Competing Risks: Modelling Crude Cumulative Incidence Functions

The clinical course of a disease is often characterized by the possible occurrence of several types of events, each one having a specific role for the evaluation of the therapeutical strategies. The event occurring as first is of particular interest, since it could be considered as ’’treatment failure’’ or ’’response to treatment’’. The measure of concern is the crude cumulative incidence, i.e. the probability of developing a specific event as first accounting for the competing action of the other events. A widespread approach to infer on this quantity, accounting for the effect of covariates, is the semi-parametric Cox regression model on the cause specific hazard. However, it has to be pointed out as the inference on the prognostic impact of a covariate on the cause specific hazard cannot be extended to the crude cumulative incidence. To face this issue, Fine and Gray (1999) observed as the crude cumulative incidence can be thought as the incidence associated to a quantity referred as subdistribution hazard. They proposed a semi-parametric regression model, accounting for the covariate effects. Despite the crude cumulative incidence is of interest in several clinical applications, Fine and Gray’s model has not been routinely applied in the medical literature. Aiming at promoting the application of this model, the present note emphasizes the differences between the Cox model on the cause specific hazard and Fine and Gray’s model on the subdistribution hazard, resorting to a standard probabilistic formalism. To enlighten the differences between the results of the two inferences, themodels are applied on two historical data sets; a carcinogenesis experiment on mice and a clinical trial on breast cancer patients.