Nonparametric tests of association between survival time and continuously measured covariates: the logit-rank and associated procedures.

O'Brien's logit-rank procedure (1978, Biometrics 34, 243-250) is shown to arise as a score test based on the partial likelihood for a proportional hazards model provided the covariate structure is suitably defined. Within this framework the asymptotic properties claimed by O'Brien can be readily deduced and can be seen to be valid under a more general model of censoring than that considered in his paper. More important, perhaps, it is now possible to make a more natural and interpretable generalization to the multiple regression problem than that suggested by O'Brien as a means of accounting for the effects of nuisance covariates. This can be achieved either by modelling or stratification. The proportional hazards framework is also helpful in that it enables us to recognize the logit-rank procedure as being one member of a class of contending procedures. One consequence of this is that the relative efficiencies of any two procedures can be readily evaluated using the results of Lagakos (1988, Biometrika 75, 156-160). Our own evaluations suggest that, for non-time-dependent covariates, a simplification of the logit-rank procedure, leading to considerable reduction in computational complexity, is to be preferred to the procedure originally outlined by O'Brien.