Diagnostic tests for frailty

A common way of allowing heterogeneity between individuals in models for lifetime data is to introduce an unobservable individual random effect Z. In a proportional hazards framework, the individual’s hazard becomes zhb(t) where hb(t) is the baseline hazard. The random variable Z is often assumed to follow the Gamma or Inverse Gaussian distribution. We develop here diagnostic tests for these assumptions. One simple graphical diagnostic is based on the form of the unconditional survival function when hb(t) is assumed to be Weibull. Another plot uses a closure property of a family of frailty distributions, which implies that the frailty among survivors at time t has the same form as the original distribution of Z, with the same shape parameter but different scale parameter. In this method, we estimate the shape parameter at different times t and examine graphically whether it is constant. We give simulation results and examples to illustrate these methods.