A Graphical Method for Assessing Goodness of Fit in Cox's Proportional Hazards Model

Abstract Suggested here is a simple graphical method for studying the goodness of fit in Cox's regression model for survival data. The method is easy to use, as it does not require the estimation of alternative models and only involves quantities similar to those already appearing in the partial likelihood expression that is needed in the parameter estimation. The rationale behind the graphs is very intuitive: They make a direct comparison between observed and expected failure frequencies, as estimated from the model. In a correctly specified model one anticipates an approximate balance between such frequencies; otherwise there will typically be groups of individuals for which the expected frequencies are systematically too high or too low to match with the data, and this shows in the graphs introduced here. In the concrete applications of the method the individuals are stratified in a way that depends on what aspect of the model is being checked against data. There is always one graph for each stratum. Simulated and real data are used to illustrate the method. In the simulations two types of defect that can come up in a Cox's model are considered: (a) an influential covariate has been deleted from the model, and (b) a common baseline hazard for all individuals has been assumed in a case in which the individuals should be stratified according to baseline hazard. Serious defects in the model are relatively easy to detect from the diagnostic graphs. As concrete applications of the method, studied briefly are the fitting of Cox's model to the well-known Stanford heart transplant data and to a data set describing the survival of malignant melanoma patients after operation. The article concludes with some general observations concerning the randomness in the graphs.