Z ‐residuals diagnostics for Cox models: Functional‐form misspecification versus nonproportional hazards

Abstract

Primary biliary cholangitis (PBC) is a progressive autoimmune cholestatic disease for which modelling death, with transplantation as a competing event, is clinically important. Cox proportional hazards (PH) models are widely used, but standard diagnostics detect lack of fit without distinguishing non‐PH from misspecified covariate functions. We address this limitation using Z‐residuals. Comparing the Shapiro–Wilk test on Z‐residuals with an ANOVA test grouped by the linear predictor helps distinguish time‐varying effects from functional‐form misspecification; covariate‐specific grouped diagnostics localise the source of misfit. In the analysis of a PBC dataset, these diagnostics identified bilirubin functional‐form misspecification, not non‐proportional hazards; log‐transforming bilirubin improved model fit. Simulations show that Z‐residual‐based tests detect non‐PH comparably to score tests while better distinguishing them from functional‐form misspecification.