Comparing nonnested Cox models

We derive the limiting distribution of the partial likelihood ratio under general conditions. The multiplicative hazards models being fitted may be nonnested and misspecified. The true model is not assumed to contain either model under consideration. The null hypothesis is that the models are equidistant in Kullback-Leibler metric applied to the rank likelihood. The statistic is consistent for the model which is closer to the truth. Its distribution depends on the unknown data-generating mechanism. A sequential testing procedure is proposed for nonnested comparisons which is valid regardless of the true model. This involves a novel statistic for the equality of the fitted models which is separate from the partial likelihood. The methodology has important applications in model assessment. Simulations and a real example demonstrate its utility in selecting the functional forms of covariates and relative risks.