The influence of random effects on the unconditional hazard rate and survival functions

Abstract

SUMMARY The consequences of introducing random effects have been a major issue in models for survival analysis. A general result is derived to establish the underestimation of the unconditional hazard rate when a random effect is present. Also, the survival time is shown to be underestimated. No distributional assumption regarding the random effect is required. The 'random effects survival model' has been a major focus in the recent literature (Heckman & Singer, 1984; Littlewood, 1984; Clayton & Cuzick, 1985; Heckman, Robb & Walker, 1990; Honore, 1990). When a hazard rate function is estimated, it is conventional practice to assume that observations are taken from a homogeneous group. If they are based on several heterogeneous subgroups, we should estimate their hazard rate functions separately. However, if we could assume that the hazard rate function from one group is parallel to those of other groups, we can deal with all observations at the same time by taking account of covariate variables by setting AN(t) = AO(t) exp (fl'x), where AO(t) is a specified hazard rate, x is a vector of fixed covariates, and the subscript N is attached to emphasize that this is a nominal hazard rate, i.e. without heterogeneity. This helps to describe the differences among subgroups. But, if we want to describe all differences exactly, we have to use too many covariates. This is not realistic since the number of observations available is limited. Thus we choose several important covariates and regard the remaining covariates as random. In order to introduce heterogeneity into the population via a random effect, we let the hazard rate for an individual be VAN(t) or V2O(t) exp (fl'x), where V is a random variable such that pr (V> 0) = 1, E(V) = 1 and pr (V= 1) O and AN(t) >0. Using FU(t) and FC(t Iv), by taking an expected value of fc(t I)