Nonparametric inference on bivariate survival data with interval sampling: association estimation and testing

In many biomedical applications, interest focuses on the occurrence of two or more consecutive failure events and the relationship between event times, such as age of disease onset and residual lifetime. Bivariate survival data with interval sampling arise frequently when disease registries or surveillance systems collect data based on disease incidence occurring within a specific calendar time interval. The initial event is then retrospectively confirmed and the subsequent failure event may be observed during follow-up. In life history studies, the initial and two consecutive failure events could correspond to birth, disease onset and death. The statistical features and bias of observed data in relation to interval sampling were discussed by Zhu & Wang (2012). Here we propose nonparametric estimation of the association between bivariate failure times based on Kendall's tau for data collected with interval sampling. A nonparametric estimator is given, where the contribution of each comparable and orderable pair is weighted by the inverse of the associated selection probability. Analysis methods for bivariate survival data with interval sampling rely on the assumption of quasi-independence, i.e., that bivariate failure times and the time of the initial event are independent in the observable region. This paper develops a nonparametric test of quasi-independence based on a bivariate conditional Kendall's tau for such data. Simulation studies demonstrate that the association estimator and testing procedure perform well with moderate sample sizes. Illustrations with two real datasets are provided.