Semiparametric estimation of an accelerated failure time model with time-dependent covariates

Abstract

SUMMARY A class of semiparametric accelerated failure time models is introduced that are useful for modelling the relationship of survival distributions to time dependent covariates. A class of semiparametric rank estimators is derived for the parameters in this model when the survival data are right censored. These estimates are shown to be consistent, and asymptotically normal with variances that can be consistently estimated. It is also shown that estimators within this class achieve the semiparametric efficiency bound. We consider an accelerated failure time model for time-to-event data in which the survival distribution is scaled by a factor which may be a function of time-dependent covariates. These models directly model the effect of covariates on the length of survival. This is in contrast to the more commonly-used proportional hazards model which models the hazard rate itself. In some applications it may be easier to visualize the concept that a treatment intervention or exposure to an environmental contaminant increases or decreases the length of survival by a certain proportion, as compared to the concept that the hazard rate is changed. For time independent covariates, say, Z = (Z1,.. , Zp)', the class of accelerated failure time models assumes that the survival distribution at time t, given a set of covariates Z = z, is given by F(t I z) = F0{ h (z) t}. The proportionality constant h (z) can be interpreted as the scale factor by which lifetime is decreased as a function of the covariates z. Often h(z) is taken to be log linear or h(z) = exp (,B'z). In such cases, if z = 0, we can interpret Fo(t) as the 'baseline' survival distribution or the survival distribution for individuals with covariates all equal to zero. A useful way of envisioning this model is to consider a hypothetical random variable, U, that corresponds to an individual's survival time if that individual had covariate values all equal to zero. This baseline failure time, U, would be modified for values of the covariate different than zero. For example, if we denote the actual survival time by T, then for an individual with covariate value Z we have T =- e'ZU. The parameters, ,B, in this model have a direct interpretation in terms of the increase or decrease in lifespan as a function of the covariates. The assumption that U is independent of Z in this hypothetical construct, induces the probabilistic model