Nonparametric Inference for Copulas and Measures of Dependence Under Length-Biased Sampling and Informative Censoring

Abstract

Length-biased data are often encountered in cross-sectional surveys and prevalent-cohort studies on disease durations. Under length-biased sampling subjects with longer disease durations have greater chance to be observed. As a result, covariate values linked to the longer survivors are favored by the sampling mechanism. When the sampled durations are also subject to right censoring, the censoring is informative. Modeling dependence structure without adjusting for these issues leads to biased results. In this article, we consider copulas for modeling dependence when the collected data are length-biased and account for both informative censoring and covariate bias that are naturally linked to length-biased sampling. We address nonparametric estimation of the bivariate distribution, copula function and its density, and Kendall and Spearman measures for right-censored length-biased data. The proposed estimator for the bivariate cdf is a Hadamard-differentiable functional of two MLEs (Kaplan–Meier and empirical cdf) and inherits their efficiency. Based on this estimator, we devise two estimators for copula function and a local-polynomial estimator for copula density that accounts for boundary bias. The limiting processes of the estimators are established by deriving their iid representations. As a by-product, we establish the oscillation behavior of the bivariate cdf estimator. In addition, we introduce estimators for Kendall and Spearman measures and study their weak convergence. The proposed method is applied to analyze a set of right-censored length-biased data on survival with dementia, collected as part of a nationwide study in Canada.

Keywords:

• Copulas
• Length-biased sampling
• Covariate(s) bias
• Hadamard-differentiable functional
• iid representation of copula
• Local-linear kernel estimation
• Measures of dependence

Acknowledgments

The authors thank Prof.Hongyu Zhao (the editor), the associate editor, and three referees for their constructive comments. The dementia data reported in this article were collected as part of the Canadian Study of Health and Aging. The core study was funded by the Senior's Independence Research Program, through the National Health Research and Development Program (NHRDP) of Health Canada [project no. 6606-3954-MC(S)]. Additional funding was provided by Pfizer Canada Incorporated through the Medical Research Council/Pharmaceutical Manufacturers Association of CanadaHealth Activity Program, NHRDP [project no. 6603-1417-302(R)], Bayer Incorporated and the British Columbia Health Research Foundation [project no. 38(93-2) and no. 34(96-1)]. The study was coordinated through the University of Ottawa and the Division of Aging and Seniors, Health Canada. The authors thank Professor Christina Wolfson for providing the data, and Prof. Masoud Asgharian for his comments.

Additional information

Funding

This work has been supported by Canssi and Statlab-CRM (Y. Rabhi), and FQRNT andNSERC of Canada (T. Bouezmarni).