Markov Models for Covariate Dependence of Binary Sequences

Abstract

Suppose that a heterogeneous group of individuals is followed over time and that each individual can be in state 0 or state 1 at each time point. The sequence of states is assumed to follow a binary Markov chain. In this paper we model the transition probabilities for the 0 to 0 and 1 to 0 transitions by two logistic regressions, thus showing how the covariates relate to changes in state. With p covariates, there are 2(p + 1) parameters including intercepts, which we estimate by maximum likelihood. We show how to use transition probability estimates to test hypotheses about the probability of occupying state 0 at time i (i = 2, ..., T) and the equilibrium probability of state 0. These probabilities depend on the covariates. A recursive algorithm is suggested to estimate regression coefficients when some responses are missing. Extensions of the basic model which allow time-dependent covariates and nonstationary or second-order Markov chains are presented. An example shows the model applied to a study of the psychological impact of breast cancer in which women did or did not manifest distress at four time points in the year following surgery.