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<!DOCTYPE html>
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<title>Chapter 9 PS vs. Double robust methods | Understanding Propensity Score Matching</title>
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<meta name="twitter:title" content="Chapter 9 PS vs. Double robust methods | Understanding Propensity Score Matching" />
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<meta name="author" content="Ehsan Karim" />
<meta name="date" content="2023-03-19" />
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<ul class="summary">
<li><a href="./">Understanding Propensity Score Matching</a></li>
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<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Preamble</a>
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<li class="chapter" data-level="" data-path="index.html"><a href="index.html#description"><i class="fa fa-check"></i>Description</a>
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<li class="chapter" data-level="" data-path="index.html"><a href="index.html#main-references"><i class="fa fa-check"></i>Main references</a></li>
<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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<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="misspecify" class="section level1 hasAnchor" number="9">
<h1><span class="header-section-number">Chapter 9</span> PS vs. Double robust methods<a href="misspecify.html#misspecify" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>Understanding sources of bias is important</p>
<p>Bias = Average of our estimates in repeated sampling - true treatment effect = bias due to finite sample + bias due to not designing the study properly + bias due to model misspecification</p>
<p>The past part is often less talked about. Nonparametric or machine learning methods could be an advantage if used properly to reduce bias due to this source. A topic for a different <a href="https://ehsanx.github.io/TMLEworkshop/">tutorial</a>, but here we will show an example of the implication of avoiding the possibility of model-misspecification.</p>
<div id="complex-data-simulation" class="section level2 hasAnchor" number="9.1">
<h2><span class="header-section-number">9.1</span> Complex Data Simulation<a href="misspecify.html#complex-data-simulation" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Complex simulation example, so that we know the true parameter <span class="math inline">\(\theta\)</span>.</p>
<table>
<colgroup>
<col width="50%" />
<col width="50%" />
</colgroup>
<tbody>
<tr class="odd">
<td><span class="math inline">\(Y\)</span> : Outcome</td>
<td>(continuous)</td>
</tr>
<tr class="even">
<td><span class="math inline">\(A\)</span> : Exposure</td>
<td>(binary)</td>
</tr>
<tr class="odd">
<td><span class="math inline">\(C\)</span> (1-4): true confounders</td>
<td>(continuous)</td>
</tr>
<tr class="even">
<td><span class="math inline">\(L\)</span> (1-4): measured and transformed version of true confounders</td>
<td>(continuous)</td>
</tr>
</tbody>
</table>
<div id="true-exposure-model" class="section level3 unnumbered hasAnchor">
<h3>True Exposure Model<a href="misspecify.html#true-exposure-model" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p><span class="math display">\[\begin{equation}
\begin{aligned}
\pi = P(A=1|C) &= f(C) \\
&= \frac{1}{1+exp[\alpha_0 + \alpha_1 C_1 + \alpha_2 C_2 + \alpha_3 C_3 + \alpha_4 C_4 ]}\nonumber
\end{aligned}
\end{equation}\]</span></p>
</div>
<div id="true-outcome-model" class="section level3 unnumbered hasAnchor">
<h3>True Outcome Model<a href="misspecify.html#true-outcome-model" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p><span class="math display">\[\begin{equation}
\begin{aligned}
E(Y|A,C) &= g(A,C)\\
&= \beta_0 + \theta A + \beta_1 C_1 + \beta_2 C_2 + \beta_3 C_3 + \beta_4 C_4 \nonumber
\end{aligned}
\end{equation}\]</span></p>
<p>In our example, <span class="math inline">\(\theta\)</span> = 6, which is our treatment effect.</p>
</div>
<div id="outcomes-and-exposures-are-complex-functions-of-measured-covariates" class="section level3 unnumbered hasAnchor">
<h3>Outcomes and exposures are complex functions of measured covariates<a href="misspecify.html#outcomes-and-exposures-are-complex-functions-of-measured-covariates" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Using example from <span class="citation">Kang and Schafer (<a href="#ref-kang2007demystifying" role="doc-biblioref">2007</a>)</span>, <span class="citation">Naimi, Mishler, and Kennedy (<a href="#ref-naimi2017challenges" role="doc-biblioref">2017</a>)</span> and <span class="citation">Balzer and Westling (<a href="#ref-balzer2021demystifying" role="doc-biblioref">2021</a>)</span>, let us assume that we don’t observed the <span class="math inline">\(C\)</span> variables directly, and we only have access to a set of transformed covariates <span class="math inline">\(L\)</span>s.</p>
<p><span class="math display">\[\begin{equation}
\begin{aligned}
L_1 &= \exp(C1/2) \nonumber\\
L_2 &= [(C2/(1+\exp(C1)) + 10)]\nonumber\\
L_3 &= [(C1*C3/25 + 0.6)^3] \nonumber\\
L_4 &= [(C2 + C4 + 20)^2\nonumber
\end{aligned}
\end{equation}\]</span></p>
<div class="sourceCode" id="cb146"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb146-1"><a href="misspecify.html#cb146-1" aria-hidden="true" tabindex="-1"></a>create.data <span class="ot"><-</span> <span class="cf">function</span>(n, <span class="at">te =</span> <span class="dv">6</span>){</span>
<span id="cb146-2"><a href="misspecify.html#cb146-2" aria-hidden="true" tabindex="-1"></a> <span class="co"># original covariates that generates A and Y</span></span>
<span id="cb146-3"><a href="misspecify.html#cb146-3" aria-hidden="true" tabindex="-1"></a> C1 <span class="ot"><-</span> <span class="fu">rnorm</span>(n,<span class="dv">0</span>,<span class="dv">1</span>)</span>
<span id="cb146-4"><a href="misspecify.html#cb146-4" aria-hidden="true" tabindex="-1"></a> C2 <span class="ot"><-</span> <span class="fu">rnorm</span>(n,<span class="dv">0</span>,<span class="dv">1</span>)</span>
<span id="cb146-5"><a href="misspecify.html#cb146-5" aria-hidden="true" tabindex="-1"></a> C3 <span class="ot"><-</span> <span class="fu">rnorm</span>(n,<span class="dv">0</span>,<span class="dv">1</span>)</span>
<span id="cb146-6"><a href="misspecify.html#cb146-6" aria-hidden="true" tabindex="-1"></a> C4 <span class="ot"><-</span> <span class="fu">rnorm</span>(n,<span class="dv">0</span>,<span class="dv">1</span>)</span>
<span id="cb146-7"><a href="misspecify.html#cb146-7" aria-hidden="true" tabindex="-1"></a> pscore <span class="ot"><-</span> <span class="fu">plogis</span>(<span class="sc">-</span><span class="dv">1</span> <span class="sc">+</span><span class="fu">log</span>(<span class="fl">1.75</span>)<span class="sc">*</span>(C1<span class="sc">+</span>C2<span class="sc">+</span>C3<span class="sc">+</span>C4) )</span>
<span id="cb146-8"><a href="misspecify.html#cb146-8" aria-hidden="true" tabindex="-1"></a> <span class="co"># treatment generation model</span></span>
<span id="cb146-9"><a href="misspecify.html#cb146-9" aria-hidden="true" tabindex="-1"></a> A <span class="ot"><-</span> <span class="fu">rbinom</span>(n, <span class="dv">1</span>, pscore)</span>
<span id="cb146-10"><a href="misspecify.html#cb146-10" aria-hidden="true" tabindex="-1"></a> eps <span class="ot"><-</span> <span class="fu">rnorm</span>(n,<span class="dv">0</span>,<span class="dv">6</span>) <span class="co"># error term</span></span>
<span id="cb146-11"><a href="misspecify.html#cb146-11" aria-hidden="true" tabindex="-1"></a> get.Y <span class="ot"><-</span> <span class="cf">function</span>(A, <span class="at">TE=</span>te) {<span class="dv">120</span><span class="sc">+</span>TE<span class="sc">*</span>A<span class="sc">+</span><span class="dv">3</span><span class="sc">*</span>(C1<span class="sc">+</span>C2<span class="sc">+</span>C3<span class="sc">+</span>C4)<span class="sc">+</span>eps}</span>
<span id="cb146-12"><a href="misspecify.html#cb146-12" aria-hidden="true" tabindex="-1"></a> Y1 <span class="ot"><-</span> <span class="fu">get.Y</span>(<span class="at">A=</span><span class="dv">1</span>, <span class="at">TE=</span>te)</span>
<span id="cb146-13"><a href="misspecify.html#cb146-13" aria-hidden="true" tabindex="-1"></a> Y0 <span class="ot"><-</span> <span class="fu">get.Y</span>(<span class="at">A=</span><span class="dv">0</span>, <span class="at">TE=</span>te)</span>
<span id="cb146-14"><a href="misspecify.html#cb146-14" aria-hidden="true" tabindex="-1"></a> <span class="co"># potential outcomes</span></span>
<span id="cb146-15"><a href="misspecify.html#cb146-15" aria-hidden="true" tabindex="-1"></a> PO <span class="ot"><-</span> <span class="fu">data.frame</span>(<span class="fu">cbind</span>(pscore, Y1,Y0, C1, C2, C3, C4))</span>
<span id="cb146-16"><a href="misspecify.html#cb146-16" aria-hidden="true" tabindex="-1"></a> <span class="co"># outcome generation model</span></span>
<span id="cb146-17"><a href="misspecify.html#cb146-17" aria-hidden="true" tabindex="-1"></a> Y <span class="ot"><-</span> <span class="fu">get.Y</span>(<span class="at">A=</span>A, <span class="at">TE=</span>te)</span>
<span id="cb146-18"><a href="misspecify.html#cb146-18" aria-hidden="true" tabindex="-1"></a> <span class="co"># misspecified versions of covariates</span></span>
<span id="cb146-19"><a href="misspecify.html#cb146-19" aria-hidden="true" tabindex="-1"></a> L1 <span class="ot"><-</span> <span class="fu">exp</span>(C1<span class="sc">/</span><span class="dv">2</span>)</span>
<span id="cb146-20"><a href="misspecify.html#cb146-20" aria-hidden="true" tabindex="-1"></a> L2 <span class="ot"><-</span> C2<span class="sc">/</span>(<span class="dv">1</span><span class="sc">+</span><span class="fu">exp</span>(C1)) <span class="sc">+</span> <span class="dv">10</span></span>
<span id="cb146-21"><a href="misspecify.html#cb146-21" aria-hidden="true" tabindex="-1"></a> L3 <span class="ot"><-</span> (C1<span class="sc">*</span>C3<span class="sc">/</span><span class="dv">25</span> <span class="sc">+</span> <span class="fl">0.6</span>)<span class="sc">^</span><span class="dv">3</span></span>
<span id="cb146-22"><a href="misspecify.html#cb146-22" aria-hidden="true" tabindex="-1"></a> L4 <span class="ot"><-</span> (C2 <span class="sc">+</span> C4 <span class="sc">+</span> <span class="dv">20</span>)<span class="sc">^</span><span class="dv">2</span></span>
<span id="cb146-23"><a href="misspecify.html#cb146-23" aria-hidden="true" tabindex="-1"></a> obs <span class="ot"><-</span> <span class="fu">data.frame</span>(<span class="fu">cbind</span>(L1,L2,L3,L4,A,Y))</span>
<span id="cb146-24"><a href="misspecify.html#cb146-24" aria-hidden="true" tabindex="-1"></a> <span class="fu">return</span>(<span class="fu">list</span>(<span class="at">observed=</span>obs, </span>
<span id="cb146-25"><a href="misspecify.html#cb146-25" aria-hidden="true" tabindex="-1"></a> <span class="at">Potential.Outcome=</span> PO, </span>
<span id="cb146-26"><a href="misspecify.html#cb146-26" aria-hidden="true" tabindex="-1"></a> <span class="at">ate =</span> <span class="fu">mean</span>(PO<span class="sc">$</span>Y1)<span class="sc">-</span><span class="fu">mean</span>(PO<span class="sc">$</span>Y0)))</span>
<span id="cb146-27"><a href="misspecify.html#cb146-27" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb146-28"><a href="misspecify.html#cb146-28" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb146-29"><a href="misspecify.html#cb146-29" aria-hidden="true" tabindex="-1"></a><span class="fu">create.data</span>(<span class="dv">6</span>)</span></code></pre></div>
<pre><code>## $observed
## L1 L2 L3 L4 A Y
## 1 2.2374436 10.06754 0.1625330 419.8183 1 120.3914
## 2 1.7679300 10.02612 0.1537718 441.0633 1 113.7917
## 3 1.8338882 10.06432 0.2636953 489.8636 0 133.9537
## 4 1.3741692 10.04336 0.1926407 389.0893 0 124.1439
## 5 0.9565624 11.25774 0.2182527 556.0792 1 138.0037
## 6 0.8111285 10.77576 0.2057078 512.2704 0 127.6085
##
## $Potential.Outcome
## pscore Y1 Y0 C1 C2 C3 C4
## 1 0.4265123 120.3914 114.3914 1.6106679 0.4056800 -0.8423000 0.08378755
## 2 0.3564975 113.7917 107.7917 1.1396187 0.1077560 -1.4095507 0.89375158
## 3 0.7938446 139.9537 133.9537 1.2128768 0.2806335 0.8504563 1.85222880
## 4 0.2154750 130.1439 124.1439 0.6356986 0.1252420 -0.8832483 -0.39989653
## 5 0.6518198 138.0037 132.0037 -0.0888186 2.4085868 -0.5850746 1.17274430
## 6 0.6371594 133.6085 127.6085 -0.4186576 1.2861567 0.5783549 1.34723419
##
## $ate
## [1] 6</code></pre>
</div>
</div>
<div id="understanding-finite-sample-bias" class="section level2 hasAnchor" number="9.2">
<h2><span class="header-section-number">9.2</span> Understanding finite sample bias<a href="misspecify.html#understanding-finite-sample-bias" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>First try:</p>
<div class="sourceCode" id="cb148"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb148-1"><a href="misspecify.html#cb148-1" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb148-2"><a href="misspecify.html#cb148-2" aria-hidden="true" tabindex="-1"></a>result.data <span class="ot"><-</span> <span class="fu">create.data</span>(<span class="dv">100</span>)</span>
<span id="cb148-3"><a href="misspecify.html#cb148-3" aria-hidden="true" tabindex="-1"></a><span class="fu">round</span>(<span class="fu">coef</span>(<span class="fu">glm</span>(Y <span class="sc">~</span> A <span class="sc">+</span> L1 <span class="sc">+</span> L2 <span class="sc">+</span> L3 <span class="sc">+</span> L4, <span class="at">family=</span><span class="st">"gaussian"</span>, </span>
<span id="cb148-4"><a href="misspecify.html#cb148-4" aria-hidden="true" tabindex="-1"></a> <span class="at">data=</span>result.data<span class="sc">$</span>observed)),<span class="dv">2</span>)</span></code></pre></div>
<pre><code>## (Intercept) A L1 L2 L3 L4
## 91.23 8.46 3.50 0.81 -19.83 0.05</code></pre>
<p>Second try:</p>
<div class="sourceCode" id="cb150"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb150-1"><a href="misspecify.html#cb150-1" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">22</span>)</span>
<span id="cb150-2"><a href="misspecify.html#cb150-2" aria-hidden="true" tabindex="-1"></a>result.data <span class="ot"><-</span> <span class="fu">create.data</span>(<span class="dv">100</span>)</span>
<span id="cb150-3"><a href="misspecify.html#cb150-3" aria-hidden="true" tabindex="-1"></a><span class="fu">round</span>(<span class="fu">coef</span>(<span class="fu">glm</span>(Y <span class="sc">~</span> A <span class="sc">+</span> L1 <span class="sc">+</span> L2 <span class="sc">+</span> L3 <span class="sc">+</span> L4, <span class="at">family=</span><span class="st">"gaussian"</span>, </span>
<span id="cb150-4"><a href="misspecify.html#cb150-4" aria-hidden="true" tabindex="-1"></a> <span class="at">data=</span>result.data<span class="sc">$</span>observed)),<span class="dv">2</span>)</span></code></pre></div>
<pre><code>## (Intercept) A L1 L2 L3 L4
## 70.05 10.20 4.48 2.29 9.53 0.05</code></pre>
<p>Third try:</p>
<div class="sourceCode" id="cb152"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb152-1"><a href="misspecify.html#cb152-1" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">33</span>)</span>
<span id="cb152-2"><a href="misspecify.html#cb152-2" aria-hidden="true" tabindex="-1"></a>result.data <span class="ot"><-</span> <span class="fu">create.data</span>(<span class="dv">100</span>)</span>
<span id="cb152-3"><a href="misspecify.html#cb152-3" aria-hidden="true" tabindex="-1"></a><span class="fu">round</span>(<span class="fu">coef</span>(<span class="fu">glm</span>(Y <span class="sc">~</span> A <span class="sc">+</span> L1 <span class="sc">+</span> L2 <span class="sc">+</span> L3 <span class="sc">+</span> L4, <span class="at">family=</span><span class="st">"gaussian"</span>, </span>
<span id="cb152-4"><a href="misspecify.html#cb152-4" aria-hidden="true" tabindex="-1"></a> <span class="at">data=</span>result.data<span class="sc">$</span>observed)),<span class="dv">2</span>)</span></code></pre></div>
<pre><code>## (Intercept) A L1 L2 L3 L4
## 97.51 7.13 2.32 -0.40 12.16 0.05</code></pre>
</div>
<div id="estimation-using-different-methods" class="section level2 hasAnchor" number="9.3">
<h2><span class="header-section-number">9.3</span> Estimation using different methods<a href="misspecify.html#estimation-using-different-methods" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>We now work with a larger data: <span class="math inline">\(n = 10,000\)</span></p>
<div class="sourceCode" id="cb154"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb154-1"><a href="misspecify.html#cb154-1" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">123</span>)</span>
<span id="cb154-2"><a href="misspecify.html#cb154-2" aria-hidden="true" tabindex="-1"></a>result.data <span class="ot"><-</span> <span class="fu">create.data</span>(<span class="dv">10000</span>)</span></code></pre></div>
<div id="regression-1" class="section level3 hasAnchor" number="9.3.1">
<h3><span class="header-section-number">9.3.1</span> Regression<a href="misspecify.html#regression-1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div class="sourceCode" id="cb155"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb155-1"><a href="misspecify.html#cb155-1" aria-hidden="true" tabindex="-1"></a><span class="fu">round</span>(<span class="fu">coef</span>(<span class="fu">glm</span>(Y <span class="sc">~</span> A <span class="sc">+</span> L1 <span class="sc">+</span> L2 <span class="sc">+</span> L3 <span class="sc">+</span> L4, <span class="at">family=</span><span class="st">"gaussian"</span>, </span>
<span id="cb155-2"><a href="misspecify.html#cb155-2" aria-hidden="true" tabindex="-1"></a> <span class="at">data=</span>result.data<span class="sc">$</span>observed)),<span class="dv">2</span>)</span></code></pre></div>
<pre><code>## (Intercept) A L1 L2 L3 L4
## 88.23 7.70 4.40 -0.12 -3.13 0.07</code></pre>
</div>
<div id="propensity-score" class="section level3 hasAnchor" number="9.3.2">
<h3><span class="header-section-number">9.3.2</span> Propensity score<a href="misspecify.html#propensity-score" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Propensity score model fitting:</p>
<div class="sourceCode" id="cb157"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb157-1"><a href="misspecify.html#cb157-1" aria-hidden="true" tabindex="-1"></a><span class="fu">require</span>(MatchIt)</span>
<span id="cb157-2"><a href="misspecify.html#cb157-2" aria-hidden="true" tabindex="-1"></a>match.obj <span class="ot"><-</span> <span class="fu">matchit</span>(A <span class="sc">~</span> L1 <span class="sc">+</span> L2 <span class="sc">+</span> L3 <span class="sc">+</span> L4, <span class="at">method =</span> <span class="st">"nearest"</span>, </span>
<span id="cb157-3"><a href="misspecify.html#cb157-3" aria-hidden="true" tabindex="-1"></a> <span class="at">data =</span> result.data<span class="sc">$</span>observed,</span>
<span id="cb157-4"><a href="misspecify.html#cb157-4" aria-hidden="true" tabindex="-1"></a> <span class="at">distance =</span> <span class="st">'logit'</span>, </span>
<span id="cb157-5"><a href="misspecify.html#cb157-5" aria-hidden="true" tabindex="-1"></a> <span class="at">caliper =</span> <span class="fl">0.2</span>,</span>
<span id="cb157-6"><a href="misspecify.html#cb157-6" aria-hidden="true" tabindex="-1"></a> <span class="at">replace =</span> <span class="cn">FALSE</span>, </span>
<span id="cb157-7"><a href="misspecify.html#cb157-7" aria-hidden="true" tabindex="-1"></a> <span class="at">ratio =</span> <span class="dv">1</span>)</span>
<span id="cb157-8"><a href="misspecify.html#cb157-8" aria-hidden="true" tabindex="-1"></a>match.obj</span></code></pre></div>
<pre><code>## A matchit object
## - method: 1:1 nearest neighbor matching without replacement
## - distance: Propensity score [caliper]
## - estimated with logistic regression
## - caliper: <distance> (0.034)
## - number of obs.: 10000 (original), 5790 (matched)
## - target estimand: ATT
## - covariates: L1, L2, L3, L4</code></pre>
<div class="sourceCode" id="cb159"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb159-1"><a href="misspecify.html#cb159-1" aria-hidden="true" tabindex="-1"></a>matched.data <span class="ot"><-</span> <span class="fu">match.data</span>(match.obj)</span></code></pre></div>
<p>Results from step 4: crude</p>
<div class="sourceCode" id="cb160"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb160-1"><a href="misspecify.html#cb160-1" aria-hidden="true" tabindex="-1"></a><span class="fu">round</span>(<span class="fu">coef</span>(<span class="fu">glm</span>(Y <span class="sc">~</span> A, </span>
<span id="cb160-2"><a href="misspecify.html#cb160-2" aria-hidden="true" tabindex="-1"></a> <span class="at">family=</span><span class="st">"gaussian"</span>, </span>
<span id="cb160-3"><a href="misspecify.html#cb160-3" aria-hidden="true" tabindex="-1"></a> <span class="at">data=</span>matched.data)),<span class="dv">2</span>)</span></code></pre></div>
<pre><code>## (Intercept) A
## 121.51 7.93</code></pre>
<p>Results from step 4: double adjustment</p>
<div class="sourceCode" id="cb162"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb162-1"><a href="misspecify.html#cb162-1" aria-hidden="true" tabindex="-1"></a><span class="fu">round</span>(<span class="fu">coef</span>(<span class="fu">glm</span>(Y <span class="sc">~</span> A <span class="sc">+</span> L1 <span class="sc">+</span> L2 <span class="sc">+</span> L3 <span class="sc">+</span> L4, </span>
<span id="cb162-2"><a href="misspecify.html#cb162-2" aria-hidden="true" tabindex="-1"></a> <span class="at">family=</span><span class="st">"gaussian"</span>, </span>
<span id="cb162-3"><a href="misspecify.html#cb162-3" aria-hidden="true" tabindex="-1"></a> <span class="at">data=</span>matched.data)),<span class="dv">2</span>)</span></code></pre></div>
<pre><code>## (Intercept) A L1 L2 L3 L4
## 89.56 7.66 4.11 -0.36 8.38 0.07</code></pre>
</div>
<div id="double-machine-learning-method" class="section level3 hasAnchor" number="9.3.3">
<h3><span class="header-section-number">9.3.3</span> Double machine learning method<a href="misspecify.html#double-machine-learning-method" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>We are using DML with only one learner:</p>
<div class="sourceCode" id="cb164"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb164-1"><a href="misspecify.html#cb164-1" aria-hidden="true" tabindex="-1"></a>res <span class="ot"><-</span> result.data<span class="sc">$</span>observed</span>
<span id="cb164-2"><a href="misspecify.html#cb164-2" aria-hidden="true" tabindex="-1"></a><span class="fu">require</span>(DoubleML)</span>
<span id="cb164-3"><a href="misspecify.html#cb164-3" aria-hidden="true" tabindex="-1"></a>dd <span class="ot"><-</span> DoubleMLData<span class="sc">$</span><span class="fu">new</span>(res, </span>
<span id="cb164-4"><a href="misspecify.html#cb164-4" aria-hidden="true" tabindex="-1"></a> <span class="at">y_col =</span> <span class="st">"Y"</span>, </span>
<span id="cb164-5"><a href="misspecify.html#cb164-5" aria-hidden="true" tabindex="-1"></a> <span class="at">d_cols =</span> <span class="st">"A"</span>, </span>
<span id="cb164-6"><a href="misspecify.html#cb164-6" aria-hidden="true" tabindex="-1"></a> <span class="at">x_cols =</span> <span class="fu">c</span>(<span class="st">"L1"</span>,<span class="st">"L2"</span>,<span class="st">"L3"</span>,<span class="st">"L4"</span>))</span>
<span id="cb164-7"><a href="misspecify.html#cb164-7" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(mlr3)</span>
<span id="cb164-8"><a href="misspecify.html#cb164-8" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(mlr3learners)</span>
<span id="cb164-9"><a href="misspecify.html#cb164-9" aria-hidden="true" tabindex="-1"></a>lgr<span class="sc">::</span><span class="fu">get_logger</span>(<span class="st">"mlr3"</span>)<span class="sc">$</span><span class="fu">set_threshold</span>(<span class="st">"warn"</span>)</span>
<span id="cb164-10"><a href="misspecify.html#cb164-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb164-11"><a href="misspecify.html#cb164-11" aria-hidden="true" tabindex="-1"></a>learner <span class="ot">=</span> <span class="fu">lrn</span>(<span class="st">"regr.ranger"</span>, </span>
<span id="cb164-12"><a href="misspecify.html#cb164-12" aria-hidden="true" tabindex="-1"></a> <span class="at">num.trees=</span><span class="dv">500</span>, </span>
<span id="cb164-13"><a href="misspecify.html#cb164-13" aria-hidden="true" tabindex="-1"></a> <span class="at">mtry=</span><span class="fu">floor</span>(<span class="fu">sqrt</span>(<span class="dv">5</span>)), </span>
<span id="cb164-14"><a href="misspecify.html#cb164-14" aria-hidden="true" tabindex="-1"></a> <span class="at">max.depth=</span><span class="dv">5</span>, </span>
<span id="cb164-15"><a href="misspecify.html#cb164-15" aria-hidden="true" tabindex="-1"></a> <span class="at">min.node.size=</span><span class="dv">2</span>)</span>
<span id="cb164-16"><a href="misspecify.html#cb164-16" aria-hidden="true" tabindex="-1"></a>ml_g <span class="ot">=</span> learner<span class="sc">$</span><span class="fu">clone</span>() <span class="co"># outcome modelling</span></span>
<span id="cb164-17"><a href="misspecify.html#cb164-17" aria-hidden="true" tabindex="-1"></a>ml_m <span class="ot">=</span> learner<span class="sc">$</span><span class="fu">clone</span>() <span class="co"># exposure modelling</span></span>
<span id="cb164-18"><a href="misspecify.html#cb164-18" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb164-19"><a href="misspecify.html#cb164-19" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1234</span>)</span>
<span id="cb164-20"><a href="misspecify.html#cb164-20" aria-hidden="true" tabindex="-1"></a>dml.res <span class="ot"><-</span> DoubleMLPLR<span class="sc">$</span><span class="fu">new</span>(dd, <span class="at">ml_g=</span>ml_g, <span class="at">ml_m=</span>ml_m)</span>
<span id="cb164-21"><a href="misspecify.html#cb164-21" aria-hidden="true" tabindex="-1"></a>dml.res<span class="sc">$</span><span class="fu">fit</span>()</span>
<span id="cb164-22"><a href="misspecify.html#cb164-22" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(dml.res)</span></code></pre></div>
<pre><code>## ================= DoubleMLPLR Object ==================
##
##
## ------------------ Data summary ------------------
## Outcome variable: Y
## Treatment variable(s): A
## Covariates: L1, L2, L3, L4
## Instrument(s):
## No. Observations: 10000
##
## ------------------ Score & algorithm ------------------
## Score function: partialling out
## DML algorithm: dml2
##
## ------------------ Machine learner ------------------
## ml_g: regr.ranger
## ml_m: regr.ranger
##
## ------------------ Resampling ------------------
## No. folds: 5
## No. repeated sample splits: 1
## Apply cross-fitting: TRUE
##
## ------------------ Fit summary ------------------
## Estimates and significance testing of the effect of target variables
## Estimate. Std. Error t value Pr(>|t|)
## A 7.0114 0.1532 45.76 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
<div class="sourceCode" id="cb166"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb166-1"><a href="misspecify.html#cb166-1" aria-hidden="true" tabindex="-1"></a>dml.res<span class="sc">$</span><span class="fu">summary</span>()</span></code></pre></div>
<pre><code>## Estimates and significance testing of the effect of target variables
## Estimate. Std. Error t value Pr(>|t|)
## A 7.0114 0.1532 45.76 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
</div>
<div id="augmented-inverse-probability-weighting" class="section level3 hasAnchor" number="9.3.4">
<h3><span class="header-section-number">9.3.4</span> Augmented Inverse probability weighting<a href="misspecify.html#augmented-inverse-probability-weighting" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div class="sourceCode" id="cb168"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb168-1"><a href="misspecify.html#cb168-1" aria-hidden="true" tabindex="-1"></a>SL.lib<span class="ot">=</span><span class="fu">c</span>(<span class="st">'SL.glm'</span>, <span class="st">'SL.step.interaction'</span>, <span class="st">'SL.earth'</span>, <span class="st">'SL.mean'</span>)</span>
<span id="cb168-2"><a href="misspecify.html#cb168-2" aria-hidden="true" tabindex="-1"></a><span class="fu">require</span>(AIPW)</span>
<span id="cb168-3"><a href="misspecify.html#cb168-3" aria-hidden="true" tabindex="-1"></a><span class="fu">require</span>(ggplot2)</span>
<span id="cb168-4"><a href="misspecify.html#cb168-4" aria-hidden="true" tabindex="-1"></a>res <span class="ot"><-</span> result.data<span class="sc">$</span>observed</span>
<span id="cb168-5"><a href="misspecify.html#cb168-5" aria-hidden="true" tabindex="-1"></a>aipw_sl <span class="ot"><-</span> AIPW<span class="sc">$</span><span class="fu">new</span>(<span class="at">Y=</span>res<span class="sc">$</span>Y, <span class="at">A=</span>res<span class="sc">$</span>A, </span>
<span id="cb168-6"><a href="misspecify.html#cb168-6" aria-hidden="true" tabindex="-1"></a> <span class="at">W=</span>res[<span class="fu">c</span>(<span class="st">"L1"</span>,<span class="st">"L2"</span>,<span class="st">"L3"</span>,<span class="st">"L4"</span>)],</span>
<span id="cb168-7"><a href="misspecify.html#cb168-7" aria-hidden="true" tabindex="-1"></a> <span class="at">Q.SL.library=</span>SL.lib,<span class="at">g.SL.library=</span>SL.lib,</span>
<span id="cb168-8"><a href="misspecify.html#cb168-8" aria-hidden="true" tabindex="-1"></a> <span class="at">k_split=</span><span class="dv">1</span>,<span class="at">verbose=</span><span class="cn">TRUE</span>)</span>
<span id="cb168-9"><a href="misspecify.html#cb168-9" aria-hidden="true" tabindex="-1"></a>aipw_sl<span class="sc">$</span><span class="fu">fit</span>()</span>
<span id="cb168-10"><a href="misspecify.html#cb168-10" aria-hidden="true" tabindex="-1"></a>aipw_sl<span class="sc">$</span><span class="fu">summary</span>(<span class="at">g.bound =</span> <span class="fl">0.025</span>)</span></code></pre></div>
<div class="sourceCode" id="cb169"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb169-1"><a href="misspecify.html#cb169-1" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(aipw_sl<span class="sc">$</span>result, <span class="at">digits =</span> <span class="dv">2</span>)</span></code></pre></div>
<pre><code>## Estimate SE 95% LCL 95% UCL N
## Risk of exposure 126.3 0.18 126.0 126.7 3055
## Risk of control 119.9 0.10 119.7 120.1 6945
## Risk Difference 6.4 0.19 6.1 6.8 10000</code></pre>
</div>
<div id="double-robust-method-tmle" class="section level3 hasAnchor" number="9.3.5">
<h3><span class="header-section-number">9.3.5</span> Double robust method (TMLE)<a href="misspecify.html#double-robust-method-tmle" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>TMLE with superlearner</p>
<div class="sourceCode" id="cb171"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb171-1"><a href="misspecify.html#cb171-1" aria-hidden="true" tabindex="-1"></a>tmle.res <span class="ot"><-</span> <span class="fu">ltmle</span>(<span class="at">data=</span>result.data<span class="sc">$</span>observed, <span class="at">Anodes=</span><span class="st">'A'</span>, <span class="at">Ynodes=</span><span class="st">'Y'</span>, <span class="at">abar=</span><span class="fu">list</span>(<span class="dv">1</span>,<span class="dv">0</span>), </span>
<span id="cb171-2"><a href="misspecify.html#cb171-2" aria-hidden="true" tabindex="-1"></a> <span class="at">SL.library=</span><span class="fu">c</span>(<span class="st">'SL.glm'</span>, <span class="st">'SL.step.interaction'</span>, <span class="st">'SL.earth'</span>, <span class="st">'SL.mean'</span>), </span>
<span id="cb171-3"><a href="misspecify.html#cb171-3" aria-hidden="true" tabindex="-1"></a> <span class="at">estimate.time=</span>F,</span>
<span id="cb171-4"><a href="misspecify.html#cb171-4" aria-hidden="true" tabindex="-1"></a> <span class="at">SL.cvControl=</span><span class="fu">list</span>(<span class="at">V=</span><span class="dv">10</span>),</span>
<span id="cb171-5"><a href="misspecify.html#cb171-5" aria-hidden="true" tabindex="-1"></a> <span class="at">gbounds=</span><span class="fu">c</span>(<span class="fl">0.025</span>, <span class="fl">0.975</span>))</span></code></pre></div>
<div class="sourceCode" id="cb172"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb172-1"><a href="misspecify.html#cb172-1" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(tmle.res)</span></code></pre></div>
<pre><code>## Estimator: tmle
## Call:
## ltmle(data = result.data$observed, Anodes = "A", Ynodes = "Y",
## abar = list(1, 0), gbounds = c(0.025, 0.975), SL.library = c("SL.glm",
## "SL.step.interaction", "SL.earth", "SL.mean"), SL.cvControl = list(V = 10),
## estimate.time = F)
##
## Treatment Estimate:
## Parameter Estimate: 126.3
## Estimated Std Err: 0.15795
## p-value: <2e-16
## 95% Conf Interval: (125.99, 126.61)
##
## Control Estimate:
## Parameter Estimate: 119.92
## Estimated Std Err: 0.099995
## p-value: <2e-16
## 95% Conf Interval: (119.73, 120.12)
##
## Additive Treatment Effect:
## Parameter Estimate: 6.382
## Estimated Std Err: 0.16961
## p-value: <2e-16
## 95% Conf Interval: (6.0496, 6.7144)</code></pre>
<p>You can also try with more candidate learners within super learner</p>
<div class="sourceCode" id="cb174"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb174-1"><a href="misspecify.html#cb174-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Time consuming</span></span>
<span id="cb174-2"><a href="misspecify.html#cb174-2" aria-hidden="true" tabindex="-1"></a><span class="co"># tmle.res1 <- ltmle(data=result.data$observed, Anodes='A', Ynodes='Y', abar=list(1,0), </span></span>
<span id="cb174-3"><a href="misspecify.html#cb174-3" aria-hidden="true" tabindex="-1"></a><span class="co"># SL.library=c('SL.glm', 'SL.step.interaction', </span></span>
<span id="cb174-4"><a href="misspecify.html#cb174-4" aria-hidden="true" tabindex="-1"></a><span class="co"># 'SL.gam', 'SL.glmnet', 'SL.cforest',</span></span>
<span id="cb174-5"><a href="misspecify.html#cb174-5" aria-hidden="true" tabindex="-1"></a><span class="co"># "SL.ranger", "SL.xgboost",</span></span>
<span id="cb174-6"><a href="misspecify.html#cb174-6" aria-hidden="true" tabindex="-1"></a><span class="co"># "SL.rpart", "SL.rpartPrune", </span></span>
<span id="cb174-7"><a href="misspecify.html#cb174-7" aria-hidden="true" tabindex="-1"></a><span class="co"># 'SL.earth', 'SL.mean'), </span></span>
<span id="cb174-8"><a href="misspecify.html#cb174-8" aria-hidden="true" tabindex="-1"></a><span class="co"># estimate.time=F,</span></span>
<span id="cb174-9"><a href="misspecify.html#cb174-9" aria-hidden="true" tabindex="-1"></a><span class="co"># SL.cvControl=list(V=10),</span></span>
<span id="cb174-10"><a href="misspecify.html#cb174-10" aria-hidden="true" tabindex="-1"></a><span class="co"># gbounds=c(0.025, 0.975))</span></span></code></pre></div>
<div class="sourceCode" id="cb175"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb175-1"><a href="misspecify.html#cb175-1" aria-hidden="true" tabindex="-1"></a><span class="co"># summary(tmle.res1)</span></span></code></pre></div>
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<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-balzer2021demystifying" class="csl-entry">
Balzer, Laura B, and Ted Westling. 2021. <span>“Demystifying Statistical Inference When Using Machine Learning in Causal Research.”</span> <em>American Journal of Epidemiology</em>.
</div>
<div id="ref-kang2007demystifying" class="csl-entry">
Kang, Joseph DY, and Joseph L Schafer. 2007. <span>“Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data.”</span> <em>Statistical Science</em> 22 (4): 523–39.
</div>
<div id="ref-naimi2017challenges" class="csl-entry">
Naimi, Ashley I, Alan E Mishler, and Edward H Kennedy. 2017. <span>“Challenges in Obtaining Valid Causal Effect Estimates with Machine Learning Algorithms.”</span> <em>arXiv Preprint arXiv:1711.07137</em>.
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