# Handling competing risks in randomized trials

Peter Austin and Jason Fine (of Fine & Gray fame) have just published a nice review article in Statistics in Medicine on handling competing risks in randomized trials. They reviewed RCTs published in four top medical journals in the last three months of 2015. Of the 40 trials found with time to event outcomes, Austin & Gray determined that 31 were potentially susceptible to competing risks.

Arguably the most important finding was that of these 31, 24 (77.4%) presented Kaplan-Meier curves (or their complement) to describe how the probability of failure for a given cause (failure type) varied with time. This is worrying, because as Austin and Fine describe (following many others before them, e.g. Andersen et al), this gives biased estimates of the cumulative incidence in the presence of competing risks. This bias occurs because once an individual fails from a competing event (e.g. death), by definition they cannot fail from the event of interest. The Kaplan-Meier approach censors such individuals when they fail from the competing risk, assuming that they are still at risk of failing from the event of interest even after they are censored, when in fact they are now at zero risk of failure from other causes by definition.

Among other issues, Austin & Fine discuss two broad approaches to regression modelling in the competing risks setting. One is to model the cause specific hazard function(s), which is often done using Cox models. A potentially confusing aspect here is that when modelling one of the cause specific hazard functions, individuals who fail from other causes are censored in the analysis at their time of failure from the competing cause. At first sight one may think from this that we are assuming that the competing risks are independent. This is however not the case, because the cause specific hazard function is by definition the hazard of failing from the cause of interest in those who have not yet failed (see this 1978 article by Prentice et al for a thorough exposition).

The second approach is to use a Fine & Gray model for the subdistribution hazard. This allows one to model how baseline covariates predict the cumulative incidence of failure for each cause (failure type). Austin & Gray suggest that it may be advisable to use both approaches, with the subdistribution hazard being useful for estimating the absolute incidence of the primary outcome (i.e. failure from one of the causes of interest) and how this varies with covariates. One issue, at least theoretically, with this advice, is that for typical model specifications in both approach, the two sets of models cannot simultaneously be correctly specified (again see Andersen et al on this point).

It is also worth pointing out that one can use cause specific hazard models (e.g. Cox models) to estimate cumulative incidence curves for different covariate values, as described by Andersen & Keiding (page 1079). The difficulty, as described by Andersen & Keiding, is that even if one assumes Cox models of the cause specific hazard models with simple forms, the implied effects of the covariates on the cumulative incidence functions do not have a simple form. Indeed, this was the motivation for the Fine & Gray model. Lastly, as noted by Andersen & Keiding and Andersen et al though, the subdistribution hazard ratios produced by the Fine & Gray model have a somewhat problematic interpretation, and there is a danger that analysts may confer on them an interpretation that is misplaced.