A dynamic frailty model for multivariate survival data.

We consider the statistical modeling of data consisting of many study subjects with serially correlated multivariate survival responses. The (ordinary) frailty model handles the serial correlation in such data by introducing an unobserved multiplicative random effect term, called the frailty, in the hazard function. The frailties are often assumed to be identical for the survival times from the same unit. We have generalized the frailty model by allowing the frailties to vary stochastically with the indices. We have proposed a simple scheme to update the dynamic frailties. This approach assumes that the random effects are gamma distributed. At each occurrence, the two gamma parameters are updated according to the past information. In terms of their marginal distributions, the dynamic frailties form a multiplicative random walk. This approach results in a tractable likelihood. The small sample behavior of the MLE is studied via a simulation experiment. The model is then illustrated with a data set from an animal carcinogenesis experiment.