  Time-to-event semi-competing risk endpoints may be correlated when both
events are occurring on the same individual. These events and the association
between them may also be influenced by individual characteristics. In this
paper, we propose copula survival models to estimate hazard ratios of
covariates on the non-terminal and terminal events, along with the effects of
covariates on the association between the two events. We use the Normal,
Clayton, Frank and Gumbel copulas to provide a variety of association
structures between the non-terminal and terminal events. We apply the proposed
methods to model semi-competing risks of graft failure and death for kidney
transplant patients. We find that copula survival models perform better than
the Cox proportional hazards model when estimating the non-terminal event
hazard ratio of covariates. We also find that the inclusion of covariates in
the association parameter of the copula models improves the estimation of the
hazard ratios.
