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Estimation of Markov Chain Transition Probabilities and Rates from Fully and Partially Observed Data: Uncertainty Propagation, Evidence Synthesis, and Model Calibration

Author

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  • Nicky J. Welton

    (MRC Health Services Research Collaboration, Bristol, United Kingdom, nicky.welton@bristol.ac.uk)

  • A. E. Ades

    (MRC Health Services Research Collaboration, Bristol, United Kingdom)

Abstract

Markov transition models are frequently used to model disease progression. The authors show how the solution to Kolmogorov’s forward equations can be exploited to map between transition rates and probabilities from probability data in multistate models. They provide a uniform, Bayesian treatment of estimation and propagation of uncertainty of transition rates and probabilities when 1) observations are available 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 sequence 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 in WinBUGS, and they suggest diagnostics to investigate inconsistencies between evidence from different starting states. An illustrative example for a 3-state model is given, which shows how the methods extend to more complex Markov models using the software WBDiff to compute solutions. Finally, the authors illustrate how to statistically combine data from multiple 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.

Suggested Citation

  • Nicky J. Welton & A. E. Ades, 2005. "Estimation of Markov Chain Transition Probabilities and Rates from Fully and Partially Observed Data: Uncertainty Propagation, Evidence Synthesis, and Model Calibration," Medical Decision Making, , vol. 25(6), pages 633-645, November.
  • Handle: RePEc:sae:medema:v:25:y:2005:i:6:p:633-645
    DOI: 10.1177/0272989X05282637
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    10. Villacorta, Pablo J. & Verdegay, José L., 2016. "FuzzyStatProb: An R Package for the Estimation of Fuzzy Stationary Probabilities from a Sequence of Observations of an Unknown Markov Chain," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 71(i08).
    11. C. Armero & G. García‐Donato & A. López‐Quílez, 2010. "Bayesian methods in cost–effectiveness studies: objectivity, computation and other relevant aspects," Health Economics, John Wiley & Sons, Ltd., vol. 19(6), pages 629-643, June.
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    14. Steven M. Shechter & Matthew D. Bailey & Andrew J. Schaefer & Mark S. Roberts, 2008. "The Optimal Time to Initiate HIV Therapy Under Ordered Health States," Operations Research, INFORMS, vol. 56(1), pages 20-33, February.
    15. João Costa Freitas & Alberto Adrego Pinto & Óscar Felgueiras, 2024. "Game Theory for Predicting Stocks’ Closing Prices," Mathematics, MDPI, vol. 12(17), pages 1-49, August.
    16. Marta Soares & Luísa Canto e Castro, 2012. "Continuous Time Simulation and Discretized Models for Cost-Effectiveness Analysis," PharmacoEconomics, Springer, vol. 30(12), pages 1101-1117, December.
    17. Anna Divoli & Eneida A Mendonça & James A Evans & Andrey Rzhetsky, 2011. "Conflicting Biomedical Assumptions for Mathematical Modeling: The Case of Cancer Metastasis," PLOS Computational Biology, Public Library of Science, vol. 7(10), pages 1-15, October.
    18. Sixten Borg & Ulf Persson & Tine Jess & Ole Østergaard Thomsen & Tryggve Ljung & Lene Riis & Pia Munkholm, 2010. "A Maximum Likelihood Estimator of a Markov Model for Disease Activity in Crohn’s Disease and Ulcerative Colitis for Annually Aggregated Partial Observations," Medical Decision Making, , vol. 30(1), pages 132-142, January.
    19. Edmund Jones & David Epstein & Leticia García-Mochón, 2017. "A Procedure for Deriving Formulas to Convert Transition Rates to Probabilities for Multistate Markov Models," Medical Decision Making, , vol. 37(7), pages 779-789, October.

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