Bivariate Markov Chains Containing A Failure Process

This paper is directed toward the challenge to model dependencies among discrete-state processes. In an earlier motivating application, we used proportional hazards regression models with time-dependent covariates to examine the relationship between relapse following treatment for leukemia, internal biological processes fighting the leukemia, and interventions intended to fix defects in these processes or to stimulate them to behave more aggressively. Complexities in this application led to the introduction of new dependency measures derived from extending Kolmogorov's differential equations. In this paper these dependency measures are interpreted for a bivariate Markov chain where one process is a failure process and another evolves concurrently. Likelihood construction using these dependency measures and others currently used in the context of a failure process are discussed.