A Flexible Parametric Modelling Framework for Survival Analysis

We introduce a general, flexible, parametric survival modelling framework which encompasses key shapes of hazard function (constant, increasing, decreasing, up-then-down, down-then-up), various common survival distributions (log-logistic, Burr type XII, Weibull, Gompertz), and includes defective distributions (cure models). This generality is achieved using four distributional parameters: two scale-type parameters – which, respectively, relate to accelerated failure time (AFT) and proportional hazards (PH) modelling – and two shape parameters. Furthermore, we advocate “multi-parameter regression” whereby more than one distributional parameter depends on covariates – rather than the usual convention of having a single covariate-dependent (scale) parameter. This general formulation unifies the most popular survival models, allowing us to consider the practical value of possible modelling choices. In particular, we suggest introducing covariates through just one or other of the two scale parameters (covering AFT and PH models), and through a “power” shape parameter (covering more complex non-AFT/non-PH effects); the other shape parameter remains covariate-independent, and handles automatic selection of the baseline distribution. We explore inferential issues and compare with alternative models through various simulation studies, with particular focus on evidence concerning the need, or otherwise, to include both AFT and PH parameters. We illustrate the efficacy of our modelling framework using data from lung cancer, melanoma, and kidney function studies. Censoring is accommodated throughout.