Heterogeneity and varying effect in hazards regression

In the analysis of survival data, when nonproportional hazards are encountered, the Cox model is often extended to allow for a time-dependent effect by accommodating a varying coefficient. This extension, however, cannot resolve the nonproportionality caused by heterogeneity. In contrast, the heteroscedastic hazards regression (HHR) model is capable of modeling heterogeneity and thus can be applied when dealing with nonproportional hazards. In this paper, we study the application of the HHR model possibly equipped with varying coefficients. An LRR (logarithm of relative risk) plot is suggested when investigating the need to impose varying coefficients. Constancy and degeneration in the plot are used as diagnostic criteria. For the HHR model, a ‘piecewise effect’ (PE) analysis and an ‘average effect’ (AE) analysis are introduced. For the PE setting, we propose a score-type test for covariate-specific varying coefficients. The Stanford Heart Transplant data are analyzed for illustration. In the case of degeneration being destroyed by a polynomial covariate, piecewise constancy and/or monotonicity of the LRRs is considered as an alternative criterion based on the PE analysis. Finally, under the framework of the varying-coefficient HHR model, the meanings of the PE and AE analyses, along with their dynamic interpretation, are discussed.