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<ul>
<li><a href="#step-0-analytic-data" id="toc-step-0-analytic-data" class="nav-link active" data-scroll-target="#step-0-analytic-data"><span class="header-section-number">23.1</span> Step 0: Analytic data</a></li>
<li><a href="#step-6-disease-risk-score-drs" id="toc-step-6-disease-risk-score-drs" class="nav-link" data-scroll-target="#step-6-disease-risk-score-drs"><span class="header-section-number">23.2</span> Step 6: Disease risk score (DRS)</a>
<ul class="collapse">
<li><a href="#unexposed-cohort" id="toc-unexposed-cohort" class="nav-link" data-scroll-target="#unexposed-cohort"><span class="header-section-number">23.2.1</span> Unexposed cohort</a></li>
<li><a href="#create-drs-formula" id="toc-create-drs-formula" class="nav-link" data-scroll-target="#create-drs-formula"><span class="header-section-number">23.2.2</span> Create DRS formula</a></li>
<li><a href="#fit-drs-model" id="toc-fit-drs-model" class="nav-link" data-scroll-target="#fit-drs-model"><span class="header-section-number">23.2.3</span> Fit DRS model</a></li>
<li><a href="#obtain-drs" id="toc-obtain-drs" class="nav-link" data-scroll-target="#obtain-drs"><span class="header-section-number">23.2.4</span> Obtain DRS</a></li>
</ul></li>
<li><a href="#step-7-association" id="toc-step-7-association" class="nav-link" data-scroll-target="#step-7-association"><span class="header-section-number">23.3</span> Step 7: Association</a></li>
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<h1 class="title"><span class="chapter-number">23</span> <span class="chapter-title">hdDRS with a survival outcome</span></h1>
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<p>To demonstrate the high-dimensional disease risk score (hdDRS) with a survival/time-to-event outcome, we will use the same simulated data for this hdDRS demonstration as we did for the hdPS with survival outcome demonstration.</p>
<section id="step-0-analytic-data" class="level2" data-number="23.1">
<h2 data-number="23.1" class="anchored" data-anchor-id="step-0-analytic-data"><span class="header-section-number">23.1</span> Step 0: Analytic data</h2>
<div class="cell">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="fu">dim</span>(hdps.data)</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="co">#> [1] 3000 410</span></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Investigator-specified covariates</span></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a>investigator.vars</span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a><span class="co">#> [1] "age" "sex" "comorbidity"</span></span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Top 200 empirical covariates section based on Cox-LASSO</span></span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span>(empirical.vars.lasso)</span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a><span class="co">#> [1] "rec_diag_V03_once" "rec_diag_S24_once" "rec_diag_W61_once"</span></span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a><span class="co">#> [4] "rec_diag_C81_once" "rec_diag_M85_once" "rec_diag_H18_once"</span></span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a><span class="co"># Investigator-specified and empirical covariates</span></span>
<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span>(vars.hsps)</span>
<span id="cb1-15"><a href="#cb1-15" aria-hidden="true" tabindex="-1"></a><span class="co">#> [1] "age" "sex" "comorbidity" </span></span>
<span id="cb1-16"><a href="#cb1-16" aria-hidden="true" tabindex="-1"></a><span class="co">#> [4] "rec_diag_V03_once" "rec_diag_S24_once" "rec_diag_W61_once"</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</section>
<section id="step-6-disease-risk-score-drs" class="level2 page-columns page-full" data-number="23.2">
<h2 data-number="23.2" class="anchored" data-anchor-id="step-6-disease-risk-score-drs"><span class="header-section-number">23.2</span> Step 6: Disease risk score (DRS)</h2>
<p>There are at least eight approaches to estimate the disease risk score (DRS):</p>
<ul>
<li><p>hdDRS-Full-Logistic: On the full cohort (both exposed and unexposed), fit logistic regression without considering the follow-up time. This model included the exposure, investigator-specified measured confounders, and the recurrence covariates. The DRS is calculated as the probability of the outcome by setting everyone as unexposed.</p></li>
<li><p>hdDRS-Full-Survival: On the full cohort, fit the Cox-PH model with the exposure, investigator-specified measured confounders and the recurrence covariates. The DRS is calculated as the survival probability of the outcome by setting everyone as unexposed.</p></li>
<li><p>hdDRS-Full-Hazard: On the full cohort, fit the Cox-PH model with the exposure, investigator-specified measured confounders and the recurrence covariates. The DRS is calculated as the hazard of the outcome by setting everyone unexposed.</p></li>
<li><p>hdDRS-Full-Rate: On the full cohort, fit the modified Poisson regression with the exposure, an offset by the natural logarithm of follow-up time, investigator-specified measured confounders and the recurrence covariates. The DRS is calculated as the rate of the outcome by setting everyone as unexposed.</p></li>
<li><p>hdDRS-Unexposed-Logistic: On the cohort with only unexposed, fit the logistic regression with the investigator-specified measured confounders and the recurrence covariates. The DRS is calculated as the probability of the outcome on the full cohort.</p></li>
<li><p>hdDRS-Unexposed-Survival: On the cohort with only unexposed, fit the Cox-PH model with the investigator-specified measured confounders and the recurrence covariates. The DRS is calculated as the survival probability of the outcome on the full cohort.</p></li>
<li><p>hdDRS-Unexposed-Hazard: On the cohort with only unexposed, fit the Cox-PH model with the investigator-specified measured confounders and the recurrence covariates. The DRS is calculated as the hazard of the outcome on the full cohort.</p></li>
<li><p>hdDRS-Unexposed-Rate: On the cohort with only unexposed, fit the modified Poisson regression with an offset by the natural logarithm of follow-up time, investigator-specified measured confounders and the recurrence covariates. The DRS is calculated as the rate of the outcome on the full cohort.</p></li>
</ul>
<p>In this example, we will focus on the rate-based approach using the unexposed cohort (hdDRS-Unexposed-Rate). To demonstrate how to apply all these eight methods in a given scenario, reproducible R codes on a simulated dataset are provided in the <a href="https://github.com/belalanik/hdDRScausal">GitHub folder</a>.</p>
<div class="no-row-height column-margin column-container"><div class="">
<p><span class="citation" data-cites="hansen2008prognostic zhang2019use hossain2025harnessing2">(<a href="references.html#ref-hansen2008prognostic" role="doc-biblioref">Hansen 2008</a>; <a href="references.html#ref-zhang2019use" role="doc-biblioref">Zhang and Kim 2019</a>; <a href="references.html#ref-hossain2025harnessing2" role="doc-biblioref">Hossain 2025</a>)</span></p>
</div></div><section id="unexposed-cohort" class="level3" data-number="23.2.1">
<h3 data-number="23.2.1" class="anchored" data-anchor-id="unexposed-cohort"><span class="header-section-number">23.2.1</span> Unexposed cohort</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Unexposed cohort</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>dat.unexposed <span class="ot"><-</span> <span class="fu">subset</span>(hdps.data, arthritis <span class="sc">==</span> <span class="st">"No"</span>)</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Offset </span></span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a>dat.unexposed<span class="sc">$</span>log.offset <span class="ot"><-</span> <span class="fu">log</span>(<span class="dv">1</span>)</span>
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Full cohort</span></span>
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a>dat.full <span class="ot"><-</span> hdps.data</span>
<span id="cb2-9"><a href="#cb2-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-10"><a href="#cb2-10" aria-hidden="true" tabindex="-1"></a><span class="co"># Offset </span></span>
<span id="cb2-11"><a href="#cb2-11" aria-hidden="true" tabindex="-1"></a>dat.full<span class="sc">$</span>log.offset <span class="ot"><-</span> <span class="fu">log</span>(<span class="dv">1</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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</section>
<section id="create-drs-formula" class="level3" data-number="23.2.2">
<h3 data-number="23.2.2" class="anchored" data-anchor-id="create-drs-formula"><span class="header-section-number">23.2.2</span> Create DRS formula</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Covariates</span></span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>vars.hsdrs <span class="ot"><-</span> <span class="fu">c</span>(investigator.vars, empirical.vars.lasso)</span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Formula</span></span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a>drs.formula <span class="ot"><-</span> <span class="fu">as.formula</span>(<span class="fu">paste0</span>(<span class="st">"cvd ~ offset(log.offset) + "</span>, </span>
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">paste</span>(vars.hsdrs, <span class="at">collapse =</span> <span class="st">"+"</span>)))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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</section>
<section id="fit-drs-model" class="level3" data-number="23.2.3">
<h3 data-number="23.2.3" class="anchored" data-anchor-id="fit-drs-model"><span class="header-section-number">23.2.3</span> Fit DRS model</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>fit.drs <span class="ot"><-</span> <span class="fu">glm</span>(drs.formula, <span class="at">data =</span> dat.unexposed, <span class="at">family =</span> poisson)</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>fit.drs<span class="sc">$</span>coefficients[<span class="fu">is.na</span>(fit.drs<span class="sc">$</span>coefficients)] <span class="ot"><-</span> <span class="dv">0</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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</section>
<section id="obtain-drs" class="level3" data-number="23.2.4">
<h3 data-number="23.2.4" class="anchored" data-anchor-id="obtain-drs"><span class="header-section-number">23.2.4</span> Obtain DRS</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>dat.full<span class="sc">$</span>drs <span class="ot"><-</span> <span class="fu">predict</span>(fit.drs, <span class="at">type =</span> <span class="st">"response"</span>, <span class="at">newdata =</span> dat.full)</span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Sumamry</span></span>
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(dat.full<span class="sc">$</span>drs)</span>
<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a><span class="co">#> Min. 1st Qu. Median Mean 3rd Qu. Max. </span></span>
<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a><span class="co">#> 0.004198 0.089762 0.232163 0.324619 0.466890 4.085130</span></span>
<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Summary by exposure status</span></span>
<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a><span class="fu">tapply</span>(dat.full<span class="sc">$</span>drs, dat.full<span class="sc">$</span>arthritis, summary)</span>
<span id="cb5-10"><a href="#cb5-10" aria-hidden="true" tabindex="-1"></a><span class="co">#> $No</span></span>
<span id="cb5-11"><a href="#cb5-11" aria-hidden="true" tabindex="-1"></a><span class="co">#> Min. 1st Qu. Median Mean 3rd Qu. Max. </span></span>
<span id="cb5-12"><a href="#cb5-12" aria-hidden="true" tabindex="-1"></a><span class="co">#> 0.004198 0.081598 0.206067 0.293047 0.433756 1.776528 </span></span>
<span id="cb5-13"><a href="#cb5-13" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb5-14"><a href="#cb5-14" aria-hidden="true" tabindex="-1"></a><span class="co">#> $Yes</span></span>
<span id="cb5-15"><a href="#cb5-15" aria-hidden="true" tabindex="-1"></a><span class="co">#> Min. 1st Qu. Median Mean 3rd Qu. Max. </span></span>
<span id="cb5-16"><a href="#cb5-16" aria-hidden="true" tabindex="-1"></a><span class="co">#> 0.006824 0.129409 0.298506 0.403565 0.568061 4.085130</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</section>
</section>
<section id="step-7-association" class="level2" data-number="23.3">
<h2 data-number="23.3" class="anchored" data-anchor-id="step-7-association"><span class="header-section-number">23.3</span> Step 7: Association</h2>
<div class="cell">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Deciles of DRS</span></span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>dat.full<span class="sc">$</span>drs.decile <span class="ot"><-</span> <span class="fu">as.factor</span>(dplyr<span class="sc">::</span><span class="fu">ntile</span>(dat.full<span class="sc">$</span>drs, <span class="dv">10</span>))</span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Outcome analysis</span></span>
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a>fit.hddrs <span class="ot"><-</span> <span class="fu">coxph</span>(<span class="fu">Surv</span>(follow_up, cvd) <span class="sc">~</span> arthritis <span class="sc">+</span> drs.decile <span class="sc">+</span> age <span class="sc">+</span> sex <span class="sc">+</span> </span>
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a> comorbidity, <span class="at">data =</span> dat.full)</span>
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a><span class="fu">publish</span>(fit.hddrs, <span class="at">pvalue.method =</span> <span class="st">"robust"</span>, <span class="at">confint.method =</span> <span class="st">"robust"</span>, </span>
<span id="cb6-9"><a href="#cb6-9" aria-hidden="true" tabindex="-1"></a> <span class="at">print =</span> F)<span class="sc">$</span>regressionTable[<span class="dv">1</span><span class="sc">:</span><span class="dv">2</span>,]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<div data-pagedtable="false">
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<div id="refs" class="references csl-bib-body hanging-indent" data-entry-spacing="0" role="list" style="display: none">
<div id="ref-hansen2008prognostic" class="csl-entry" role="listitem">
Hansen, Ben B. 2008. <span>“The Prognostic Analogue of the Propensity Score.”</span> <em>Biometrika</em> 95 (2): 481–88.
</div>
<div id="ref-hossain2025harnessing2" class="csl-entry" role="listitem">
Hossain, Md Belal. 2025. <span>“Chapter 2: High-Dimensional Disease Risk Score for Dealing with Residual Confounding in Estimating Treatment Effects with a Survival Outcome.”</span> In. Harnessing the power of causal inference; predictive analytics for survival outcomes with health administrative data: applications to tuberculosis research.
</div>
<div id="ref-zhang2019use" class="csl-entry" role="listitem">
Zhang, Di, and Jessica Kim. 2019. <span>“Use of Propensity Score and Disease Risk Score for Multiple Treatments with Time-to-Event Outcome: A Simulation Study.”</span> <em>Journal of Biopharmaceutical Statistics</em> 29 (6): 1103–15.
</div>
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