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<!DOCTYPE html>
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<title>Chapter 1 Defining Parameter | Understanding Propensity Score Matching</title>
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<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="" 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="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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</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>
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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>
</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>
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<h1>
<i class="fa fa-circle-o-notch fa-spin"></i><a href="./">Understanding Propensity Score Matching</a>
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<section class="normal" id="section-">
<div id="terms" class="section level1 hasAnchor" number="1">
<h1><span class="header-section-number">Chapter 1</span> Defining Parameter<a href="terms.html#terms" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<div id="epidemiological-research-goals" class="section level2 hasAnchor" number="1.1">
<h2><span class="header-section-number">1.1</span> Epidemiological research goals<a href="terms.html#epidemiological-research-goals" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Two common goals for epidemiological research are prediction and causal inference:</p>
<ul>
<li><strong>Prediction goal</strong>: The primary objective of a prediction goal is to forecast the occurrence or risk of an outcome (<span class="math inline">\(Y\)</span>) based on one or more risk factors (<span class="math inline">\(A\)</span>). The focus of this goal is often on making accurate predictions.</li>
</ul>
<p><img src="images/RCT0.png" /></p>
<ul>
<li><strong>Causal goal</strong>: The causal goal focuses on understanding the causal relationship between a risk factor (often a <em>treatment</em>, <span class="math inline">\(A\)</span>) and a health outcome (<span class="math inline">\(Y\)</span>). Control for confounding factors (<span class="math inline">\(L\)</span>) is often a necessary step in understanding such a relationship. The focus of this goal is often on estimating the parameter ‘treatment effect’.</li>
</ul>
<p><img src="images/condRCT.png" /></p>
<p>We only focus on estimating treatment effect today. For that, let us define the notations first.</p>
</div>
<div id="potential-outcome" class="section level2 hasAnchor" number="1.2">
<h2><span class="header-section-number">1.2</span> Potential outcome<a href="terms.html#potential-outcome" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<ul>
<li><span class="math inline">\(A\)</span>: Exposure status
<ul>
<li><span class="math inline">\(1\)</span> = takes Rosuvastatin</li>
<li><span class="math inline">\(0\)</span> = does not take rosuvastatin</li>
</ul></li>
<li><span class="math inline">\(Y\)</span>: Outcome: Total cholesterol levels
<ul>
<li><span class="math inline">\(Y(A=1)\)</span> = potential outcome when exposed</li>
<li><span class="math inline">\(Y(A=0)\)</span> = potential outcome when not exposed</li>
</ul></li>
</ul>
<p>Relationship between <span class="math inline">\(Y\)</span> and <span class="math inline">\([Y(A=1), Y(A=0)]\)</span> can be expressed as follows:
<span class="math inline">\(Y = A \times Y(A=1) + (1-A) \times Y(A=0)\)</span></p>
</div>
<div id="parameters-of-interest" class="section level2 hasAnchor" number="1.3">
<h2><span class="header-section-number">1.3</span> Parameters of interest<a href="terms.html#parameters-of-interest" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>When assessing the effect of an exposure on an outcome, we are interested about the following estimands</p>
<ul>
<li>treatment effect for an individual (TE)</li>
<li>average treatment effect (ATE)</li>
<li>average treatment effect on the treated (ATT)</li>
</ul>
<div id="te" class="section level3 hasAnchor" number="1.3.1">
<h3><span class="header-section-number">1.3.1</span> TE<a href="terms.html#te" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<ul>
<li>John takes Rosuvastatin <span class="math inline">\((A=1)\)</span> and his total cholesterol level is = <span class="math inline">\(Y(A=1)\)</span> = <span class="math inline">\(195\)</span> mg/dL (milligrams per deciliter) after 3 months</li>
<li>John does not take Rosuvastatin <span class="math inline">\((A=0)\)</span> and his total cholesterol level is = <span class="math inline">\(Y(A=0)\)</span> = <span class="math inline">\(245\)</span> mg/dL after 3 months
Effect of Rosuvastatin on John is =</li>
</ul>
<p><span class="math inline">\(TE = Y(A=1) - Y(A=0) = 195 - 245 = - 50\)</span></p>
<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;">
TE is not estimable as we generally can’t observe outcomes under both treatment conditions.
</td>
</tr>
</tbody>
</table>
</div>
<div id="ate" class="section level3 hasAnchor" number="1.3.2">
<h3><span class="header-section-number">1.3.2</span> ATE<a href="terms.html#ate" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb2-1"><a href="terms.html#cb2-1" aria-hidden="true" tabindex="-1"></a>Person <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">"John"</span>,<span class="st">"Jim"</span>,<span class="st">"Jake"</span>,<span class="st">"Cody"</span>,<span class="st">"Luke"</span>)</span>
<span id="cb2-2"><a href="terms.html#cb2-2" aria-hidden="true" tabindex="-1"></a>Y1 <span class="ot"><-</span> <span class="fu">c</span>( <span class="dv">195</span>, <span class="dv">100</span>, <span class="dv">210</span>, <span class="dv">155</span>, <span class="dv">165</span>)</span>
<span id="cb2-3"><a href="terms.html#cb2-3" aria-hidden="true" tabindex="-1"></a>Y0 <span class="ot"><-</span> <span class="fu">c</span>(<span class="dv">245</span>, <span class="dv">160</span>, <span class="dv">270</span>, <span class="dv">210</span>, <span class="dv">230</span>)</span>
<span id="cb2-4"><a href="terms.html#cb2-4" aria-hidden="true" tabindex="-1"></a>PotentialOutcomes <span class="ot"><-</span> <span class="fu">data.frame</span>(Person, Y1, Y0, <span class="at">TE =</span> Y1<span class="sc">-</span>Y0)</span>
<span id="cb2-5"><a href="terms.html#cb2-5" aria-hidden="true" tabindex="-1"></a>mean.values <span class="ot"><-</span> <span class="fu">c</span>(<span class="cn">NA</span>, <span class="fu">mean</span>(PotentialOutcomes<span class="sc">$</span>Y1),</span>
<span id="cb2-6"><a href="terms.html#cb2-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">mean</span>(PotentialOutcomes<span class="sc">$</span>Y0),</span>
<span id="cb2-7"><a href="terms.html#cb2-7" aria-hidden="true" tabindex="-1"></a> <span class="fu">mean</span>(PotentialOutcomes<span class="sc">$</span>TE))</span>
<span id="cb2-8"><a href="terms.html#cb2-8" aria-hidden="true" tabindex="-1"></a>PotentialOutcomes <span class="ot"><-</span> <span class="fu">rbind</span>(PotentialOutcomes, mean.values)</span>
<span id="cb2-9"><a href="terms.html#cb2-9" aria-hidden="true" tabindex="-1"></a><span class="fu">kable</span>(PotentialOutcomes, <span class="at">booktabs =</span> <span class="cn">TRUE</span>, </span>
<span id="cb2-10"><a href="terms.html#cb2-10" aria-hidden="true" tabindex="-1"></a> <span class="at">col.names =</span> <span class="fu">c</span>(<span class="st">"Person"</span>, <span class="st">"Y(1)"</span>, <span class="st">"Y(0)"</span>, <span class="st">"TE"</span>)) <span class="sc">%>%</span></span>
<span id="cb2-11"><a href="terms.html#cb2-11" aria-hidden="true" tabindex="-1"></a> <span class="fu">row_spec</span>(<span class="dv">6</span>, <span class="at">bold =</span> T, <span class="at">color =</span> <span class="st">"white"</span>, <span class="at">background =</span> <span class="st">"#D7261E"</span>)</span></code></pre></div>
<table>
<thead>
<tr>
<th style="text-align:left;">
Person
</th>
<th style="text-align:right;">
Y(1)
</th>
<th style="text-align:right;">
Y(0)
</th>
<th style="text-align:right;">
TE
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
John
</td>
<td style="text-align:right;">
195
</td>
<td style="text-align:right;">
245
</td>
<td style="text-align:right;">
-50
</td>
</tr>
<tr>
<td style="text-align:left;">
Jim
</td>
<td style="text-align:right;">
100
</td>
<td style="text-align:right;">
160
</td>
<td style="text-align:right;">
-60
</td>
</tr>
<tr>
<td style="text-align:left;">
Jake
</td>
<td style="text-align:right;">
210
</td>
<td style="text-align:right;">
270
</td>
<td style="text-align:right;">
-60
</td>
</tr>
<tr>
<td style="text-align:left;">
Cody
</td>
<td style="text-align:right;">
155
</td>
<td style="text-align:right;">
210
</td>
<td style="text-align:right;">
-55
</td>
</tr>
<tr>
<td style="text-align:left;">
Luke
</td>
<td style="text-align:right;">
165
</td>
<td style="text-align:right;">
230
</td>
<td style="text-align:right;">
-65
</td>
</tr>
<tr>
<td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;">
</td>
<td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;">
165
</td>
<td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;">
223
</td>
<td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;">
-58
</td>
</tr>
</tbody>
</table>
<p><span class="math inline">\(ATE = E[Y(A=1)-Y(A=0)]\)</span></p>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb3-1"><a href="terms.html#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="fu">mean</span>(PotentialOutcomes<span class="sc">$</span>Y1 <span class="sc">-</span> PotentialOutcomes<span class="sc">$</span>Y0)</span></code></pre></div>
<pre><code>## [1] -58</code></pre>
</div>
<div id="interpretation-of-ate" class="section level3 hasAnchor" number="1.3.3">
<h3><span class="header-section-number">1.3.3</span> Interpretation of ATE<a href="terms.html#interpretation-of-ate" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>This is a treatment effect (on an average) of the following hypothetical situation</p>
<ul>
<li>having the entire population as treated, vs.</li>
<li>having the entire population as untreated.</li>
</ul>
<p>Entire population is the reference goup here.</p>
</div>
<div id="identifiability-assumptions" class="section level3 hasAnchor" number="1.3.4">
<h3><span class="header-section-number">1.3.4</span> Identifiability Assumptions<a href="terms.html#identifiability-assumptions" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Real-world scenario (both outcomes under different treatments can not be observed):</p>
<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb5-1"><a href="terms.html#cb5-1" aria-hidden="true" tabindex="-1"></a>Person <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">"John"</span>,<span class="st">"Jim"</span>,<span class="st">"Jake"</span>,<span class="st">"Cody"</span>,<span class="st">"Luke"</span>)</span>
<span id="cb5-2"><a href="terms.html#cb5-2" aria-hidden="true" tabindex="-1"></a>Y1 <span class="ot"><-</span> <span class="fu">c</span>( <span class="cn">NA</span>, <span class="dv">100</span>, <span class="cn">NA</span>, <span class="dv">155</span>, <span class="cn">NA</span>)</span>
<span id="cb5-3"><a href="terms.html#cb5-3" aria-hidden="true" tabindex="-1"></a>Y0 <span class="ot"><-</span> <span class="fu">c</span>(<span class="dv">245</span>, <span class="cn">NA</span>, <span class="dv">270</span>, <span class="cn">NA</span>, <span class="dv">230</span>)</span>
<span id="cb5-4"><a href="terms.html#cb5-4" aria-hidden="true" tabindex="-1"></a>PotentialOutcomes <span class="ot"><-</span> <span class="fu">data.frame</span>(Person, Y1, Y0, <span class="at">TE =</span> Y1<span class="sc">-</span>Y0)</span>
<span id="cb5-5"><a href="terms.html#cb5-5" aria-hidden="true" tabindex="-1"></a>mean.values <span class="ot"><-</span> <span class="fu">c</span>(<span class="cn">NA</span>, <span class="fu">mean</span>(PotentialOutcomes<span class="sc">$</span>Y1, <span class="at">na.rm =</span> <span class="cn">TRUE</span>),</span>
<span id="cb5-6"><a href="terms.html#cb5-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">mean</span>(PotentialOutcomes<span class="sc">$</span>Y0, <span class="at">na.rm =</span> <span class="cn">TRUE</span>),</span>
<span id="cb5-7"><a href="terms.html#cb5-7" aria-hidden="true" tabindex="-1"></a> <span class="fu">mean</span>(PotentialOutcomes<span class="sc">$</span>TE))</span>
<span id="cb5-8"><a href="terms.html#cb5-8" aria-hidden="true" tabindex="-1"></a>PotentialOutcomes <span class="ot"><-</span> <span class="fu">rbind</span>(PotentialOutcomes, <span class="fu">round</span>(mean.values,<span class="dv">1</span>))</span>
<span id="cb5-9"><a href="terms.html#cb5-9" aria-hidden="true" tabindex="-1"></a>PotentialOutcomes[<span class="dv">6</span>,<span class="dv">4</span>] <span class="ot"><-</span> <span class="fu">round</span>(<span class="fu">mean</span>(PotentialOutcomes<span class="sc">$</span>Y1, <span class="at">na.rm =</span> <span class="cn">TRUE</span>)<span class="sc">-</span> </span>
<span id="cb5-10"><a href="terms.html#cb5-10" aria-hidden="true" tabindex="-1"></a> <span class="fu">mean</span>(PotentialOutcomes<span class="sc">$</span>Y0, <span class="at">na.rm =</span> <span class="cn">TRUE</span>),<span class="dv">1</span>)</span>
<span id="cb5-11"><a href="terms.html#cb5-11" aria-hidden="true" tabindex="-1"></a><span class="fu">kable</span>(PotentialOutcomes, <span class="at">booktabs =</span> <span class="cn">TRUE</span>, </span>
<span id="cb5-12"><a href="terms.html#cb5-12" aria-hidden="true" tabindex="-1"></a> <span class="at">col.names =</span> <span class="fu">c</span>(<span class="st">"Person"</span>, <span class="st">"Y(1)"</span>, <span class="st">"Y(0)"</span>, <span class="st">"TE"</span>)) <span class="sc">%>%</span></span>
<span id="cb5-13"><a href="terms.html#cb5-13" aria-hidden="true" tabindex="-1"></a> <span class="fu">row_spec</span>(<span class="dv">6</span>, <span class="at">bold =</span> T, <span class="at">color =</span> <span class="st">"white"</span>, <span class="at">background =</span> <span class="st">"#D7261E"</span>)</span></code></pre></div>
<table>
<thead>
<tr>
<th style="text-align:left;">
Person
</th>
<th style="text-align:right;">
Y(1)
</th>
<th style="text-align:right;">
Y(0)
</th>
<th style="text-align:right;">
TE
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
John
</td>
<td style="text-align:right;">
</td>
<td style="text-align:right;">
245.0
</td>
<td style="text-align:right;">
</td>
</tr>
<tr>
<td style="text-align:left;">
Jim
</td>
<td style="text-align:right;">
100.0
</td>
<td style="text-align:right;">
</td>
<td style="text-align:right;">
</td>
</tr>
<tr>
<td style="text-align:left;">
Jake
</td>
<td style="text-align:right;">
</td>
<td style="text-align:right;">
270.0
</td>
<td style="text-align:right;">
</td>
</tr>
<tr>
<td style="text-align:left;">
Cody
</td>
<td style="text-align:right;">
155.0
</td>
<td style="text-align:right;">
</td>
<td style="text-align:right;">
</td>
</tr>
<tr>
<td style="text-align:left;">
Luke
</td>
<td style="text-align:right;">
</td>
<td style="text-align:right;">
230.0
</td>
<td style="text-align:right;">
</td>
</tr>
<tr>
<td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;">
</td>
<td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;">
127.5
</td>
<td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;">
248.3
</td>
<td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;">
-120.8
</td>
</tr>
</tbody>
</table>
<p>We can rearrange it as follows:</p>
<div class="sourceCode" id="cb6"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb6-1"><a href="terms.html#cb6-1" aria-hidden="true" tabindex="-1"></a>Person <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">"John"</span>,<span class="st">"Jim"</span>,<span class="st">"Jake"</span>,<span class="st">"Cody"</span>,<span class="st">"Luke"</span>)</span>
<span id="cb6-2"><a href="terms.html#cb6-2" aria-hidden="true" tabindex="-1"></a>A <span class="ot"><-</span> <span class="fu">c</span>( <span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">0</span>)</span>
<span id="cb6-3"><a href="terms.html#cb6-3" aria-hidden="true" tabindex="-1"></a>Y <span class="ot"><-</span> <span class="fu">c</span>(<span class="dv">245</span>, <span class="dv">100</span>, <span class="dv">270</span>, <span class="dv">155</span>, <span class="dv">230</span>)</span>
<span id="cb6-4"><a href="terms.html#cb6-4" aria-hidden="true" tabindex="-1"></a>RealOutcomes <span class="ot"><-</span> <span class="fu">data.frame</span>(Person, A, Y)</span>
<span id="cb6-5"><a href="terms.html#cb6-5" aria-hidden="true" tabindex="-1"></a><span class="fu">kable</span>(RealOutcomes, <span class="at">booktabs =</span> <span class="cn">TRUE</span>, </span>
<span id="cb6-6"><a href="terms.html#cb6-6" aria-hidden="true" tabindex="-1"></a> <span class="at">col.names =</span> <span class="fu">c</span>(<span class="st">"Person"</span>, <span class="st">"A"</span>, <span class="st">"Y"</span>)) </span></code></pre></div>
<table>
<thead>
<tr>
<th style="text-align:left;">
Person
</th>
<th style="text-align:right;">
A
</th>
<th style="text-align:right;">
Y
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
John
</td>
<td style="text-align:right;">
0
</td>
<td style="text-align:right;">
245
</td>
</tr>
<tr>
<td style="text-align:left;">
Jim
</td>
<td style="text-align:right;">
1
</td>
<td style="text-align:right;">
100
</td>
</tr>
<tr>
<td style="text-align:left;">
Jake
</td>
<td style="text-align:right;">
0
</td>
<td style="text-align:right;">
270
</td>
</tr>
<tr>
<td style="text-align:left;">
Cody
</td>
<td style="text-align:right;">
1
</td>
<td style="text-align:right;">
155
</td>
</tr>
<tr>
<td style="text-align:left;">
Luke
</td>
<td style="text-align:right;">
0
</td>
<td style="text-align:right;">
230
</td>
</tr>
</tbody>
</table>
<p>If we can compute a causal quantity, such as <span class="math inline">\(ATE = E[Y(A=1)-Y(A=0)]\)</span> or <code>mean(PotentialOutcomes$Y1 - PotentialOutcomes$Y0)</code> using a statistical quantity, such as <span class="math inline">\(E[Y|A=1]-E[Y|A=0]\)</span> or <code>mean(Y[A=1]) - mean(Y[A=0])</code>, we say that the causal quantity is identifiable. For such identifiability, we need to meet the following assumptions:</p>
<table>
<colgroup>
<col width="33%" />
<col width="33%" />
<col width="33%" />
</colgroup>
<tbody>
<tr class="odd">
<td>Exchangeability</td>
<td><span class="math inline">\(Y(1), Y(0) \perp A\)</span></td>
<td>Treatment assignment is independent of the potential outcome</td>
</tr>
<tr class="even">
<td>Positivity</td>
<td><span class="math inline">\(0 < P(A=1) < 1\)</span></td>
<td>Subjects are eligible to receive both treatment</td>
</tr>
<tr class="odd">
<td>Consistency</td>
<td><span class="math inline">\(Y = Y(a) \forall A=a\)</span></td>
<td>No multiple version of the treatment</td>
</tr>
<tr class="even">
<td>No interference</td>
<td></td>
<td>Treated one patient will not impact outcome for others</td>
</tr>
</tbody>
</table>
<p>Note here, from data we get the estimate of average TE is <code>(100+155)/2 - (245+270+230)/3 = -120.8</code>. Alternatively, we can calculate the beta coefficient associated with <span class="math inline">\(A\)</span> as follows:</p>
<div class="sourceCode" id="cb7"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb7-1"><a href="terms.html#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="fu">round</span>(<span class="fu">coef</span>(<span class="fu">lm</span>(Y<span class="sc">~</span>A)),<span class="dv">1</span>)</span></code></pre></div>
<pre><code>## (Intercept) A
## 248.3 -120.8</code></pre>
<p>Here, beta coefficient associated with <span class="math inline">\(A\)</span> is -120.8, which is different than average TE <code>-58</code> that we obtained from the potential outcome data table above. Part of it is because of finite sample bias (having only 5 data points) instead of infinite population. If we had a large enough sample, we would expect the estimate to be close to the true average TE.</p>
<p>You can find more detailed exploration of estimation in a <a href="https://ehsanx.github.io/TMLEworkshop/g-computation.html">different tutorial</a> using a real data.</p>
<p><img src="images/condRCT.png" /></p>
<p>Extending these assumptions when confounders exist:</p>
<table>
<colgroup>
<col width="33%" />
<col width="33%" />
<col width="33%" />
</colgroup>
<tbody>
<tr class="odd">
<td>Conditional Exchangeability</td>
<td><span class="math inline">\(Y(1), Y(0) \perp A | L\)</span></td>
<td>Treatment assignment is independent of the potential outcome, given L</td>
</tr>
<tr class="even">
<td>Positivity</td>
<td><span class="math inline">\(0 < P(A=1 | L) < 1\)</span></td>
<td>Subjects are eligible to receive both treatment, given L</td>
</tr>
</tbody>
</table>
<p>Here, - <span class="math inline">\(L\)</span>: Confounder: Age, could be an example</p>
</div>
<div id="att" class="section level3 hasAnchor" number="1.3.5">
<h3><span class="header-section-number">1.3.5</span> ATT<a href="terms.html#att" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<ul>
<li>Assume that the following are the confounders that impact the relationship between rosuvastatin and cholesterol levels
<ul>
<li>race</li>
<li>sex</li>
<li>age</li>
</ul></li>
<li>We have 5 Rosuvastatin-treated subjects who are all
<ul>
<li>white,</li>
<li>male,</li>
<li>50 years of age</li>
</ul></li>
<li>We recruited additional 5 subjects (same characteristics) to non-rosuvastatin group.</li>
</ul>
<p><strong>Treated group</strong>:</p>
<div class="sourceCode" id="cb9"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb9-1"><a href="terms.html#cb9-1" aria-hidden="true" tabindex="-1"></a>Person <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">"John"</span>,<span class="st">"Jim"</span>,<span class="st">"Jake"</span>,<span class="st">"Cody"</span>,<span class="st">"Luke"</span>)</span>
<span id="cb9-2"><a href="terms.html#cb9-2" aria-hidden="true" tabindex="-1"></a>Y1 <span class="ot"><-</span> <span class="fu">c</span>( <span class="dv">195</span>, <span class="dv">100</span>, <span class="dv">210</span>, <span class="dv">155</span>, <span class="dv">165</span>)</span>
<span id="cb9-3"><a href="terms.html#cb9-3" aria-hidden="true" tabindex="-1"></a>Y0 <span class="ot"><-</span> <span class="fu">rep</span>(<span class="cn">NA</span>, <span class="fu">length</span>(Y1))</span>
<span id="cb9-4"><a href="terms.html#cb9-4" aria-hidden="true" tabindex="-1"></a>Treated <span class="ot"><-</span> <span class="fu">data.frame</span>(Person, Y1, Y0, <span class="at">TE =</span> Y1<span class="sc">-</span>Y0)</span>
<span id="cb9-5"><a href="terms.html#cb9-5" aria-hidden="true" tabindex="-1"></a>Treated[<span class="dv">6</span>,<span class="dv">2</span>] <span class="ot"><-</span> <span class="fu">mean</span>(Treated<span class="sc">$</span>Y1)</span>
<span id="cb9-6"><a href="terms.html#cb9-6" aria-hidden="true" tabindex="-1"></a><span class="fu">kable</span>(Treated, <span class="at">booktabs =</span> <span class="cn">TRUE</span>, </span>
<span id="cb9-7"><a href="terms.html#cb9-7" aria-hidden="true" tabindex="-1"></a> <span class="at">col.names =</span> <span class="fu">c</span>(<span class="st">"Person"</span>, <span class="st">"Y(1)"</span>, <span class="st">"Y(0)"</span>, <span class="st">"TE"</span>))<span class="sc">%>%</span></span>
<span id="cb9-8"><a href="terms.html#cb9-8" aria-hidden="true" tabindex="-1"></a> <span class="fu">row_spec</span>(<span class="dv">6</span>, <span class="at">bold =</span> T, <span class="at">color =</span> <span class="st">"white"</span>, <span class="at">background =</span> <span class="st">"#D7261E"</span>)</span></code></pre></div>
<table>
<thead>
<tr>
<th style="text-align:left;">
Person
</th>
<th style="text-align:right;">
Y(1)
</th>
<th style="text-align:left;">
Y(0)
</th>
<th style="text-align:right;">
TE
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
John
</td>
<td style="text-align:right;">
195
</td>
<td style="text-align:left;">
</td>
<td style="text-align:right;">
</td>
</tr>
<tr>
<td style="text-align:left;">
Jim
</td>
<td style="text-align:right;">
100
</td>
<td style="text-align:left;">
</td>
<td style="text-align:right;">
</td>
</tr>
<tr>
<td style="text-align:left;">
Jake
</td>
<td style="text-align:right;">
210
</td>
<td style="text-align:left;">
</td>
<td style="text-align:right;">
</td>
</tr>
<tr>
<td style="text-align:left;">
Cody
</td>
<td style="text-align:right;">
155
</td>
<td style="text-align:left;">
</td>
<td style="text-align:right;">
</td>
</tr>
<tr>
<td style="text-align:left;">
Luke
</td>
<td style="text-align:right;">
165
</td>
<td style="text-align:left;">
</td>
<td style="text-align:right;">
</td>
</tr>
<tr>
<td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;">
</td>
<td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;">
165
</td>
<td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;">
</td>
<td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;">
</td>
</tr>
</tbody>
</table>
<p><strong>Untreated group</strong>: New folks with characteristics similar to the treated group.</p>
<div class="sourceCode" id="cb10"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb10-1"><a href="terms.html#cb10-1" aria-hidden="true" tabindex="-1"></a>Person <span class="ot"><-</span> <span class="fu">c</span>( <span class="st">"Jack"</span>, <span class="st">"Dustin"</span>, <span class="st">"Cole"</span>, <span class="st">"Lucas"</span>, <span class="st">"Dylan"</span>)</span>
<span id="cb10-2"><a href="terms.html#cb10-2" aria-hidden="true" tabindex="-1"></a>Y0 <span class="ot"><-</span> <span class="fu">c</span>( <span class="dv">245</span>, <span class="dv">160</span>, <span class="dv">270</span>, <span class="dv">210</span>, <span class="dv">165</span>)</span>
<span id="cb10-3"><a href="terms.html#cb10-3" aria-hidden="true" tabindex="-1"></a>Y1 <span class="ot"><-</span> <span class="fu">rep</span>(<span class="cn">NA</span>, <span class="fu">length</span>(Y0))</span>
<span id="cb10-4"><a href="terms.html#cb10-4" aria-hidden="true" tabindex="-1"></a>Untreated <span class="ot"><-</span> <span class="fu">data.frame</span>(Person, Y1, Y0, <span class="at">TE =</span> Y1<span class="sc">-</span>Y0)</span>
<span id="cb10-5"><a href="terms.html#cb10-5" aria-hidden="true" tabindex="-1"></a>Untreated[<span class="dv">6</span>,<span class="dv">3</span>] <span class="ot"><-</span> <span class="fu">mean</span>(Untreated<span class="sc">$</span>Y0)</span>
<span id="cb10-6"><a href="terms.html#cb10-6" aria-hidden="true" tabindex="-1"></a><span class="fu">kable</span>(Untreated, <span class="at">booktabs =</span> <span class="cn">TRUE</span>, </span>
<span id="cb10-7"><a href="terms.html#cb10-7" aria-hidden="true" tabindex="-1"></a> <span class="at">col.names =</span> <span class="fu">c</span>(<span class="st">"Person"</span>, <span class="st">"Y(1)"</span>, <span class="st">"Y(0)"</span>, <span class="st">"TE"</span>))<span class="sc">%>%</span></span>
<span id="cb10-8"><a href="terms.html#cb10-8" aria-hidden="true" tabindex="-1"></a> <span class="fu">row_spec</span>(<span class="dv">6</span>, <span class="at">bold =</span> T, <span class="at">color =</span> <span class="st">"white"</span>, <span class="at">background =</span> <span class="st">"#D7261E"</span>)</span></code></pre></div>
<table>
<thead>
<tr>
<th style="text-align:left;">
Person
</th>
<th style="text-align:left;">
Y(1)
</th>
<th style="text-align:right;">
Y(0)
</th>
<th style="text-align:right;">
TE
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
Jack
</td>
<td style="text-align:left;">
</td>
<td style="text-align:right;">
245
</td>
<td style="text-align:right;">
</td>
</tr>
<tr>
<td style="text-align:left;">
Dustin
</td>
<td style="text-align:left;">
</td>
<td style="text-align:right;">
160
</td>
<td style="text-align:right;">
</td>
</tr>
<tr>
<td style="text-align:left;">
Cole
</td>
<td style="text-align:left;">
</td>
<td style="text-align:right;">
270
</td>
<td style="text-align:right;">
</td>
</tr>
<tr>
<td style="text-align:left;">
Lucas
</td>
<td style="text-align:left;">
</td>
<td style="text-align:right;">
210
</td>
<td style="text-align:right;">
</td>
</tr>
<tr>
<td style="text-align:left;">
Dylan
</td>
<td style="text-align:left;">
</td>
<td style="text-align:right;">
165
</td>
<td style="text-align:right;">
</td>