Partial orders with respect to continuous covariates and tests for the proportional hazards model

Several omnibus tests of the proportional hazards assumption have been proposed in the literature. In the two-sample case, tests have also been developed against ordered alternatives like monotone hazard ratio and monotone ratio of cumulative hazards. Here we propose a natural extension of these partial orders to the case of continuous covariates. The work is motivated by applications in biomedicine and economics where covariate effects often decay over lifetime. We develop tests for the proportional hazards assumption against ordered alternatives and propose a graphical method to identify the nature of departures from proportionality. The proposed tests do not make restrictive assumptions on the underlying regression model, and are applicable in the presence of multiple covariates and frailty. Small sample performance and applications to real data highlight the usefulness of the framework and methodology.