshop on Mediation Analysis🎉</h2>
</section>
<section id="outlines" class="slide level2">
<h2>Outlines 📌</h2>
<p>In this course, you will:</p>
<ul>
<li><p>Understand the fundementals of mediation analysis</p>
<p><em>Apply What is mediation analysis? When and why should you use this?</em></p></li>
<li><p>Master the theoretical framework</p>
<p><em>Understand traditional mediation concepts; Learn principles of causal mediation analysis</em></p></li>
<li><p>Conduct practical analyses using R software</p>
<p><em>Apply appropriate statistical methods to perform mediation analysis</em></p></li>
<li><p>Interpret and communicate results</p>
<p><em>Analyze outputs from mediation analyses and draw meaningful conclusions from results</em></p></li>
</ul>
</section>
<section id="what-is-mediation" class="slide level2">
<h2>What is mediation?</h2>
<p>Mediation analysis is the study of pathways and mechanisms through which an <em>exposure</em> or <em>intervention</em> impacts an outcome.</p>
<p>In clinical and epidemiological research, the primary focus is often on determining whether a specific intervention has an effect on a disease or health outcome. Once this effect is established, the next natural question is to explore the “black box”—the underlying mechanisms that explain how the intervention (or exposure) leads to the observed outcome. As we are not only interested in whether an intervention works, but <em>how</em> it works.</p>
</section>
<section id="mediation-analyses" class="slide level2">
<h2>Mediation analyses</h2>
<p>The techniques to assess the relative magnitude of the direct and indirect effects is referred to as ‘mediation analysis’.</p>
</section>
<section id="mediation-analyses-1" class="slide level2">
<h2>Mediation analyses</h2>
<p>The purpose of mediation analysis is to determine if the effect of a treatment (A) on an outcome (Y) can be explained by a third mediating variable (M). Thus, mediation analysis not only answers whether two variables are related, but also <strong>why and how</strong>.</p>
</section>
<section id="motivation-for-mediation-analysis" class="slide level2">
<h2>Motivation for mediation analysis</h2>
<ol type="1">
<li><p><strong>Explanation and Understanding</strong></p></li>
<li><p><strong>Confirmation and Refutation of Theory</strong></p></li>
<li><p><strong>Refining Interventions</strong></p></li>
</ol>
</section>
<section id="mediation-analysis" class="slide level2">
<h2>Mediation analysis</h2>
<p>Mediation analysis is becoming more popular. <a href="https://pmc.ncbi.nlm.nih.gov/articles/PMC8496983/#F1">Fig. 1</a> shows that both the number of entries in Google Scholar and the number of peer-reviewed articles in PsycINFO that have “mediation analysis” in the title or text have been growing exponentially. </p>
<img data-src="images/trend_mediation.jpg" class="quarto-figure quarto-figure-center r-stretch" width="525"><p class="caption">
Figure 1: Trend Mediation Analysis
</p></section>
<section id="mediation" class="slide level2">
<h2>Mediation</h2>
<p>Overall relationship between A and Y:</p>
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="images/a-y.png" style="width:80.0%;height:50.0%"></p>
</figure>
</div>
</div>
</div>
<p>Relationship between A and Y through M:</p>
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="images/a-m-y.png" style="width:80.0%;height:50.0%"></p>
</figure>
</div>
</div>
</div>
</section>
<section id="mediation-1" class="slide level2">
<h2>Mediation</h2>
<p>How much does BMI explain the relation between physical activity and the risk of diabetes?</p>
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="images/example1.jpg" width="606"></p>
</figure>
</div>
</div>
</div>
<p>How much does inflammation explain the relation between physical activity and the risk of diabetes?</p>
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="images/example2.png" width="611"></p>
</figure>
</div>
</div>
</div>
</section>
<section id="traditional-approaches-for-mediation-analysis" class="slide level2">
<h2>Traditional approaches for mediation analysis</h2>
<p>The two traditional approaches to mediation analysis are <strong>the difference method</strong> and <strong>the product method</strong> (also known as the Baron & Kenny-method).</p>
</section>
<section id="method-1-baron-kenny-the-product-method" class="slide level2">
<h2>Method 1: Baron & Kenny (the product method)</h2>
<p>The following criteria need to be satisfied for a variable to be considered a mediator:</p>
<ol type="1">
<li>The exposure should be associated with the mediator.</li>
<li>The mediator should be associated with the outcome.</li>
<li>The exposure should be associated with the outcome.</li>
<li>When controlling for the mediator, the association between the exposure and outcome should be reduced (up to debate).</li>
</ol>
</section>
<section id="baronkenny" class="slide level2">
<h2>Baron&Kenny</h2>
<h3 id="the-four-steps">The four steps:</h3>
<p><strong>STEP 1:</strong> test association between exposure and outcome (a)</p>
<p><strong>STEP 2:</strong> test association between exposure and mediator (b)</p>
<p><strong>STEP 3:</strong> test association between mediator and outcome, controlling for exposure (c)</p>
<p><strong>STEP 4:</strong> mediation is shown if association between X and Y is reduced to non-significance when M is controlled (d)</p>
</section>
<section id="method-2-difference-approach" class="slide level2">
<h2>Method 2: Difference approach</h2>
<p>Total effect: regress Y on X</p>
<p>Direct effect: regress Y on X and M</p>
<p>If a mediation effect exists, the effect of A on Y will be attenuated when M is included in the regression, indicating the effect of A on Y goes through M. If the effect of A on Y completely disappears, M fully mediates between A and Y (full mediation). If the effect of A on Y still exists, but in a smaller magnitude, M partially mediates between A and Y (partial mediation).</p>
</section>
<section id="build-models" class="slide level2">
<h2>Build models</h2>
<p>Mediator-model:</p>
<p><span class="math inline">\(E(M|A=a, C=c) = \beta_0 + \beta_1a + \beta_2c\)</span> (1.1)</p>
<p>Outcome model with adjustment for mediator:</p>
<p><span class="math inline">\(E(Y|A=a, M=m, C=c) = \theta_0 + \theta_1a + \theta_2m + \theta4c\)</span> (1.2)</p>
<p>Outcome model without adjustment for mediator:</p>
<p><span class="math inline">\(E(Y|A=a, C=c) = \theta_0' + \theta_1'a + \theta4'c\)</span> (1.3)</p>
</section>
<section id="summary" class="slide level2">
<h2>Summary</h2>
<ul>
<li><p><strong>Total effect</strong> = <span class="math inline">\(\theta_1'\)</span>, the total effect of the independent variable on the dependent variable.</p></li>
<li><p><strong>Direct effect</strong> = <span class="math inline">\(\theta_1\)</span>, the effect of the independent variable on the dependent variable that is not mediated by the mediator.</p></li>
<li><p><strong>Mediation effect</strong> = <span class="math inline">\(\beta_1 \cdot \theta_2\)</span> (product method).</p></li>
<li><p><strong>Mediation effect</strong> = <span class="math inline">\(\theta_1' - \theta_1\)</span> (difference method).</p>
<p><em>The terms mediation effect and indirect effect are used synonymously</em>.</p></li>
</ul>
</section>
<section id="product-method-vs-difference-method" class="slide level2">
<h2>Product method VS Difference method</h2>
<p>The algebraic equivalence of the indirect effect using the product method and the difference method will coincide for a continuous outcome on the difference scale. However, the two methods diverge when using a binary outcome and logistic regression.</p>
</section>
<section id="limitations-of-the-traditional-approach" class="slide level2">
<h2>Limitations of the traditional approach</h2>
<ul>
<li>Non-linearity</li>
<li>Interactions</li>
<li>Multiple mediators</li>
</ul>
</section>
<section id="further-limitations-of-the-traditional-approach" class="slide level2">
<h2>Further limitations of the traditional approach</h2>
<p>⚠️ Unmeasured confounding of the Mediator-Outcome path</p>
<p>This assumption can be violated in both observational studies as well as RCTs because while the exposure can sometimes be randomized, it is often not the case that both exposure and mediator are randomized.</p>
</section>
<section id="problem-1-intermediate-confounding" class="slide level2">
<h2>Problem 1-Intermediate confounding</h2>
<img data-src="images/intermediate_eg.jpg" width="1333" class="r-stretch"></section>
<section id="problem-1-intermediate-confounding-1" class="slide level2">
<h2>Problem 1-Intermediate confounding</h2>
<ul>
<li><p>In an <strong>RCT study</strong>, pregnant women are randomized to receive a <strong>lifestyle intervention (A)</strong> aimed at weight loss. While <strong>A (lifestyle intervention)</strong> can be randomized, <strong>M (birth weight)</strong> cannot, as it is a pre-existing condition that results from the pregnancy.</p></li>
<li><p>Gestational diabetes is a descent of A (lifestyle intervention on pregnant women), and a cause of Y (child adiposity), you cannot condition on it because it is a mediator of A-Y(lifestyle intervention-child adiposity);</p></li>
<li><p>Gestational diabetes is also a confounder of M-Y relationship (birth weight-child adiposity), it biases the M-Y path if you do not condition on it.</p></li>
</ul>
</section>
<section id="problem-2-collider-bias" class="slide level2">
<h2>Problem 2-Collider bias</h2>
<img data-src="images/intermediate_eg2.jpg" width="634" class="r-stretch"></section>
<section id="problem-2-collider-bias-1" class="slide level2">
<h2>Problem 2-Collider bias</h2>
<p>In observational studies, the situation might get even more complex.</p>
<ul>
<li><p>C1, C2 represent a series of confounders on the pathways.</p></li>
<li><p>Notably, in the presence of <strong>C2</strong>, <strong>gestational diabetes (L)</strong> acts as a <strong>collider</strong> on the pathway <strong>A → L ← C2 → Y</strong> (maternal pre-pregnancy BMI → gestational diabetes <strong>←</strong> C2 → child adiposity). If we adjust for <strong>gestational diabetes (L)</strong>, it will <strong>open a backdoor path</strong>, introducing bias in the estimation of both <strong>direct and indirect effects</strong>. This could distort the causal interpretation of mediation.</p></li>
</ul>
</section>
<section id="when-can-we-use-the-traditional-approach" class="slide level2">
<h2>When can we use the traditional approach</h2>
<p>When fulfill the criteria, simple tools like regression can be used to estimate a causal mediation effect:</p>
<ul>
<li>no unmeasured confounding</li>
<li>no exposure-mediator interaction</li>
<li>linear relationship</li>
<li>rare binary outcome</li>
</ul>
</section>
<section>
<section id="introduction-to-causal-mediation-analysis" class="title-slide slide level1 center">
<h1>Introduction to causal mediation analysis</h1>
</section>
<section id="concept-of-cause" class="slide level2">
<h2>Concept of Cause</h2>
<h3 id="what-is-a-cause">What is a cause?</h3>
<p>If I do A, then Y will happen. If I press the switch, the light will come on.</p>
</section>
<section id="individual-causal-effect" class="slide level2">
<h2>Individual causal effect</h2>
<p>When investigating health outcomes, we would ideally want to know if <strong>you</strong> do X, then Y will happen. We could have a specific question:</p>
<blockquote>
<p>Will eating more red meat give me higher blood glucose in 1 year?</p>
</blockquote>
</section>
<section id="individual-causal-effect-1" class="slide level2">
<h2>Individual causal effect</h2>
<blockquote>
<p>Will eating more red meat give me higher blood glucose in 1 year?</p>
</blockquote>
<p>To answer this question we would ideally have you consume more red meat over 1 year and measure your blood glucose levels. Then, we would turn back time, and make you eat something else over 1 year and then measure your blood glucose levels again. If there is a <strong>difference</strong> between your two outcomes, then we say there is a causal effect.</p>
<p>But we can never do this in the real world. Only ever observe one of Y(0) or Y(1) the other is counterfactual.</p>
</section>
<section id="average-causal-effect" class="slide level2">
<h2>Average causal effect</h2>
<p>Instead, we can perform a randomized controlled trial. We now ask a slightly different question:</p>
<blockquote>
<p>Will eating more red meat give adults higher blood glucose in 1 year?</p>
</blockquote>
<p>We can randomely assigning one group to consume more red meat and the other group to consume more of something else over 1 year. Then we compare the average blood glucose levels after 1 year in each of the groups. If there is a difference, we could say there is an average causal effect.</p>
</section>
<section id="association-vs-causation" class="slide level2">
<h2><strong>Association vs causation</strong></h2>
<p>What makes it complicated to estimate a causal effect is that we cannot observe the outcome under different treatments.</p>
<p>When we only have a subset of the outcomes, we have an association. This is illustrated in <a href="#/fig-causation-association" class="quarto-xref">Figure 2</a>.</p>
<p></p>
<img data-src="images/causation-association.png" width="562" class="r-stretch quarto-figure-center"><p class="caption">
Figure 2: Relationship between causation and association
</p></section>
<section id="key-assumptions" class="slide level2">
<h2><a href="images/filename.png">Key As</a>sumptions</h2>
<p>If we want to infer a causal effect (i.e., what would have happened, had everyone done A=1 vs A=0), we need three assumptions to be fulfilled:</p>
<ul>
<li><p>Exchangeability</p></li>
<li><p>Consistency</p></li>
<li><p>Positivity</p></li>
</ul>
</section>
<section id="notation" class="slide level2">
<h2>Notation</h2>
<p>A = received treatment/intervention/exposure (e.g., 1 = intervention, 0 = no intervention)</p>
<p>Y = observed outcome (e.g., 1 = developed the outcome, 0 = no outcome)</p>
<p><span class="math inline">\(Y^{a=1}\)</span> = Counterfactual outcome under treatment a =1(i.e., the outcome had everyone, counter to the fact, received treatment a = 1)</p>
<p><span class="math inline">\(Y^{a=0}\)</span> = Counterfactual outcome under treatment a=0 (i.e., the outcome had everyone, counter to the fact, received treatment a = 0)</p>
</section>
<section id="modify-the-terms-a-little-for-average-causal-effect" class="slide level2">
<h2>Modify the terms a little for average causal effect:</h2>
<p><span class="math inline">\(E[Y^{a=1}]\)</span> the average counterfactual outcome, had all subjects in the population received treatment a = 1.</p>
<p><span class="math inline">\(Pr[Y^{a=1}]\)</span> the proportion of subjects that would have developed the outcome Y had all subjects in the population of interest received treatment a = 1.</p>
<p><span class="math inline">\(E[Y^{a=0}]\)</span> the average counterfactual outcome, had no subjects in the population received treatment a = 0.</p>
<p><span class="math inline">\(Pr[Y^{a=0}]\)</span> the proportion of subjects that would have developed the outcome Y had no subjects in the population of interest received treatment a = 0.</p>
</section>
<section id="definition-of-a-causal-effect" class="slide level2">
<h2>Definition of a causal effect</h2>
<p>More formally we can now define a causal effect:</p>
<p><span class="math inline">\(E[Y^{a=1} = 1] - E[Y^{a=0} = 1] \ne 0\)</span></p>
</section>
<section id="causal-mediation-analysis" class="slide level2">
<h2>Causal mediation analysis</h2>
<p>Causal mediation analyses help you establish whether treatment <em>causes</em> the outcome because it <em>causes</em> the mediator.</p>
<p>To do this, causal mediation seek to understand how the paths behave under circumstances different from the observed circumstances (e.g., interventions).</p>
</section>
<section id="why-use-causal-mediation-analysis" class="slide level2">
<h2>Why use causal mediation analysis?</h2>
<p>Causal mediation analysis is an extension of the traditional approach by:</p>
<ul>
<li>outlining all confounding assumptions needed</li>
<li>handling non-linearity and interaction</li>
<li>clearly defining estimands of interest</li>
</ul>
</section>
<section id="causal-approach-brings-alternative-parameters" class="slide level2">
<h2>Causal approach brings alternative parameters</h2>
<ul>
<li>Stand Mediation Analysis
<ul>
<li><p>Total Effect</p></li>
<li><p>Direct Effect</p></li>
<li><p>Indirect Effect</p></li>
</ul></li>
<li>Counterfactual-based Causal Mediation Analysis
<ul>
<li><p>Controlled Direct Effect (CDE(m))</p></li>
<li><p>Natural Direct Effect (NDE)</p></li>
<li><p>Natural Indirect Effect (NIE)</p></li>
</ul></li>
</ul>
</section>
<section id="key-strengths" class="slide level2">
<h2>Key Strengths</h2>
<ul>
<li><p>can provide causal estimates even when mediator or outcome are binary</p></li>
<li><p>can deal with interaction between exposure and mediator</p></li>
<li><p>can deal with intermediate confounding</p></li>
</ul>
</section>
<section id="non-linearity-and-interactions" class="slide level2">
<h2>Non-linearity and interactions</h2>
<p>Neither the product method nor the difference method can take interaction and non-linearity into account.</p>
<p>Causal mediation analysis can take this into account. It can do this using a regression-based approach. It can also use other causal inference analysis methods such as g-computation, that are different from the traditional regression approach in that:</p>
<ul>
<li>it builds a causal model. This model can include non-linearity and interactions</li>
<li>then artificially manipulate the data to set the treatment <strong>and</strong> the mediator to certain values</li>
<li>then predict the outcome using the causal model and contrast the outcomes</li>
</ul>
</section>
<section id="defining-estimands" class="slide level2">
<h2>Defining estimands</h2>
<p>Imagine we have a hypothetical randomized controlled trial where we give participants treatment or no treatment on a specific outcome Y.</p>
<p><span class="math inline">\(Y^{a=1} - Y^{a=0}\)</span></p>
<p>For mediation, we are also interested in the effect of a mediator on this pathway. Now image that we also intervene on the mediator in a new hypothetical randomized controlled trial.</p>
<p><span class="math inline">\(Y^{m=1} - Y^{m=0}\)</span></p>
<p>Now consider if we, in the same trial, could intervene on both because we are interested in whether treatment <em>causes</em> the outcome because it <em>causes</em> the mediator.</p>
</section>
<section id="notation-1" class="slide level2">
<h2>Notation</h2>
<p><span class="math inline">\(Y^a\)</span> = a subject’s outcome if treatment A were set, possible contrary to fact, to a</p>
<p><span class="math inline">\(M^a\)</span> = a subject’s value of the mediator if the exposure A were set to the value of a</p>
<p><span class="math inline">\(Y^{a,m}\)</span> = a subject’s outcome if A were set to a and M were set to m</p>
<p><span class="math inline">\(Y^{a,M_a}\)</span> = a subject’s outcome if A were set to a and M were set the value m would have had had a been set to a. Note, this is a nested counterfactual</p>
</section>
<section id="effect-decomposition" class="slide level2">
<h2>Effect Decomposition</h2>
<p>We can now define these estimands:</p>
<ul>
<li>the controlled direct effect (CDE)</li>
<li>natural direct effect (NDE)</li>
<li>natural indirect effect (NIE)</li>
</ul>
</section></section>
<section>
<section id="estimation-of-effects-using-causal-mediation-analysis" class="title-slide slide level1 center">
<h1>Estimation of effects using causal mediation analysis</h1>
</section>
<section id="natural-direct-effect" class="slide level2">
<h2>Natural direct effect</h2>
<p>The NDE is how much the outcome would change if the treatment a, was set at its natural value versus 0 but for each individual the mediator was kept at the level it would have taken, for that individual, in the absence of the exposure.</p>
</section>
<section id="controlled-direct-effect" class="slide level2">
<h2>Controlled direct effect</h2>
<p>For the controlled direct effect we set m to a specific value. The CDE answer the question, what would be the effect of A on Y, when fixing M at a specific value for everyone in the population.</p>
</section>
<section id="natural-indirect-effect" class="slide level2">
<h2>Natural indirect effect</h2>
<p>The NIE is how much the outcome would change on average if the treatment was fixed at level a but the mediator was changed from the level it would take if a* = 0 to the level it would take if a =1 .</p>
<p>Note that exposure has to have an effect on M otherwise this will be zero.</p>
<p>The NIE asks the question: the effect of exposure that ‘would be prevented if the exposure did not cause the mediator’ (i.e., the portion of the effect for which mediation is ‘necessary’)</p>
<p>This is often the effect we are interested in in biomedical research for questions regarding mediation.</p>
</section>
<section id="proportion-mediation" class="slide level2">
<h2>Proportion mediation</h2>
<p>From this, we can calculate the proportion mediated.</p>
<p><span class="math inline">\(PM = \frac{TNIE}{TE}\)</span></p>
</section>
<section id="total-effect" class="slide level2">
<h2>Total effect</h2>
<p>The total effect can be decomposed as:</p>
<p><span class="math inline">\(TE = PNDE + TNIE\)</span></p>
<p>This is the overall effect of x on y.</p>
<h3 id="section"></h3>
</section>
<section id="further-decomposition" class="slide level2">
<h2>Further decomposition</h2>
<img data-src="images/a-i-y.jpg" width="606" class="r-stretch"></section>
<section id="effect-decomposition-robins-and-greenland" class="slide level2">
<h2>Effect Decomposition (Robins and Greenland)</h2>
<p>When there are interaction and non-linearity, different ways of accounting for the interaction:</p>
<ul>
<li>Pure natural direct effect (PNDE)-indirect effect due to mediator alone</li>
<li>Total natural indirect effect (TNIE)-indirect effect due to mediator and its interaction with the exposure</li>
<li>Pure natural indirect effect (PNIE)</li>
<li>Total nature direct effect (TNDE)</li>
</ul>
<p>TE = PNDE + TNIE = TNDE + PNIE</p>
<h3 id="controlled-direct-effect-1">Controlled direct effect</h3>
<p>The effect of A on Y not mediated through M. Fixing the value of M to m.</p>
<p><span class="math inline">\(Y^{a=1,m}\)</span> - <span class="math inline">\(Y^{a=0,m}\)</span></p>
<p>We intervene on <span class="math inline">\(a\)</span> but fix <span class="math inline">\(m\)</span> to a certain value. The CDE is how much the outcome would change on average if the mediator were fixed at level m uniformly in the population but the treatment were changed from 0 to 1.</p>
<p>This could be relevant in the context of a change in a policy that impacted the mediator for everyone. For instance, if air pollution was a mediator between physical activity and cardiovascular disease risk. If a new policy would change the level of air pollution for all while we implement an intervention to increase biking in the city.</p>
<p>This effect is not used that often. But can be highly relevant in some situations.</p>
</section>
<section id="pure-natural-direct-effect" class="slide level2">
<h2>(Pure) Natural direct effect</h2>
<p>The effect that would remain, if we were to disable the pathway from exposure to mediator.</p>
<p><span class="math inline">\(Y^{a=1,M_a=0}\)</span> - <span class="math inline">\(Y^{a=0,M_a=0}\)</span></p>
<p>The PNDE is how much the outcome would change if the exposure was set at a = 1 versus a* = 0 but for each individual the mediator was kept at the level it would have taken, for that individual, in the absence of the exposure.</p>
<p>Note that the word “natural” refers to the nested counterfactual, the level the mediator would have taken in the absence of exposure. What it would naturally have been in the absence of exposure.</p>
</section>
<section id="total-natural-direct-effect" class="slide level2">
<h2>Total natural direct effect</h2>
<p><span class="math inline">\(Y^{a=1,M_a=1}\)</span> - <span class="math inline">\(Y^{a=0,M_a=1}\)</span></p>
<p>Note, different from above in that the mediator is kept at the level it would have taken in the <strong>presence</strong> of the exposure.</p>
</section>
<section id="total-natural-indirect-effect" class="slide level2">
<h2>(Total) Natural indirect effect</h2>
<p>The effect of the mediator pathway.</p>
<p><span class="math inline">\(Y^{a=1,M_a=1}\)</span> - <span class="math inline">\(Y^{a=1,M_a=0}\)</span></p>
<p>The NIE is how much the outcome would change on average if the exposure were fixed at level a = 1 but the mediator were changed from the level it would take if a* = 0 to the level it would take if a = 1.</p>
<p>Note that exposure has to have an effect on M otherwise this will be zero.</p>
</section>
<section id="pure-natural-indirect-effect" class="slide level2">
<h2>Pure natural indirect effect</h2>
<p><span class="math inline">\(Y^{a=0,M_a=1}\)</span> - <span class="math inline">\(Y^{a=0,M_a=0}\)</span></p>
<p>Note, this is different from the TNIE in that the exposure is set to no intervention.</p>
</section>
<section id="interaction-effects" class="slide level2">
<h2>Interaction effects</h2>
<p><span class="math inline">\(INT_{ref} = PNDE - CDE\)</span></p>
<p>Mediation interaction:</p>
<p><span class="math inline">\(INT_{med} = TNIE - PNIE\)</span></p>
<h3 id="proportions">Proportions</h3>
<p>Proportion CDE:</p>
<p><span class="math inline">\(prop^{CDE} = CDE / TE\)</span></p>
<p>Proportion <span class="math inline">\(INT_{ref}\)</span></p>
<p><span class="math inline">\(prop^{INT_{ref}} = INT_{ref} / TE\)</span></p>
<p>Proportion <span class="math inline">\(INT_{med}\)</span></p>
<p><span class="math inline">\(prop^{INT_{med}} = INT_{med} / TE\)</span></p>
<p>Proportion pure natural indirect effect:</p>
<p><span class="math inline">\(prop^{PNIE} = PNIE / TE\)</span></p>
<p>Proportion mediated:</p>
<p><span class="math inline">\(PM = TNIE / TE\)</span></p>
<p>Proportion attributable to interaction:</p>
<p><span class="math inline">\(INT = (INT_{ref} + INT_{med}) / TE\)</span></p>
<p>Proportion eliminated:</p>
<p><span class="math inline">\(PE = (INT_{ref} + INT_{med} + PNIE) / TE\)</span></p>
</section>
<section id="code-along-practice" class="slide level2">
<h2>Code-along practice</h2>
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