Characterizations of Competing Risks in Terms of Independent-Risks Proxy Models

In competing risks a failure time T and a cause C, one of p possible, are observed. A traditional representation is via a vector (T₁, ..., T_p) of latent failure times such that T = min(T₁, ..., T_p); C is defined by T = T_C in the basic situation of failure from a single cause. There are several results in the literature to the effect that a joint distribution for (T₁, ..., T_p), in which the T_j are independent, can always be constructed to yield any given bivariate distribution for (C, T). For this reason the prevailing wisdom is that independence cannot be assessed from competing risks data, not even with arbitrarily large sample sizes (e.g. Prentice et al., 1978). A result was given by Crowder (1996) which shows that, under certain circumstances, independence can be assessed. The various results will be drawn together and a complete characterization can now be given in terms of independent-risks proxy models.