Continuous time Markov chain approaches for analyzing transtheoretical models of health behavioral change: A case study and comparison of model estimations

Continuous time Markov chain models are frequently employed in medical research to study the disease progression but are rarely applied to the transtheoretical model, a psychosocial model widely used in the studies of health-related outcomes. The transtheoretical model often includes more than three states and conceptually allows for all possible instantaneous transitions (referred to as general continuous time Markov chain). This complicates the likelihood function because it involves calculating a matrix exponential that may not be simplified for general continuous time Markov chain models. We undertook a Bayesian approach wherein we numerically evaluated the likelihood using ordinary differential equation solvers available from the gnu scientific library. We compared our Bayesian approach with the maximum likelihood method implemented with the R package MSM. Our simulation study showed that the Bayesian approach provided more accurate point and interval estimates than the maximum likelihood method, especially in complex continuous time Markov chain models with five states. When applied to data from a four-state transtheoretical model collected from a nutrition intervention study in the next step trial, we observed results consistent with the results of the simulation study. Specifically, the two approaches provided comparable point estimates and standard errors for most parameters, but the maximum likelihood offered substantially smaller standard errors for some parameters. Comparable estimates of the standard errors are obtainable from package MSM, which works only when the model estimation algorithm converges.