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<h1 class="quarto-secondary-nav-title"><span class="chapter-number">7</span> <span class="chapter-title">Step 4: Prioritize</span></h1>
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<h2 id="toc-title">Table of contents</h2>
<ul>
<li><a href="#bross-formula" id="toc-bross-formula" class="nav-link active" data-scroll-target="#bross-formula"><span class="toc-section-number">7.1</span> Bross formula</a></li>
<li><a href="#calculating-bias-from-a-recurrence-covariate" id="toc-calculating-bias-from-a-recurrence-covariate" class="nav-link" data-scroll-target="#calculating-bias-from-a-recurrence-covariate"><span class="toc-section-number">7.2</span> Calculating bias from a recurrence covariate</a></li>
<li><a href="#calculating-bias-from-all-recurrence-covariates" id="toc-calculating-bias-from-all-recurrence-covariates" class="nav-link" data-scroll-target="#calculating-bias-from-all-recurrence-covariates"><span class="toc-section-number">7.3</span> Calculating bias from all recurrence covariates</a></li>
<li><a href="#obtain-log-of-absolute-bias" id="toc-obtain-log-of-absolute-bias" class="nav-link" data-scroll-target="#obtain-log-of-absolute-bias"><span class="toc-section-number">7.4</span> Obtain log of absolute-bias</a></li>
<li><a href="#convert-to-absolute-log-of-multiplicative-bias" id="toc-convert-to-absolute-log-of-multiplicative-bias" class="nav-link" data-scroll-target="#convert-to-absolute-log-of-multiplicative-bias"><span class="toc-section-number">7.5</span> Convert to Absolute log of multiplicative bias</a></li>
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<h1 class="title d-none d-lg-block"><span class="chapter-number">7</span> <span class="chapter-title">Step 4: Prioritize</span></h1>
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<div class="cell" data-hash="step4_cache/html/unnamed-chunk-1_a903b1cdd345077a5496b4455f55c46a">
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<section id="bross-formula" class="level2 page-columns page-full" data-number="7.1">
<h2 data-number="7.1" class="anchored" data-anchor-id="bross-formula"><span class="header-section-number">7.1</span> Bross formula</h2>
<p>We need to make an educated guess about 3 components (i.e., <mark>make an assumption</mark>), that are used in the calculation of <mark>bias contributed by not adjusting for a covariate</mark> based on <a href="https://pubmed.ncbi.nlm.nih.gov/5966011/">Bross (1966)</a> formula:</p>
<div class="no-row-height column-margin column-container"><div class="">
<p>Bross formula <span class="citation" data-cites="bross1966spurious schneeweiss2006sensitivity">(<a href="references.html#ref-bross1966spurious" role="doc-biblioref">Bross 1966</a>; <a href="references.html#ref-schneeweiss2006sensitivity" role="doc-biblioref">Schneeweiss 2006</a>)</span> for the Bias Multiplier considers both the imbalance in the prevalence of the unmeasured confounder between the exposure groups and the association between the confounder and the outcome to assess the potential bias.</p>
</div></div><ul>
<li>prevalence of a binary unmeasured confounder (<span class="math inline">\(U\)</span>) among exposed (<span class="math inline">\(P_{UA_1}\)</span>)</li>
<li>prevalence of that binary unmeasured confounder among unexposed (<span class="math inline">\(P_{UA_0}\)</span>)</li>
<li>association between that binary unmeasured confounder and the outcome (<span class="math inline">\(RR_{UY} = \frac{P_{UY_1}}{P_{UY_1}}\)</span>)</li>
</ul>
<p>The above components can help us calculate <span class="math inline">\(bias\)</span> amount (known as <mark>‘Bias Multiplier’</mark>) using the Bross formula when we omit adjusting for <span class="math inline">\(U\)</span>:</p>
<p><span class="math display">\[\text{Bias}_U = \frac{P_{UA_1} (RR_{UY} - 1) + 1}{P_{UA_0} (RR_{UY} - 1) + 1}\]</span></p>
<div class="no-row-height column-margin column-container"><div class="">
<p>These are the ingredients of the Bross formula. This formula is helpful for understanding the impact of unmeasured confounding of a binary variable. We have to put assumed prevalence and risk ratio associated with an unmeasured confounder.</p>
</div></div></section>
<section id="calculating-bias-from-a-recurrence-covariate" class="level2 page-columns page-full" data-number="7.2">
<h2 data-number="7.2" class="anchored" data-anchor-id="calculating-bias-from-a-recurrence-covariate"><span class="header-section-number">7.2</span> Calculating bias from a recurrence covariate</h2>
<p>For recurrence covariates (<span class="math inline">\(R\)</span>), we do not need to assume, we just <mark>plug-in <span class="math inline">\(R\)</span> instead of <span class="math inline">\(U\)</span></mark> in the following calculations:</p>
<ul>
<li>prevalence of a binary recurrence variable among exposed (<span class="math inline">\(P_{RA_1}\)</span>)</li>
<li>prevalence of that binary recurrence variable among unexposed (<span class="math inline">\(P_{RA_0}\)</span>)</li>
<li>association between that binary recurrence variable and the outcome (<span class="math inline">\(RR_{RY} = \frac{P_{RY_1}}{P_{RY_1}}\)</span>)</li>
</ul>
<p>These components can help us empirically calculate <span class="math inline">\(bias\)</span> amount:</p>
<p><span class="math display">\[\text{Bias}_R = \frac{P_{RA_1} (RR_{RY} - 1) + 1}{P_{RA_0} (RR_{RY} - 1) + 1}\]</span></p>
<p>Here, <span class="math inline">\(RR_{RY}\)</span> is the crude risk ratio between the recurrence covariate and the outcome, <span class="math inline">\(Y\)</span> is the outcome, <span class="math inline">\(A\)</span> is the exposure, and <span class="math inline">\(R\)</span> is a recurrence covariate.</p>
<div class="no-row-height column-margin column-container"><div class="">
<p>For recurrence covariates, we do not need to assume, we can basically calculate these numbers (<span class="math inline">\(log-absolute-bias\)</span>) for all of the recurrence covariates <span class="citation" data-cites="schneeweiss2009high">(<a href="references.html#ref-schneeweiss2009high" role="doc-biblioref">Schneeweiss et al. 2009</a>)</span>. For each data dimension, we can rank each of the recurrence covariates based on the amount of bias (confounding or imbalance) it could likely adjust.</p>
</div></div></section>
<section id="calculating-bias-from-all-recurrence-covariates" class="level2" data-number="7.3">
<h2 data-number="7.3" class="anchored" data-anchor-id="calculating-bias-from-all-recurrence-covariates"><span class="header-section-number">7.3</span> Calculating bias from all recurrence covariates</h2>
<p>In our example, we simply plug-in each recurrence covariates one-by-one to calculate <span class="math inline">\(log-absolute-bias\)</span>:</p>
<div class="cell" data-hash="step4_cache/html/unnamed-chunk-2_6e98ea6b50ebc4e8af3bb2bdbe82b256">
<div class="cell-output-display">
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<tbody>
<tr class="odd">
<td style="text-align: center;">R=rec_dx_D64_once</td>
</tr>
<tr class="even">
<td style="text-align: center;">R=rec_dx_D75_sporadic</td>
</tr>
<tr class="odd">
<td style="text-align: center;">…</td>
</tr>
<tr class="even">
<td style="text-align: center;">R=rec_dx_E07_frequent</td>
</tr>
</tbody>
</table>
</div>
</div>
</section>
<section id="obtain-log-of-absolute-bias" class="level2 page-columns page-full" data-number="7.4">
<h2 data-number="7.4" class="anchored" data-anchor-id="obtain-log-of-absolute-bias"><span class="header-section-number">7.4</span> Obtain log of absolute-bias</h2>
<p>We calculate <span class="math inline">\(log-absolute-bias\)</span> for all recurrence covariates.</p>
<div class="no-row-height column-margin column-container"><div class="">
<p>Absolute log of the Bias Multiplier, <span class="math inline">\(log-absolute-bias\)</span>, is a symmetric measure of the potential bias introduced by the recurrence covariate, making it easier to compare and rank recurrence covariates.</p>
</div></div><div class="cell" data-hash="step4_cache/html/unnamed-chunk-3_6993dbca467e242da7b998b520b5043d">
<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>out3 <span class="ot"><-</span> <span class="fu">get_prioritised_covariates</span>(<span class="at">df =</span> out2,</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a> <span class="at">patientIdVarname =</span> <span class="st">"idx"</span>, </span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a> <span class="at">exposureVector =</span> basetable<span class="sc">$</span>exposure,</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a> <span class="at">outcomeVector =</span> basetable<span class="sc">$</span>outcome,</span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a> <span class="at">patientIdVector =</span> patientIds, </span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a> <span class="at">k =</span> <span class="dv">50</span>)</span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a>sorted_values <span class="ot"><-</span> <span class="fu">sort</span>(out3<span class="sc">$</span>multiplicative_bias, </span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a> <span class="at">decreasing =</span> <span class="cn">TRUE</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>This would return absolute log of the multiplicative bias for each recurrence covariate (by univariate Bross formula). We can use this information to prioritize recurrence covariates in the next step.</p>
</section>
<section id="convert-to-absolute-log-of-multiplicative-bias" class="level2 page-columns page-full" data-number="7.5">
<h2 data-number="7.5" class="anchored" data-anchor-id="convert-to-absolute-log-of-multiplicative-bias"><span class="header-section-number">7.5</span> Convert to Absolute log of multiplicative bias</h2>
<p>Here are the few covariates and associated Absolute log of the multiplicative bias:</p>
<div class="cell" data-hash="step4_cache/html/unnamed-chunk-4_ae400e36504b02783987a6c53b5abc43">
<div class="cell-output-display">
<table class="table table-striped table-hover" style="width: auto !important; margin-left: auto; margin-right: auto;">
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<td style="text-align:center;"> rec_dx_I10_once : 0.124 </td>
</tr>
<tr>
<td style="text-align:center;"> rec_dx_R73_once : 0.078 </td>
</tr>
<tr>
<td style="text-align:center;"> rec_dx_I10_frequent : 0.065 </td>
</tr>
<tr>
<td style="text-align:center;"> rec_dx_R60_once : 0.038 </td>
</tr>
<tr>
<td style="text-align:center;"> rec_dx_E78_once : 0.036 </td>
</tr>
<tr>
<td style="text-align:center;"> rec_dx_M79_once : 0.033 </td>
</tr>
<tr>
<td style="text-align:center;"> rec_dx_I51_once : 0.019 </td>
</tr>
<tr>
<td style="text-align:center;"> rec_dx_M10_once : 0.017 </td>
</tr>
<tr>
<td style="text-align:center;"> rec_dx_I50_once : 0.016 </td>
</tr>
</tbody>
</table>
</div>
</div>
<p>And here are translated table with description:</p>
<div class="cell" data-hash="step4_cache/html/unnamed-chunk-5_f0c9d19103f22d9051a4cc7f2b1c1ba5">
<div class="cell-output-display">
<table class="table table-striped table-hover" style="width: auto !important; margin-left: auto; margin-right: auto;">
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<th style="text-align:center;"> </th>
</tr>
</thead>
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<tr>
<td style="text-align:center;"> Hypertension : 0.115 </td>
</tr>
<tr>
<td style="text-align:center;"> Elevated blood glucose level : 0.088 </td>
</tr>
<tr>
<td style="text-align:center;"> Hypertension : 0.068 </td>
</tr>
<tr>
<td style="text-align:center;"> Edema : 0.054 </td>
</tr>
<tr>
<td style="text-align:center;"> Pure hypercholesterolemia : 0.038 </td>
</tr>
<tr>
<td style="text-align:center;"> musculoskeletal pain : 0.017 </td>
</tr>
<tr>
<td style="text-align:center;"> Hypokalemia : 0.015 </td>
</tr>
<tr>
<td style="text-align:center;"> Heart disease : 0.013 </td>
</tr>
<tr>
<td style="text-align:center;"> Heart failure : 0.011 </td>
</tr>
</tbody>
</table>
</div>
</div>
<div class="no-row-height column-margin column-container"><div class="">
<p>Some of the empirical covariates with top Absolute log of the multiplicative bias are actually relevant to the outcome (diabetes): Hypertension, Elevated blood glucose level , etc. <span class="citation" data-cites="choi2001risk">(<a href="references.html#ref-choi2001risk" role="doc-biblioref">Choi and Shi 2001</a>)</span></p>
</div></div><div class="cell" data-hash="step4_cache/html/unnamed-chunk-6_ba8e213ee3c38ab1ac122a11cb59247d">
</div>
<div class="callout-tip callout callout-style-default callout-captioned">
<div class="callout-header d-flex align-content-center">
<div class="callout-icon-container">
<i class="callout-icon"></i>
</div>
<div class="callout-caption-container flex-fill">
SMD vs Bias multiplier
</div>
</div>
<div class="callout-body-container callout-body">
<p>Standardized mean difference (SMD) is useful for assessing the balance in the propensity score literature. However, Bross formula incorporates outcome information. In the investigation of empirical covariates or recurrence covariates where interpretations of these covariates are unknown, it may seem more safe to use the multiplicative bias term from the Bross formula to identify proxy covariates that are helpful in predicting the outcome.</p>
</div>
</div>
<div class="no-row-height column-margin column-container"><div class="">
<p><span class="citation" data-cites="stuart2013prognostic">(<a href="references.html#ref-stuart2013prognostic" role="doc-biblioref">Stuart, Lee, and Leacy 2013</a>)</span></p>
</div></div><div class="cell" data-hash="step4_cache/html/unnamed-chunk-7_05d346fd4909e25461522c6f6bc957ae">
</div>
<div id="refs" class="references csl-bib-body hanging-indent" role="doc-bibliography" style="display: none">
<div id="ref-bross1966spurious" class="csl-entry" role="doc-biblioentry">
Bross, Irwin DJ. 1966. <span>“Spurious Effects from an Extraneous Variable.”</span> <em>Journal of Chronic Diseases</em> 19 (6): 637–47.
</div>
<div id="ref-choi2001risk" class="csl-entry" role="doc-biblioentry">
Choi, BCK, and F Shi. 2001. <span>“Risk Factors for Diabetes Mellitus by Age and Sex: Results of the National Population Health Survey.”</span> <em>Diabetologia</em> 44: 1221–31.
</div>
<div id="ref-schneeweiss2006sensitivity" class="csl-entry" role="doc-biblioentry">
Schneeweiss, Sebastian. 2006. <span>“Sensitivity Analysis and External Adjustment for Unmeasured Confounders in Epidemiologic Database Studies of Therapeutics.”</span> <em>Pharmacoepidemiology and Drug Safety</em> 15 (5): 291–303.
</div>
<div id="ref-schneeweiss2009high" class="csl-entry" role="doc-biblioentry">
Schneeweiss, Sebastian, Jeremy A Rassen, Robert J Glynn, Jerry Avorn, Helen Mogun, and M Alan Brookhart. 2009. <span>“High-Dimensional Propensity Score Adjustment in Studies of Treatment Effects Using Health Care Claims Data.”</span> <em>Epidemiology (Cambridge, Mass.)</em> 20 (4): 512.
</div>
<div id="ref-stuart2013prognostic" class="csl-entry" role="doc-biblioentry">
Stuart, Elizabeth A, Brian K Lee, and Finbarr P Leacy. 2013. <span>“Prognostic Score–Based Balance Measures Can Be a Useful Diagnostic for Propensity Score Methods in Comparative Effectiveness Research.”</span> <em>Journal of Clinical Epidemiology</em> 66 (8): S84–90.
</div>
</div>
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