Estimation of Markov Chain Transition Probabilities and Rates from Fully and Partially Observed Data

Markov transition models are frequently used to model dis-ease progression. The authors show how the solution to Kolmogorov’s forward equations can be exploited to map be-tween transition rates and probabilities fromprobability data inmultistatemodels. Theyprovideauniform,Bayesian treat-ment of estimation and propagation of uncertainty of transi-tion rates and probabilities when 1) observations are avail-able on all transitions and exact time at risk in each state (fully observed data) and 2) observations are on initial state and final state after a fixed interval of time but not on the se-quence of transitions (partially observed data). The authors show how underlying transition rates can be recovered from partially observed data using Markov chain Monte Carlo methods inWinBUGS, and they suggest diagnostics to inves-tigate inconsistencies between evidence from different start-ing states.An illustrative example for a3-statemodel is given, which shows how the methods extend to more complex Markov models using the software WBDiff to compute solu-tions. Finally, the authors illustrate how to statistically com-bine data frommultiple sources, including partially observed data at several follow-up times and also how to calibrate a Markov model to be consistent with data from one specific study. Key words: rate; risk; probability; transitions; Markov models; uncertainty propagation; BayesianMCMC; evidenc