Semi-competing risks modeling for sequential non-terminal events with bathtub-shaped hazards

Motivated by clinical data from leukemia patients undergoing bone marrow transplantation, we focus on three potential post-transplant events: graft-versus-host disease (GVHD), leukemia relapse, and death. Using a semi-competing risks framework with sequential non-terminal events, we treat GVHD and leukemia relapse as two non-terminal events and death as the terminal event, allowing GVHD to progress to leukemia relapse prior to death. Modeling the time to these events is of great clinical importance. We propose a novel statistical approach that semi-competing model such data via a three-dimensional copula to account for dependence structure, while the marginal distribution of each event follows a bathtub-shaped hazard function. We separately discuss two scenarios: one in which the three events are independent, and another in which they are dependent. The unknown parameters within these functions are estimated using both pseudo-likelihood and Bayesian methods. In addition, we estimate the conditional survival and hazard functions for death, given the occurrence of one of the non-terminal events. Finally, we extend the model from two non-terminal events to K non-terminal events and provide the corresponding likelihood framework.