Perturbation analysis of continuous‐time absorbing Markov chains

Abstract

SUMMARYThe behavior of Markov chains depends on the transition intensities or probabilities; in applications, these transitions are often written as functions of parameters. In this paper, matrix calculus is used to provide the sensitivities and elasticities of the behavior of continuous time‐absorbing Markov chains to arbitrary parameters, which could include the transition intensities themselves or parameters in which those intensities depend. The results include the sensitivity and elasticity of the moments, variances, standard deviations, and coefficients of variation of the time spent in transient states and the time to absorption. The number of visits to transient states is also analyzed, using the embedded discrete‐time chain. For models with multiple absorbing states, the sensitivity of the probability of absorption in each absorbing state is presented. As an example, a published model for the progress of colorectal cancer is analyzed in detail. Transient states in this model are defined by disease status, and the model includes two absorbing states: death from cancer and death from other causes. The perturbation analysis provides a quantitative comparison of the effect of changing transition and mortality rates. Such information may be useful for evaluating treatment or screening programs. Copyright © 2011 John Wiley & Sons, Ltd.