Quantile regression for censored data with a cure fraction

Quantile regression is emerging as a valuable statistical tool for exploring complete information on the exposure-outcome relationship. This paper investigates a novel inference procedure for quantile regression models to analyze survival data with a cure fraction. A two-component mixture cure model is used, in which the cure fraction follows a logistic model and the uncured individuals follow a linear quantile regression model. Our main interest lies in the estimation of regression coefficients in the latter model. This is achieved by minimizing a loss function that has three building blocks: (1) the check function, commonly used in quantile regression, (2) a first weight function that redistributes the weight of censored observations, and (3) a second weight function that accounts for the latent cure status. The minimization of this loss function leads to a computationally simple estimation procedure for quantile regression parameters. The approach of dimension reduction is employed to circumvent the curse of dimensionality. Asymptotic properties of the resulting estimators are established, including the consistency and weak convergence. Simulation studies, including homogeneous and heterogeneous error cases, demonstrate the advantageous empirical performance of the proposed method. The analysis of a breast cancer study illustrates the practical utility of our cure rate quantile regression.