Iterated residuals and time-varying covariate effects in Cox regression

The Cox proportional hazards model, which is widely used for the analysis of treatment and prognostic effects with censored survival data, makes the assumption that the hazard ratio is constant over time. Nonparametric estimators have been developed for an extended model in which the hazard ratio is allowed to change over time. Estimators based on residuals are appealing as they are easy to use and relate in a simple way to the more restricted Cox model estimator. After fitting a Cox model and calculating the residuals, one can obtain a crude estimate of the time-varying coefficients by adding a smooth of the residuals to the initial (constant) estimate. Treating the crude estimate as the fit, one can re-estimate the residuals. Iteration leads to consistent estimation of the nonparametric time-varying coefficients. This approach leads to clear guidelines for residual analysis in applications. The results are illustrated by an analysis of the Medical Research Council's myeloma trials, and by simulation.