Checking for the Gamma Frailty Distribution under the Marginal Proportional Hazards Frailty Model

The marginal proportional hazards frailty model for multivariate failure time data characterizes the intracluster dependency with the frailty model while formulating the marginal distributions with the proportional hazards model. The gamma frailty distribution has been widely used to model intracluster dependency because of its simple interpretation and mathematical tractability. Glidden (2000) proposed a two-stage method for estimating the dependence parameter under the marginal proportional hazards frailty model when the frailty follows a gamma distribution. The goodness of fit test for the marginal proportional hazards model has been proposed by Spiekerman and Lin (1996). In this paper, we provide a graphical as well as a numerical method for checking the adequacy of the gamma frailty distribution. The test process is derived from the posterior expectation of the frailty given the observable data. The critical value can be obtained by a Monte Carlo simulation. Two examples from genetics studies are provided to illustrate the proposed testing procedure.