Evaluation of long-term survival: use of diagnostics and robust estimators with Cox's proportional hazards model

Abstract

We consider methodological problems in evaluating long-term survival in clinical trials. In particular we examine the use of several methods that extend the basic Cox regression analysis. In the presence of a long term observation, the proportional hazard (PH) assumption may easily be violated and a few long term survivors may have a large effect on parameter estimates. We consider both model selection and robust estimation in a data set of 474 ovarian cancer patients enrolled in a clinical trial and followed for between 7 and 12 years after randomization. Two diagnostic plots for assessing goodness-of-fit are introduced. One shows the variation in time of parameter estimates and is an alternative to PH checking based on time-dependent covariates. The other takes advantage of the martingale residual process in time to represent the lack of fit with a metric of the type 'observed minus expected' number of events. Robust estimation is carried out by maximizing a weighted partial likelihood which downweights the contribution to estimation of influential observations. This type of complementary analysis of long-term results of clinical studies is useful in assessing the soundness of the conclusions on treatment effect. In the example analysed here, the difference in survival between treatments was mostly confined to those individuals who survived at least two years beyond randomization.