Quantile Regression Adjusting for Dependent Censoring from Semi-Competing Risks

Quantile regression under dependent censoring with unknown association

The study of survival data often requires taking proper care of the censoring mechanism that prohibits complete observation of the data. Under right censoring, only the first occurring event is observed: either the event of interest, or a competing event like withdrawal of a subject from the study. The corresponding identifiability difficulties led many authors to imposing (conditional) independence or a fully known dependence between survival and censoring times, both of which are not always realistic. However, recent results in survival literature showed that parametric copula models allow identification of all model parameters, including the association parameter, under appropriately chosen marginal distributions. The present paper is the first one to apply such models in a quantile regression context, hence benefiting from its well-known advantages in terms of e.g. robustness and richer inference results. The parametric copula is supplemented with a likewise parametric, yet flexible, enriched asymmetric Laplace distribution for the survival times conditional on the covariates. Its asymmetric Laplace basis provides its close connection to quantiles, while the extension with Laguerre orthogonal polynomials ensures sufficient flexibility for increasing polynomial degrees. The distributional flavour of the quantile regression presented, comes with advantages of both theoretical and computational nature. All model parameters are proven to be identifiable, consistent, and asymptotically normal. Finally, performance of the model and of the proposed estimation procedure is assessed through extensive simulation studies as well as an application on liver transplant data.

Modeling semi-competing risks data as a longitudinal bivariate process

As individuals age, death is a competing risk for Alzheimer's disease (AD) but the reverse is not the case. As such, studies of AD can be placed within the semi-competing risks framework. Central to semi-competing risks, and in contrast to standard competing risks , is that one can learn about the dependence structure between the two events. To-date, however, most methods for semi-competing risks treat dependence as a nuisance and not a potential source of new clinical knowledge. We propose a novel regression-based framework that views the two time-to-event outcomes through the lens of a longitudinal bivariate process on a partition of the time scales of the two events. A key innovation of the framework is that dependence is represented in two distinct forms, local and global dependence, both of which have intuitive clinical interpretations. Estimation and inference are performed via penalized maximum likelihood, and can accommodate right censoring, left truncation, and time-varying covariates. An important consequence of the partitioning of the time scale is that an ambiguity regarding the specific form of the likelihood contribution may arise; a strategy for sensitivity analyses regarding this issue is described. The framework is then used to investigate the role of gender and having ≥1 apolipoprotein E (APOE) ε4 allele on the joint risk of AD and death using data from the Adult Changes in Thought study.

Quantile regression for censored data in haematopoietic cell transplant research

SERIES EDITORS' NOTE

One of the most important endpoints in haematopoietic cell transplant research is survival. A common objective is to interrogate which, if any, co-variates correlate with these endpoints. The most common statistical approach uses the Cox proportional hazards model. However, there are several problems and limitations of using this model including assumptions of proportional hazards and homogenous effects. In contrast, results of transplant studies often show non-proportional hazards because of early transplant-related mortality such that there is a survival disadvantage to transplants early on followed by a benefit. Even when a transplant proves better than a comparator not all transplant recipients benefit equally and some may be disadvantaged. Also, the favourable or unfavourable impact of a co-variate may vary in different time intervals. The accelerated failure time model which directly evaluates the association between survival and co-variates has similar limitations. Also, these models confer only a static view of the treatment effect. Several articles in our statistics series such as that by Zhen-Huan Hu and us (Bone Marrow Transplant. 2021 Aug 19. doi: 10.1038/s41409-021-01435-2), by Zhen-Huan Hu, Hai-Lin Wang and us and forthcoming articles by Megan Othus and by Liesbeth C. de Wreede, Johannes Schetelig and Hein Putter discuss issues in proper analyses of survival data from transplant studies including observational databases and randomized controlled trials. Are there better alternatives? A new popular model is quantile regression. In this typescript Bo Wei concisely introduce the quantile regression model for right censored data. He uses data from a Center for International Blood and Marrow Transplant Research (CIBMTR) registry study to show how to use the quantile regression and interpret the results. He also discusses use of quantile regression in complex survival analyses such as competing risk data or non-compliant data. Quantile regression is a natural, powerful approach for analyzing censored data with heterogenous co-variate effects. It has advantages compared with other survival models in depicting the dynamic association between survival outcome and co-variates. It can be applied to other transplant outcomes such as cumulative incidence of relapse, event-free and relapse-free survivals. There is an equation, but only one. Remember: The only thing to fear is fear itself (FDR). Please stick with it and you will be rewarded.Robert Peter Gale MD, PhD, DSc(hc), FACP, FRCP, FRCPI(hon), LHD, DPS, Mei-Jie Zhang PhD.

Quantile Regression for Survival Data

Quantile regression offers a useful alternative strategy for analyzing survival data. Compared to traditional survival analysis methods, quantile regression allows for comprehensive and flexible evaluations of covariate effects on a survival outcome of interest, while providing simple physical interpretations on the time scale. Moreover, many quantile regression methods enjoy easy and stable computation. These appealing features make quantile regression a valuable practical tool for delivering in-depth analyses of survival data. In this paper, I review a comprehensive set of statistical methods for performing quantile regression with different types of survival data. This review covers various survival scenarios, including randomly censored data, data subject to left truncation or censoring, competing risks and semi-competing risks data, and recurrent events data. Two real examples are presented to illustrate the utility of quantile regression for practical survival data analyses.

Smoothed quantile regression analysis of competing risks

Censored quantile regression models, which offer great flexibility in assessing covariate effects on event times, have attracted considerable research interest. In this study, we consider flexible estimation and inference procedures for competing risks quantile regression, which not only provides meaningful interpretations by using cumulative incidence quantiles but also extends the conventional accelerated failure time model by relaxing some of the stringent model assumptions, such as global linearity and unconditional independence. Current method for censored quantile regressions often involves the minimization of the L₁ -type convex function or solving the nonsmoothed estimating equations. This approach could lead to multiple roots in practical settings, particularly with multiple covariates. Moreover, variance estimation involves an unknown error distribution and most methods rely on computationally intensive resampling techniques such as bootstrapping. We consider the induced smoothing procedure for censored quantile regressions to the competing risks setting. The proposed procedure permits the fast and accurate computation of quantile regression parameter estimates and standard variances by using conventional numerical methods such as the Newton-Raphson algorithm. Numerical studies show that the proposed estimators perform well and the resulting inference is reliable in practical settings. The method is finally applied to data from a soft tissue sarcoma study.

Quantile association for bivariate survival data

Bivariate survival data arise frequently in familial association studies of chronic disease onset, as well as in clinical trials and observational studies with multiple time to event endpoints. The association between two event times is often scientifically important. In this article, we examine the association via a novel quantile association measure, which describes the dynamic association as a function of the quantile levels. The quantile association measure is free of marginal distributions, allowing direct evaluation of the underlying association pattern at different locations of the event times. We propose a nonparametric estimator for quantile association, as well as a semiparametric estimator that is superior in smoothness and efficiency. The proposed methods possess desirable asymptotic properties including uniform consistency and root-n convergence. They demonstrate satisfactory numerical performances under a range of dependence structures. An application of our methods suggests interesting association patterns between time to myocardial infarction and time to stroke in an atherosclerosis study.

A new flexible dependence measure for semi-competing risks

Semi-competing risks data are often encountered in chronic disease follow-up studies that record both nonterminal events (e.g., disease landmark events) and terminal events (e.g., death). Studying the relationship between the nonterminal event and the terminal event can provide insightful information on disease progression. In this article, we propose a new sensible dependence measure tailored to addressing such an interest. We develop a nonparametric estimator, which is general enough to handle both independent right censoring and left truncation. Our strategy of connecting the new dependence measure with quantile regression enables a natural extension to adjust for covariates with minor additional assumptions imposed. We establish the asymptotic properties of the proposed estimators and develop inferences accordingly. Simulation studies suggest good finite-sample performance of the proposed methods. Our proposals are illustrated via an application to Denmark diabetes registry data.