Semiparametric model for bivariate survival data subject to biased sampling

To understand better the relationship between patient characteristics and their residual survival after an intermediate event such as the local recurrence of cancer, it is of interest to identify patients with the intermediate event and then to analyse their residual survival data. One challenge in analysing such data is that the observed residual survival times tend to be longer than those in the target population, since patients who die before experiencing the intermediate event are excluded from the cohort identified. We propose to model jointly the ordered bivariate survival data by using a copula model and appropriately adjusting for the sampling bias. We develop an estimating procedure to estimate simultaneously the parameters for the marginal survival functions and the association parameter in the copula model, and we use a two‐stage expectation–maximization algorithm. Using empirical process theory, we prove that the estimators have strong consistency and asymptotic normality. We conduct simulation studies to evaluate the finite sample performance of the method proposed. We apply the method to two cohort studies to evaluate the association between patient characteristics and residual survival.