Stochastic Network Models for Survival Analysis

Abstract

We present methodology giving highly accurate approximations for Bayesian predictive densities and distribution functions of first passage times between states of a semi-Markov process with a finite number of states. When the states describe a degenerative disorder with an absorbing end state, such predictive distributions are the survival distributions of a patient. We illustrate these methods with a variety of examples, including data from the San Francisco AIDS study. We achieve our approximations using a three-step sequence. First, we introduce advanced concepts of flowgraph theory, which allow us to compute the moment generating function of the first passage time given the model parameters. Next, we use saddlepoint approximations to convert this into a density or distribution function conditional on the model parameter. Finally, we use Monte Carlo methods to remove dependence on the model parameter. These methods apply quite generally to all finite-state semi-Markov models in discrete or continuous time. Currently, there are no competing alternative methods that can achieve the saddlepoint accuracy of these computations.