Censored Interquantile Regression Model with Time-Dependent Covariates

Abstract

Conventionally, censored quantile regression stipulates a specific, pointwise conditional quantile of the survival time given covariates. Despite its model flexibility and straightforward interpretation, the pointwise formulation oftentimes yields rather unstable estimates across neighboring quantile levels with large variances. In view of this phenomenon, we propose a new class of quantile-based regression models with time-dependent covariates for censored data. The models proposed aim to capture the relationship between the failure time and the covariate processes of a target population that falls within a specific quantile bracket. The pooling of information within a homogeneous neighborhood facilitates more efficient estimates hence, more consistent conclusion on statistical significances of the variables concerned. This new formulation can also be regarded as a generalization of the accelerated failure time model for survival data in the sense that it relaxes the assumption of global homogeneity for the error at all quantile levels. By introducing a class of weighted rank-based estimation procedure, our framework allows a quantile-based inference on the covariate effect with a less restrictive set of assumptions. Numerical studies demonstrate that the proposed estimator outperforms existing alternatives under various settings in terms of smaller empirical biases and standard deviations. A perturbation-based resampling method is also developed to reconcile the asymptotic distribution of the parameter estimates. Finally, consistency and weak convergence of the proposed estimator are established via empirical process theory. Supplementary materials for this article are available online.

KEYWORDS:

• Accelerated failure time model
• Censored data
• Empirical process
• Quantile inference
• Survival analysis
• Time-dependent covariate

Supplementary Materials

The supplementary materials contain additional details about the asymptotic results and the simulation studies. Section A to C of the appendix present respectively the proofs of Theorems 1–3 discussed in Sections 3 and 4. Section D summarizes the numerical results for the alternative iterative algorithm discussed in Remark 1 of Section 5.

Acknowledgments

The authors would like to thank the editor, the associate editor and the anonymous referees for their constructive comments and valuable suggestions with which the manuscript have been substantially improved.

Additional information

Funding

Sit’s work was partially supported by Hong Kong Research Grant Council RGC-14301618, RGC-14301920, and RGC-14307221.