Using Markov chain to describe the progression of chronic disease

Abstract

A discrete-time Markov chain with stationary transition probabilities is often used for the purpose of investigating treatment programs and health care protocols for chronic disease. Suppose the patients of a certain chronic disease are observed over equally spaced time intervals. If we classify the chronic disease into n distinct health states, the movement through these health states over time then represents a patient’s disease history. We can use a discrete-time Markov chain to describe such movement using the transition probabilities between the health states.

The purpose of this study was to investigate the case when the observation interval coincided with the cycle length of the Markov chain as well as the case when the observational interval and the cycle length did not coincide. In particular, we are interested in how the estimated transition matrix behaves as the ratio of observation interval and cycle length changes.

Our results suggest that more estimation problems arose for small sample sizes as the length of observational interval increased, and that the deviation from the known transition probability matrix got larger as the length of observational interval increased. With increasing sample size, there were fewer estimation problems and the deviation from the known transition probability matrix was reduced.