Cure Rate Quantile Regression for Censored Data With a Survival Fraction

Abstract

Censored quantile regression offers a valuable complement to the traditional Cox proportional hazards model for survival analysis. Survival times tend to be right-skewed, particularly when there exists a substantial fraction of long-term survivors who are either cured or immune to the event of interest. For survival data with a cure possibility, we propose cure rate quantile regression under the common censoring scheme that survival times and censoring times are conditionally independent given the covariates. In a mixture formulation, we apply censored quantile regression to model the survival times of susceptible subjects and logistic regression to model the indicators of whether patients are susceptible. We develop two estimation methods using martingale-based equations: One approach fully uses all regression quantiles by iterating estimation between the cure rate and quantile regression parameters; and the other separates the two via a nonparametric kernel smoothing estimator. We establish the uniform consistency and weak convergence properties for the estimators obtained from both methods. The proposed model is evaluated through extensive simulation studies and illustrated with a bone marrow transplantation data example. Technical proofs of key theorems are given in Appendices A, B, and C, while those of lemmas and additional simulation studies on model misspecification and comparisons with other models are provided in the online Supplementary Materials A and B.

KEY WORDS:

• Cure rate model
• Empirical process
• Long-term survivor
• Martingale
• Random censoring
• Regression quantile
• Survival analysis
• Volterra integral equation

Acknowledgments

We thank the associate editor and three anonymous referees for their careful reading and many insightful suggestions, which strengthened the work immensely. Wu’s research was supported in part by National Natural Science Foundation of China grant 11201350, and Yin’s research was supported in part by a grant (784010) from the Research Grants Council of Hong Kong.