Generic solution of the heterogeneity-induced competing risk problem in survival analysis

Most papers implicitly assume competing risks to be induced by residual cohort heterogeneity, i.e. heterogeneity that is not captured by the recorded covariates. Based on this observation we develop a generic statistical description of competing risks that unifies the main schools of thought. Assuming heterogeneity-induced competing risks is much weaker than assuming risk independence. However, we show that it still imposes sufficient constraints to solve the competing risk problem, and derive exact formulae for decontaminated primary risk hazard rates and cause-specific survival functions. The canonical description is in terms of a cohort's covariate-constrained functional distribution of individual hazard rates of all risks. Assuming proportional hazards at the level of individuals leads to a natural parametrisation of this distribution, from which Cox regression, frailty and random effects models, and latent class models can all be recovered in special limits, and which also generates parametrised cumulative incidence functions (the language of Fine and Gray). We demonstrate with synthetic data how the generic method can uncover and map a cohort's substructure, if such substructure exists, and remove heterogeneity-induced false protectivity and false exposure effects. Application to real survival data from the ULSAM study, with prostate cancer as the primary risk, is found to give plausible alternative explanations for previous counter-intuitive inferences.