Bounds on the covariate-time transformation for competing-risks survival analysis

A fundamental problem with the latent-time framework in competing risks is the lack of identifiability of the joint distribution. Given observed covariates along with assumptions as to the form of their effect, then identifiability may obtain. However it is difficult to check any assumptions about form since a more general model may lose identifiability. This paper considers a general framework for modelling the effect of covariates, with the single assumption that the copula dependency structure of the latent times is invariant to the covariates. This framework consists of a set of functions: the covariate-time transformations. The main result produces bounds on these functions, which are derived solely from the crude incidence functions. These bounds are a useful model checking tool when considering the covariate-time transformation resulting from any particular set of further assumptions. An example is given where the widely-used assumption of independent competing risks is checked.