Inference for the Cox Model under Proportional and Non-Proportional Hazards

John O'Quigley and Ronghui Xu

We consider some new ideas concerning inference in the proportional hazards model and the broader non proportional hazards model. Recall that bivariate regression models generally focus in an explicit way on the conditional distribution of one variable given the other. There are always two ways of doing this but, typically, it is most natural to condition on the explanatory or design variable. However, as far as concerns inference for proportional hazards regression, it is in fact more natural to condition the other way around. This simple observation leads to many results. Among these are a natural estimator of average effects under non-proportional hazards, a straightforward and natural way to assess fit and a new estimator of the survivorship function conditonal on some set of covariates. These ideas all stem from a main theorem which we describe below. All the ideas for the bivariate case, i.e. a single explanatory variable in the model, generalize readily to the multivariate case.