Dependency in multivariate markov chains

Abstract

The absence of a general multivariate model for discrete-state processes has been a major barrier for applications. This paper is motivated by the challenge to understand the relationship between relapse following treatment for leukemia, the biological processes by which a patient fights disease internally, and interventions intede to assist the patient by fixing defects in these processes or stimulating them to behave more aggressively. A multivariate version of the Chapman-Kolmogorov equations is defined to form a basis for developing two classes of association among discrete state random processes. A conditional intensity function measures the association between two sets of multivariate processes, U( t) and V( s), as a function of both times s ⩽ t. The individual processes in U and V need not be distinct. A cross-intensity function not only measures the association between U( t) and V( s), but also controls for another vector-valued process W( t). Toward this end, we propose a robust group of measures of association.