On semi-competing risks data

We consider a variation of the competing risks problem in which a terminal event censors a non-terminal event, but not vice versa. The joint distribution of the events is formulated via a gamma frailty model in the upper wedge where data are observable (Day et al., 1997), with the marginal distributions unspecified. An estimator for the association parameter is obtained from a concordance estimating function. A novel plug-in estimator for the marginal distribution of the non-terminal event is shown to be uniformly consistent and to converge weakly to a Gaussian process. The assumptions on the joint distribution outside the upper wedge are weaker than those usually made in competing risks analyses. Simulations demonstrate that the methods work well with practical sample sizes. The proposals are illustrated with data on morbidity and mortality in leukaemia patients.