Estimation and Inference of Quantile Regression for Survival Data Under Biased Sampling

ABSTRACT

Biased sampling occurs frequently in economics, epidemiology, and medical studies either by design or due to data collecting mechanism. Failing to take into account the sampling bias usually leads to incorrect inference. We propose a unified estimation procedure and a computationally fast resampling method to make statistical inference for quantile regression with survival data under general biased sampling schemes, including but not limited to the length-biased sampling, the case-cohort design, and variants thereof. We establish the uniform consistency and weak convergence of the proposed estimator as a process of the quantile level. We also investigate more efficient estimation using the generalized method of moments and derive the asymptotic normality. We further propose a new resampling method for inference, which differs from alternative procedures in that it does not require to repeatedly solve estimating equations. It is proved that the resampling method consistently estimates the asymptotic covariance matrix. The unified framework proposed in this article provides researchers and practitioners a convenient tool for analyzing data collected from various designs. Simulation studies and applications to real datasets are presented for illustration. Supplementary materials for this article are available online.

KEYWORDS:

• Case-cohort sampling
• Censored quantile regression
• Length-biased data
• Resampling
• Stratified case-cohort sampling
• Survival time

Supplementary Materials

The supplementary material contains the verification of Equation (11) and the proofs of main theorems.

Acknowledgments

The first two authors contributed equally to this work. The authors thank the editors, the associate editor and two anonymous referees for their constructive comments that led to substantial improvements.

Funding

Xu’s work was partially supported by IES Grant R305D160010 and NSA H98230-16-1-0299; Sit’s work was partially supported by ECS-24300514 and GRF-14317716; Wang’s work was partially supported by NSF DMS-1308960; Huang’s work was sponsored by National Institutes of Health grant 1R01CA193888. The authors also express gratitude to Professors Ian McDowell, Masoud Asgharian, and Christina Wolfson for kindly sharing the Canadian Study of Health and Aging (CSHA) data. The core study (CSHA) was funded by the National Health Research and Development Program (NHRDP) of Health Canada Project 6606-3954-MC(S). Additional funding was provided by Pfizer Canada Incorporated through the Medical Research Council/Pharmaceutical Manufacturers Association of Canada Health Activity Program, NHRDP Project 6603-1417-302(R), Bayer Incorporated, and the British Columbia Health Research Foundation Projects 38 (93-2) and 34 (96-1).