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In disease registries, bivariate survival data are typically collected under interval sampling. It refers to a situation when entry into a registry is at the time of the first failure event (i.e., HIV infection) within a calendar time window. For all the cases in the registry, time of the initiating event (i.e., birth) is retrospectively identified, and subsequently the second failure event (i.e., death) is observed during follow-up. In this paper we discuss how interval sampling introduces bias into the data. Given the sampling design that the first event occurs within a specific time interval, the first failure time is doubly truncated, and the second failure time is possibly informatively right censored. Consider semi-stationary condition that the disease progression is independent of when the initiating event occurs. Under this condition, this paper adopts copula models to assess association between the bivariate survival times with interval sampling. We first obtain bias-corrected estimators of marginal survival functions, and estimate association parameter of copula model by a two-stage procedure. In the second part of the work, covariates are incorporated into the survival distributions via the proportional hazards models. Inference of the association measure in copula model is established, where the association is allowed to depend on covariates. Asymptotic properties of proposed estimators are established, and finite sample performance is evaluated by simulation studies. The method is applied to a community-based AIDS study in Rakai to investigate dependence between age at infection and residual lifetime without and with adjustment for HIV subtype.

A Semi-stationary Copula Model Approach for Bivariate Survival Data with Interval Sampling