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<!DOCTYPE html>
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<title>Chapter 6 Step 3: Balance and overlap | Understanding Propensity Score Matching</title>
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<meta name="author" content="Ehsan Karim" />
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<ul class="summary">
<li><a href="./">Understanding Propensity Score Matching</a></li>
<li class="divider"></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Preamble</a>
<ul>
<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>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#version-history"><i class="fa fa-check"></i>Version history</a></li>
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<li class="chapter" data-level="" data-path="index.html"><a href="index.html#prerequisites"><i class="fa fa-check"></i>Prerequisites</a>
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<li class="chapter" data-level="" data-path="index.html"><a href="index.html#license"><i class="fa fa-check"></i>License</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>
</ul></li>
</ul></li>
<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>
<ul>
<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>
<ul>
<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>
</ul></li>
<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>
<ul>
<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>
</ul></li>
<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>
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<i class="fa fa-circle-o-notch fa-spin"></i><a href="./">Understanding Propensity Score Matching</a>
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<div id="s3" class="section level1 hasAnchor" number="6">
<h1><span class="header-section-number">Chapter 6</span> Step 3: Balance and overlap<a href="s3.html#s3" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p><strong>Balance is more important than prediction</strong>!</p>
<ul>
<li>Criteria to assess success of step 2: PS estimation
<ul>
<li>better balance</li>
<li>better overlap [no extrapolation!]</li>
<li>PS = 0 or PS = 1 needs close inspection</li>
</ul></li>
</ul>
<div id="assessment-of-balance-by-smd" class="section level2 hasAnchor" number="6.1">
<h2><span class="header-section-number">6.1</span> Assessment of Balance by SMD<a href="s3.html#assessment-of-balance-by-smd" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<ul>
<li>balance = similarity of the covariate distributions</li>
</ul>
<p><strong>Full data</strong>:</p>
<div class="sourceCode" id="cb55"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb55-1"><a href="s3.html#cb55-1" aria-hidden="true" tabindex="-1"></a>tab1e <span class="ot"><-</span> <span class="fu">CreateTableOne</span>(<span class="at">vars =</span> baselinevars,</span>
<span id="cb55-2"><a href="s3.html#cb55-2" aria-hidden="true" tabindex="-1"></a> <span class="at">data =</span> analytic, </span>
<span id="cb55-3"><a href="s3.html#cb55-3" aria-hidden="true" tabindex="-1"></a> <span class="at">strata =</span> <span class="st">"diabetes"</span>,</span>
<span id="cb55-4"><a href="s3.html#cb55-4" aria-hidden="true" tabindex="-1"></a> <span class="at">includeNA =</span> <span class="cn">TRUE</span>,</span>
<span id="cb55-5"><a href="s3.html#cb55-5" aria-hidden="true" tabindex="-1"></a> <span class="at">test =</span> <span class="cn">TRUE</span>, <span class="at">smd =</span> <span class="cn">TRUE</span>)</span>
<span id="cb55-6"><a href="s3.html#cb55-6" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(tab1e, <span class="at">showAllLevels =</span> <span class="cn">FALSE</span>, <span class="at">smd =</span> <span class="cn">TRUE</span>, <span class="at">test =</span> <span class="cn">TRUE</span>)</span></code></pre></div>
<pre><code>## Stratified by diabetes
## 0 1 p test SMD
## n 1232 330
## gender = Male (%) 738 (59.9) 221 (67.0) 0.023 0.147
## age (mean (SD)) 50.54 (17.23) 63.04 (12.87) <0.001 0.822
## race (%) 0.110 0.151
## Black 253 (20.5) 71 (21.5)
## Hispanic 220 (17.9) 64 (19.4)
## Other 169 (13.7) 59 (17.9)
## White 590 (47.9) 136 (41.2)
## education (%) 0.005 0.186
## College 649 (52.7) 157 (47.6)
## High.School 518 (42.0) 140 (42.4)
## School 65 ( 5.3) 33 (10.0)
## married (%) <0.001 0.282
## Married 727 (59.0) 194 (58.8)
## Never.married 201 (16.3) 27 ( 8.2)
## Previously.married 304 (24.7) 109 (33.0)
## bmi (mean (SD)) 29.14 (7.03) 33.01 (7.65) <0.001 0.526</code></pre>
<p><strong>Matched data</strong>:</p>
<div class="sourceCode" id="cb57"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb57-1"><a href="s3.html#cb57-1" aria-hidden="true" tabindex="-1"></a>matched.data <span class="ot"><-</span> <span class="fu">match.data</span>(match.obj)</span>
<span id="cb57-2"><a href="s3.html#cb57-2" aria-hidden="true" tabindex="-1"></a>tab1m <span class="ot"><-</span> <span class="fu">CreateTableOne</span>(<span class="at">vars =</span> baselinevars, </span>
<span id="cb57-3"><a href="s3.html#cb57-3" aria-hidden="true" tabindex="-1"></a> <span class="at">strata =</span> <span class="st">"diabetes"</span>, </span>
<span id="cb57-4"><a href="s3.html#cb57-4" aria-hidden="true" tabindex="-1"></a> <span class="at">data =</span> matched.data, </span>
<span id="cb57-5"><a href="s3.html#cb57-5" aria-hidden="true" tabindex="-1"></a> <span class="at">includeNA =</span> <span class="cn">TRUE</span>, </span>
<span id="cb57-6"><a href="s3.html#cb57-6" aria-hidden="true" tabindex="-1"></a> <span class="at">test =</span> <span class="cn">TRUE</span>, <span class="at">smd =</span> <span class="cn">TRUE</span>)</span></code></pre></div>
<p>Compare the similarity of baseline characteristics between treated and untreated subjects in a the propensity score-matched sample.</p>
<ul>
<li>In this case, we will compare SMD < 0.1 or not.</li>
<li>In some literature, other generous values (0.25) are proposed. <span class="citation">(<a href="#ref-austin2011introduction" role="doc-biblioref">Austin 2011b</a>)</span></li>
</ul>
<div class="sourceCode" id="cb58"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb58-1"><a href="s3.html#cb58-1" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(tab1m, <span class="at">showAllLevels =</span> <span class="cn">FALSE</span>, <span class="at">smd =</span> <span class="cn">TRUE</span>, <span class="at">test =</span> <span class="cn">FALSE</span>) </span></code></pre></div>
<pre><code>## Stratified by diabetes
## 0 1 SMD
## n 316 316
## gender = Male (%) 218 (69.0) 212 (67.1) 0.041
## age (mean (SD)) 63.03 (13.48) 62.67 (12.87) 0.027
## race (%) 0.105
## Black 79 (25.0) 68 (21.5)
## Hispanic 58 (18.4) 61 (19.3)
## Other 44 (13.9) 53 (16.8)
## White 135 (42.7) 134 (42.4)
## education (%) 0.007
## College 153 (48.4) 152 (48.1)
## High.School 133 (42.1) 134 (42.4)
## School 30 ( 9.5) 30 ( 9.5)
## married (%) 0.099
## Married 183 (57.9) 186 (58.9)
## Never.married 20 ( 6.3) 27 ( 8.5)
## Previously.married 113 (35.8) 103 (32.6)
## bmi (mean (SD)) 32.38 (7.62) 32.63 (7.20) 0.035</code></pre>
<div class="sourceCode" id="cb60"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb60-1"><a href="s3.html#cb60-1" aria-hidden="true" tabindex="-1"></a><span class="fu">require</span>(cobalt)</span>
<span id="cb60-2"><a href="s3.html#cb60-2" aria-hidden="true" tabindex="-1"></a><span class="fu">bal.plot</span>(match.obj, </span>
<span id="cb60-3"><a href="s3.html#cb60-3" aria-hidden="true" tabindex="-1"></a> <span class="at">var.name =</span> <span class="st">"distance"</span>, </span>
<span id="cb60-4"><a href="s3.html#cb60-4" aria-hidden="true" tabindex="-1"></a> <span class="at">which =</span> <span class="st">"both"</span>, </span>
<span id="cb60-5"><a href="s3.html#cb60-5" aria-hidden="true" tabindex="-1"></a> <span class="at">type =</span> <span class="st">"histogram"</span>, </span>
<span id="cb60-6"><a href="s3.html#cb60-6" aria-hidden="true" tabindex="-1"></a> <span class="at">mirror =</span> <span class="cn">TRUE</span>)</span></code></pre></div>
<p><img src="UnderstandingPropensityScore_files/figure-html/pscobalt-1.png" width="672" /></p>
<div class="sourceCode" id="cb61"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb61-1"><a href="s3.html#cb61-1" aria-hidden="true" tabindex="-1"></a><span class="fu">bal.tab</span>(match.obj, <span class="at">un =</span> <span class="cn">TRUE</span>, </span>
<span id="cb61-2"><a href="s3.html#cb61-2" aria-hidden="true" tabindex="-1"></a> <span class="at">thresholds =</span> <span class="fu">c</span>(<span class="at">m =</span> .<span class="dv">1</span>))</span></code></pre></div>
<pre><code>## Call
## matchit(formula = ps.formula, data = analytic, method = "nearest",
## distance = "logit", replace = FALSE, caliper = 0.2 * sd(logitPS),
## ratio = 1)
##
## Balance Measures
## Type Diff.Un Diff.Adj M.Threshold
## distance Distance 0.9493 0.0276 Balanced, <0.1
## gender_Male Binary 0.0707 -0.0190 Balanced, <0.1
## age Contin. 0.9715 -0.0278 Balanced, <0.1
## race_Black Binary 0.0098 -0.0348 Balanced, <0.1
## race_Hispanic Binary 0.0154 0.0095 Balanced, <0.1
## race_Other Binary 0.0416 0.0285 Balanced, <0.1
## race_White Binary -0.0668 -0.0032 Balanced, <0.1
## education_College Binary -0.0510 -0.0032 Balanced, <0.1
## education_High.School Binary 0.0038 0.0032 Balanced, <0.1
## education_School Binary 0.0472 0.0000 Balanced, <0.1
## married_Married Binary -0.0022 0.0095 Balanced, <0.1
## married_Never.married Binary -0.0813 0.0222 Balanced, <0.1
## married_Previously.married Binary 0.0835 -0.0316 Balanced, <0.1
## bmi Contin. 0.5051 0.0338 Balanced, <0.1
##
## Balance tally for mean differences
## count
## Balanced, <0.1 14
## Not Balanced, >0.1 0
##
## Variable with the greatest mean difference
## Variable Diff.Adj M.Threshold
## race_Black -0.0348 Balanced, <0.1
##
## Sample sizes
## Control Treated
## All 1232 330
## Matched 316 316
## Unmatched 916 14</code></pre>
<div class="sourceCode" id="cb63"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb63-1"><a href="s3.html#cb63-1" aria-hidden="true" tabindex="-1"></a><span class="fu">love.plot</span>(match.obj, <span class="at">binary =</span> <span class="st">"std"</span>, </span>
<span id="cb63-2"><a href="s3.html#cb63-2" aria-hidden="true" tabindex="-1"></a> <span class="at">thresholds =</span> <span class="fu">c</span>(<span class="at">m =</span> .<span class="dv">1</span>)) </span></code></pre></div>
<p><img src="UnderstandingPropensityScore_files/figure-html/pscobalt-2.png" width="672" /></p>
</div>
<div id="smd-vs.-p-values" class="section level2 hasAnchor" number="6.2">
<h2><span class="header-section-number">6.2</span> SMD vs. P-values<a href="s3.html#smd-vs.-p-values" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Possible to get p-values to check balance: but strongly discouraged</p>
<ul>
<li>P-value based balance assessment can be influenced by sample size</li>
</ul>
<div class="sourceCode" id="cb64"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb64-1"><a href="s3.html#cb64-1" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(tab1m, <span class="at">showAllLevels =</span> <span class="cn">FALSE</span>, <span class="at">smd =</span> <span class="cn">FALSE</span>, <span class="at">test =</span> <span class="cn">TRUE</span>) </span></code></pre></div>
<pre><code>## Stratified by diabetes
## 0 1 p test
## n 316 316
## gender = Male (%) 218 (69.0) 212 (67.1) 0.670
## age (mean (SD)) 63.03 (13.48) 62.67 (12.87) 0.733
## race (%) 0.629
## Black 79 (25.0) 68 (21.5)
## Hispanic 58 (18.4) 61 (19.3)
## Other 44 (13.9) 53 (16.8)
## White 135 (42.7) 134 (42.4)
## education (%) 0.996
## College 153 (48.4) 152 (48.1)
## High.School 133 (42.1) 134 (42.4)
## School 30 ( 9.5) 30 ( 9.5)
## married (%) 0.465
## Married 183 (57.9) 186 (58.9)
## Never.married 20 ( 6.3) 27 ( 8.5)
## Previously.married 113 (35.8) 103 (32.6)
## bmi (mean (SD)) 32.38 (7.62) 32.63 (7.20) 0.662</code></pre>
<p>Assessment of balance in the matched data</p>
<div class="sourceCode" id="cb66"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb66-1"><a href="s3.html#cb66-1" aria-hidden="true" tabindex="-1"></a>smd.res <span class="ot"><-</span> <span class="fu">ExtractSmd</span>(tab1m)</span>
<span id="cb66-2"><a href="s3.html#cb66-2" aria-hidden="true" tabindex="-1"></a><span class="fu">t</span>(<span class="fu">round</span>(smd.res,<span class="dv">2</span>))</span></code></pre></div>
<pre><code>## gender age race education married bmi
## 1 vs 2 0.04 0.03 0.11 0.01 0.1 0.03</code></pre>
</div>
<div id="vizualization-for-overlap" class="section level2 hasAnchor" number="6.3">
<h2><span class="header-section-number">6.3</span> Vizualization for Overlap<a href="s3.html#vizualization-for-overlap" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<div class="sourceCode" id="cb68"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb68-1"><a href="s3.html#cb68-1" aria-hidden="true" tabindex="-1"></a><span class="fu">boxplot</span>(PS <span class="sc">~</span> diabetes, <span class="at">data =</span> analytic, </span>
<span id="cb68-2"><a href="s3.html#cb68-2" aria-hidden="true" tabindex="-1"></a> <span class="at">lwd =</span> <span class="dv">2</span>, <span class="at">ylab =</span> <span class="st">'PS'</span>)</span>
<span id="cb68-3"><a href="s3.html#cb68-3" aria-hidden="true" tabindex="-1"></a><span class="fu">stripchart</span>(PS <span class="sc">~</span> diabetes, <span class="at">vertical =</span> <span class="cn">TRUE</span>, </span>
<span id="cb68-4"><a href="s3.html#cb68-4" aria-hidden="true" tabindex="-1"></a> <span class="at">data =</span> analytic, <span class="at">method =</span> <span class="st">"jitter"</span>, </span>
<span id="cb68-5"><a href="s3.html#cb68-5" aria-hidden="true" tabindex="-1"></a> <span class="at">add =</span> <span class="cn">TRUE</span>, <span class="at">pch =</span> <span class="dv">20</span>, <span class="at">col =</span> <span class="st">'blue'</span>)</span></code></pre></div>
<p><img src="UnderstandingPropensityScore_files/figure-html/ps3vv-1.png" width="30%" /></p>
<div class="sourceCode" id="cb69"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb69-1"><a href="s3.html#cb69-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(match.obj, <span class="at">type =</span> <span class="st">"jitter"</span>)</span></code></pre></div>
<p><img src="UnderstandingPropensityScore_files/figure-html/ps8vv-1.png" width="50%" /></p>
<pre><code>## [1] "To identify the units, use first mouse button; to stop, use second."</code></pre>
<pre><code>## integer(0)</code></pre>
<p>Vizualization for assessing overlap issues</p>
<div class="sourceCode" id="cb72"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb72-1"><a href="s3.html#cb72-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(match.obj, <span class="at">type =</span> <span class="st">"hist"</span>)</span></code></pre></div>
<p><img src="UnderstandingPropensityScore_files/figure-html/ps9vv-1.png" width="50%" /></p>
</div>
<div id="variance-ratio-1" class="section level2 hasAnchor" number="6.4">
<h2><span class="header-section-number">6.4</span> Variance ratio<a href="s3.html#variance-ratio-1" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<ul>
<li>Variance ratios <span class="math inline">\(\sim\)</span> 1 means:
<ul>
<li>equal variances in groups</li>
<li>group balance</li>
<li>could vary from 1/2 to 2</li>
<li>other cut-points are suggested as well (0.8 to 1.2)</li>
</ul></li>
</ul>
<p>See <span class="citation">Stuart (<a href="#ref-stuart2010matching" role="doc-biblioref">2010</a>)</span> and <span class="citation">Austin (<a href="#ref-austin2009balance" role="doc-biblioref">2009</a>)</span></p>
<div class="sourceCode" id="cb73"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb73-1"><a href="s3.html#cb73-1" aria-hidden="true" tabindex="-1"></a><span class="fu">require</span>(cobalt)</span>
<span id="cb73-2"><a href="s3.html#cb73-2" aria-hidden="true" tabindex="-1"></a>baltab.res <span class="ot"><-</span> <span class="fu">bal.tab</span>(<span class="at">x =</span> match.obj, <span class="at">data =</span> analytic, </span>
<span id="cb73-3"><a href="s3.html#cb73-3" aria-hidden="true" tabindex="-1"></a> <span class="at">treat =</span> analytic<span class="sc">$</span>diabetes, </span>
<span id="cb73-4"><a href="s3.html#cb73-4" aria-hidden="true" tabindex="-1"></a> <span class="at">disp.v.ratio =</span> <span class="cn">TRUE</span>)</span>
<span id="cb73-5"><a href="s3.html#cb73-5" aria-hidden="true" tabindex="-1"></a>baltab.res</span></code></pre></div>
<pre><code>## Call
## matchit(formula = ps.formula, data = analytic, method = "nearest",
## distance = "logit", replace = FALSE, caliper = 0.2 * sd(logitPS),
## ratio = 1)
##
## Balance Measures
## Type Diff.Adj V.Ratio.Adj
## distance Distance 0.0276 1.0992
## gender_Male Binary -0.0190 .
## age Contin. -0.0278 0.9114
## race_Black Binary -0.0348 .
## race_Hispanic Binary 0.0095 .
## race_Other Binary 0.0285 .
## race_White Binary -0.0032 .
## education_College Binary -0.0032 .
## education_High.School Binary 0.0032 .
## education_School Binary 0.0000 .
## married_Married Binary 0.0095 .
## married_Never.married Binary 0.0222 .
## married_Previously.married Binary -0.0316 .
## bmi Contin. 0.0338 0.8928
##
## Sample sizes
## Control Treated
## All 1232 330
## Matched 316 316
## Unmatched 916 14</code></pre>
</div>
<div id="close-inspection-of-boundaries" class="section level2 hasAnchor" number="6.5">
<h2><span class="header-section-number">6.5</span> Close inspection of boundaries<a href="s3.html#close-inspection-of-boundaries" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<div class="sourceCode" id="cb75"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb75-1"><a href="s3.html#cb75-1" aria-hidden="true" tabindex="-1"></a><span class="fu">boxplot</span>(PS <span class="sc">~</span> diabetes, <span class="at">data =</span> matched.data, </span>
<span id="cb75-2"><a href="s3.html#cb75-2" aria-hidden="true" tabindex="-1"></a> <span class="at">lwd =</span> <span class="dv">2</span>, <span class="at">ylab =</span> <span class="st">'PS'</span>, <span class="at">ylim=</span><span class="fu">c</span>(<span class="dv">0</span>,<span class="dv">1</span>))</span>
<span id="cb75-3"><a href="s3.html#cb75-3" aria-hidden="true" tabindex="-1"></a><span class="fu">stripchart</span>(PS <span class="sc">~</span> diabetes, <span class="at">vertical =</span> <span class="cn">TRUE</span>, </span>
<span id="cb75-4"><a href="s3.html#cb75-4" aria-hidden="true" tabindex="-1"></a> <span class="at">data =</span> matched.data, <span class="at">method =</span> <span class="st">"jitter"</span>, </span>
<span id="cb75-5"><a href="s3.html#cb75-5" aria-hidden="true" tabindex="-1"></a> <span class="at">add =</span> <span class="cn">TRUE</span>, <span class="at">pch =</span> <span class="dv">20</span>, <span class="at">col =</span> <span class="st">'blue'</span>)</span>
<span id="cb75-6"><a href="s3.html#cb75-6" aria-hidden="true" tabindex="-1"></a><span class="fu">abline</span>(<span class="at">h=</span><span class="fu">c</span>(<span class="dv">0</span><span class="fl">+0.05</span>,<span class="dv">1</span><span class="fl">-0.05</span>), <span class="at">col =</span> <span class="st">"red"</span>, <span class="at">lty =</span> <span class="dv">2</span>)</span></code></pre></div>
<p><img src="UnderstandingPropensityScore_files/figure-html/ps3cgvv-1.png" width="30%" /></p>
<ul>
<li>Sensitivity analysis should be done with trimming.</li>
<li>Have consequences in interpretation
<ul>
<li>target population may be unclear</li>
</ul></li>
</ul>
</div>
<div id="unsatirfactory-balance" class="section level2 hasAnchor" number="6.6">
<h2><span class="header-section-number">6.6</span> Unsatirfactory balance<a href="s3.html#unsatirfactory-balance" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<ul>
<li>Iterative process
<ul>
<li>Best strategy is to go back to step 2, and make changes in the PS model specification</li>
</ul></li>
</ul>
</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-austin2009balance" class="csl-entry">
———. 2009. <span>“Balance Diagnostics for Comparing the Distribution of Baseline Covariates Between Treatment Groups in Propensity-Score Matched Samples.”</span> <em>Statistics in Medicine</em> 28 (25): 3083–3107.
</div>
<div id="ref-austin2011introduction" class="csl-entry">
———. 2011b. <span>“An Introduction to Propensity Score Methods for Reducing the Effects of Confounding in Observational Studies.”</span> <em>Multivariate Behavioral Research</em> 46 (3): 399–424.
</div>
<div id="ref-stuart2010matching" class="csl-entry">
Stuart, Elizabeth A. 2010. <span>“Matching Methods for Causal Inference: A Review and a Look Forward.”</span> <em>Statistical Science: A Review Journal of the Institute of Mathematical Statistics</em> 25 (1): 1.
</div>
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