Smooth Goodness-of-Fit Tests for the Baseline Hazard in Cox's Proportional Hazards Model

Abstract

For the Cox proportional hazards model, the goodness-of-fit problem of testing whether the baseline hazard rate function λ(·) equals some specified hazard rate function λ0(·) in the presence of incomplete data is considered. Extending Neyman's idea, the goodness-of-fit tests are score tests obtained by embedding λ(·) in a larger family of hazard rate functions developed through smooth and possibly random transformations of λ0(·). But whereas their smooth alternatives were formulated through the density function, my formulation is through the hazard rate function, which is more natural in failure time models where model specification is now done through hazard rates. Furthermore, my smooth alternatives are allowed to be data driven or dynamic. Through this machinery, classes of omnibus, as well as directional goodness-of-fit tests are obtained. Moreover, this machinery provides a theoretically sound method for combining varied directional goodness-of-fit tests into omnibus tests. Some of the resulting tests are specialized to the randomly censored model. Finite and asymptotic properties of the resulting tests are presented. The asymptotic results shed light into properties of generalized residuals typically used in model validation and diagnostics. The theoretical results indicate the adjustments that need to be made to properly use these residuals for goodness of fit and model validation. Two medical datasets are reanalyzed using the proposed procedures.