Residuals for relative risk regression

Abstract

SUMMARY Several possible definitions of residuals are given for relative risk regression with time-varying covariates. Each such residual has a representation as an estimator of a stochastic integral with respect to the martingale arising from a subject's failure time counting process. Previously proposed residuals for individual study subjects and for specific time points are shown to be special cases of this definition, as are previously derived regression diagnostics. An illustration and various generalizations are also given. able methods are required to detect various departures from modelling assumptions. Suitably defined residuals may play an important role in such identification. However, the nonparametric aspect of the model, the possibility that modelled regression variables may be varying with follow-up time and, most importantly, the usual presence of right censorship, implies that specialized residual definitions are required. The class of residuals considered here is most easily formulated using counting process notation for the failure time data. Let Ni(t), Yi(t) and Zi(t) represent, respectively, for the ith subject the values of counting, censoring and covariate processes at follow-up time t (i = 1, . . ., n), while {Ni(u), Yi(u), Zi(u); 0C u < t} specifies the corresponding histories for the ith subject prior to time t. Thus in a typical univariate failure time application, Ni, with right- continuous sample paths, will take value zero prior to the time of failure on the ith subject and value one thereafter, while Yi with left-continuous sample paths will take value one at times at which the ith subject is 'at risk' for an observed failure, and value zero otherwise. The counting process Ni can be uniquely decomposed so that for all (t, i)