A Nested Copula Model for Recurrent Gap Times With a Dependent Terminal Event

In many clinical trials, subjects often experience recurrent events, and some individuals may have their follow-up terminated by a dependent terminal event, such as death. We are particularly interested in the effect of covariates on recurrent gap times and correlations among recurrent gap times and the terminal event. In this article, we propose a novel nested copula model with a two-layer structure: the inner layer uses an Archimedean copula to capture the internal correlations among recurrent gap times, while the outer layer employs a general bivariate copula to model the dependence between recurrent events and the terminal event. The nested copula framework serves as a copula-based counterpart to the frailty model, providing improved interpretability and enabling straightforward extensions to accommodate heterogeneous dependence structures. Both parametric and semiparametric methods are developed for parameter estimation, and the resulting estimators are shown to be consistent and asymptotically normal. In addition, a likelihood-based copula selection procedure is provided to select a suitable model for a given dataset. Extensive simulation studies are conducted to investigate the finite sample properties of the proposed methods. Finally, an application to colorectal cancer data from a clinical study conducted at Bellvitge Hospital is presented.