Martingale-based residuals for survival models

Abstract

SUMMARY Graphical methods based on the analysis of residuals are considered for the setting of the highly-used Cox (1972) regression model and for the Andersen-Gill (1982) generalization of that model. We start with a class of martingale-based residuals as proposed by Barlow & Prentice (1988). These residuals and/or their transforms are useful for investigating the functional form of a covariate, the proportional hazards assumption, the leverage of each subject upon the estimates of 13, and the lack of model fit to a given subject. 1 1. Model Consider a set of n independent subjects such that the counting process Ni {Ni(t), t } O} for the ith subject in the set indicates the number of observed events experienced over time t. The sample paths of the Ni are step functions with jumps of size +1 and with Ni(0) =0. We assume that the intensity function for Ni(t) is given by Yi(t)dA{t, Zi(t)} = Yi(t) eP'Z(t) dAO(t), (1) where Yi(t) is a 0-1 process indicating whether the ith subject is a risk at time t, 13 is a vector of regression coefficients, Zi(t) is a p dimensional vector of covariate processes, and Ao is the baseline cumulative hazard function. Several familar survival models fit into this framework. The Andersen & Gill (1982) generalization of the Cox (1972) model arises when AO(t) is completely unspecified. The further restriction that Yi(t) = 1 until the first event or censoring, and is 0 thereafter yields the Cox model. The parametric form AO(t) = t yields a Poisson model, or an exponential if restricted to a single event per subject, and AO(t) = tP a Weibull model. Our attention will focus primarily on the Andersen-Gill and Cox models; however, the