A General Framework for Random Effects Survival Analysis in the Cox Proportional Hazards Setting

The use of random effects modeling in statistics has increased greatly in recent years. The introduction of such modeling into event-time analysis has proceeded more slowly, however. Previously, random effects models for survival data have either required assumptions regarding the form of the baseline hazard function or restrictions on the classes of models that can be fit. In this paper, we develop a method of random effect analysis of survival data, the hierarchical Cox model, that is an extension of Cox's original formulation in that the baseline hazard function remains unspecified. This method also allows an arbitrary distribution for the random effects. We accomplish this using Markov chain Monte Carlo methods in a Bayesian setting. The method is illustrated with three models for a dataset with times to multiple occurrences of mammory tumors for 48 rats treated with a carcinogen and then randomized to either treatment or control. This analysis is more satisfying than standard approaches, such as studying the first event for each subject, which does not fully use the data, or assuming independence, which in this case would overestimate the precision.