The landscape of survival analysis is constantly being revolutionized to answer biomedical challenges, most recently the statistical challenge of censored covariates rather than outcomes. There are many promising strategies to tackle censored covariates, including weighting, imputation, maximum likelihood, and Bayesian methods. Still, this is a relatively fresh area of research, different from the areas of censored outcomes (i.e., survival analysis) or missing covariates. In this review, we discuss the unique statistical challenges encountered when handling censored covariates and provide an in-depth review of existing methods designed to address those challenges. We emphasize each method\'s relative strengths and weaknesses, providing recommendations to help investigators pinpoint the best approach to handling censored covariates in their data.

BACKGROUND: Innovations in single cell technologies have lead to a flurry of datasets and computational tools to process and interpret them, including analyses of cell composition changes and transition in cell states. The diffcyt workflow for differential discovery in cytometry data consist of several steps, including preprocessing, cell population identification and differential testing for an association with a binary or continuous covariate. However, the commonly measured quantity of survival time in clinical studies often results in a censored covariate where classical differential testing is inapplicable.
RESULTS: To overcome this limitation, multiple methods to directly include censored covariates in differential abundance analysis were examined with the use of simulation studies and a case study. Results show that multiple imputation based methods offer on-par performance with the Cox proportional hazards model in terms of sensitivity and error control, while offering flexibility to account for covariates. The tested methods are implemented in the R package censcyt as an extension of diffcyt and are available at https://bioconductor.org/packages/censcyt .
CONCLUSIONS: Methods for the direct inclusion of a censored variable as a predictor in GLMMs are a valid alternative to classical survival analysis methods, such as the Cox proportional hazard model, while allowing for more flexibility in the differential analysis.

This paper deals with a Cox proportional hazards regression model, where some covariates of interest are randomly right-censored. While methods for censored outcomes have become ubiquitous in the literature, methods for censored covariates have thus far received little attention and, for the most part, dealt with the issue of limit-of-detection. For randomly censored covariates, an often-used method is the inefficient complete-case analysis (CCA) which consists in deleting censored observations in the data analysis. When censoring is not completely independent, the CCA leads to biased and spurious results. Methods for missing covariate data, including type I and type II covariate censoring as well as limit-of-detection do not readily apply due to the fundamentally different nature of randomly censored covariates. We develop a novel method for censored covariates using a conditional mean imputation based on either Kaplan-Meier estimates or a Cox proportional hazards model to estimate the effects of these covariates on a time-to-event outcome. We evaluate the performance of the proposed method through simulation studies and show that it provides good bias reduction and statistical efficiency. Finally, we illustrate the method using data from the Framingham Heart Study to assess the relationship between offspring and parental age of onset of cardiovascular events.

This research is motivated by studying the progression of age-related macular degeneration where both a covariate and the response variable are subject to censoring. We develop a general framework to handle regression with censored covariate where the response can be different types and the censoring can be random or subject to (constant) detection limits. Multiple imputation is a popular technique to handle missing data that requires compatibility between the imputation model and the substantive model to obtain valid estimates. With censored covariate, we propose a novel multiple imputation-based approach, namely, the semiparametric two-step importance sampling imputation (STISI) method, to impute the censored covariate. Specifically, STISI imputes the missing covariate from a semiparametric accelerated failure time model conditional on fully observed covariates (Step 1) with the acceptance probability derived from the substantive model (Step 2). The 2-step procedure automatically ensures compatibility and takes full advantage of the relaxed semiparametric assumption in the imputation. Extensive simulations demonstrate that the STISI method yields valid estimates in all scenarios and outperforms some existing methods that are commonly used in practice. We apply STISI on data from the Age-related Eye Disease Study, to investigate the association between the progression time of the less severe eye and that of the more severe eye. We also illustrate the method by analyzing the urine arsenic data for patients from National Health and Nutrition Examination Survey (2003-2004) where the response is binary and 1 covariate is subject to detection limit.

This article describes a nonparametric conditional imputation analytic method for randomly censored covariates in linear regression. While some existing methods make assumptions about the distribution of covariates or underestimate standard error due to lack of imputation error, the proposed approach is distribution-free and utilizes resampling to correct for variance underestimation. The performance of the novel method is assessed using simulations, and results are contrasted with methods currently used for a limit of detection censored design, including the complete case approach and other nonparametric approaches. Theoretical justifications for the proposed method are provided, and its application is demonstrated through a study of association between lipoprotein cholesterol in offspring and parental history of cardiovascular disease.

Frailty models are multiplicative hazard models for studying association between survival time and important clinical covariates. When some values of a clinical covariate are unobserved but known to be below a threshold called the limit of detection (LOD), naive approaches ignoring this problem, such as replacing the undetected value by the LOD or half of the LOD, often produce biased parameter estimate with larger mean squared error of the estimate. To address the LOD problem in a frailty model, we propose a flexible smooth nonparametric density estimator along with Simpson\'s numerical integration technique. This is an extension of an existing method in the likelihood framework for the estimation and inference of the model parameters. The proposed new method shows the estimators are asymptotically unbiased and gives smaller mean squared error of the estimates. Compared with the existing method, the proposed new method does not require distributional assumptions for the underlying covariates. Simulation studies were conducted to evaluate the performance of the new method in realistic scenarios. We illustrate the use of the proposed method with a data set from Genetic and Inflammatory Markers of Sepsis study in which interlekuin-10 was subject to LOD.