Abstract

Semi-competing risks data arise frequently in survival analysis when individuals are subject to two related events: a non-terminal event and a terminal event, where the occurrence of the terminal event prevents the subsequent observation of the non-terminal event, whereas the reverse is not true. This asymmetric dependence structure is particularly relevant in medical studies, including bone marrow transplant (BMT) data, where the onset of acute graft-versus-host disease (acute GVHD) may be associated with the time to death. Conventional survival models often analyze each event time separately and therefore may fail to capture the underlying dependence structure adequately, especially when the marginal distributions differ in form and the dependence is nonlinear. To address this limitation, the present study proposes a new joint distribution based on the Clayton copula combined with Gamma and Weibull marginal for modeling semi-competing risks data. Specifically, the non-terminal event time T₁​is assumed to follow a Gamma distribution, while the terminal event time T₂​is assumed to follow a Weibull distribution, and their dependence is modeled through the Clayton copula. The study derives the main statistical properties of the proposed Clayton Copula Gamma–Weibull model, including the joint cumulative distribution function, joint probability density function, survival function, and hazard function. Model parameters are estimated using the maximum likelihood method. The proposed model is then applied to bone marrow transplant data from the Semi Comp Risks package, involving (137) patients, where acute GVHD is treated as the non-terminal event and death as the terminal event. Goodness-of-fit results for the marginal distributions indicate that the Gamma distribution provides an adequate fit for T₁​, while the Weibull distribution adequately fits T₂​. Furthermore, comparison among alternative copula-based specifications shows that the proposed Gamma–Weibull Clayton copula model achieves the best performance, with the highest log-likelihood and the lowest AIC and BIC values. The Cramér–von Mises copula goodness-of-fit test also supports the adequacy of the proposed model. The results demonstrate that the Clayton copula effectively captures positive lower-tail dependence between acute GVHD occurrence and death time, providing a more realistic representation of the temporal association between the two events. In addition, the close agreement between theoretical and maximum likelihood estimated survival values confirms the stability and practical suitability of the model. Overall, the proposed Clayton Copula Gamma–Weibull distribution offers a flexible and efficient framework for analyzing semi-competing risks data and provides a valuable statistical tool for studying complex dependent event-time outcomes in medical applications.