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<!DOCTYPE html>
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<title>Chapter 2 Balance and Overlap | Understanding Propensity Score Matching</title>
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<meta name="twitter:title" content="Chapter 2 Balance and Overlap | 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>
<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>
<ul>
<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>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#comments"><i class="fa fa-check"></i>Comments</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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<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="balance" class="section level1 hasAnchor" number="2">
<h1><span class="header-section-number">Chapter 2</span> Balance and Overlap<a href="balance.html#balance" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<div id="balance-1" class="section level2 hasAnchor" number="2.1">
<h2><span class="header-section-number">2.1</span> Balance<a href="balance.html#balance-1" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Table 1 in any RCT paper is very important to assess the balance of the baseline characteristics between the two treatment arms.</p>
<p><strong>Balance in RCT</strong>:</p>
<table>
<tbody>
<tr class="odd">
<td><img src="images/RCT.png" /></td>
<td><img src="images/balance.png" /></td>
</tr>
</tbody>
</table>
<p>Compare the proportions in each age categories by eye-balling in making an evaluation.</p>
<p><strong>In absence of randomization</strong>:</p>
<table>
<tbody>
<tr class="odd">
<td><img src="images/condRCT.png" /></td>
<td><img src="images/imbalance.png" /></td>
</tr>
</tbody>
</table>
<div id="measures-of-balance" class="section level3 hasAnchor" number="2.1.1">
<h3><span class="header-section-number">2.1.1</span> Measures of Balance<a href="balance.html#measures-of-balance" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>To compare baseline characteristics between the two treatment groups, we use</p>
<ul>
<li>t-test (for continuous variables) and</li>
<li>chi-square test (for categorical variables)</li>
</ul>
<p>Today we will introduce a new concept, known as standardized mean differences (SMD) that can be also used to compare baseline characteristics between the two treatment groups.</p>
<div id="smd" class="section level4 hasAnchor" number="2.1.1.1">
<h4><span class="header-section-number">2.1.1.1</span> SMD<a href="balance.html#smd" class="anchor-section" aria-label="Anchor link to header"></a></h4>
<p><span class="citation">Austin (<a href="#ref-austin2011introduction" role="doc-biblioref">2011b</a>)</span></p>
<ul>
<li>For continuous confounders:
<ul>
<li><span class="math inline">\(SDM_{continuous} = \frac{\bar{L}_{A=1} - \bar{L}_{A=0}}{\sqrt{\frac{(n_1 - 1) \cdot s^2_{A=1} + (n_0 - 1) \cdot s^2_{A=0}}{n_1 + n_0 - 2}}}\)</span>
<ul>
<li><span class="math inline">\(\bar{L}_{A=1}\)</span>: mean of the continuous variable L for the treated group (A=1)</li>
<li><span class="math inline">\(\bar{L}_{A=0}\)</span>: mean of the continuous variable L for the untreated group (A=0)</li>
<li><span class="math inline">\(n_1\)</span>: number of treated individuals (A=1)</li>
<li><span class="math inline">\(n_0\)</span>: number of untreated individuals (A=0)</li>
<li><span class="math inline">\(s^2_{A=1}\)</span>: variance of the continuous variable L for the treated group (A=1)</li>
<li><span class="math inline">\(s^2_{A=0}\)</span>: variance of the continuous variable L for the untreated group (A=0)</li>
</ul></li>
</ul></li>
<li>For binary confounders:
<ul>
<li><span class="math inline">\(SDM_{binary} = \frac{\hat{p}_{A=1} - \hat{p}_{A=0}}{\sqrt{\frac{\hat{p}_{A=1} \times (1 - \hat{p}_{A=1}) + \hat{p}_{A=0} \times (1 - \hat{p}_{A=0})}{2}}}\)</span> where <span class="math inline">\(\hat{p}{A=1}\)</span> and <span class="math inline">\(\hat{p}{A=0}\)</span> are the proportions of the two binary groups.</li>
</ul></li>
<li>For categorical confounders (more than 2 categories):
<ul>
<li>Calculating SMD for categorical variables is slightly more complicated, requiring some additional calculation and averaging out the results from category-specific SMDs. See formulas in <span class="citation">Yang and Dalton (<a href="#ref-yang2012unified" role="doc-biblioref">2012</a>)</span>. It is also possible to report SMD for each category separately (similar to binary).</li>
</ul></li>
</ul>
<p><img src="images/SMD.png" /></p>
<p>Generally, <span class="math inline">\(0.1\)</span> is used as a cut-point. But some suggest more liberal cut-points. More on that later.</p>
<p><strong>COVID example</strong> from <span class="citation">Gautret et al. (<a href="#ref-gautret2020hydroxychloroquine" role="doc-biblioref">2020</a>)</span></p>
<p>p-value vs. SMD</p>
<p><img src="images/hql.png" /></p>
<ul>
<li>Statistical tests are affected by sample size
<ul>
<li>t-test</li>
<li>McNemar tests</li>
<li>Wilcoxon rank test</li>
</ul></li>
<li>Balance of what?
<ul>
<li>statistical tests make inference about balance at the population level</li>
<li>but we are really interested in balance at the sample level</li>
</ul></li>
</ul>
</div>
<div id="variance-ratio" class="section level4 hasAnchor" number="2.1.1.2">
<h4><span class="header-section-number">2.1.1.2</span> Variance ratio<a href="balance.html#variance-ratio" class="anchor-section" aria-label="Anchor link to header"></a></h4>
<p>Variances of baseline characteristics between comparator groups under consideration. Suggested cut-point rages are (0.5 to 2). More liberal cut-points are also used in the literature. More on this later.</p>
</div>
</div>
</div>
<div id="adjustment" class="section level2 hasAnchor" number="2.2">
<h2><span class="header-section-number">2.2</span> Adjustment<a href="balance.html#adjustment" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<div id="why-adjust" class="section level3 hasAnchor" number="2.2.1">
<h3><span class="header-section-number">2.2.1</span> Why adjust?<a href="balance.html#why-adjust" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>In absence of randomization, treatment effect estimate ATE = <span class="math inline">\(E[Y|A=1] - E[Y|A=0]\)</span> includes</p>
<ul>
<li>Treatment effect</li>
<li>Systematic differences in 2 groups (‘confounding’)
<ul>
<li>Doctors may prescribe tx more to frail and older age patients.</li>
<li>In here, <span class="math inline">\(L\)</span> = age is a confounder.</li>
</ul></li>
</ul>
<p>In absence of randomization, if age is a known confounder, conditioning can solve the problem:</p>
<table>
<colgroup>
<col width="50%" />
<col width="50%" />
</colgroup>
<tbody>
<tr class="odd">
<td>Causal effect for young (<span class="math inline">\(<50\)</span>)</td>
<td><span class="math inline">\(E[Y|A=1, L =\)</span> <code>younger age</code><span class="math inline">\(]\)</span> - <span class="math inline">\(E[Y|A=0, L =\)</span> <code>younger age</code><span class="math inline">\(]\)</span></td>
</tr>
<tr class="even">
<td>Causal effect for old (<span class="math inline">\(\ge 50\)</span>)</td>
<td><span class="math inline">\(E[Y|A=1, L =\)</span> <code>older age</code><span class="math inline">\(]\)</span> - <span class="math inline">\(E[Y|A=0, L =\)</span> <code>older age</code><span class="math inline">\(]\)</span></td>
</tr>
</tbody>
</table>
<p>Conditional exchangeability; only works if <span class="math inline">\(L\)</span> is measured.</p>
</div>
<div id="adjustment-methods" class="section level3 hasAnchor" number="2.2.2">
<h3><span class="header-section-number">2.2.2</span> Adjustment Methods<a href="balance.html#adjustment-methods" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Adjustment of imbalance could mean</p>
<ul>
<li>exact matching <span class="citation">(<a href="#ref-rubin1973matching" role="doc-biblioref">Rubin 1973</a>)</span></li>
<li>stratification, restriction</li>
</ul>
<table>
<tbody>
<tr>
<td style="text-align:left;">
<img src="images/info.png" />
</td>
<td style="text-align:left;color: white !important;background-color: #3A3B3C !important;">
When L includes a large number of covariates, matching method would result in a small sample size.
</td>
</tr>
</tbody>
</table>
<p>Regression and machine learning methods are popular adjustment methods. Below is a list of adjustment methods that uses different combinations of exposure and outcome modelling.</p>
<table>
<colgroup>
<col width="32%" />
<col width="39%" />
<col width="28%" />
</colgroup>
<thead>
<tr class="header">
<th>Method</th>
<th align="center">Exposure model <span class="math inline">\((\color{green}{\text{A}} \sim ...)\)</span></th>
<th align="center">Outcome Model <span class="math inline">\((Y \sim ...)\)</span></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>Regression (Gauss 1821 <span class="math inline">\(\dagger\)</span>)</td>
<td align="center">No <span class="math inline">\(\color{green}{\text{A}}\)</span> modelling</td>
<td align="center">Yes <span class="math inline">\((Y \sim \color{green}{\text{A}}+\color{red}{\text{L}})\)</span></td>
</tr>
<tr class="even">
<td>Instrumental variable methods (2SLS) <span class="math inline">\(\dagger\dagger\)</span></td>
<td align="center"><span class="math inline">\(A \sim IV + L\)</span>, but explicitly to obtain <span class="math inline">\(\hat{A}\)</span></td>
<td align="center"><span class="math inline">\(Y \sim \hat{A} + L\)</span></td>
</tr>
<tr class="odd">
<td>Propensity score matching <span class="citation">(<a href="#ref-rosenbaum1983central" role="doc-biblioref">Rosenbaum and Rubin 1983</a>)</span></td>
<td align="center">Yes <span class="math inline">\((\color{green}{\text{A}} \sim \color{red}{\text{L}})\)</span></td>
<td align="center">Crude comparison on matched data <span class="math inline">\((Y \sim \color{green}{\text{A}})\)</span></td>
</tr>
<tr class="even">
<td>Propensity score Weighting <span class="citation">(<a href="#ref-rosenbaum1983central" role="doc-biblioref">Rosenbaum and Rubin 1983</a>)</span></td>
<td align="center">Yes <span class="math inline">\((\color{green}{\text{A}} \sim \color{red}{\text{L}})\)</span></td>
<td align="center">Crude comparison on weighted data <span class="math inline">\((Y \sim \color{green}{\text{A}})\)</span></td>
</tr>
<tr class="odd">
<td>Propensity score double adjustment</td>
<td align="center">Yes <span class="math inline">\((\color{green}{\text{A}} \sim \color{red}{\text{L}})\)</span></td>
<td align="center">Yes <span class="math inline">\((Y \sim \color{green}{\text{A}}+ \color{red}{\text{L}})\)</span></td>
</tr>
<tr class="even">
<td>Decision tree-based method <span class="citation">(<a href="#ref-breiman1984classification" role="doc-biblioref">Breiman et al. 1984</a>)</span></td>
<td align="center">No <span class="math inline">\(\color{green}{\text{A}}\)</span> modelling</td>
<td align="center">Yes <span class="math inline">\((Y \sim \color{green}{\text{A}}+\color{red}{\text{L}})\)</span></td>
</tr>
<tr class="odd">
<td>G-Computation <span class="citation">(<a href="#ref-robins1986new" role="doc-biblioref">J. Robins 1986</a>)</span></td>
<td align="center">No <span class="math inline">\(\color{green}{\text{A}}\)</span> modelling</td>
<td align="center">Yes <span class="math inline">\((Y \sim \color{green}{\text{A=1 vs. 0}}+\color{red}{\text{L}})\)</span></td>
</tr>
<tr class="even">
<td>Random Forest <span class="citation">(<a href="#ref-ho1995random" role="doc-biblioref">Ho 1995</a>)</span></td>
<td align="center">No <span class="math inline">\(\color{green}{\text{A}}\)</span> modelling</td>
<td align="center">Yes <span class="math inline">\((Y \sim \color{green}{\text{A}}+\color{red}{\text{L}})\)</span></td>
</tr>
<tr class="odd">
<td>Double robust <span class="citation">(<a href="#ref-rrr" role="doc-biblioref">J. M. Robins and Rotnitzky 2001</a>)</span>, <span class="citation">(<a href="#ref-van2006targeted" role="doc-biblioref">Van Der Laan and Rubin 2006</a>)</span> (augmented weighted, or TMLE), causal forest <span class="citation">(<a href="#ref-athey2019estimating" role="doc-biblioref">Athey and Wager 2019</a>)</span>, double machine learning (DML) <span class="citation">(<a href="#ref-chernozhukov2017double" role="doc-biblioref">Chernozhukov et al. 2017</a>)</span>, potentially using machine learning</td>
<td align="center">Yes <span class="math inline">\((\color{green}{\text{A}} \sim \color{red}{\text{L}})\)</span></td>
<td align="center">Yes <span class="math inline">\((Y \sim \color{green}{\text{A}}+ \color{red}{\text{L}})\)</span></td>
</tr>
</tbody>
</table>
<p><span class="math inline">\(\dagger\)</span> <span class="citation">(<a href="#ref-stanton2001galton" role="doc-biblioref">Stanton 2001</a>)</span>.
<span class="math inline">\(\dagger\dagger\)</span> other economic models also exists: Structural equation modeling, Difference-in-differences, Regression discontinuity, Interrupted time series</p>
</div>
</div>
<div id="lack-of-overlap" class="section level2 hasAnchor" number="2.3">
<h2><span class="header-section-number">2.3</span> Lack of overlap<a href="balance.html#lack-of-overlap" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<ul>
<li>“Lack of complete overlap” happens if there is a baseline covariate space where there are exposed patients, but no control or vice versa.
<ul>
<li>Region of ‘no overlap’ is an inherent limitation of the data.</li>
</ul></li>
</ul>
<p><img src="images/overlap.png" width="305" /></p>
<ul>
<li><strong>Regression adjustment</strong> usually do not offer any solution to this.
<ul>
<li>Consequently, inference is not generalizable beyond the region of overlap.</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-athey2019estimating" class="csl-entry">
Athey, Susan, and Stefan Wager. 2019. <span>“Estimating Treatment Effects with Causal Forests: An Application.”</span> <em>Observational Studies</em> 5 (2): 37–51.
</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-breiman1984classification" class="csl-entry">
Breiman, L, JH Friedman, R Olshen, and CJ Stone. 1984. <span>“Classification and Regression Trees.”</span>
</div>
<div id="ref-chernozhukov2017double" class="csl-entry">
Chernozhukov, Victor, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, and Whitney Newey. 2017. <span>“Double/Debiased/Neyman Machine Learning of Treatment Effects.”</span> <em>American Economic Review</em> 107 (5): 261–65.
</div>
<div id="ref-gautret2020hydroxychloroquine" class="csl-entry">
Gautret, Philippe, Jean-Christophe Lagier, Philippe Parola, Line Meddeb, Morgane Mailhe, Barbara Doudier, Johan Courjon, et al. 2020. <span>“Hydroxychloroquine and Azithromycin as a Treatment of COVID-19: Results of an Open-Label Non-Randomized Clinical Trial.”</span> <em>International Journal of Antimicrobial Agents</em> 56 (1): 105949.
</div>
<div id="ref-ho1995random" class="csl-entry">
Ho, Tin Kam. 1995. <span>“Random Decision Forests.”</span> In <em>Proceedings of 3rd International Conference on Document Analysis and Recognition</em>, 1:278–82. IEEE.
</div>
<div id="ref-rrr" class="csl-entry">
Robins, J. M., and A G Rotnitzky. 2001. <span>“Comment on the Bickel and Kwon Article, ’Inference for Semiparametric Models: Some Questions and an Answer’.”</span> <em>Statistica Sinica</em> 11 (January): 920–36.
</div>
<div id="ref-robins1986new" class="csl-entry">
Robins, James. 1986. <span>“A New Approach to Causal Inference in Mortality Studies with a Sustained Exposure Period—Application to Control of the Healthy Worker Survivor Effect.”</span> <em>Mathematical Modelling</em> 7 (9-12): 1393–1512.
</div>
<div id="ref-rosenbaum1983central" class="csl-entry">
Rosenbaum, Paul R, and Donald B Rubin. 1983. <span>“The Central Role of the Propensity Score in Observational Studies for Causal Effects.”</span> <em>Biometrika</em> 70 (1): 41–55.
</div>
<div id="ref-rubin1973matching" class="csl-entry">
Rubin, Donald B. 1973. <span>“Matching to Remove Bias in Observational Studies.”</span> <em>Biometrics</em>, 159–83.
</div>
<div id="ref-stanton2001galton" class="csl-entry">
Stanton, Jeffrey M. 2001. <span>“Galton, Pearson, and the Peas: A Brief History of Linear Regression for Statistics Instructors.”</span> <em>Journal of Statistics Education</em> 9 (3).
</div>
<div id="ref-van2006targeted" class="csl-entry">
Van Der Laan, Mark J, and Daniel Rubin. 2006. <span>“Targeted Maximum Likelihood Learning.”</span> <em>The International Journal of Biostatistics</em> 2 (1).
</div>
<div id="ref-yang2012unified" class="csl-entry">
Yang, Dongsheng, and Jarrod E Dalton. 2012. <span>“A Unified Approach to Measuring the Effect Size Between Two Groups Using SAS.”</span> In <em>SAS Global Forum</em>, 335:1–6. Citeseer.
</div>
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