The missing indicator approach for censored covariates subject to limit of detection in logistic regression models

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H. Chan School of Public Health, Boston, MA"> <meta name="citation_author" content="Rebecca A Betensky"> <meta name="citation_author_institution" content="Department of Biostatistics, Harvard T. H. Chan School of Public Health, Boston, MA"> <meta name="citation_author" content="Raji Balasubramanian"> <meta name="citation_author_institution" content="Department of Biostatistics and Epidemiology, University of Massachusetts – Amherst, Amherst MA 01003"> <meta name="citation_publication_date" content="2019 Aug 13"> <meta name="citation_volume" content="38"> <meta name="citation_firstpage" content="57"> <meta name="citation_doi" content="10.1016/j.annepidem.2019.07.014"> <meta name="citation_pmid" content="31604610"> <meta name="citation_abstract_html_url" content="https://pmc.ncbi.nlm.nih.gov/articles/PMC6812630/"> <meta name="citation_fulltext_html_url" content="https://pmc.ncbi.nlm.nih.gov/articles/PMC6812630/"> <meta name="citation_pdf_url" content="https://pmc.ncbi.nlm.nih.gov/articles/PMC6812630/pdf/nihms-1537397.pdf"> <meta name="description" content="In several biomedical studies, one or more exposures of interest may be subject to non-random missingness due to the failure of the measurement assay at levels below its limit of detection. 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Author manuscript; available in PMC: 2020 Oct 1.</div> <div> <em>Published in final edited form as: </em>Ann Epidemiol. 2019 Aug 13;38:57–64. doi: <a href="https://doi.org/10.1016/j.annepidem.2019.07.014" class="usa-link usa-link--external" data-ga-action="click_feat_suppl" target="_blank" rel="noopener noreferrer">10.1016/j.annepidem.2019.07.014</a> </div> <nav id="journal_context_menu" hidden="hidden"><ul class="menu-list font-family-ui" role="menu"> <li role="presentation"><a href="https://www.ncbi.nlm.nih.gov/pmc/?term=%22Ann%20Epidemiol%22%5Bjour%5D" class="usa-link" role="menuitem">Search in PMC</a></li> <li role="presentation"><a href="https://pubmed.ncbi.nlm.nih.gov/?term=%22Ann%20Epidemiol%22%5Bjour%5D" lang="en" class="usa-link" role="menuitem">Search in PubMed</a></li> <li role="presentation"><a href="https://www.ncbi.nlm.nih.gov/nlmcatalog?term=%22Ann%20Epidemiol%22%5BTitle%20Abbreviation%5D" class="usa-link" role="menuitem">View in NLM Catalog</a></li> <li role="presentation"><a href="?term=%22Ann%20Epidemiol%22%5Bjour%5D" class="usa-link" role="menuitem" data-add-to-search="true">Add to search</a></li> </ul></nav></section><section class="front-matter"><div class="ameta p font-secondary font-xs"> <hgroup><h1>The missing indicator approach for censored covariates subject to limit of detection in logistic regression models</h1></hgroup><div class="cg p"> <a href="https://pubmed.ncbi.nlm.nih.gov/?term=%22Chiou%20SH%22%5BAuthor%5D" class="usa-link" aria-describedby="id1"><span class="name western">Sy Han Chiou</span></a><div hidden="hidden" id="id1"> <h3><span class="name western">Sy Han Chiou</span></h3> <div class="p"> <sup>1</sup>Department of Biostatistics, Harvard T. H. Chan School of Public Health, Boston, MA</div> <div class="p">Find articles by <a href="https://pubmed.ncbi.nlm.nih.gov/?term=%22Chiou%20SH%22%5BAuthor%5D" class="usa-link"><span class="name western">Sy Han Chiou</span></a> </div> </div> <sup>1</sup>, <a href="https://pubmed.ncbi.nlm.nih.gov/?term=%22Betensky%20RA%22%5BAuthor%5D" class="usa-link" aria-describedby="id2"><span class="name western">Rebecca A Betensky</span></a><div hidden="hidden" id="id2"> <h3><span class="name western">Rebecca A Betensky</span></h3> <div class="p"> <sup>1</sup>Department of Biostatistics, Harvard T. H. Chan School of Public Health, Boston, MA</div> <div class="p">Find articles by <a href="https://pubmed.ncbi.nlm.nih.gov/?term=%22Betensky%20RA%22%5BAuthor%5D" class="usa-link"><span class="name western">Rebecca A Betensky</span></a> </div> </div> <sup>1</sup>, <a href="https://pubmed.ncbi.nlm.nih.gov/?term=%22Balasubramanian%20R%22%5BAuthor%5D" class="usa-link" aria-describedby="id3"><span class="name western">Raji Balasubramanian</span></a><div hidden="hidden" id="id3"> <h3><span class="name western">Raji Balasubramanian</span></h3> <div class="p"> <sup>2</sup>Department of Biostatistics and Epidemiology, University of Massachusetts – Amherst, Amherst MA 01003</div> <div class="p">Find articles by <a href="https://pubmed.ncbi.nlm.nih.gov/?term=%22Balasubramanian%20R%22%5BAuthor%5D" class="usa-link"><span class="name western">Raji Balasubramanian</span></a> </div> </div> <sup>2</sup> </div> <ul class="d-buttons inline-list"> <li><button class="d-button" aria-controls="aip_a" aria-expanded="false">Author information</button></li> <li><button class="d-button" aria-controls="anp_a" aria-expanded="false">Article notes</button></li> <li><button class="d-button" aria-controls="clp_a" aria-expanded="false">Copyright and License information</button></li> </ul> <div class="d-panels font-secondary-light"> <div id="aip_a" class="d-panel p" style="display: none"> <div class="p" id="A1"> <sup>1</sup>Department of Biostatistics, Harvard T. H. Chan School of Public Health, Boston, MA</div> <div id="A2"> <sup>2</sup>Department of Biostatistics and Epidemiology, University of Massachusetts – Amherst, Amherst MA 01003</div> </div> <div id="anp_a" class="d-panel p" style="display: none"><div class="notes p"><section id="historyarticle-meta1" class="history"><p>Issue date 2019 Oct.</p></section></div></div> <div id="clp_a" class="d-panel p" style="display: none"><div class="p"><a href="/about/copyright/" class="usa-link">PMC Copyright notice</a></div></div> </div> <div>PMCID: PMC6812630 NIHMSID: NIHMS1537397 PMID: <a href="https://pubmed.ncbi.nlm.nih.gov/31604610/" class="usa-link">31604610</a> </div> <div class="ra xbox p" role="complementary" aria-label="Related or updated information about this article"><div>The publisher's version of this article is available at <a href="https://doi.org/10.1016/j.annepidem.2019.07.014" class="usa-link usa-link--external" data-ga-action="click_feat_suppl" target="_blank" rel="noopener noreferrer">Ann Epidemiol</a> </div></div> </div></section></section><section aria-label="Article content"><section class="body main-article-body"><section class="abstract" id="ABS1"><h2>Abstract</h2> <p id="P1">In several biomedical studies, one or more exposures of interest may be subject to non-random missingness due to the failure of the measurement assay at levels below its limit of detection. This issue is commonly encountered in studies of the metabolome employing tandem mass spectrometry-based technologies. Due to a large number of metabolites measured in these studies, preserving statistical power is of utmost interest. In these settings, the missing indicator model minimizes loss of information and thus provides an attractive alternative to the oft-used complete case analysis and other imputation approaches. In this paper, we evaluate the small sample properties of the missing indicator approach in logistic and conditional logistic regression models. We show that under a variety of settings, the missing indicator approach outperforms complete case analysis and a variety of imputation approaches with regard to bias, mean squared error and power. We compare the results from the missing indicator model to that from a complete case analysis using data from a Cardiovascular Disease Biomarker study employing metabolomic technologies.</p> <section id="kwd-group1" class="kwd-group"><p><strong>Keywords:</strong> limit of detection, logistics regression, matched design, metabolomics</p></section></section><hr class="headless"> <p id="P2">We consider the setting in which one or more covariates of interest may be subject to a limit of detection associated with the measurement assay. This issue arises in high-throughput òmics’ technologies, such as metabolomics that involve the measurement of several hundred metabolites per specimen. Several methods are commonly used for handling covariates that are subject to limit of detection. These include complete case analysis, models including a missing indicator, ad hoc substitution methods, parametric, likelihood-based and imputation-based approaches [<a href="#R1" class="usa-link" aria-describedby="R1">1</a>–<a href="#R3" class="usa-link" aria-describedby="R3">3</a>]. Here, in the presence of a binary outcome, we compare the performance of the simple missing indicator model [<a href="#R4" class="usa-link" aria-describedby="R4">4</a>–<a href="#R6" class="usa-link" aria-describedby="R6">6</a>] to complete case analysis and other imputation approaches to handle covariates with non-random missingness and show that this approach provides an attractive alternative that is especially useful in high dimensional data settings.</p> <p id="P3">Metabolomic technologies are characterized by detection limits that affect the measurement of low abundance metabolites. In many clinical applications, low abundance metabolites are also likely to be implicated in disease. One widely used treatment of censored covariates is to discard subjects who have censored covariates or the <em>complete case</em> (CC) analysis. This approach is inefficient under moderate to heavy censoring. Ad hoc substitution methods, likelihood-based methods under the assumption of a parametric distribution for the covariate and several imputation techniques have been proposed [<a href="#R1" class="usa-link" aria-describedby="R1">1</a>–<a href="#R3" class="usa-link" aria-describedby="R3">3</a>] [<a href="#R7" class="usa-link" aria-describedby="R7">7</a>–<a href="#R12" class="usa-link" aria-describedby="R12">12</a>] [<a href="#R13" class="usa-link" aria-describedby="R13">13</a>, <a href="#R14" class="usa-link" aria-describedby="R14">14</a>]. Parametric and imputation approaches offer considerable improvements, but are computationally intensive and/or require stringent assumptions. A recently published comprehensive paper compared 31 imputation frameworks for handling missing values in metabolomics data, in which they concluded that multiple imputation using predictive mean matching and K-nearest neighbors had optimal performance [<a href="#R15" class="usa-link" aria-describedby="R15">15</a>]. Additional papers have considered linear and time-to-event regression and have obtained similar findings [<a href="#R16" class="usa-link" aria-describedby="R16">16</a>–<a href="#R21" class="usa-link" aria-describedby="R21">21</a>]. However, to the best of our knowledge, none of these works have evaluated the performance of missing indicator approach to handle covariates subject to non-random missingness due to the limit of detection of the measurement technique.</p> <p id="P4">A simple approach for handling missing covariates is the missing data indicator (MDI) model, in which an indicator variable for whether the explanatory variable is observed is included as a covariate in the model, along with the continuous measurements of the explanatory variable of interest for those for whom it is observed [<a href="#R4" class="usa-link" aria-describedby="R4">4</a>–<a href="#R6" class="usa-link" aria-describedby="R6">6</a>]. The theoretical properties of the missing indicator model were examined in the context of linear regression for continuous outcomes under various mechanisms of missingness, including when missingness depends on the true value of the covariate as in settings affected by limit of detection [<a href="#R4" class="usa-link" aria-describedby="R4">4</a>]. In a MDI model including a completely observed covariate and a censored covariate, the corresponding regression coefficients are unbiased when the covariates are uncorrelated [<a href="#R4" class="usa-link" aria-describedby="R4">4</a>]. However, in general, the asymptotic bias of the regression coefficient associated with the censored covariate increases with increasing magnitude of the correlation between the two covariates and the proportion of the censored covariate that falls below the limit of detection. Moreover, these theoretical results do not apply directly to small sample settings, to models with binary outcomes, or to matched studies [<a href="#R4" class="usa-link" aria-describedby="R4">4</a>].</p> <p id="P5">Here, in a numerical analysis, we extend the results presented by Jones, M. P. (1996) to the analysis of binary outcomes with a focus on studies of modest sample size (n < 400) when missing values of covariates are due to a non-random process resulting from the limit of detection of the measurement technique. Through simulations, we compare the MDI approach to a CC analysis and other imputation approaches (i.e. substitution, imputation using predictive mean matching, iterative Random Forests based imputation) previously studied in logistic regression models. For each method, we evaluate the bias and mean squared error associated with the estimation of the regression parameter of interest and power/Type 1 error associated with the corresponding hypothesis tests. We also consider the setting of matched studies, in which a conditional logistic regression is used. We apply the MDI and CC approaches to a cardiovascular disease biomarker study and compare the results. These results could provide useful guidance to investigators involved in the analyses of covariate sets that may be subject to missingness due to limit of detection associated with the measurement technology.</p> <section id="S1"><h2 class="pmc_sec_title">METHODS</h2> <p id="P6">Let Y denote the outcome of interest and <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M1" display="inline" overflow="linebreak"><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo stretchy="true">˜</mo></mover></mrow></math></span> denote a p-dimensional covariate vector, where each component of <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M2" display="inline" overflow="linebreak"><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo stretchy="true">˜</mo></mover></mrow></math></span> is subject to a different level of left-censoring due to limit of detection. In addition, we assume there exists a q-dimensional covariate vector, <strong>U</strong>, that is fully observed and included in the model.</p> <p id="P7">For simplicity, we set p = q = 1 and assume the generalized linear model:</p> <p id="P8"><span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M3" display="inline" overflow="linebreak"><mrow><mi>g</mi><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>1</mn></msub><mo>+</mo><msub><mi>β</mi><mn>2</mn></msub><msub><mi>U</mi><mn>1</mn></msub></mrow></math></span>, where g(.) is a link function and (<em>β</em><sub>0</sub>, <em>β</em><sub>1</sub>, <em>β</em><sub>2</sub>) are regression coefficients. The parameter of interest is <em>β</em><sub>1</sub>, the regression coefficient reflecting the main effect of <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M4" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>1</mn></msub></mrow></math></span>. In the presence of limit of detection, the observed data are (<em>Y</em><sub>{<em>i</em>}</sub><span class="font-variant-small-caps">,</span><em>X</em><sub>{1<em>i</em>}</sub><span class="font-variant-small-caps">,</span><em>U</em><sub>{1<em>i</em>},</sub> <span class="font-variant-small-caps">Δ</span><sub>{1<em>i</em>}</sub><span class="font-variant-small-caps">),</span> for i = 1,..., n, where <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M5" display="inline" overflow="linebreak"><mrow><msub><mi>X</mi><mrow><mo>{</mo><mn>1</mn><mi>i</mi><mo>}</mo></mrow></msub><mo>=</mo><mi mathvariant="normal">max</mi><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mrow><mo>{</mo><mn>1</mn><mi>i</mi><mo>}</mo></mrow></msub><mo>,</mo><msub><mi>α</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow></mrow></math></span>, <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M6" display="inline" overflow="linebreak"><mrow><msub><mi mathvariant="normal">Δ</mi><mrow><mo>{</mo><mn>1</mn><mi>i</mi><mo>}</mo></mrow></msub><mo>=</mo><mi>I</mi><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mrow><mo>{</mo><mn>1</mn><mi>i</mi><mo>}</mo></mrow></msub><mo>≤</mo><msub><mi>α</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow></mrow></math></span>, <em>α</em><sub>1</sub> is the limit of detection for <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M7" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>1</mn></msub></mrow></math></span> and I(.) is the indicator function. With the missing indicator Δ<sub>1</sub>, the CC model can be expressed as a modification of the model above, as follows: </p> <table class="disp-formula p" id="FD1"><tr><td class="formula"><math id="M8" display="block" overflow="linebreak"><mrow><mi>g</mi><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><msub><mi>β</mi><mrow><mi>c</mi><mn>0</mn></mrow></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi mathvariant="normal">Δ</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>β</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><msub><mi>X</mi><mn>1</mn></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi mathvariant="normal">Δ</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>β</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub><msub><mi>U</mi><mn>1</mn></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi mathvariant="normal">Δ</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow></mrow></math></td></tr></table> <p id="P9">It can be shown that the CC estimators are consistent estimators for true parameters [<a href="#R21" class="usa-link" aria-describedby="R21">21</a>].</p> <p id="P10">Despite its desirable asymptotic properties, when the censoring rate is high, the CC approach likely suffers from loss of information. As an alternate approach, we consider the MDI model defined as follows: </p> <table class="disp-formula p" id="FD2"><tr><td class="formula"><math id="M9" display="block" overflow="linebreak"><mrow><mi>g</mi><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>0</mn></mrow></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>1</mn></mrow></msub><msub><mi>X</mi><mn>1</mn></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi mathvariant="normal">Δ</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>2</mn></mrow></msub><msub><mi>U</mi><mn>1</mn></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>3</mn></mrow></msub><msub><mi mathvariant="normal">Δ</mi><mn>1</mn></msub><mo>.</mo></mrow></math></td></tr></table> <p id="P11">In the context of linear regression, the least squares estimators of (<em>β</em><sub><em>m</em>0</sub>, <em>β</em><sub><em>m</em>1</sub>, <em>β</em><sub><em>m</em>2</sub>) are asymptotically unbiased for (<em>β</em><sub>0</sub><em>, β</em><sub>1</sub><em>, β</em><sub>2</sub>) in <a href="#FD3" class="usa-link">Equation (1)</a>, if X<sub>1</sub> and U<sub>1</sub> are uncorrelated [<a href="#R4" class="usa-link" aria-describedby="R4">4</a>].</p> <p id="P12">Considering the setting in which there are two predictors subject to limits of detection, denoted <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M10" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>1</mn></msub></mrow></math></span> and <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M11" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>2</mn></msub></mrow></math></span>, and a fully observed predictor U<sub>1</sub>, the true model is assumed to follow: </p> <table class="disp-formula p" id="FD3"><tr> <td class="formula"><math id="M12" display="block" overflow="linebreak"><mrow><mi>g</mi><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>1</mn></msub><mo>+</mo><msub><mi>β</mi><mn>2</mn></msub><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>2</mn></msub><mo>+</mo><msub><mi>β</mi><mn>3</mn></msub><msub><mi>U</mi><mn>1</mn></msub><mo>.</mo></mrow></math></td> <td class="label">(1)</td> </tr></table> <p id="P13">For this setting, the MDI model is given by: </p> <table class="disp-formula p" id="FD4"><tr> <td class="formula"><math id="M13" display="block" overflow="linebreak"><mrow><mi>g</mi><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>0</mn></mrow></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>1</mn></mrow></msub><msub><mi>X</mi><mn>1</mn></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi mathvariant="normal">Δ</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>2</mn></mrow></msub><msub><mi>X</mi><mn>2</mn></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi mathvariant="normal">Δ</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>3</mn></mrow></msub><msub><mi>U</mi><mn>1</mn></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>4</mn></mrow></msub><msub><mi mathvariant="normal">Δ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>5</mn></mrow></msub><msub><mi mathvariant="normal">Δ</mi><mn>2</mn></msub><mo>,</mo></mrow></math></td> <td class="label">(2)</td> </tr></table> <p> where (Δ<sub>1</sub>, Δ<sub>2</sub>) are the missing indicators for the predictors <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M14" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>1</mn></msub></mrow></math></span> and <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M15" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>2</mn></msub></mrow></math></span>, respectively. The estimates of <em>β</em><sub><em>m</em>0</sub>, <em>β</em><sub><em>m</em>1</sub>, <em>β</em><sub><em>m</em>2</sub>, <em>β</em><sub><em>m</em>3</sub>) in <a href="#FD4" class="usa-link">Equation (2)</a> are used to estimate the parameters (<em>β</em><sub>0</sub><em>, β</em><sub>1</sub><em>, β</em><sub>2</sub>, <em>β</em><sub>3</sub>) in <a href="#FD3" class="usa-link">Equation (1)</a>.</p> <p id="P14">In addition, we consider the expanded MDI model with interactions between the fully observed covariate and the missing data indicators as follows: </p> <table class="disp-formula p" id="FD5"><tr> <td class="formula"><math id="M16" display="block" overflow="linebreak"><mi>g</mi><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>0</mn></mrow></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>1</mn></mrow></msub><msub><mi>X</mi><mn>1</mn></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi mathvariant="normal">Δ</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>2</mn></mrow></msub><msub><mi>X</mi><mn>2</mn></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi mathvariant="normal">Δ</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>3</mn></mrow></msub><msub><mi>U</mi><mn>1</mn></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>4</mn></mrow></msub><msub><mi mathvariant="normal">Δ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>5</mn></mrow></msub><msub><mi mathvariant="normal">Δ</mi><mn>2</mn></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>m</mi><mn>6</mn></mrow></msub><msub><mi mathvariant="normal">Δ</mi><mn>1</mn></msub><msub><mi>U</mi><mn>1</mn></msub><mo>+</mo><mspace linebreak="newline"></mspace><msub><mi>β</mi><mrow><mi>m</mi><mn>7</mn></mrow></msub><msub><mi mathvariant="normal">Δ</mi><mn>2</mn></msub><msub><mi>U</mi><mn>1</mn></msub><mo>.</mo></math></td> <td class="label">(3)</td> </tr></table> <p id="P15">Model (3) is useful in settings in which the censored covariates have interaction effects with the fully observed covariates.</p></section><section id="S2"><h2 class="pmc_sec_title">SIMULATION</h2> <p id="P16">We present results from simulation to assess the performance of the MDI approach incorporated in logistic and conditional logistic regression models. The latter is appropriate for the matched case-control study design of the Cardiovascular Disease Biomarker study. In all simulation studies, we considered three possibly correlated covariates, denoted <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M17" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>1</mn></msub></mrow></math></span>, <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M18" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>2</mn></msub></mrow></math></span>, <em>U</em><sub>1</sub> as specified in <a href="#FD3" class="usa-link">Equation (1)</a>, where U<sub>1</sub> is fully observed, and <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M19" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>1</mn></msub></mrow></math></span>, <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M20" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>2</mn></msub></mrow></math></span> are left-censored. Details of the simulation are included in the <a href="#SD1" class="usa-link">Supplement</a>.</p> <p id="P17">Using results obtained in the setting in the absence of censoring as the gold standard (M1), we compared the performance of six approaches to handling missing data. These include</p> <ul id="L1" class="list" style="list-style-type:disc"> <li><p id="P18">M2: complete case (CC) analysis;</p></li> <li><p id="P19">M3: the MDI model in <a href="#FD4" class="usa-link">equation (2)</a>;</p></li> <li><p id="P20">M4: the expanded MDI model in <a href="#FD5" class="usa-link">equation (3)</a>;</p></li> <li><p id="P21">M5: substitution of the missing value by one-half the observed minimum;</p></li> <li><p id="P22">M6: imputation of the missing value using the predictive mean matching (PMM) algorithm implemented in the R package <em>mice.</em> Here, imputed values are selected randomly from among the five observed values of the covariate whose regression-predicted values are closest to the regression-predicted value of the missing covariate [<a href="#R14" class="usa-link" aria-describedby="R14">14</a>, <a href="#R22" class="usa-link" aria-describedby="R22">22</a>, <a href="#R23" class="usa-link" aria-describedby="R23">23</a>]. This procedure produces imputed covariate values that lie in the range of the observed covariate values.</p></li> <li><p id="P23">M7: imputation of the missing value using the MissForest algorithm implemented in the R package <em>missForest</em> [<a href="#R13" class="usa-link" aria-describedby="R13">13</a>]. In this case, the missing values are directly predicted using a Random Forests model that is trained on the observed parts of the dataset. This approach does not make distributional assumptions inherent in the PMM algorithm in M6. Default parameter values of 100 trees and mtry=1 were assumed.</p></li> </ul> <section id="S3"><h3 class="pmc_sec_title">Logistic Regression</h3> <p id="P24">Assuming the model in <a href="#FD3" class="usa-link">Equation (1)</a>, we implemented two simulation settings corresponding to (1) Multivariate normal; and (2) Non-normal distribution for <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M21" display="inline" overflow="linebreak"><mrow><mrow><mo>(</mo><mrow><msub><mrow><mover accent="true"><mtext>X</mtext><mo stretchy="true">˜</mo></mover></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mover accent="true"><mtext>X</mtext><mo stretchy="true">˜</mo></mover></mrow><mn>2</mn></msub><mo>,</mo><msub><mtext>U</mtext><mn>1</mn></msub></mrow><mo>)</mo></mrow></mrow></math></span>. In both settings, Spearman’s rho (ρ) was used to specify the strength of dependence between <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M22" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi mathvariant="normal">X</mi><mo>˜</mo></mover><mn>1</mn></msub></mrow></math></span>, <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M23" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi mathvariant="normal">X</mi><mo>˜</mo></mover><mn>2</mn></msub></mrow></math></span>, U<sub>1</sub>. We simulate data by setting the coefficients β<sub>0</sub> = β<sub>1</sub> = β<sub>2</sub> = β<sub>3</sub> =1 in the model given in <a href="#FD3" class="usa-link">Equation (1)</a>. We compare the bias and mean squared error (MSE) associated with the maximum likelihood estimates (MLE) of β<sub>1</sub> and β<sub>2</sub>. Total sample sizes of n=100 and 200 with equal numbers of cases and controls were considered. All models incorporated the bias reduction method for reduction in finite sample bias as implemented in the <em>brglm</em> R package [<a href="#R24" class="usa-link" aria-describedby="R24">24</a>]. The simulation was repeated 100,000 times and results averaged.</p></section><section id="S4"><h3 class="pmc_sec_title">Bias and MSE</h3> <p id="P25"><a href="#F1" class="usa-link">Figure 1</a> (<a href="#SD1" class="usa-link">Table 2 in Supplement</a>) summarizes the average bias and MSE associated with the MLEs of <em>β</em><sub>1</sub><em>, β</em><sub>2</sub> in <a href="#FD3" class="usa-link">Equation (1)</a>, when n=100 (50 cases). The MDI (M3) and expanded MDI (M4) approaches had among the smallest bias across all settings considered for logistic regression (<a href="#F1" class="usa-link">Figure 1</a>, <a href="#SD1" class="usa-link">Table 2 in Supplement</a>). These general trends persisted for a larger sample size of n=200 (100 cases) (<a href="#SD1" class="usa-link">Supplemental Table 3</a>).</p> <figure class="fig xbox font-sm" id="F1"><h4 class="obj_head">Figure 1. Bias and MSE associated with estimates of regression coefficients for logistic regression models.</h4> <p class="img-box line-height-none margin-x-neg-2 tablet:margin-x-0 text-center"><a class="tileshop" target="_blank" href="https://www.ncbi.nlm.nih.gov/core/lw/2.0/html/tileshop_pmc/tileshop_pmc_inline.html?title=Click%20on%20image%20to%20zoom&p=PMC3&id=6812630_nihms-1537397-f0001.jpg"><img class="graphic zoom-in" src="https://cdn.ncbi.nlm.nih.gov/pmc/blobs/02b9/6812630/45a9b8df13b2/nihms-1537397-f0001.jpg" loading="lazy" height="396" width="800" alt="Figure 1"></a></p> <div class="p text-right font-secondary"><a href="figure/F1/" class="usa-link" target="_blank" rel="noopener noreferrer">Open in a new tab</a></div> <figcaption><p id="P58">The sample size considered here is n = 100 (50 cases). Results are based on 100,000 converged replications. Approaches with mean bias reduction to estimate the true regression coefficients <em>β</em><sub>1</sub>, <em>β</em><sub>2</sub> are as follows: M2 denotes the complete case analysis; M3 denotes the missing data indicator (MDI) model; M4 denotes the expanded missing data indicator (MDI) model; M5 denotes imputation using one half the observed minimum value; M6 denotes predictive mean matching (PMM) imputation implemented in R package mice; M7 denotes the MissForest algorithm implemented in the R package missForest.</p></figcaption></figure><p id="P26">The MDI approach (M3) has a clear advantage over the CC approach (M2) for small and moderate sample sizes, with respect to MSE. The MDI and CC approaches had generally equivalent bias across all settings considered.</p> <p id="P27">Estimates of bias were largest for imputation using <em>mice</em> (M6) and <em>missForest</em> (M7). The distributions of the difference between the true and imputed covariate values from <em>mice</em> (M6) and <em>missForest</em> (M7) were examined for X<sub>1</sub> and X<sub>2</sub> under each simulation setting. The imputed values were uniformly larger than the corresponding true covariate values that fall below the limit of detection - see <a href="#SD1" class="usa-link">Figure 1 in Supplement</a> for a representative distribution. Imputed values from <em>mice</em> (M6) are always randomly selected from among a set of observed values of the covariate whose regression-predicted values are closest to the regression-predicted value of the missing covariate. Imputed values from <em>missForest</em> (M7) are based on predicted values from a Random Forests classifier trained on the observed parts of the dataset, resulting in imputed values that were in the range of the observed covariate values. Despite large bias, the MSE of the imputation approaches using <em>mice</em> and <em>MissForest</em> were comparable and sometimes smaller than that of MDI and expanded MDI.</p> <p id="P28">The imputation approach based on one half the minimum observed value (M5) had larger bias when compared to the MDI approaches overall; however, the bias associated with this ad-hoc approach was substantially lower in the non-normal setting. This is driven by the fact that the conditional expectation, E(X | X < limit of detection), is well approximated by one half the minimum observed for the non-normal setting, but not in the normal setting (<a href="#SD1" class="usa-link">Table 1 in the Supplement</a>).</p> <p id="P29"><a href="#F1" class="usa-link">Figure 1</a> (<a href="#SD1" class="usa-link">Table 2 in Supplement</a>) results also suggest that higher ρ is generally associated with larger MSE in CC (M2), MDI (M3) and expanded MDI (M4) models. However, the effect of ρ on the magnitude of the bias is less noticeable. As expected, since the censoring rate is higher for <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M24" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>2</mn></msub></mrow></math></span> when compared to <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M25" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>1</mn></msub></mrow></math></span>, the corresponding regression coefficient estimate is associated with larger bias and MSE. The non-normal distribution setting was observed to be associated with lower bias and MSE when compared to the Gaussian setting.</p> <p id="P30">Bias and MSE estimates for all approaches are presented for a larger sample size of n=200 (100 cases) subjects in <a href="#SD1" class="usa-link">Supplemental Table 3</a>.</p></section><section id="S5"><h3 class="pmc_sec_title">Type I error and Power</h3> <p id="P31">corresponding to the individual hypothesis tests for <em>H</em><sub>0</sub>: <em>β</em><sub>1</sub> = 0, <em>H</em><sub>0</sub>: <em>β</em><sub>2</sub> <em>=</em> 0 are summarized in <a href="#T1" class="usa-link">Table 1</a>. We estimated power and type-I error by computing the proportion (out of 100,000 replicates) of P-values ≤ 0.05. For the MDI approaches (M3, M4), the P-values were obtained from a likelihood ratio test of the composite null hypothesis of the coefficients associated with <em>X</em><sub><em>k</em></sub>, Δ<sub><em>k</em></sub> for <em>k</em> = 1,2. For the CC approach (M2) and other imputation models (M5, M6, M7), the P-values were computed based on a Wald test for each of the parameters of interest. To obtain type-I error, we set <em>β</em><sub>1</sub> <em>= β</em><sub>2</sub> <em>= β</em><sub>3</sub> <em>=</em> 0 in the data generating model.</p> <section class="tw xbox font-sm" id="T1"><h4 class="obj_head">Table 1:</h4> <div class="caption p"><p id="P60">Summary of power and type-I error for logistic regression models. Each entry represents the proportion of p-value less than or equal to 0.05 out of 100,000 replicates. The sample size is n = 100 (50 cases). M1 denotes the true model before censoring; M2 denotes the complete case analysis; M3 denotes the missing data indicator (MDI) model; M4 denotes the expanded missing data indicator (MDI) model; M5 denotes imputation using one half the observed minimum value; M6 denotes predictive mean matching (PMM) imputation implemented in R package mice; M7 denotes the MissForest algorithm implemented in the R package missForest.</p></div> <div class="tbl-box p" tabindex="0"><table class="content" frame="hsides" rules="groups"> <colgroup span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> </colgroup> <thead><tr> <th align="left" valign="top" rowspan="1" colspan="1">ρ</th> <th align="left" valign="top" rowspan="1" colspan="1"></th> <th align="left" valign="top" rowspan="1" colspan="1">M1</th> <th align="left" valign="top" rowspan="1" colspan="1">M2</th> <th align="left" valign="top" rowspan="1" colspan="1">M3</th> <th align="left" valign="top" rowspan="1" colspan="1">M4</th> <th align="left" valign="top" rowspan="1" colspan="1">M5</th> <th align="left" valign="top" rowspan="1" colspan="1">M6</th> <th align="left" valign="top" rowspan="1" colspan="1">M7</th> </tr></thead> <tbody> <tr> <td align="center" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td colspan="7" align="center" valign="top" rowspan="1">Normal margins</td> </tr> <tr> <td align="center" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td colspan="7" align="center" valign="top" rowspan="1">Power</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.969</td> <td align="left" valign="top" rowspan="1" colspan="1">0.284</td> <td align="left" valign="top" rowspan="1" colspan="1">0.931</td> <td align="left" valign="top" rowspan="1" colspan="1">0.897</td> <td align="left" valign="top" rowspan="1" colspan="1">0.944</td> <td align="left" valign="top" rowspan="1" colspan="1">0.472</td> <td align="left" valign="top" rowspan="1" colspan="1">0.536</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.970</td> <td align="left" valign="top" rowspan="1" colspan="1">0.179</td> <td align="left" valign="top" rowspan="1" colspan="1">0.907</td> <td align="left" valign="top" rowspan="1" colspan="1">0.868</td> <td align="left" valign="top" rowspan="1" colspan="1">0.926</td> <td align="left" valign="top" rowspan="1" colspan="1">0.188</td> <td align="left" valign="top" rowspan="1" colspan="1">0.225</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.2</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.940</td> <td align="left" valign="top" rowspan="1" colspan="1">0.261</td> <td align="left" valign="top" rowspan="1" colspan="1">0.898</td> <td align="left" valign="top" rowspan="1" colspan="1">0.854</td> <td align="left" valign="top" rowspan="1" colspan="1">0.919</td> <td align="left" valign="top" rowspan="1" colspan="1">0.506</td> <td align="left" valign="top" rowspan="1" colspan="1">0.582</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.941</td> <td align="left" valign="top" rowspan="1" colspan="1">0.155</td> <td align="left" valign="top" rowspan="1" colspan="1">0.861</td> <td align="left" valign="top" rowspan="1" colspan="1">0.808</td> <td align="left" valign="top" rowspan="1" colspan="1">0.887</td> <td align="left" valign="top" rowspan="1" colspan="1">0.174</td> <td align="left" valign="top" rowspan="1" colspan="1">0.211</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.4</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.867</td> <td align="left" valign="top" rowspan="1" colspan="1">0.236</td> <td align="left" valign="top" rowspan="1" colspan="1">0.819</td> <td align="left" valign="top" rowspan="1" colspan="1">0.757</td> <td align="left" valign="top" rowspan="1" colspan="1">0.850</td> <td align="left" valign="top" rowspan="1" colspan="1">0.502</td> <td align="left" valign="top" rowspan="1" colspan="1">0.579</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.866</td> <td align="left" valign="top" rowspan="1" colspan="1">0.131</td> <td align="left" valign="top" rowspan="1" colspan="1">0.751</td> <td align="left" valign="top" rowspan="1" colspan="1">0.684</td> <td align="left" valign="top" rowspan="1" colspan="1">0.793</td> <td align="left" valign="top" rowspan="1" colspan="1">0.145</td> <td align="left" valign="top" rowspan="1" colspan="1">0.180</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.6</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.707</td> <td align="left" valign="top" rowspan="1" colspan="1">0.203</td> <td align="left" valign="top" rowspan="1" colspan="1">0.664</td> <td align="left" valign="top" rowspan="1" colspan="1">0.581</td> <td align="left" valign="top" rowspan="1" colspan="1">0.703</td> <td align="left" valign="top" rowspan="1" colspan="1">0.467</td> <td align="left" valign="top" rowspan="1" colspan="1">0.525</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.711</td> <td align="left" valign="top" rowspan="1" colspan="1">0.108</td> <td align="left" valign="top" rowspan="1" colspan="1">0.571</td> <td align="left" valign="top" rowspan="1" colspan="1">0.501</td> <td align="left" valign="top" rowspan="1" colspan="1">0.615</td> <td align="left" valign="top" rowspan="1" colspan="1">0.114</td> <td align="left" valign="top" rowspan="1" colspan="1">0.134</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td colspan="7" align="center" valign="top" rowspan="1">Type-I error</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.043</td> <td align="left" valign="top" rowspan="1" colspan="1">0.015</td> <td align="left" valign="top" rowspan="1" colspan="1">0.045</td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.034</td> <td align="left" valign="top" rowspan="1" colspan="1">0.037</td> <td align="left" valign="top" rowspan="1" colspan="1">0.039</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.042</td> <td align="left" valign="top" rowspan="1" colspan="1">0.015</td> <td align="left" valign="top" rowspan="1" colspan="1">0.053</td> <td align="left" valign="top" rowspan="1" colspan="1">0.048</td> <td align="left" valign="top" rowspan="1" colspan="1">0.031</td> <td align="left" valign="top" rowspan="1" colspan="1">0.036</td> <td align="left" valign="top" rowspan="1" colspan="1">0.041</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.2</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.044</td> <td align="left" valign="top" rowspan="1" colspan="1">0.017</td> <td align="left" valign="top" rowspan="1" colspan="1">0.046</td> <td align="left" valign="top" rowspan="1" colspan="1">0.039</td> <td align="left" valign="top" rowspan="1" colspan="1">0.036</td> <td align="left" valign="top" rowspan="1" colspan="1">0.037</td> <td align="left" valign="top" rowspan="1" colspan="1">0.039</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.042</td> <td align="left" valign="top" rowspan="1" colspan="1">0.016</td> <td align="left" valign="top" rowspan="1" colspan="1">0.053</td> <td align="left" valign="top" rowspan="1" colspan="1">0.047</td> <td align="left" valign="top" rowspan="1" colspan="1">0.031</td> <td align="left" valign="top" rowspan="1" colspan="1">0.036</td> <td align="left" valign="top" rowspan="1" colspan="1">0.040</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.4</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.043</td> <td align="left" valign="top" rowspan="1" colspan="1">0.019</td> <td align="left" valign="top" rowspan="1" colspan="1">0.047</td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.036</td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.039</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.043</td> <td align="left" valign="top" rowspan="1" colspan="1">0.018</td> <td align="left" valign="top" rowspan="1" colspan="1">0.054</td> <td align="left" valign="top" rowspan="1" colspan="1">0.049</td> <td align="left" valign="top" rowspan="1" colspan="1">0.034</td> <td align="left" valign="top" rowspan="1" colspan="1">0.037</td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.6</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.042</td> <td align="left" valign="top" rowspan="1" colspan="1">0.021</td> <td align="left" valign="top" rowspan="1" colspan="1">0.048</td> <td align="left" valign="top" rowspan="1" colspan="1">0.039</td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.039</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.042</td> <td align="left" valign="top" rowspan="1" colspan="1">0.019</td> <td align="left" valign="top" rowspan="1" colspan="1">0.053</td> <td align="left" valign="top" rowspan="1" colspan="1">0.049</td> <td align="left" valign="top" rowspan="1" colspan="1">0.035</td> <td align="left" valign="top" rowspan="1" colspan="1">0.035</td> <td align="left" valign="top" rowspan="1" colspan="1">0.035</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td colspan="7" align="center" valign="top" rowspan="1">Power</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.970</td> <td align="left" valign="top" rowspan="1" colspan="1">0.460</td> <td align="left" valign="top" rowspan="1" colspan="1">0.950</td> <td align="left" valign="top" rowspan="1" colspan="1">0.923</td> <td align="left" valign="top" rowspan="1" colspan="1">0.958</td> <td align="left" valign="top" rowspan="1" colspan="1">0.734</td> <td align="left" valign="top" rowspan="1" colspan="1">0.810</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.832</td> <td align="left" valign="top" rowspan="1" colspan="1">0.241</td> <td align="left" valign="top" rowspan="1" colspan="1">0.759</td> <td align="left" valign="top" rowspan="1" colspan="1">0.701</td> <td align="left" valign="top" rowspan="1" colspan="1">0.790</td> <td align="left" valign="top" rowspan="1" colspan="1">0.259</td> <td align="left" valign="top" rowspan="1" colspan="1">0.334</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.2</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.944</td> <td align="left" valign="top" rowspan="1" colspan="1">0.477</td> <td align="left" valign="top" rowspan="1" colspan="1">0.920</td> <td align="left" valign="top" rowspan="1" colspan="1">0.886</td> <td align="left" valign="top" rowspan="1" colspan="1">0.929</td> <td align="left" valign="top" rowspan="1" colspan="1">0.729</td> <td align="left" valign="top" rowspan="1" colspan="1">0.804</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">V.777</td> <td align="left" valign="top" rowspan="1" colspan="1">0.224</td> <td align="left" valign="top" rowspan="1" colspan="1">0.698</td> <td align="left" valign="top" rowspan="1" colspan="1">0.633</td> <td align="left" valign="top" rowspan="1" colspan="1">0.731</td> <td align="left" valign="top" rowspan="1" colspan="1">0.240</td> <td align="left" valign="top" rowspan="1" colspan="1">0.302</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.4</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.877</td> <td align="left" valign="top" rowspan="1" colspan="1">0.457</td> <td align="left" valign="top" rowspan="1" colspan="1">0.844</td> <td align="left" valign="top" rowspan="1" colspan="1">0.794</td> <td align="left" valign="top" rowspan="1" colspan="1">0.853</td> <td align="left" valign="top" rowspan="1" colspan="1">0.692</td> <td align="left" valign="top" rowspan="1" colspan="1">0.762</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.665</td> <td align="left" valign="top" rowspan="1" colspan="1">0.201</td> <td align="left" valign="top" rowspan="1" colspan="1">0.578</td> <td align="left" valign="top" rowspan="1" colspan="1">0.515</td> <td align="left" valign="top" rowspan="1" colspan="1">0.613</td> <td align="left" valign="top" rowspan="1" colspan="1">0.203</td> <td align="left" valign="top" rowspan="1" colspan="1">0.250</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.6</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.719</td> <td align="left" valign="top" rowspan="1" colspan="1">0.397</td> <td align="left" valign="top" rowspan="1" colspan="1">0.680</td> <td align="left" valign="top" rowspan="1" colspan="1">0.619</td> <td align="left" valign="top" rowspan="1" colspan="1">0.686</td> <td align="left" valign="top" rowspan="1" colspan="1">0.610</td> <td align="left" valign="top" rowspan="1" colspan="1">0.648</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.490</td> <td align="left" valign="top" rowspan="1" colspan="1">0.163</td> <td align="left" valign="top" rowspan="1" colspan="1">0.412</td> <td align="left" valign="top" rowspan="1" colspan="1">0.363</td> <td align="left" valign="top" rowspan="1" colspan="1">0.433</td> <td align="left" valign="top" rowspan="1" colspan="1">0.161</td> <td align="left" valign="top" rowspan="1" colspan="1">0.188</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td colspan="7" align="center" valign="top" rowspan="1">Type-I error</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.034</td> <td align="left" valign="top" rowspan="1" colspan="1">0.015</td> <td align="left" valign="top" rowspan="1" colspan="1">0.045</td> <td align="left" valign="top" rowspan="1" colspan="1">0.037</td> <td align="left" valign="top" rowspan="1" colspan="1">0.033</td> <td align="left" valign="top" rowspan="1" colspan="1">0.034</td> <td align="left" valign="top" rowspan="1" colspan="1">0.035</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.037</td> <td align="left" valign="top" rowspan="1" colspan="1">0.015</td> <td align="left" valign="top" rowspan="1" colspan="1">0.053</td> <td align="left" valign="top" rowspan="1" colspan="1">0.046</td> <td align="left" valign="top" rowspan="1" colspan="1">0.032</td> <td align="left" valign="top" rowspan="1" colspan="1">0.034</td> <td align="left" valign="top" rowspan="1" colspan="1">0.039</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.2</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.035</td> <td align="left" valign="top" rowspan="1" colspan="1">0.016</td> <td align="left" valign="top" rowspan="1" colspan="1">0.045</td> <td align="left" valign="top" rowspan="1" colspan="1">0.036</td> <td align="left" valign="top" rowspan="1" colspan="1">0.034</td> <td align="left" valign="top" rowspan="1" colspan="1">0.033</td> <td align="left" valign="top" rowspan="1" colspan="1">0.034</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.017</td> <td align="left" valign="top" rowspan="1" colspan="1">0.053</td> <td align="left" valign="top" rowspan="1" colspan="1">0.046</td> <td align="left" valign="top" rowspan="1" colspan="1">0.034</td> <td align="left" valign="top" rowspan="1" colspan="1">0.035</td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.4</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.033</td> <td align="left" valign="top" rowspan="1" colspan="1">0.017</td> <td align="left" valign="top" rowspan="1" colspan="1">0.045</td> <td align="left" valign="top" rowspan="1" colspan="1">0.036</td> <td align="left" valign="top" rowspan="1" colspan="1">0.032</td> <td align="left" valign="top" rowspan="1" colspan="1">0.032</td> <td align="left" valign="top" rowspan="1" colspan="1">0.033</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.018</td> <td align="left" valign="top" rowspan="1" colspan="1">0.052</td> <td align="left" valign="top" rowspan="1" colspan="1">0.047</td> <td align="left" valign="top" rowspan="1" colspan="1">0.033</td> <td align="left" valign="top" rowspan="1" colspan="1">0.033</td> <td align="left" valign="top" rowspan="1" colspan="1">0.036</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.6</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.033</td> <td align="left" valign="top" rowspan="1" colspan="1">0.018</td> <td align="left" valign="top" rowspan="1" colspan="1">0.048</td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.031</td> <td align="left" valign="top" rowspan="1" colspan="1">0.031</td> <td align="left" valign="top" rowspan="1" colspan="1">0.030</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.018</td> <td align="left" valign="top" rowspan="1" colspan="1">0.054</td> <td align="left" valign="top" rowspan="1" colspan="1">0.048</td> <td align="left" valign="top" rowspan="1" colspan="1">0.034</td> <td align="left" valign="top" rowspan="1" colspan="1">0.033</td> <td align="left" valign="top" rowspan="1" colspan="1">0.032</td> </tr> </tbody> </table></div> <div class="p text-right font-secondary"><a href="table/T1/" class="usa-link" target="_blank" rel="noopener noreferrer">Open in a new tab</a></div></section><p id="P32">For all approaches, larger correlation (ρ) results in lower power, mirrored by the corresponding increase in MSE seen in <a href="#F1" class="usa-link">Figure 1</a> (<a href="#SD1" class="usa-link">Supplemental Table 2</a>). Imputation using one-half the minimum observed value (M5) was associated with the highest power – however, this advantage is offset by the relatively large bias in the normal distribution setting when the imputed value is not a good estimate of E(X | X < limit of detection). The power associated with MDI (M3) was among the highest, while that of the expanded MDI model (M4) was somewhat lower. This is expected as the expanded MDI model includes two additional parameters, resulting in a corresponding loss of power. The power associated with imputation using <em>mice</em> (M6) and <em>missForest</em> (M7) was considerably lower than that of the two MDI approaches. Similar trends were observed for both the normal margins and the non-normal margins.</p> <p id="P33">These observations demonstrate a clear advantage of the MDI approaches over the CC approach and other imputation approaches in small to modest sample size settings. The type-I error associated with the MDI model is in good agreement with the nominal value of 0.05.</p> <p id="P34">Power and Type 1 error rate estimates for all approaches are presented for a larger sample size of n=200 (100 cases) subjects in <a href="#SD1" class="usa-link">Supplemental Tables 4</a>.</p></section><section id="S6"><h3 class="pmc_sec_title">Conditional logistic regression</h3> <p id="P35">We considered two settings with respect to the joint distribution of <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M26" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>1</mn></msub></mrow></math></span>, <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M27" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>2</mn></msub></mrow></math></span>, <em>U</em><sub>1</sub>: (1) Multivariate normal distribution; and (2) Non-normal distribution; the Spearman’s rho, ρ, was used to specify the strength of dependence between <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M28" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>1</mn></msub></mrow></math></span>, <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M29" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>2</mn></msub></mrow></math></span>, <em>U</em><sub>1</sub>. To mimic a matched study design, for each subject, an unobserved variable ε ~ N(1<span class="font-variant-small-caps">,</span>σ<span class="font-variant-small-caps">=</span>1.5) was simulated. The deciles of the distribution of ε were used to determine the matching stratum <em>g</em> for every subject. For each subject, the covariates <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M30" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>1</mn></msub></mrow></math></span>, <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M31" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>2</mn></msub></mrow></math></span>, <em>U</em><sub>1</sub> were simulated according to a multivariate distribution with stratum specific parameters. In the multivariate normal setting, both mean and variance parameters were assumed to depend on matching stratum. In the non-normal setting, the rate parameters of the Exponential and Weibull distributed covariates were assumed to depend on matching stratum. Details of the simulations are presented in the <a href="#SD1" class="usa-link">Supplement</a>.</p> <p id="P36">The binary outcome was simulated according to: </p> <table class="disp-formula p" id="FD6"><tr><td class="formula"><math id="M32" display="block" overflow="linebreak"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mrow><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>1</mn></msub><mo>+</mo><msub><mi>β</mi><mn>2</mn></msub><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>2</mn></msub><mo>+</mo><msub><mi>β</mi><mn>3</mn></msub><msub><mi>U</mi><mn>1</mn></msub><mo>+</mo><msub><mi>β</mi><mn>4</mn></msub><mi>ε</mi></mrow></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>e</mi><mrow><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>1</mn></msub><mo>+</mo><msub><mi>β</mi><mn>2</mn></msub><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>2</mn></msub><mo>+</mo><msub><mi>β</mi><mn>3</mn></msub><msub><mi>U</mi><mn>1</mn></msub><mo>+</mo><msub><mi>β</mi><mn>4</mn></msub><mi>ε</mi></mrow></msup></mrow></mfrac><mo>,</mo><mspace width="0.25em"></mspace><mtext>where</mtext><mspace width="0.25em"></mspace><msub><mi>β</mi><mn>0</mn></msub><mo>=</mo><mo>−</mo><mn>3</mn><mo>,</mo><msub><mi>β</mi><mn>1</mn></msub><mo>=</mo><msub><mi>β</mi><mn>2</mn></msub><mo>=</mo><msub><mi>β</mi><mn>3</mn></msub><mo>=</mo><msub><mi>β</mi><mn>4</mn></msub><mo>=</mo><mn>1.</mn></mrow></math></td></tr></table> <p id="P37">A matched dataset was generated by selecting m cases and m matching controls, where the matching was done within group <em>g.</em></p></section><section id="S7"><h3 class="pmc_sec_title">Bias and MSE</h3> <p id="P38">corresponding to MLEs of <em>β</em><sub>1</sub>, <em>β</em>2 in <a href="#FD3" class="usa-link">Equation (1)</a> are shown in <a href="#F2" class="usa-link">Figure 2</a> (<a href="#SD1" class="usa-link">Table 5 of the Supplement</a>) for n=136 (68 matched pairs). The MDI approach (M3) shows a clear advantage over the CC (M2) and <em>mice</em> imputation (M6) approaches in terms of bias and MSE. The expanded MDI model (M4) has a consistently larger bias and MSE when compared to the MDI approach (M3). The bias and MSE reduction in the MDI model is more substantial when compared to the CC approach in the context of a matched study because the CC approach drops study pairs when at least one of its pairs is missing. For the multivariate normal distribution setting, the substitution with one-half the minimum value (M5) achieves comparable bias and MSE to that of the MDI approach (M3), when n=136 (<a href="#F2" class="usa-link">Figure 2</a>, <a href="#SD1" class="usa-link">Table 5 of Supplement</a>). However, the trends reverse in favor of the MDI approach for larger sample sizes such as n=400 (<a href="#SD1" class="usa-link">Supplemental Table 8</a>). Imputation with <em>missForest</em> (M7) appears to have a somewhat larger bias when compared to the MDI (M3) in settings of low ρ – however, this trend is reversed in favor of <em>missForest</em> imputation (M7) when ρ is large.</p> <figure class="fig xbox font-sm" id="F2"><h4 class="obj_head">Figure 2: Bias and MSE associated with estimates of regression coefficients for conditional logistic regression models. The sample size considered here is n = 136 (68 matched pairs). Results are based on 100,000 converged replications.</h4> <p class="img-box line-height-none margin-x-neg-2 tablet:margin-x-0 text-center"><a class="tileshop" target="_blank" href="https://www.ncbi.nlm.nih.gov/core/lw/2.0/html/tileshop_pmc/tileshop_pmc_inline.html?title=Click%20on%20image%20to%20zoom&p=PMC3&id=6812630_nihms-1537397-f0002.jpg"><img class="graphic zoom-in" src="https://cdn.ncbi.nlm.nih.gov/pmc/blobs/02b9/6812630/ba97a6ab61ef/nihms-1537397-f0002.jpg" loading="lazy" height="396" width="800" alt="Figure 2:"></a></p> <div class="p text-right font-secondary"><a href="figure/F2/" class="usa-link" target="_blank" rel="noopener noreferrer">Open in a new tab</a></div> <figcaption><p id="P59">Approaches with mean bias reduction to estimate the true regression coefficients <em>β<sub>1</sub></em>, <em>β<sub>2</sub></em> are as follows: M2 denotes the complete case analysis; M3 denotes the missing data indicator (MDI) model; M4 denotes the expanded missing data indicator (MDI) model; M5 denotes imputation using one half the observed minimum value; M6 denotes predictive mean matching (PMM) imputation implemented in R package mice; M7 denotes the MissForest algorithm implemented in the R package missForest.</p></figcaption></figure><p id="P39">Higher ρ, reflecting higher correlation between covariates is associated with larger MSE for the MDI model (M3); as in logistic regression, the effect of ρ on bias is modest. Since the censoring percentage is higher for <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M33" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>2</mn></msub></mrow></math></span> when compared to <span xmlns:mml="http://www.w3.org/1998/Math/MathML"><math id="M34" display="inline" overflow="linebreak"><mrow><msub><mover accent="true"><mi>X</mi><mo>˜</mo></mover><mn>1</mn></msub></mrow></math></span>, the estimate of β<sub>2</sub> has larger bias and MSE when compared to β<sub>1</sub> in the MDI model. In the CC approach, the bias associated with estimates of <em>β</em><sub>1</sub>, <em>β</em><sub>2</sub> are comparable since the study subjects are discarded at the cluster level; however, the MSE corresponding to the estimate of β<sub>2</sub> is still observed to be larger than that for β<sub>1</sub>. In most cases, similar trends are observed when comparing the various approaches in the non-normal distribution setting. An exception was in the case of substitution with one half the minimum (M5) – under non-normal covariate distributions, this approach achieves minimum bias and MSE, with a substantial advantage over other approaches when ρ is large. This is driven by the fact that the imputed values are good approximations to the expectation E(X | X < limit of detection) (See <a href="#SD1" class="usa-link">Table 1 in Supplement</a>).</p> <p id="P40">Bias and MSE estimates for all approaches are presented for larger sample sizes of n=200 (100 matched pairs) and n=400 (200 matched pairs) subjects in <a href="#SD1" class="usa-link">Supplemental Tables 6</a> and <a href="#SD1" class="usa-link">8</a>.</p></section><section id="S8"><h3 class="pmc_sec_title">Type I error and Power</h3> <p id="P41">are summarized in <a href="#T2" class="usa-link">Table 2</a> corresponding to the hypothesis tests <em>H</em><sub>0</sub>: <em>β</em><sub>1</sub> <em>=</em> 0, <em>H</em><sub>0</sub>: <em>β</em><sub>2</sub> <em>=</em> 0. Trends observed here were similar to that for logistic regression - the MDI approach (M3) yields substantial higher power when compared to the CC (M2) and imputation using <em>mice</em> (M6) and <em>missForest</em> (M7) approaches, in all scenarios considered. The expanded MDI (M4) and substitution using one half the observed minimum (M5) had comparable but lower power than the MDI approach.</p> <section class="tw xbox font-sm" id="T2"><h4 class="obj_head">Table 2:</h4> <div class="caption p"><p id="P61">Summary of power and type-I error for conditional logistic regression models. Each entry represents the proportion of p-value less than or equal to 0.05 out of 100,000 replicates. The sample size is n = 68 matched pairs. M1 denotes the true model before censoring; M2 denotes the complete case analysis; M3 denotes the missing data indicator (MDI) model; M4 denotes the expanded missing data indicator (MDI) model; M5 denotes imputation using one half the observed minimum value; M6 denotes predictive mean matching (PMM) imputation implemented in R package mice; M7 denotes the MissForest algorithm implemented in the R package missForest.</p></div> <div class="tbl-box p" tabindex="0"><table class="content" frame="hsides" rules="groups"> <colgroup span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> </colgroup> <thead><tr> <th align="left" valign="top" rowspan="1" colspan="1">ρ</th> <th align="left" valign="top" rowspan="1" colspan="1"></th> <th align="left" valign="top" rowspan="1" colspan="1">M1</th> <th align="left" valign="top" rowspan="1" colspan="1">M2</th> <th align="left" valign="top" rowspan="1" colspan="1">M3</th> <th align="left" valign="top" rowspan="1" colspan="1">M4</th> <th align="left" valign="top" rowspan="1" colspan="1">M5</th> <th align="left" valign="top" rowspan="1" colspan="1">M6</th> <th align="left" valign="top" rowspan="1" colspan="1">M7</th> </tr></thead> <tbody> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td colspan="7" align="center" valign="top" rowspan="1">Normal margins</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td colspan="7" align="center" valign="top" rowspan="1">Power</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.889</td> <td align="left" valign="top" rowspan="1" colspan="1">0.423</td> <td align="left" valign="top" rowspan="1" colspan="1">0.906</td> <td align="right" valign="top" rowspan="1" colspan="1">0.873</td> <td align="right" valign="top" rowspan="1" colspan="1">0.880</td> <td align="left" valign="top" rowspan="1" colspan="1">0.431</td> <td align="left" valign="top" rowspan="1" colspan="1">0.500</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.302</td> <td align="left" valign="top" rowspan="1" colspan="1">0.104</td> <td align="left" valign="top" rowspan="1" colspan="1">0.320</td> <td align="left" valign="top" rowspan="1" colspan="1">0.290</td> <td align="left" valign="top" rowspan="1" colspan="1">0.270</td> <td align="left" valign="top" rowspan="1" colspan="1">0.053</td> <td align="left" valign="top" rowspan="1" colspan="1">0.064</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.2</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.793</td> <td align="left" valign="top" rowspan="1" colspan="1">0.412</td> <td align="left" valign="top" rowspan="1" colspan="1">0.839</td> <td align="left" valign="top" rowspan="1" colspan="1">0.799</td> <td align="left" valign="top" rowspan="1" colspan="1">0.793</td> <td align="left" valign="top" rowspan="1" colspan="1">0.385</td> <td align="left" valign="top" rowspan="1" colspan="1">0.448</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.234</td> <td align="left" valign="top" rowspan="1" colspan="1">0.105</td> <td align="left" valign="top" rowspan="1" colspan="1">0.279</td> <td align="left" valign="top" rowspan="1" colspan="1">0.252</td> <td align="left" valign="top" rowspan="1" colspan="1">0.234</td> <td align="left" valign="top" rowspan="1" colspan="1">0.057</td> <td align="left" valign="top" rowspan="1" colspan="1">0.055</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.4</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.652</td> <td align="left" valign="top" rowspan="1" colspan="1">0.351</td> <td align="left" valign="top" rowspan="1" colspan="1">0.721</td> <td align="left" valign="top" rowspan="1" colspan="1">0.670</td> <td align="left" valign="top" rowspan="1" colspan="1">0.652</td> <td align="left" valign="top" rowspan="1" colspan="1">0.306</td> <td align="left" valign="top" rowspan="1" colspan="1">0.365</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.173</td> <td align="left" valign="top" rowspan="1" colspan="1">0.106</td> <td align="left" valign="top" rowspan="1" colspan="1">0.222</td> <td align="left" valign="top" rowspan="1" colspan="1">0.207</td> <td align="left" valign="top" rowspan="1" colspan="1">0.173</td> <td align="left" valign="top" rowspan="1" colspan="1">0.048</td> <td align="left" valign="top" rowspan="1" colspan="1">0.049</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.6</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.419</td> <td align="left" valign="top" rowspan="1" colspan="1">0.373</td> <td align="left" valign="top" rowspan="1" colspan="1">0.526</td> <td align="left" valign="top" rowspan="1" colspan="1">0.489</td> <td align="left" valign="top" rowspan="1" colspan="1">0.419</td> <td align="left" valign="top" rowspan="1" colspan="1">0.217</td> <td align="left" valign="top" rowspan="1" colspan="1">0.261</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.115</td> <td align="left" valign="top" rowspan="1" colspan="1">0.105</td> <td align="left" valign="top" rowspan="1" colspan="1">0.167</td> <td align="left" valign="top" rowspan="1" colspan="1">0.167</td> <td align="left" valign="top" rowspan="1" colspan="1">0.115</td> <td align="left" valign="top" rowspan="1" colspan="1">0.043</td> <td align="left" valign="top" rowspan="1" colspan="1">0.044</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td colspan="7" align="center" valign="top" rowspan="1">Type-I error</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.043</td> <td align="left" valign="top" rowspan="1" colspan="1">0.004</td> <td align="left" valign="top" rowspan="1" colspan="1">0.048</td> <td align="left" valign="top" rowspan="1" colspan="1">0.072</td> <td align="left" valign="top" rowspan="1" colspan="1">0.041</td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.041</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.040</td> <td align="left" valign="top" rowspan="1" colspan="1">0.004</td> <td align="left" valign="top" rowspan="1" colspan="1">0.047</td> <td align="left" valign="top" rowspan="1" colspan="1">0.064</td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.042</td> <td align="left" valign="top" rowspan="1" colspan="1">0.039</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.2</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.044</td> <td align="left" valign="top" rowspan="1" colspan="1">0.006</td> <td align="left" valign="top" rowspan="1" colspan="1">0.043</td> <td align="left" valign="top" rowspan="1" colspan="1">0.071</td> <td align="left" valign="top" rowspan="1" colspan="1">0.044</td> <td align="left" valign="top" rowspan="1" colspan="1">0.045</td> <td align="left" valign="top" rowspan="1" colspan="1">0.046</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.039</td> <td align="left" valign="top" rowspan="1" colspan="1">0.005</td> <td align="left" valign="top" rowspan="1" colspan="1">0.042</td> <td align="left" valign="top" rowspan="1" colspan="1">0.066</td> <td align="left" valign="top" rowspan="1" colspan="1">0.039</td> <td align="left" valign="top" rowspan="1" colspan="1">0.043</td> <td align="left" valign="top" rowspan="1" colspan="1">0.040</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.4</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.045</td> <td align="left" valign="top" rowspan="1" colspan="1">0.010</td> <td align="left" valign="top" rowspan="1" colspan="1">0.047</td> <td align="left" valign="top" rowspan="1" colspan="1">0.071</td> <td align="left" valign="top" rowspan="1" colspan="1">0.045</td> <td align="left" valign="top" rowspan="1" colspan="1">0.042</td> <td align="left" valign="top" rowspan="1" colspan="1">0.044</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.045</td> <td align="left" valign="top" rowspan="1" colspan="1">0.008</td> <td align="left" valign="top" rowspan="1" colspan="1">0.048</td> <td align="left" valign="top" rowspan="1" colspan="1">0.068</td> <td align="left" valign="top" rowspan="1" colspan="1">0.045</td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.042</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.6</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.045</td> <td align="left" valign="top" rowspan="1" colspan="1">0.013</td> <td align="left" valign="top" rowspan="1" colspan="1">0.041</td> <td align="left" valign="top" rowspan="1" colspan="1">0.074</td> <td align="left" valign="top" rowspan="1" colspan="1">0.045</td> <td align="left" valign="top" rowspan="1" colspan="1">0.046</td> <td align="left" valign="top" rowspan="1" colspan="1">0.044</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.042</td> <td align="left" valign="top" rowspan="1" colspan="1">0.013</td> <td align="left" valign="top" rowspan="1" colspan="1">0.047</td> <td align="left" valign="top" rowspan="1" colspan="1">0.065</td> <td align="left" valign="top" rowspan="1" colspan="1">0.042</td> <td align="left" valign="top" rowspan="1" colspan="1">0.043</td> <td align="left" valign="top" rowspan="1" colspan="1">0.041</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td colspan="7" align="center" valign="top" rowspan="1">Power</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.874</td> <td align="left" valign="top" rowspan="1" colspan="1">0.547</td> <td align="left" valign="top" rowspan="1" colspan="1">0.923</td> <td align="left" valign="top" rowspan="1" colspan="1">0.885</td> <td align="left" valign="top" rowspan="1" colspan="1">0.873</td> <td align="left" valign="top" rowspan="1" colspan="1">0.545</td> <td align="left" valign="top" rowspan="1" colspan="1">0.651</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.255</td> <td align="left" valign="top" rowspan="1" colspan="1">0.279</td> <td align="left" valign="top" rowspan="1" colspan="1">0.386</td> <td align="left" valign="top" rowspan="1" colspan="1">0.344</td> <td align="left" valign="top" rowspan="1" colspan="1">0.253</td> <td align="left" valign="top" rowspan="1" colspan="1">0.105</td> <td align="left" valign="top" rowspan="1" colspan="1">0.121</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.2</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.807</td> <td align="left" valign="top" rowspan="1" colspan="1">0.564</td> <td align="left" valign="top" rowspan="1" colspan="1">0.872</td> <td align="left" valign="top" rowspan="1" colspan="1">0.833</td> <td align="left" valign="top" rowspan="1" colspan="1">0.807</td> <td align="left" valign="top" rowspan="1" colspan="1">0.509</td> <td align="left" valign="top" rowspan="1" colspan="1">0.594</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">^0.196</td> <td align="left" valign="top" rowspan="1" colspan="1">0.240</td> <td align="left" valign="top" rowspan="1" colspan="1">0.307</td> <td align="left" valign="top" rowspan="1" colspan="1">0.277</td> <td align="left" valign="top" rowspan="1" colspan="1">0.196</td> <td align="left" valign="top" rowspan="1" colspan="1">0.093</td> <td align="left" valign="top" rowspan="1" colspan="1">0.095</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.4</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.668</td> <td align="left" valign="top" rowspan="1" colspan="1">0.530</td> <td align="left" valign="top" rowspan="1" colspan="1">0.766</td> <td align="left" valign="top" rowspan="1" colspan="1">0.708</td> <td align="left" valign="top" rowspan="1" colspan="1">0.668</td> <td align="left" valign="top" rowspan="1" colspan="1">0.433</td> <td align="left" valign="top" rowspan="1" colspan="1">0.500</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.151</td> <td align="left" valign="top" rowspan="1" colspan="1">0.211</td> <td align="left" valign="top" rowspan="1" colspan="1">0.254</td> <td align="left" valign="top" rowspan="1" colspan="1">0.230</td> <td align="left" valign="top" rowspan="1" colspan="1">0.151</td> <td align="left" valign="top" rowspan="1" colspan="1">0.078</td> <td align="left" valign="top" rowspan="1" colspan="1">0.082</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.6</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.461</td> <td align="left" valign="top" rowspan="1" colspan="1">0.460</td> <td align="left" valign="top" rowspan="1" colspan="1">0.589</td> <td align="left" valign="top" rowspan="1" colspan="1">0.533</td> <td align="left" valign="top" rowspan="1" colspan="1">0.461</td> <td align="left" valign="top" rowspan="1" colspan="1">0.323</td> <td align="left" valign="top" rowspan="1" colspan="1">0.362</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.091</td> <td align="left" valign="top" rowspan="1" colspan="1">0.190</td> <td align="left" valign="top" rowspan="1" colspan="1">0.171</td> <td align="left" valign="top" rowspan="1" colspan="1">0.165</td> <td align="left" valign="top" rowspan="1" colspan="1">0.091</td> <td align="left" valign="top" rowspan="1" colspan="1">0.060</td> <td align="left" valign="top" rowspan="1" colspan="1">0.056</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td colspan="7" align="center" valign="top" rowspan="1">Type-I error</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.031</td> <td align="left" valign="top" rowspan="1" colspan="1">0.048</td> <td align="left" valign="top" rowspan="1" colspan="1">0.066</td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.040</td> <td align="left" valign="top" rowspan="1" colspan="1">0.037</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.033</td> <td align="left" valign="top" rowspan="1" colspan="1">0.011</td> <td align="left" valign="top" rowspan="1" colspan="1">0.045</td> <td align="left" valign="top" rowspan="1" colspan="1">0.065</td> <td align="left" valign="top" rowspan="1" colspan="1">0.032</td> <td align="left" valign="top" rowspan="1" colspan="1">0.042</td> <td align="left" valign="top" rowspan="1" colspan="1">0.037</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.2</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.037</td> <td align="left" valign="top" rowspan="1" colspan="1">0.032</td> <td align="left" valign="top" rowspan="1" colspan="1">0.043</td> <td align="left" valign="top" rowspan="1" colspan="1">0.067</td> <td align="left" valign="top" rowspan="1" colspan="1">0.037</td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.012</td> <td align="left" valign="top" rowspan="1" colspan="1">0.041</td> <td align="left" valign="top" rowspan="1" colspan="1">0.067</td> <td align="left" valign="top" rowspan="1" colspan="1">0.038</td> <td align="left" valign="top" rowspan="1" colspan="1">0.032</td> <td align="left" valign="top" rowspan="1" colspan="1">0.035</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.4</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.037</td> <td align="left" valign="top" rowspan="1" colspan="1">0.034</td> <td align="left" valign="top" rowspan="1" colspan="1">0.048</td> <td align="left" valign="top" rowspan="1" colspan="1">0.072</td> <td align="left" valign="top" rowspan="1" colspan="1">0.037</td> <td align="left" valign="top" rowspan="1" colspan="1">0.037</td> <td align="left" valign="top" rowspan="1" colspan="1">0.037</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.037</td> <td align="left" valign="top" rowspan="1" colspan="1">0.013</td> <td align="left" valign="top" rowspan="1" colspan="1">0.047</td> <td align="left" valign="top" rowspan="1" colspan="1">0.064</td> <td align="left" valign="top" rowspan="1" colspan="1">0.037</td> <td align="left" valign="top" rowspan="1" colspan="1">0.040</td> <td align="left" valign="top" rowspan="1" colspan="1">0.034</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">0.6</td> <td align="left" valign="top" rowspan="1" colspan="1">β<sub>1</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.042</td> <td align="left" valign="top" rowspan="1" colspan="1">0.039</td> <td align="left" valign="top" rowspan="1" colspan="1">0.043</td> <td align="left" valign="top" rowspan="1" colspan="1">0.078</td> <td align="left" valign="top" rowspan="1" colspan="1">0.042</td> <td align="left" valign="top" rowspan="1" colspan="1">0.039</td> <td align="left" valign="top" rowspan="1" colspan="1">0.040</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1"></td> <td align="right" valign="top" rowspan="1" colspan="1">β<sub>2</sub> </td> <td align="right" valign="top" rowspan="1" colspan="1">0.039</td> <td align="left" valign="top" rowspan="1" colspan="1">0.018</td> <td align="left" valign="top" rowspan="1" colspan="1">0.047</td> <td align="left" valign="top" rowspan="1" colspan="1">0.067</td> <td align="left" valign="top" rowspan="1" colspan="1">0.039</td> <td align="left" valign="top" rowspan="1" colspan="1">0.037</td> <td align="left" valign="top" rowspan="1" colspan="1">0.036</td> </tr> </tbody> </table></div> <div class="p text-right font-secondary"><a href="table/T2/" class="usa-link" target="_blank" rel="noopener noreferrer">Open in a new tab</a></div></section><p id="P42">Power and Type 1 error rate estimates for all approaches are presented for larger sample sizes of n=200 (100 matched pairs) and n=400 (200 matched pairs) subjects in <a href="#SD1" class="usa-link">Supplemental Tables 7</a>, <a href="#SD1" class="usa-link">9</a>.</p></section></section><section id="S9"><h2 class="pmc_sec_title">CARDIOVASCULAR DISEASE BIOMARKER STUDY</h2> <p id="P43">This matched case-control study was conducted by the High Risk Plaque Initiative [BG Medicine Inc. (Waltham, MA) and other partners] to discover prognostic biomarkers in blood plasma for near-term cardiovascular events. Matched cases and controls were selected from the CATHGEN study, in which peripheral blood samples were collected from consenting research subjects undergoing cardiac catheterization at Duke University Medical Center from 2001 through 2011[<a href="#R25" class="usa-link" aria-describedby="R25">25</a>]. 68 cases were selected from among individuals who had a major adverse cardiac event (MACE) within two years following the time of their sample collection. 68 controls were selected from individuals who were MACE-free for the two years following sample collection and were matched to cases on age, gender, race/ethnicity and severity of coronary artery disease. High-content mass spectrometry based techniques were employed to quantify 472 metabolites from each subject’s serum specimen [<a href="#R26" class="usa-link" aria-describedby="R26">26</a>, <a href="#R27" class="usa-link" aria-describedby="R27">27</a>]. The identities of the measured metabolites and proteins are masked due to a data confidentiality agreement.</p> <p id="P44">Of the 472 quantified metabolites, 99 metabolites have at least one missing value. Among these 99 metabolites, the median number of pairs missing at least one measurement was 16. Each metabolite was analyzed in a MDI model: </p> <table class="disp-formula p" id="FD7"><tr><td class="formula"><math id="M35" display="block" overflow="linebreak"><mrow><mi>g</mi><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><mi>X</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi mathvariant="normal">Δ</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>β</mi><mn>2</mn></msub><mi mathvariant="normal">Δ</mi></mrow></math></td></tr></table> <p> where X is the metabolite that is subject to limit of detection, and Δ is the missing indicator defined as Δ = I(X ≤ α) for some limit of detection threshold α. For comparison, we also analyzed the data using the CC approach. We used a Wald test to test for the significance of the main effect of metabolite in the CC model, and a likelihood ratio test to test for the joint significance of both the main effect and the indicator term in the MDI model. Since the observed X is highly correlated with missing indicator, Δ, we also fit a separate model for each metabolite including the missing indicator as the only covariate, referred to as the Δ-model below. The Δ-model enables us to test whether the metabolite level falling below the limit of detection is associated with the outcome, MACE.</p> <p id="P45">At the 0.05 level of significance, our analysis identified 15 metabolites for which the CC and MDI models both converged and yielded discordant results based on a p value threshold of 0.05; i.e., one model has a P-value less than 0.05 while the other has a P-value greater than 0.05. There was one additional metabolite for which the CC model did not converge but the MDI model converged. This metabolite had extensive censoring where at least one member of 81% of the matched pairs had an undetectable (missing) value. The results for the 15 metabolites with discordant results when comparing the CC and MDI models are presented in <a href="#T3" class="usa-link">Table 3</a>.</p> <section class="tw xbox font-sm" id="T3"><h3 class="obj_head">Table 3: Cardiovascular Disease Biomarker Study.</h3> <div class="caption p"><p id="P62">Summary of results for 15 metabolites with discordant results from the complete case (CC) and missing indicator (MDI) models. We use cen % to denote the censoring proportion at cluster level; a cluster is considered censored if at least one of its element is censored. EST denotes the estimate of the log odds ratio. The P-values presented under the CC model and the Δ-model were obtained by Wald tests. The P-values presented under the MDI model was obtained by a likelihood ratio test of the joint association of (X, Δ)</p></div> <div class="tbl-box p" tabindex="0"><table class="content" frame="hsides" rules="groups"> <colgroup span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> <col align="left" valign="middle" span="1"> </colgroup> <thead> <tr> <th align="left" valign="top" rowspan="1" colspan="1"></th> <th align="left" valign="top" rowspan="1" colspan="1"></th> <th colspan="3" align="center" valign="top" rowspan="1">Complete case (CC) model</th> <th colspan="3" align="center" valign="top" rowspan="1">Missing indicator (MDI) model</th> <th align="left" valign="top" rowspan="1" colspan="1">Δ-model<a href="#TFN1" class="usa-link"><sup>*</sup></a> </th> </tr> <tr> <th align="left" valign="top" rowspan="1" colspan="1">metabolites</th> <th align="left" valign="top" rowspan="1" colspan="1">cen %</th> <th align="left" valign="top" rowspan="1" colspan="1">EST</th> <th align="left" valign="top" rowspan="1" colspan="1">95% CI</th> <th align="center" valign="top" rowspan="1" colspan="1"> <em>p</em> -value <em>X</em> </th> <th align="left" valign="top" rowspan="1" colspan="1">EST</th> <th align="left" valign="top" rowspan="1" colspan="1">95% CI</th> <th align="center" valign="top" rowspan="1" colspan="1"> <em>p</em> -value (<em>X</em>, Δ)</th> <th align="center" valign="top" rowspan="1" colspan="1"> <em>p</em> -value Δ</th> </tr> </thead> <tbody> <tr> <td align="left" valign="top" rowspan="1" colspan="1">Mx 1</td> <td align="left" valign="top" rowspan="1" colspan="1">0.02</td> <td align="left" valign="top" rowspan="1" colspan="1">0.43</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.19, 0.97)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.03</td> <td align="left" valign="top" rowspan="1" colspan="1">0.43</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.19, 0.97)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.05</td> <td align="center" valign="top" rowspan="1" colspan="1">-</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">Mx 2</td> <td align="left" valign="top" rowspan="1" colspan="1">0.10</td> <td align="left" valign="top" rowspan="1" colspan="1">2.32</td> <td align="center" valign="top" rowspan="1" colspan="1">(1.03, 5.26)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.03</td> <td align="left" valign="top" rowspan="1" colspan="1">2.08</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.96, 4.55)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.08</td> <td align="center" valign="top" rowspan="1" colspan="1">0.27</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">Mx 3</td> <td align="left" valign="top" rowspan="1" colspan="1">0.10</td> <td align="left" valign="top" rowspan="1" colspan="1">1.97</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.96, 4.03)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.05</td> <td align="left" valign="top" rowspan="1" colspan="1">1.87</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.94, 3.73)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.09</td> <td align="center" valign="top" rowspan="1" colspan="1">0.27</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">Mx 4</td> <td align="left" valign="top" rowspan="1" colspan="1">0.81</td> <td align="left" valign="top" rowspan="1" colspan="1">0.18</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.01, 2.73)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.09</td> <td align="left" valign="top" rowspan="1" colspan="1">0.17</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.03, 1.02)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.05</td> <td align="center" valign="top" rowspan="1" colspan="1">0.72</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">Mx 5</td> <td align="left" valign="top" rowspan="1" colspan="1">0.04</td> <td align="left" valign="top" rowspan="1" colspan="1">0.68</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.40, 1.16)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.14</td> <td align="left" valign="top" rowspan="1" colspan="1">0.68</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.40, 1.16)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.04</td> <td align="center" valign="top" rowspan="1" colspan="1">-</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">Mx 6</td> <td align="left" valign="top" rowspan="1" colspan="1">0.09</td> <td align="left" valign="top" rowspan="1" colspan="1">0.59</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.01, 25.61)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.78</td> <td align="left" valign="top" rowspan="1" colspan="1">0.59</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.01, 25.61)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.02</td> <td align="center" valign="top" rowspan="1" colspan="1">-</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">Mx 7</td> <td align="left" valign="top" rowspan="1" colspan="1">0.18</td> <td align="left" valign="top" rowspan="1" colspan="1">1.71</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.45, 6.45)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.43</td> <td align="left" valign="top" rowspan="1" colspan="1">1.50</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.41, 5.52)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.01</td> <td align="center" valign="top" rowspan="1" colspan="1">0.02</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">Mx 8</td> <td align="left" valign="top" rowspan="1" colspan="1">0.18</td> <td align="left" valign="top" rowspan="1" colspan="1">1.63</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.46, 5.79)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.45</td> <td align="left" valign="top" rowspan="1" colspan="1">1.45</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.42, 5.02)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.01</td> <td align="center" valign="top" rowspan="1" colspan="1">0.02</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">Mx 9</td> <td align="left" valign="top" rowspan="1" colspan="1">0.22</td> <td align="left" valign="top" rowspan="1" colspan="1">1.21</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.92, 1.59)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.17</td> <td align="left" valign="top" rowspan="1" colspan="1">1.17</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.90, 1.52)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.01</td> <td align="center" valign="top" rowspan="1" colspan="1">0.01</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">Mx 10</td> <td align="left" valign="top" rowspan="1" colspan="1">0.29</td> <td align="left" valign="top" rowspan="1" colspan="1">0.79</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.27, 2.30)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.67</td> <td align="left" valign="top" rowspan="1" colspan="1">0.77</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.29, 2.08)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.03</td> <td align="center" valign="top" rowspan="1" colspan="1">0.02</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">Mx 11</td> <td align="left" valign="top" rowspan="1" colspan="1">0.24</td> <td align="left" valign="top" rowspan="1" colspan="1">2.52</td> <td align="center" valign="top" rowspan="1" colspan="1">(1.07, 5.97)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.02</td> <td align="left" valign="top" rowspan="1" colspan="1">2.18</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.98, 4.86)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.12</td> <td align="center" valign="top" rowspan="1" colspan="1">0.62</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">Mx 12</td> <td align="left" valign="top" rowspan="1" colspan="1">0.24</td> <td align="left" valign="top" rowspan="1" colspan="1">4.29</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.99, 18.54)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.04</td> <td align="left" valign="top" rowspan="1" colspan="1">3.50</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.95, 12.92)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.13</td> <td align="center" valign="top" rowspan="1" colspan="1">0.62</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">Mx 13</td> <td align="left" valign="top" rowspan="1" colspan="1">0.37</td> <td align="left" valign="top" rowspan="1" colspan="1">0.68</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.48, 0.98)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.02</td> <td align="left" valign="top" rowspan="1" colspan="1">0.83</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.63, 1.08)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.34</td> <td align="center" valign="top" rowspan="1" colspan="1">0.68</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">Mx 14</td> <td align="left" valign="top" rowspan="1" colspan="1">0.79</td> <td align="left" valign="top" rowspan="1" colspan="1">0.29</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.06, 1.49)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.05</td> <td align="left" valign="top" rowspan="1" colspan="1">0.76</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.41, 1.42)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.63</td> <td align="center" valign="top" rowspan="1" colspan="1">0.72</td> </tr> <tr> <td align="left" valign="top" rowspan="1" colspan="1">Mx 15</td> <td align="left" valign="top" rowspan="1" colspan="1">0.81</td> <td align="left" valign="top" rowspan="1" colspan="1">0.09</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.01, 1.67)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.02</td> <td align="left" valign="top" rowspan="1" colspan="1">0.41</td> <td align="center" valign="top" rowspan="1" colspan="1">(0.12, 1.45)</td> <td align="center" valign="top" rowspan="1" colspan="1">0.29</td> <td align="center" valign="top" rowspan="1" colspan="1">0.72</td> </tr> </tbody> </table></div> <div class="p text-right font-secondary"><a href="table/T3/" class="usa-link" target="_blank" rel="noopener noreferrer">Open in a new tab</a></div> <div class="tw-foot p"><div class="fn" id="TFN1"> <sup>*:</sup><p class="display-inline" id="P63">The Δ model did not converge for Mx 1, Mx 5 and Mx 6 due to perfect separation.</p> </div></div></section><p id="P46">As expected, when the censoring percent is < 10% of the matched pairs, the point estimates and the 95% confidence intervals from the CC and MDI models are almost identical. As the censoring proportion increases, the results from the two models begin to diverge, due to increasing differences in sample sizes. Of the 15 metabolites shown in <a href="#T3" class="usa-link">Table 3</a>, four metabolites (Mx 1 - Mx 4) have similar results from the two models; i.e., one model has a significant association with <em>P</em> < 0.05 and the other model has a marginally significant association (0.05 < P < 0.1). Six other metabolites (Mx 5-Mx 10) have statistically significant <em>P</em>-values (<em>P</em> < 0.05) resulting from the MDI model but non-significant <em>P</em>-values (<em>P</em> > 0.1) from the CC model. These six metabolites have censoring levels ranging from 4% (Mx 5) to 29% (Mx 10). For each of these six metabolites, the CC coefficient estimate is comparable to that from the MDI model. In addition, four of these metabolites (Mx 7-Mx 10) have a statistically significant association in the Δ-model. In the case of metabolites Mx 5 and Mx 6, the Δ-model did not converge due to perfect separation. For metabolites Mx 5 - Mx 10, the missing indicator Δ is strongly associated with outcome. These results are consistent with the observations in the simulations that indicate that the more efficient use of information in the MDI model yields increased power. Three metabolites (Mx 11 – Mx 13) had statistically significant (P < 0.05) associations in the CC model but insignificant <em>P</em>-values in the MDI model. These metabolites had censoring levels between 24% and 37% - in all three cases, the Δ-model results are insignificant with <em>P</em>-values exceeding 0.5, thereby resulting in insignificant <em>P</em>-values from the MDI approach. Lastly, two metabolites (Mx 14 - Mx 15) with heavy censoring levels of approximately 80%−81% have significant associations (<em>P</em> < 0.05) in the CC model but non-significant associations (<em>P</em> > 0.1) in the MDI model. Since these metabolites have extreme levels of censoring, it is difficult to determine which model is appropriate (if any).</p></section><section id="S10"><h2 class="pmc_sec_title">DISCUSSION</h2> <p id="P47">Missing covariates are commonly encountered in many biomedical investigations. In studies employing high-throughput metabolomic technologies, missing values can arise due to a combination of limit of detection issues as well as random instrument failure. In these settings, investigators often consider discarding data from subjects with incomplete covariate measurements and/or various imputation techniques. Previous literature has shown that ad hoc imputation techniques can result in severe bias [<a href="#R1" class="usa-link" aria-describedby="R1">1</a>, <a href="#R2" class="usa-link" aria-describedby="R2">2</a>]. More complex likelihood-based methods rely on stringent parametric assumptions and can be computationally intensive. Discarding subjects with partially missing values in a complete case analysis can result in a dramatic reduction in power. In these settings, the missing data indicator or MDI approach is an attractive alternative as all available information remains in the analysis to maintain statistical power. In this paper, for settings of moderate sample size, a binary outcome, and where the missingness in the covariate is non-random and is a result of falling below the limit of detection, we evaluated the bias, MSE and statistical power associated with the MDI model when compared to the CC analysis and three different imputation approaches.</p> <p id="P48">In our simulation study, the MDI approach shows a clear advantage over the CC approach when there are two censored covariates. Imputation using two other strategies, namely the MissForest algorithm and the PMM algorithm implemented in the <em>mice</em> R package did not show improved performance when compared to MDI approach. In particular, these approaches resulted in imputed values in the range of the observed covariate distributions, resulting in large bias due to the non-random censoring mechanism (<a href="#SD1" class="usa-link">Figure 1 in Supplement</a>). The advantage of the MDI approach was preserved in settings of normal and non-normal covariate distributions and in the presence of modest correlation between the censored covariates. The advantage of the MDI model over the CC approach was more substantial for the analysis of matched case-control studies in conditional logistic regression models. The MDI approach can be easily adopted in multivariable models that include several censored covariates jointly, in which setting the CC approach would be severely impacted as it discards the union of subjects with at least one missing value.</p> <p id="P49">In our study, the performance of logistic regression and conditional logistic regression models were evaluated solely in the context of missing covariate values that arise due to limit of detection limitations of the assay. The MDI approach can be used in datasets where missingness is due to other mechanisms, including data that is missing completely at random. In these settings, we also expect approaches such as the PMM based imputation (<em>mice</em>) and MissForest algorithms to have better performance. However, a comparative evaluation of the MDI approach relative to PMM and MissForest algorithms for more general missing data mechanisms was not studied in this paper.</p> <p id="P50">For the logistic regression analyses, the imputation approach based on one half the minimum observed value had larger bias relative to the MDI model in the normal distribution setting; however, the bias associated with this ad-hoc approach was substantially lower in the non-normal setting. For the simulation settings considered in this paper, the imputed value based on by one half the minimum observed value closely approximated the conditional expectation, E(X | X < limit of detection), in the non-normal setting, but not in the normal setting (<a href="#SD1" class="usa-link">Table 1 in the Supplement</a>). For the conditional logistic regression analyses, the imputation approach based on one half the minimum observed value did as well or better than the MDI model. As discussed in Cole, S. R. et. al. (2009) [<a href="#R2" class="usa-link" aria-describedby="R2">2</a>], the observed bias is a function of the imputed value. Thus, when the ad-hoc substitution approach closely approximates the conditional expectation of the covariate, the resulting bias is low.</p> <p id="P51">To the best of our knowledge, theoretical asymptotic properties for the missing indicator model have been derived only for linear regression and are not yet available for more general settings [<a href="#R4" class="usa-link" aria-describedby="R4">4</a>]. We acknowledge that the MDI approach could be asymptotically biased for general link functions. Here, we assumed that the censoring of the covariates is independent of the outcome when conditioning on the covariates. The performance of the missing indicator approach under informative censoring warrants further investigation. It would also be useful to evaluate the properties of the MDI approach in models for count data and survival outcomes.</p></section><section id="SM1"><h2 class="pmc_sec_title">Supplementary Material</h2> <section class="sm xbox font-sm" id="SD1"><div class="caption p"><span>1</span></div> <div class="media p" id="d36e2224"><div class="caption"> <a href="/articles/instance/6812630/bin/NIHMS1537397-supplement-1.pdf" data-ga-action="click_feat_suppl" class="usa-link">NIHMS1537397-supplement-1.pdf</a><sup> (197.5KB, pdf) </sup> </div></div></section></section><section id="S11" class="ack"><h2 class="pmc_sec_title">ACKNOWLEDGEMENTS</h2> <p id="P52">This work was supported in part by the Harvard NeuroDiscovery Center, National Institutes of Health grants T32NS048005 and R01HL122241.</p></section><section id="fn-group1" class="fn-group"><h2 class="pmc_sec_title">Footnotes</h2> <div class="fn-group p font-secondary-light font-sm"> <div class="fn p" id="FN1"> <p id="P53">All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.</p> <p id="P54">This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.</p> <p id="P55">The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript</p> <p id="P56">The following authors have affiliations with organizations with direct or indirect financial interest in the subject matter discussed in the manuscript: No conflicts of interests to declare.</p> </div> <div class="fn p" id="FN2"><p id="P57"><strong>Publisher's Disclaimer:</strong> This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.</p></div> </div></section><section id="ref-list1" class="ref-list"><h2 class="pmc_sec_title">References</h2> <section id="ref-list1_sec2"><ul class="ref-list font-sm" style="list-style-type:none"> <li id="R1"> <span class="label">1.</span><cite>Schisterman EF, et al. , The limitations due to exposure detection limits for regression models. 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The missing indicator approach for censored covariates subject to limit of detection in logistic regression models - PMC