Quantile Regression Adjusting for Dependent Censoring from Semi-Competing Risks

Abstract

In this work, we study quantile regression when the response is an event time subject to potentially dependent censoring. We consider the semi-competing risks setting, where time to censoring remains observable after the occurrence of the event of interest. While such a scenario frequently arises in biomedical studies, most of current quantile regression methods for censored data are not applicable because they generally require the censoring time and the event time be independent. By imposing rather mild assumptions on the association structure between the time-to-event response and the censoring time variable, we propose quantile regression procedures, which allow us to garner a comprehensive view of the covariate effects on the event time outcome as well as to examine the informativeness of censoring. An efficient and stable algorithm is provided for implementing the new method. We establish the asymptotic properties of the resulting estimators including uniform consistency and weak convergence. The theoretical development may serve as a useful template for addressing estimating settings that involve stochastic integrals. Extensive simulation studies suggest that the proposed method performs well with moderate sample sizes. We illustrate the practical utility of our proposals through an application to a bone marrow transplant trial.

Keywords: Copula; Dependent censoring; Quantile regression; Semi-competing risks; Stochastic integral equation.

Fig. 1

Assessment of the robustness of $\widehat{\beta }\left(\tau \right)$ with data generated from set-up S2.C (the first row) and from set-up S2.F (the second row). Solid lines represent true regression quantiles, dashed lines represent empirical averages of naive estimates, dotted lines represent empirical averages of the proposed estimates based on Clayton's copula, and dashed dotted lines represent empirical averages of the proposed estimates based on Frank's copula.

Fig. 2

Estimated regression coefficients for the BMT dataset, where the x-axis represents τ and the y-axis represents the regression coefficients. Bold solid line represents the proposed regression coefficients as a function of τ, dashed line represents the estimator based on Peng and Huang's method, and dotted lines represent the percentile-based 95% intervals.

Fig. 3

Estimated quantiles of time to the GVHD endpoint, where x-axis represents quantiles, and y-axis represents predicted quantiles in months. The solid line and dashed line correspond to the proposed estimator and Peng and Huang (2008)'s estimator, respectively.