Competing risks (CR) data arise when an occurrence of an event precludes other type of events from being observed. Two broad classes of models for analyzing the CR data have been developed based on Cox’s proportional hazards (PH) models; one is to model the cause-specific hazard of the different event types (Prentice et al., 1978) and the other is to model the subhazard (i.e., the hazard function of a subdistribution) for the event of interest (Fine and Gray, 1999).
The frailty model, an extension of the PH model, is often used to model clustered survival data. Here, the frailty is an unobserved random effect in the hazard model and it is useful to model correlation within clusters and/or heterogeneity among clusters. However, some extension of the ordinary frailty model is required when there exist competing risks within a cluster (or center). Under competing risks, we consider the subhazard frailty models and cause-specific hazard frailty models. The inferences are presented based on hierarchical likelihood (or h-likelihood; Lee and Nelder, 1996; Ha, Lee and Song, 2001); the h-likelihood obviates the need for intractable integrations over the frailty terms, whereas marginal likelihood usually does not. We compare and discuss both modelling approaches using the CR time-to-event data from clustered clinical studies (e.g. multi-center clinical trials). Furthermore, we also discuss extensions of both models for semi-competing risks data.