The Mizon–Richard Encompassing Test for the Cox and Aalen Additive Hazards Models

Summary .   The Cox hazards model ( Cox, 1972 , Journal of the Royal Statistical Society, Series B 34, 187–220) for survival data is routinely used in many applied fields, sometimes, however, with too little emphasis on the fit of the model. A useful alternative to the Cox model is the Aalen additive hazards model ( Aalen, 1980 , in Lecture Notes in Statistics-2 , 1–25) that can easily accommodate time changing covariate effects. It is of interest to decide which of the two models that are most appropriate to apply in a given application. This is a nontrivial problem as these two classes of models are nonnested except only for special cases. In this article we explore the Mizon–Richard encompassing test for this particular problem. It turns out that it corresponds to fitting of the Aalen model to the martingale residuals obtained from the Cox regression analysis. We also consider a variant of this method, which relates to the proportional excess model ( Martinussen and Scheike, 2002 , Biometrika 89, 283–298). Large sample properties of the suggested methods under the two rival models are derived. The finite-sample properties of the proposed procedures are assessed through a simulation study. The methods are further applied to the well-known primary biliary cirrhosis data set.