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<!DOCTYPE html>
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<title>Chapter 7 Step 4: Outcome modelling | Understanding Propensity Score Matching</title>
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<meta name="twitter:title" content="Chapter 7 Step 4: Outcome modelling | Understanding Propensity Score Matching" />
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<meta name="author" content="Ehsan Karim" />
<meta name="date" content="2023-03-19" />
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<ul class="summary">
<li><a href="./">Understanding Propensity Score Matching</a></li>
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<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Preamble</a>
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<li class="chapter" data-level="" data-path="index.html"><a href="index.html#description"><i class="fa fa-check"></i>Description</a>
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<li class="chapter" data-level="" data-path="index.html"><a href="index.html#main-references"><i class="fa fa-check"></i>Main references</a></li>
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<li class="chapter" data-level="1" data-path="terms.html"><a href="terms.html"><i class="fa fa-check"></i><b>1</b> Defining Parameter</a>
<ul>
<li class="chapter" data-level="1.1" data-path="terms.html"><a href="terms.html#epidemiological-research-goals"><i class="fa fa-check"></i><b>1.1</b> Epidemiological research goals</a></li>
<li class="chapter" data-level="1.2" data-path="terms.html"><a href="terms.html#potential-outcome"><i class="fa fa-check"></i><b>1.2</b> Potential outcome</a></li>
<li class="chapter" data-level="1.3" data-path="terms.html"><a href="terms.html#parameters-of-interest"><i class="fa fa-check"></i><b>1.3</b> Parameters of interest</a>
<ul>
<li class="chapter" data-level="1.3.1" data-path="terms.html"><a href="terms.html#te"><i class="fa fa-check"></i><b>1.3.1</b> TE</a></li>
<li class="chapter" data-level="1.3.2" data-path="terms.html"><a href="terms.html#ate"><i class="fa fa-check"></i><b>1.3.2</b> ATE</a></li>
<li class="chapter" data-level="1.3.3" data-path="terms.html"><a href="terms.html#interpretation-of-ate"><i class="fa fa-check"></i><b>1.3.3</b> Interpretation of ATE</a></li>
<li class="chapter" data-level="1.3.4" data-path="terms.html"><a href="terms.html#identifiability-assumptions"><i class="fa fa-check"></i><b>1.3.4</b> Identifiability Assumptions</a></li>
<li class="chapter" data-level="1.3.5" data-path="terms.html"><a href="terms.html#att"><i class="fa fa-check"></i><b>1.3.5</b> ATT</a></li>
<li class="chapter" data-level="1.3.6" data-path="terms.html"><a href="terms.html#interpretation-of-att"><i class="fa fa-check"></i><b>1.3.6</b> Interpretation of ATT</a></li>
<li class="chapter" data-level="1.3.7" data-path="terms.html"><a href="terms.html#att-vs.-ate"><i class="fa fa-check"></i><b>1.3.7</b> ATT vs. ATE</a></li>
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<li class="chapter" data-level="2" data-path="balance.html"><a href="balance.html"><i class="fa fa-check"></i><b>2</b> Balance and Overlap</a>
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<li class="chapter" data-level="2.1" data-path="balance.html"><a href="balance.html#balance-1"><i class="fa fa-check"></i><b>2.1</b> Balance</a>
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<li class="chapter" data-level="2.1.1" data-path="balance.html"><a href="balance.html#measures-of-balance"><i class="fa fa-check"></i><b>2.1.1</b> Measures of Balance</a></li>
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<li class="chapter" data-level="2.2" data-path="balance.html"><a href="balance.html#adjustment"><i class="fa fa-check"></i><b>2.2</b> Adjustment</a>
<ul>
<li class="chapter" data-level="2.2.1" data-path="balance.html"><a href="balance.html#why-adjust"><i class="fa fa-check"></i><b>2.2.1</b> Why adjust?</a></li>
<li class="chapter" data-level="2.2.2" data-path="balance.html"><a href="balance.html#adjustment-methods"><i class="fa fa-check"></i><b>2.2.2</b> Adjustment Methods</a></li>
</ul></li>
<li class="chapter" data-level="2.3" data-path="balance.html"><a href="balance.html#lack-of-overlap"><i class="fa fa-check"></i><b>2.3</b> Lack of overlap</a></li>
</ul></li>
<li class="chapter" data-level="3" data-path="ps.html"><a href="ps.html"><i class="fa fa-check"></i><b>3</b> Propensity score</a>
<ul>
<li class="chapter" data-level="3.1" data-path="ps.html"><a href="ps.html#motivating-problem"><i class="fa fa-check"></i><b>3.1</b> Motivating problem</a></li>
<li class="chapter" data-level="3.2" data-path="ps.html"><a href="ps.html#defining-propensity-score"><i class="fa fa-check"></i><b>3.2</b> Defining Propensity score</a>
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<li class="chapter" data-level="3.2.1" data-path="ps.html"><a href="ps.html#theoretical-result"><i class="fa fa-check"></i><b>3.2.1</b> Theoretical result</a></li>
<li class="chapter" data-level="3.2.2" data-path="ps.html"><a href="ps.html#assumptions"><i class="fa fa-check"></i><b>3.2.2</b> Assumptions</a></li>
<li class="chapter" data-level="3.2.3" data-path="ps.html"><a href="ps.html#ways-to-use-ps"><i class="fa fa-check"></i><b>3.2.3</b> Ways to use PS</a></li>
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<li class="chapter" data-level="3.3" data-path="ps.html"><a href="ps.html#ps-matching-steps"><i class="fa fa-check"></i><b>3.3</b> PS Matching Steps</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="s1.html"><a href="s1.html"><i class="fa fa-check"></i><b>4</b> Step 1: Exposure modelling</a>
<ul>
<li class="chapter" data-level="4.1" data-path="s1.html"><a href="s1.html#model-specification"><i class="fa fa-check"></i><b>4.1</b> Model specification</a>
<ul>
<li class="chapter" data-level="4.1.1" data-path="s1.html"><a href="s1.html#updating-model-specification"><i class="fa fa-check"></i><b>4.1.1</b> Updating model specification</a></li>
<li class="chapter" data-level="4.1.2" data-path="s1.html"><a href="s1.html#stability-of-ps"><i class="fa fa-check"></i><b>4.1.2</b> Stability of PS</a></li>
</ul></li>
<li class="chapter" data-level="4.2" data-path="s1.html"><a href="s1.html#variables-to-adjust"><i class="fa fa-check"></i><b>4.2</b> Variables to adjust</a>
<ul>
<li class="chapter" data-level="4.2.1" data-path="s1.html"><a href="s1.html#best-approach"><i class="fa fa-check"></i><b>4.2.1</b> Best approach</a></li>
<li class="chapter" data-level="4.2.2" data-path="s1.html"><a href="s1.html#general-guideline-of-type-of-variables"><i class="fa fa-check"></i><b>4.2.2</b> General guideline of type of variables</a></li>
<li class="chapter" data-level="4.2.3" data-path="s1.html"><a href="s1.html#what-not-to-include"><i class="fa fa-check"></i><b>4.2.3</b> What NOT to include</a></li>
<li class="chapter" data-level="4.2.4" data-path="s1.html"><a href="s1.html#mediators"><i class="fa fa-check"></i><b>4.2.4</b> Mediators</a></li>
<li class="chapter" data-level="4.2.5" data-path="s1.html"><a href="s1.html#unmeasured-confounding"><i class="fa fa-check"></i><b>4.2.5</b> Unmeasured confounding</a></li>
</ul></li>
<li class="chapter" data-level="4.3" data-path="s1.html"><a href="s1.html#model-selection"><i class="fa fa-check"></i><b>4.3</b> Model selection</a>
<ul>
<li class="chapter" data-level="4.3.1" data-path="s1.html"><a href="s1.html#based-on-association-with-outcome"><i class="fa fa-check"></i><b>4.3.1</b> Based on association with outcome</a></li>
<li class="chapter" data-level="4.3.2" data-path="s1.html"><a href="s1.html#based-on-association-with-exposure"><i class="fa fa-check"></i><b>4.3.2</b> Based on association with exposure</a></li>
</ul></li>
<li class="chapter" data-level="4.4" data-path="s1.html"><a href="s1.html#alternative-modelling-strategies"><i class="fa fa-check"></i><b>4.4</b> Alternative modelling strategies</a></li>
<li class="chapter" data-level="4.5" data-path="s1.html"><a href="s1.html#ps-estimation"><i class="fa fa-check"></i><b>4.5</b> PS estimation</a></li>
</ul></li>
<li class="chapter" data-level="5" data-path="s2.html"><a href="s2.html"><i class="fa fa-check"></i><b>5</b> Step 2: Propensity score Matching</a>
<ul>
<li class="chapter" data-level="5.1" data-path="s2.html"><a href="s2.html#matching-method-nn"><i class="fa fa-check"></i><b>5.1</b> Matching method NN</a></li>
<li class="chapter" data-level="5.2" data-path="s2.html"><a href="s2.html#initial-fit"><i class="fa fa-check"></i><b>5.2</b> Initial fit</a></li>
<li class="chapter" data-level="5.3" data-path="s2.html"><a href="s2.html#fine-tuning-add-caliper"><i class="fa fa-check"></i><b>5.3</b> Fine tuning: add caliper</a></li>
<li class="chapter" data-level="5.4" data-path="s2.html"><a href="s2.html#things-to-keep-track-of"><i class="fa fa-check"></i><b>5.4</b> Things to keep track of</a></li>
<li class="chapter" data-level="5.5" data-path="s2.html"><a href="s2.html#matches"><i class="fa fa-check"></i><b>5.5</b> Matches</a></li>
<li class="chapter" data-level="5.6" data-path="s2.html"><a href="s2.html#other-matching-algorithms"><i class="fa fa-check"></i><b>5.6</b> Other matching algorithms</a></li>
</ul></li>
<li class="chapter" data-level="6" data-path="s3.html"><a href="s3.html"><i class="fa fa-check"></i><b>6</b> Step 3: Balance and overlap</a>
<ul>
<li class="chapter" data-level="6.1" data-path="s3.html"><a href="s3.html#assessment-of-balance-by-smd"><i class="fa fa-check"></i><b>6.1</b> Assessment of Balance by SMD</a></li>
<li class="chapter" data-level="6.2" data-path="s3.html"><a href="s3.html#smd-vs.-p-values"><i class="fa fa-check"></i><b>6.2</b> SMD vs. P-values</a></li>
<li class="chapter" data-level="6.3" data-path="s3.html"><a href="s3.html#vizualization-for-overlap"><i class="fa fa-check"></i><b>6.3</b> Vizualization for Overlap</a></li>
<li class="chapter" data-level="6.4" data-path="s3.html"><a href="s3.html#variance-ratio-1"><i class="fa fa-check"></i><b>6.4</b> Variance ratio</a></li>
<li class="chapter" data-level="6.5" data-path="s3.html"><a href="s3.html#close-inspection-of-boundaries"><i class="fa fa-check"></i><b>6.5</b> Close inspection of boundaries</a></li>
<li class="chapter" data-level="6.6" data-path="s3.html"><a href="s3.html#unsatirfactory-balance"><i class="fa fa-check"></i><b>6.6</b> Unsatirfactory balance</a></li>
</ul></li>
<li class="chapter" data-level="7" data-path="s4.html"><a href="s4.html"><i class="fa fa-check"></i><b>7</b> Step 4: Outcome modelling</a>
<ul>
<li class="chapter" data-level="7.1" data-path="s4.html"><a href="s4.html#crude-outcome-model"><i class="fa fa-check"></i><b>7.1</b> Crude outcome model</a></li>
<li class="chapter" data-level="7.2" data-path="s4.html"><a href="s4.html#double-adjustment"><i class="fa fa-check"></i><b>7.2</b> Double-adjustment</a></li>
<li class="chapter" data-level="7.3" data-path="s4.html"><a href="s4.html#adjusted-outcome-model"><i class="fa fa-check"></i><b>7.3</b> Adjusted outcome model</a></li>
<li class="chapter" data-level="7.4" data-path="s4.html"><a href="s4.html#variance-considerations"><i class="fa fa-check"></i><b>7.4</b> Variance considerations</a>
<ul>
<li class="chapter" data-level="7.4.1" data-path="s4.html"><a href="s4.html#cluster-option"><i class="fa fa-check"></i><b>7.4.1</b> Cluster option</a></li>
<li class="chapter" data-level="7.4.2" data-path="s4.html"><a href="s4.html#bootstrap"><i class="fa fa-check"></i><b>7.4.2</b> Bootstrap</a></li>
</ul></li>
<li class="chapter" data-level="7.5" data-path="s4.html"><a href="s4.html#estimate-obtained"><i class="fa fa-check"></i><b>7.5</b> Estimate obtained</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="compare.html"><a href="compare.html"><i class="fa fa-check"></i><b>8</b> PS vs. Regression</a>
<ul>
<li class="chapter" data-level="8.1" data-path="compare.html"><a href="compare.html#data-simulation"><i class="fa fa-check"></i><b>8.1</b> Data Simulation</a></li>
<li class="chapter" data-level="8.2" data-path="compare.html"><a href="compare.html#treatment-effect-from-counterfactuals"><i class="fa fa-check"></i><b>8.2</b> Treatment effect from counterfactuals</a></li>
<li class="chapter" data-level="8.3" data-path="compare.html"><a href="compare.html#treatment-effect-from-regression"><i class="fa fa-check"></i><b>8.3</b> Treatment effect from Regression</a></li>
<li class="chapter" data-level="8.4" data-path="compare.html"><a href="compare.html#treatment-effect-from-ps"><i class="fa fa-check"></i><b>8.4</b> Treatment effect from PS</a></li>
<li class="chapter" data-level="8.5" data-path="compare.html"><a href="compare.html#non-linear-model"><i class="fa fa-check"></i><b>8.5</b> Non-linear Model</a>
<ul>
<li class="chapter" data-level="8.5.1" data-path="compare.html"><a href="compare.html#data-generation"><i class="fa fa-check"></i><b>8.5.1</b> Data generation</a></li>
<li class="chapter" data-level="8.5.2" data-path="compare.html"><a href="compare.html#regression"><i class="fa fa-check"></i><b>8.5.2</b> Regression</a></li>
<li class="chapter" data-level="8.5.3" data-path="compare.html"><a href="compare.html#ps-1"><i class="fa fa-check"></i><b>8.5.3</b> PS</a></li>
<li class="chapter" data-level="8.5.4" data-path="compare.html"><a href="compare.html#machine-learning"><i class="fa fa-check"></i><b>8.5.4</b> Machine learning</a></li>
<li class="chapter" data-level="8.5.5" data-path="compare.html"><a href="compare.html#regression-is-doomed"><i class="fa fa-check"></i><b>8.5.5</b> Regression is doomed?</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="9" data-path="misspecify.html"><a href="misspecify.html"><i class="fa fa-check"></i><b>9</b> PS vs. Double robust methods</a>
<ul>
<li class="chapter" data-level="9.1" data-path="misspecify.html"><a href="misspecify.html#complex-data-simulation"><i class="fa fa-check"></i><b>9.1</b> Complex Data Simulation</a>
<ul>
<li class="chapter" data-level="" data-path="misspecify.html"><a href="misspecify.html#true-exposure-model"><i class="fa fa-check"></i>True Exposure Model</a></li>
<li class="chapter" data-level="" data-path="misspecify.html"><a href="misspecify.html#true-outcome-model"><i class="fa fa-check"></i>True Outcome Model</a></li>
<li class="chapter" data-level="" data-path="misspecify.html"><a href="misspecify.html#outcomes-and-exposures-are-complex-functions-of-measured-covariates"><i class="fa fa-check"></i>Outcomes and exposures are complex functions of measured covariates</a></li>
</ul></li>
<li class="chapter" data-level="9.2" data-path="misspecify.html"><a href="misspecify.html#understanding-finite-sample-bias"><i class="fa fa-check"></i><b>9.2</b> Understanding finite sample bias</a></li>
<li class="chapter" data-level="9.3" data-path="misspecify.html"><a href="misspecify.html#estimation-using-different-methods"><i class="fa fa-check"></i><b>9.3</b> Estimation using different methods</a>
<ul>
<li class="chapter" data-level="9.3.1" data-path="misspecify.html"><a href="misspecify.html#regression-1"><i class="fa fa-check"></i><b>9.3.1</b> Regression</a></li>
<li class="chapter" data-level="9.3.2" data-path="misspecify.html"><a href="misspecify.html#propensity-score"><i class="fa fa-check"></i><b>9.3.2</b> Propensity score</a></li>
<li class="chapter" data-level="9.3.3" data-path="misspecify.html"><a href="misspecify.html#double-machine-learning-method"><i class="fa fa-check"></i><b>9.3.3</b> Double machine learning method</a></li>
<li class="chapter" data-level="9.3.4" data-path="misspecify.html"><a href="misspecify.html#augmented-inverse-probability-weighting"><i class="fa fa-check"></i><b>9.3.4</b> Augmented Inverse probability weighting</a></li>
<li class="chapter" data-level="9.3.5" data-path="misspecify.html"><a href="misspecify.html#double-robust-method-tmle"><i class="fa fa-check"></i><b>9.3.5</b> Double robust method (TMLE)</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="10" data-path="guide.html"><a href="guide.html"><i class="fa fa-check"></i><b>10</b> Reporting Guidelines</a>
<ul>
<li class="chapter" data-level="10.1" data-path="guide.html"><a href="guide.html#discipline-specific-reviews"><i class="fa fa-check"></i><b>10.1</b> Discipline-specific Reviews</a></li>
<li class="chapter" data-level="10.2" data-path="guide.html"><a href="guide.html#suggested-guidelines"><i class="fa fa-check"></i><b>10.2</b> Suggested Guidelines</a></li>
<li class="chapter" data-level="10.3" data-path="guide.html"><a href="guide.html#additional-topics"><i class="fa fa-check"></i><b>10.3</b> Additional topics</a></li>
</ul></li>
<li class="chapter" data-level="11" data-path="final.html"><a href="final.html"><i class="fa fa-check"></i><b>11</b> Final Words</a>
<ul>
<li class="chapter" data-level="11.1" data-path="final.html"><a href="final.html#common-misconception"><i class="fa fa-check"></i><b>11.1</b> Common misconception</a></li>
<li class="chapter" data-level="11.2" data-path="final.html"><a href="final.html#benifits-of-ps"><i class="fa fa-check"></i><b>11.2</b> Benifits of PS</a></li>
<li class="chapter" data-level="11.3" data-path="final.html"><a href="final.html#limitations-of-ps"><i class="fa fa-check"></i><b>11.3</b> Limitations of PS</a></li>
<li class="chapter" data-level="11.4" data-path="final.html"><a href="final.html#when-ps-may-not-be-useful"><i class="fa fa-check"></i><b>11.4</b> When PS may not be useful?</a></li>
<li class="chapter" data-level="11.5" data-path="final.html"><a href="final.html#software"><i class="fa fa-check"></i><b>11.5</b> Software</a></li>
<li class="chapter" data-level="11.6" data-path="final.html"><a href="final.html#further-resources"><i class="fa fa-check"></i><b>11.6</b> Further Resources</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="references.html"><a href="references.html"><i class="fa fa-check"></i>References</a></li>
<li class="divider"></li>
<li><a href="https://ehsank.com/" target="blank">Ehsan Karim</a></li>
</ul>
</nav>
</div>
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<div class="body-inner">
<div class="book-header" role="navigation">
<h1>
<i class="fa fa-circle-o-notch fa-spin"></i><a href="./">Understanding Propensity Score Matching</a>
</h1>
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<section class="normal" id="section-">
<div id="s4" class="section level1 hasAnchor" number="7">
<h1><span class="header-section-number">Chapter 7</span> Step 4: Outcome modelling<a href="s4.html#s4" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<ul>
<li>Some flexibility in choosing outcome model
<ul>
<li>considered independent of exposure modelling</li>
<li>some propose double robust approach</li>
<li>adjusting imbalanced covariates only?
<ul>
<li>double-adjustment may address residual confounding <span class="citation">(<a href="#ref-nguyen2017double" role="doc-biblioref">Nguyen et al. 2017</a>)</span></li>
</ul></li>
</ul></li>
</ul>
<div id="crude-outcome-model" class="section level2 hasAnchor" number="7.1">
<h2><span class="header-section-number">7.1</span> Crude outcome model<a href="s4.html#crude-outcome-model" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Estimate the effect of treatment on outcomes using propensity score-matched sample</p>
<div class="sourceCode" id="cb76"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb76-1"><a href="s4.html#cb76-1" aria-hidden="true" tabindex="-1"></a>fit3 <span class="ot"><-</span> <span class="fu">glm</span>(cholesterol<span class="sc">~</span>diabetes,</span>
<span id="cb76-2"><a href="s4.html#cb76-2" aria-hidden="true" tabindex="-1"></a> <span class="at">family=</span>binomial, <span class="at">data =</span> matched.data)</span>
<span id="cb76-3"><a href="s4.html#cb76-3" aria-hidden="true" tabindex="-1"></a><span class="fu">publish</span>(fit3)</span></code></pre></div>
<pre><code>## Variable Units OddsRatio CI.95 p-value
## diabetes 0.90 [0.54;1.50] 0.6984</code></pre>
</div>
<div id="double-adjustment" class="section level2 hasAnchor" number="7.2">
<h2><span class="header-section-number">7.2</span> Double-adjustment<a href="s4.html#double-adjustment" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Estimate the effect of treatment on outcomes using propensity score-matched sample, and adjust for imbalanced covariate</p>
<div class="sourceCode" id="cb78"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb78-1"><a href="s4.html#cb78-1" aria-hidden="true" tabindex="-1"></a>fit3r <span class="ot"><-</span> <span class="fu">glm</span>(cholesterol<span class="sc">~</span>diabetes <span class="sc">+</span> race,</span>
<span id="cb78-2"><a href="s4.html#cb78-2" aria-hidden="true" tabindex="-1"></a> <span class="at">family=</span>binomial, <span class="at">data =</span> matched.data)</span>
<span id="cb78-3"><a href="s4.html#cb78-3" aria-hidden="true" tabindex="-1"></a><span class="fu">publish</span>(fit3r)</span></code></pre></div>
<pre><code>## Variable Units OddsRatio CI.95 p-value
## diabetes 0.89 [0.54;1.49] 0.6657
## race Black Ref
## Hispanic 0.96 [0.46;2.02] 0.9165
## Other 1.32 [0.63;2.78] 0.4581
## White 0.58 [0.30;1.13] 0.1095</code></pre>
</div>
<div id="adjusted-outcome-model" class="section level2 hasAnchor" number="7.3">
<h2><span class="header-section-number">7.3</span> Adjusted outcome model<a href="s4.html#adjusted-outcome-model" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Adjust for all covariates, again! (suggested)</p>
<div class="sourceCode" id="cb80"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb80-1"><a href="s4.html#cb80-1" aria-hidden="true" tabindex="-1"></a>out.formula <span class="ot"><-</span> <span class="fu">as.formula</span>(<span class="fu">paste</span>(<span class="st">"cholesterol"</span>, <span class="st">"~ diabetes +"</span>, </span>
<span id="cb80-2"><a href="s4.html#cb80-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">paste</span>(baselinevars, </span>
<span id="cb80-3"><a href="s4.html#cb80-3" aria-hidden="true" tabindex="-1"></a> <span class="at">collapse =</span> <span class="st">"+"</span>)))</span>
<span id="cb80-4"><a href="s4.html#cb80-4" aria-hidden="true" tabindex="-1"></a>out.formula</span></code></pre></div>
<pre><code>## cholesterol ~ diabetes + gender + age + race + education + married +
## bmi</code></pre>
<div class="sourceCode" id="cb82"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb82-1"><a href="s4.html#cb82-1" aria-hidden="true" tabindex="-1"></a>fit3b <span class="ot"><-</span> <span class="fu">glm</span>(out.formula,</span>
<span id="cb82-2"><a href="s4.html#cb82-2" aria-hidden="true" tabindex="-1"></a> <span class="at">family=</span>binomial, <span class="at">data =</span> matched.data)</span>
<span id="cb82-3"><a href="s4.html#cb82-3" aria-hidden="true" tabindex="-1"></a><span class="fu">publish</span>(fit3b)</span></code></pre></div>
<pre><code>## Variable Units OddsRatio CI.95 p-value
## diabetes 0.86 [0.51;1.46] 0.5794126
## gender Female Ref
## Male 0.38 [0.21;0.69] 0.0012767
## age 0.95 [0.93;0.97] < 1e-04
## race Black Ref
## Hispanic 0.72 [0.31;1.65] 0.4346787
## Other 0.77 [0.34;1.73] 0.5224157
## White 0.51 [0.25;1.04] 0.0649791
## education College Ref
## High.School 0.70 [0.39;1.24] 0.2215142
## School 0.93 [0.35;2.43] 0.8791455
## married Married Ref
## Never.married 0.48 [0.15;1.54] 0.2173180
## Previously.married 0.84 [0.45;1.57] 0.5900732
## bmi 0.93 [0.89;0.97] 0.0005547</code></pre>
<p>The above analysis do not take matched pair into consideration while regressing.</p>
</div>
<div id="variance-considerations" class="section level2 hasAnchor" number="7.4">
<h2><span class="header-section-number">7.4</span> Variance considerations<a href="s4.html#variance-considerations" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Literature proposes different strategies:</p>
<ul>
<li>do not control for pairs / clusters
<ul>
<li>use <code>glm</code> as is</li>
</ul></li>
<li>control for pairs / clusters
<ul>
<li>use <code>cluster</code> option (preferred)</li>
<li>use GEE or</li>
<li>use conditional logistic</li>
</ul></li>
</ul>
<div id="cluster-option" class="section level3 hasAnchor" number="7.4.1">
<h3><span class="header-section-number">7.4.1</span> Cluster option<a href="s4.html#cluster-option" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Here is an example using cluster option:</p>
<div class="sourceCode" id="cb84"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb84-1"><a href="s4.html#cb84-1" aria-hidden="true" tabindex="-1"></a><span class="fu">require</span>(jtools)</span>
<span id="cb84-2"><a href="s4.html#cb84-2" aria-hidden="true" tabindex="-1"></a><span class="fu">summ</span>(fit3b, <span class="at">rubust =</span> <span class="st">"HC0"</span>, <span class="at">confint =</span> <span class="cn">TRUE</span>, <span class="at">digists =</span> <span class="dv">3</span>, </span>
<span id="cb84-3"><a href="s4.html#cb84-3" aria-hidden="true" tabindex="-1"></a> <span class="at">cluster =</span> <span class="st">"subclass"</span>, <span class="at">model.info =</span> <span class="cn">FALSE</span>, </span>
<span id="cb84-4"><a href="s4.html#cb84-4" aria-hidden="true" tabindex="-1"></a> <span class="at">model.fit =</span> <span class="cn">FALSE</span>, <span class="at">exp =</span> <span class="cn">TRUE</span>)</span></code></pre></div>
<table class="table table-striped table-hover table-condensed table-responsive" style="width: auto !important; margin-left: auto; margin-right: auto;border-bottom: 0;">
<thead>
<tr>
<th style="text-align:left;">
</th>
<th style="text-align:right;">
exp(Est.)
</th>
<th style="text-align:right;">
2.5%
</th>
<th style="text-align:right;">
97.5%
</th>
<th style="text-align:right;">
z val.
</th>
<th style="text-align:right;">
p
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;font-weight: bold;">
(Intercept)
</td>
<td style="text-align:right;">
100.02
</td>
<td style="text-align:right;">
8.74
</td>
<td style="text-align:right;">
1144.55
</td>
<td style="text-align:right;">
3.70
</td>
<td style="text-align:right;">
0.00
</td>
</tr>
<tr>
<td style="text-align:left;font-weight: bold;">
diabetes
</td>
<td style="text-align:right;">
0.86
</td>
<td style="text-align:right;">
0.51
</td>
<td style="text-align:right;">
1.46
</td>
<td style="text-align:right;">
-0.55
</td>
<td style="text-align:right;">
0.58
</td>
</tr>
<tr>
<td style="text-align:left;font-weight: bold;">
genderMale
</td>
<td style="text-align:right;">
0.38
</td>
<td style="text-align:right;">
0.21
</td>
<td style="text-align:right;">
0.69
</td>
<td style="text-align:right;">
-3.22
</td>
<td style="text-align:right;">
0.00
</td>
</tr>
<tr>
<td style="text-align:left;font-weight: bold;">
age
</td>
<td style="text-align:right;">
0.95
</td>
<td style="text-align:right;">
0.93
</td>
<td style="text-align:right;">
0.97
</td>
<td style="text-align:right;">
-4.47
</td>
<td style="text-align:right;">
0.00
</td>
</tr>
<tr>
<td style="text-align:left;font-weight: bold;">
raceHispanic
</td>
<td style="text-align:right;">
0.72
</td>
<td style="text-align:right;">
0.31
</td>
<td style="text-align:right;">
1.65
</td>
<td style="text-align:right;">
-0.78
</td>
<td style="text-align:right;">
0.43
</td>
</tr>
<tr>
<td style="text-align:left;font-weight: bold;">
raceOther
</td>
<td style="text-align:right;">
0.77
</td>
<td style="text-align:right;">
0.34
</td>
<td style="text-align:right;">
1.73
</td>
<td style="text-align:right;">
-0.64
</td>
<td style="text-align:right;">
0.52
</td>
</tr>
<tr>
<td style="text-align:left;font-weight: bold;">
raceWhite
</td>
<td style="text-align:right;">
0.51
</td>
<td style="text-align:right;">
0.25
</td>
<td style="text-align:right;">
1.04
</td>
<td style="text-align:right;">
-1.85
</td>
<td style="text-align:right;">
0.06
</td>
</tr>
<tr>
<td style="text-align:left;font-weight: bold;">
educationHigh.School
</td>
<td style="text-align:right;">
0.70
</td>
<td style="text-align:right;">
0.39
</td>
<td style="text-align:right;">
1.24
</td>
<td style="text-align:right;">
-1.22
</td>
<td style="text-align:right;">
0.22
</td>
</tr>
<tr>
<td style="text-align:left;font-weight: bold;">
educationSchool
</td>
<td style="text-align:right;">
0.93
</td>
<td style="text-align:right;">
0.35
</td>
<td style="text-align:right;">
2.43
</td>
<td style="text-align:right;">
-0.15
</td>
<td style="text-align:right;">
0.88
</td>
</tr>
<tr>
<td style="text-align:left;font-weight: bold;">
marriedNever.married
</td>
<td style="text-align:right;">
0.48
</td>
<td style="text-align:right;">
0.15
</td>
<td style="text-align:right;">
1.54
</td>
<td style="text-align:right;">
-1.23
</td>
<td style="text-align:right;">
0.22
</td>
</tr>
<tr>
<td style="text-align:left;font-weight: bold;">
marriedPreviously.married
</td>
<td style="text-align:right;">
0.84
</td>
<td style="text-align:right;">
0.45
</td>
<td style="text-align:right;">
1.57
</td>
<td style="text-align:right;">
-0.54
</td>
<td style="text-align:right;">
0.59
</td>
</tr>
<tr>
<td style="text-align:left;font-weight: bold;">
bmi
</td>
<td style="text-align:right;">
0.93
</td>
<td style="text-align:right;">
0.89
</td>
<td style="text-align:right;">
0.97
</td>
<td style="text-align:right;">
-3.45
</td>
<td style="text-align:right;">
0.00
</td>
</tr>
</tbody>
<tfoot>
<tr>
<td style="padding: 0; " colspan="100%">
<sup></sup> Standard errors: MLE
</td>
</tr>
</tfoot>
</table>
</div>
<div id="bootstrap" class="section level3 hasAnchor" number="7.4.2">
<h3><span class="header-section-number">7.4.2</span> Bootstrap<a href="s4.html#bootstrap" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<ul>
<li>Bootstrap for matched pair for WOR <span class="citation">(<a href="#ref-austin2014use" role="doc-biblioref">Austin and Small 2014</a>)</span>
<ul>
<li>straightforward method to estimate SE</li>
<li>may not be appropriate for WR</li>
</ul></li>
<li>Following is an example of block bootstrap; see MatchIt package <a href="https://cran.r-project.org/web/packages/MatchIt/vignettes/estimating-effects.html">vignettes</a> for additional details.</li>
</ul>
<div class="sourceCode" id="cb85"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb85-1"><a href="s4.html#cb85-1" aria-hidden="true" tabindex="-1"></a><span class="co"># For binary outcomes</span></span>
<span id="cb85-2"><a href="s4.html#cb85-2" aria-hidden="true" tabindex="-1"></a><span class="co"># with covariate adjustment and bootstrapping</span></span>
<span id="cb85-3"><a href="s4.html#cb85-3" aria-hidden="true" tabindex="-1"></a>pair_ids <span class="ot"><-</span> <span class="fu">levels</span>(matched.data<span class="sc">$</span>subclass)</span>
<span id="cb85-4"><a href="s4.html#cb85-4" aria-hidden="true" tabindex="-1"></a>est_fun <span class="ot"><-</span> <span class="cf">function</span>(pairs, i) {</span>
<span id="cb85-5"><a href="s4.html#cb85-5" aria-hidden="true" tabindex="-1"></a> <span class="co"># See MatchIt vignettes</span></span>
<span id="cb85-6"><a href="s4.html#cb85-6" aria-hidden="true" tabindex="-1"></a> <span class="co">#Compute number of times each pair is present</span></span>
<span id="cb85-7"><a href="s4.html#cb85-7" aria-hidden="true" tabindex="-1"></a> numreps <span class="ot"><-</span> <span class="fu">table</span>(pairs[i])</span>
<span id="cb85-8"><a href="s4.html#cb85-8" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb85-9"><a href="s4.html#cb85-9" aria-hidden="true" tabindex="-1"></a> <span class="co">#For each pair p, copy corresponding matched.data row indices numreps[p] times</span></span>
<span id="cb85-10"><a href="s4.html#cb85-10" aria-hidden="true" tabindex="-1"></a> ids <span class="ot"><-</span> <span class="fu">unlist</span>(<span class="fu">lapply</span>(pair_ids[pair_ids <span class="sc">%in%</span> <span class="fu">names</span>(numreps)],</span>
<span id="cb85-11"><a href="s4.html#cb85-11" aria-hidden="true" tabindex="-1"></a> <span class="cf">function</span>(p) <span class="fu">rep</span>(<span class="fu">which</span>(matched.data<span class="sc">$</span>subclass <span class="sc">==</span> p), </span>
<span id="cb85-12"><a href="s4.html#cb85-12" aria-hidden="true" tabindex="-1"></a> numreps[p])))</span>
<span id="cb85-13"><a href="s4.html#cb85-13" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb85-14"><a href="s4.html#cb85-14" aria-hidden="true" tabindex="-1"></a> <span class="co">#Subset matched.data with block bootstrapped ids</span></span>
<span id="cb85-15"><a href="s4.html#cb85-15" aria-hidden="true" tabindex="-1"></a> matched.data_boot <span class="ot"><-</span> matched.data[ids,]</span>
<span id="cb85-16"><a href="s4.html#cb85-16" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb85-17"><a href="s4.html#cb85-17" aria-hidden="true" tabindex="-1"></a> <span class="co">#Fitting outcome the model</span></span>
<span id="cb85-18"><a href="s4.html#cb85-18" aria-hidden="true" tabindex="-1"></a> out.formula <span class="ot"><-</span> <span class="fu">as.formula</span>(<span class="fu">paste</span>(<span class="st">"cholesterol"</span>, <span class="st">"~ diabetes +"</span>, </span>
<span id="cb85-19"><a href="s4.html#cb85-19" aria-hidden="true" tabindex="-1"></a> <span class="fu">paste</span>(baselinevars, </span>
<span id="cb85-20"><a href="s4.html#cb85-20" aria-hidden="true" tabindex="-1"></a> <span class="at">collapse =</span> <span class="st">"+"</span>)))</span>
<span id="cb85-21"><a href="s4.html#cb85-21" aria-hidden="true" tabindex="-1"></a> fit_boot <span class="ot"><-</span> <span class="fu">glm</span>(out.formula,</span>
<span id="cb85-22"><a href="s4.html#cb85-22" aria-hidden="true" tabindex="-1"></a> <span class="at">family =</span> <span class="fu">binomial</span>(<span class="at">link =</span> <span class="st">"logit"</span>), </span>
<span id="cb85-23"><a href="s4.html#cb85-23" aria-hidden="true" tabindex="-1"></a> <span class="at">weights =</span> weights,</span>
<span id="cb85-24"><a href="s4.html#cb85-24" aria-hidden="true" tabindex="-1"></a> <span class="at">data =</span> matched.data_boot)</span>
<span id="cb85-25"><a href="s4.html#cb85-25" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb85-26"><a href="s4.html#cb85-26" aria-hidden="true" tabindex="-1"></a> <span class="co">#Estimate potential outcomes for each unit</span></span>
<span id="cb85-27"><a href="s4.html#cb85-27" aria-hidden="true" tabindex="-1"></a> <span class="co">#Under control</span></span>
<span id="cb85-28"><a href="s4.html#cb85-28" aria-hidden="true" tabindex="-1"></a> matched.data_boot<span class="sc">$</span>diabetes <span class="ot"><-</span> <span class="dv">0</span></span>
<span id="cb85-29"><a href="s4.html#cb85-29" aria-hidden="true" tabindex="-1"></a> P0 <span class="ot"><-</span> <span class="fu">weighted.mean</span>(<span class="fu">predict</span>(fit_boot, matched.data_boot, <span class="at">type =</span> <span class="st">"response"</span>),</span>
<span id="cb85-30"><a href="s4.html#cb85-30" aria-hidden="true" tabindex="-1"></a> <span class="at">w =</span> matched.data_boot<span class="sc">$</span>weights)</span>
<span id="cb85-31"><a href="s4.html#cb85-31" aria-hidden="true" tabindex="-1"></a> Odds0 <span class="ot"><-</span> P0 <span class="sc">/</span> (<span class="dv">1</span> <span class="sc">-</span> P0)</span>
<span id="cb85-32"><a href="s4.html#cb85-32" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb85-33"><a href="s4.html#cb85-33" aria-hidden="true" tabindex="-1"></a> <span class="co">#Under treatment</span></span>
<span id="cb85-34"><a href="s4.html#cb85-34" aria-hidden="true" tabindex="-1"></a> matched.data_boot<span class="sc">$</span>diabetes <span class="ot"><-</span> <span class="dv">1</span></span>
<span id="cb85-35"><a href="s4.html#cb85-35" aria-hidden="true" tabindex="-1"></a> P1 <span class="ot"><-</span> <span class="fu">weighted.mean</span>(<span class="fu">predict</span>(fit_boot, matched.data_boot, <span class="at">type =</span> <span class="st">"response"</span>),</span>
<span id="cb85-36"><a href="s4.html#cb85-36" aria-hidden="true" tabindex="-1"></a> <span class="at">w =</span> matched.data_boot<span class="sc">$</span>weights)</span>
<span id="cb85-37"><a href="s4.html#cb85-37" aria-hidden="true" tabindex="-1"></a> Odds1 <span class="ot"><-</span> P1 <span class="sc">/</span> (<span class="dv">1</span> <span class="sc">-</span> P1)</span>
<span id="cb85-38"><a href="s4.html#cb85-38" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb85-39"><a href="s4.html#cb85-39" aria-hidden="true" tabindex="-1"></a> <span class="co">#Return marginal odds ratio</span></span>
<span id="cb85-40"><a href="s4.html#cb85-40" aria-hidden="true" tabindex="-1"></a> <span class="fu">return</span>(Odds1 <span class="sc">/</span> Odds0)</span>
<span id="cb85-41"><a href="s4.html#cb85-41" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb85-42"><a href="s4.html#cb85-42" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb85-43"><a href="s4.html#cb85-43" aria-hidden="true" tabindex="-1"></a>boot_est <span class="ot"><-</span> <span class="fu">boot</span>(pair_ids, est_fun, <span class="at">R =</span> <span class="dv">1000</span>)</span>
<span id="cb85-44"><a href="s4.html#cb85-44" aria-hidden="true" tabindex="-1"></a><span class="co"># R must be larger than # of data rows</span></span>
<span id="cb85-45"><a href="s4.html#cb85-45" aria-hidden="true" tabindex="-1"></a>boot_est</span></code></pre></div>
<pre><code>##
## ORDINARY NONPARAMETRIC BOOTSTRAP
##
##
## Call:
## boot(data = pair_ids, statistic = est_fun, R = 1000)
##
##
## Bootstrap Statistics :
## original bias std. error
## t1* 0.8706933 0.04061083 0.2376491</code></pre>
<div class="sourceCode" id="cb87"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb87-1"><a href="s4.html#cb87-1" aria-hidden="true" tabindex="-1"></a><span class="fu">boot.ci</span>(boot_est, <span class="at">type =</span> <span class="st">"bca"</span>)</span></code></pre></div>
<pre><code>## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
## Based on 1000 bootstrap replicates
##
## CALL :
## boot.ci(boot.out = boot_est, type = "bca")
##
## Intervals :
## Level BCa
## 95% ( 0.5239, 1.3936 )
## Calculations and Intervals on Original Scale</code></pre>
</div>
</div>
<div id="estimate-obtained" class="section level2 hasAnchor" number="7.5">
<h2><span class="header-section-number">7.5</span> Estimate obtained<a href="s4.html#estimate-obtained" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<ul>
<li>The example compared <code>diabetic</code> (a treated group; target) vs <code>Not diabetic</code> (untreated).</li>
<li>Thc corresponding treatment effect estimate is known as
<ul>
<li>Average Treatment Effects on the Treated (<strong>ATT</strong>)</li>
</ul></li>
<li>Other estimates from PS analysis (e.g., PS weighting) are possible that compared the whole population
<ul>
<li>what if everyone treated vs. what if nobody was treated (ATE)</li>
</ul></li>
</ul>
<p>You can see further comparison of results <a href="https://ehsanx.github.io/TMLEworkshop/rhc-data-description.html">elsewhere</a>.</p>
</div>
</div>
<h3>References<a href="references.html#references" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div id="refs" class="references csl-bib-body hanging-indent">
<div id="ref-austin2014use" class="csl-entry">
Austin, Peter C, and Dylan S Small. 2014. <span>“The Use of Bootstrapping When Using Propensity-Score Matching Without Replacement: A Simulation Study.”</span> <em>Statistics in Medicine</em> 33 (24): 4306–19.
</div>
<div id="ref-nguyen2017double" class="csl-entry">
Nguyen, Tri-Long, Gary S Collins, Jessica Spence, Jean-Pierre Daurès, PJ Devereaux, Paul Landais, and Yannick Le Manach. 2017. <span>“Double-Adjustment in Propensity Score Matching Analysis: Choosing a Threshold for Considering Residual Imbalance.”</span> <em>BMC Medical Research Methodology</em> 17 (1): 1–8.
</div>
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