Quantile calculus and censored regression

TL;DR

This paper introduces a novel quantile calculus framework for censored regression that overcomes limitations of existing methods, providing a reliable, efficient estimation procedure applicable to survival analysis with censored data.

Contribution

It develops a general quantile calculus on the probability scale and proposes a new estimation method with a specialized algorithm for censored quantile regression.

Findings

The estimator is uniformly consistent and converges to a Gaussian process.

Simulation studies demonstrate good statistical and computational performance.

The method effectively applies to real clinical survival data.

Abstract

Quantile regression has been advocated in survival analysis to assess evolving covariate effects. However, challenges arise when the censoring time is not always observed and may be covariate-dependent, particularly in the presence of continuously-distributed covariates. In spite of several recent advances, existing methods either involve algorithmic complications or impose a probability grid. The former leads to difficulties in the implementation and asymptotics, whereas the latter introduces undesirable grid dependence. To resolve these issues, we develop fundamental and general quantile calculus on cumulative probability scale in this article, upon recognizing that probability and time scales do not always have a one-to-one mapping given a survival distribution. These results give rise to a novel estimation procedure for censored quantile regression, based on estimating integral equations. A numerically reliable and efficient Progressive Localized Minimization (PLMIN) algorithm is proposed for the computation. This procedure reduces exactly to the Kaplan--Meier method in the $k$-sample problem, and to standard uncensored quantile regression in the absence of censoring. Under regularity conditions, the proposed quantile coefficient estimator is uniformly consistent and converges weakly to a Gaussian process. Simulations show good statistical and algorithmic performance. The proposal is illustrated in the application to a clinical study.