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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
Nicky J. Welton, PhD
MRC Health Services Research Collaboration, Bristol, United Kingdom, nicky.welton{at}bristol.ac.uk
A. E. Ades, PhD
MRC Health Services Research Collaboration, Bristol, United Kingdom
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.
Key Words: rate • risk • probability • transitions • Markov models • uncertainty propagation • Bayesian MCMC • evidence synthesis • calibration
References
- Beck JR, Pauker S. The Markov process in medical prognosis. Med Decis Making. 1983;3:419-458.[Abstract/Free Full Text]
- Sonnenberg FA, Beck Jr. Markov models in medical decision making: a practical guide. Health Econ. 1983;13:322-338.
- Hazen G. Preference factoring for stochastic trees. Management Science. 2000;46:389-403.
- Hazen GB. Factored stochastic trees: a tool for solving complex temporalmedical decision models. Med Decis Making. 1993;13:227-236.[Abstract/Free Full Text]
- Hazen GB. Stochastic trees: a new technique for temporal medical decision modeling. Med Decis Making. 1992;12:163-178.[Abstract/Free Full Text]
- Hazen GB, Pellissier JM. Recursive utility for stochastic trees. Operations Research. 1996;44:788-809.[Abstract/Free Full Text]
- Hazen GB, Pellissier JM, Sounderpandian J. Stochastic-tree models in medical decision making. Interfaces. 1998;28:64-80.[Abstract/Free Full Text]
- Miller DK, Homan SM. Determining transition probabilities: confusion and suggestions. Med Decis Making. 1994;14:52-58.[Abstract/Free Full Text]
- Parmigiani G. Modeling in Medical Decision Making: a Bayesian Approach. New York: Wiley; 2002.
- Craig BA, Sendi PP. Estimation of the transition matrix of a discrete-time Markov chain. Health Econ. 2002;11:33-42.[CrossRef][Web of Science][Medline]
[Order article via Infotrieve]
- Craig BA, Fryback DG, Klein R, Klein BEK. A Bayesian approach to modelling the natural history of a chronic condition from observations with intervention. Statistics in Medicine. 1999;18:1355-1371.[CrossRef][Web of Science][Medline]
[Order article via Infotrieve]
- Sendi PP, Craig BA, Pfluger D, Gafni A, Bucher HC. Systematic validation of disease models for pharmacoeconomic evaluations. Journal of Evaluation in Clinical Practice. 1999;5:283-295.[CrossRef][Web of Science][Medline]
[Order article via Infotrieve]
- Fryback DG, Chinnis JO, Ulvila JW. Bayesian cost-effectiveness analysis: an example using the GUSTO trial. Med Decis Making. 2001;17:83-97.
- Grimmett GDR, Stirzaker DR. Probability and Random Processes. 2nd ed. Oxford: Oxford University Press, 1992.
- Cinlar E. Introduction to stochastic processes. Englewood Cliffs (New Jersey): Prentice-Hall; 1975.
- Maple 8. Available from: http://www.maplesoft.com/products/Maple8/index.shtml. 2002.
- Congdon P. Bayesian Statistical Modelling. Chichester: Wiley; 2001.
- Cox DR, Oakes D. Analysis of Survival Data. London: Chapman and Hall; 1984.
- Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data. New York: Wiley, 1980.
- Parmigiani G, Samsa GA, Ancukiewicz M, Lipscomb J, Hasselblad V, Matchar D. Assessing uncertainty in cost-effectiveness analyses: application to a complex decision model. Med Decis Making. 1997;17:390-401.[Abstract/Free Full Text]
- Ades AE, Cliffe S. Markov Chain Monte Carlo estimation of a multi-parameter decision model: consistency of evidence and the accurate assessment of uncertainty. Med Decis Making. 2002;22:359-371.[Abstract/Free Full Text]
- Ades AE. A chain of evidence with mixed comparisons: models for multi-parameter evidence synthesis and consistency of evidence. Statistics in Medicine. 2003;22:2995-3016.[CrossRef][Web of Science][Medline]
[Order article via Infotrieve]
- Eddy DM, Hasselblad V, Shachter R. Meta-analysis by the Confidence Profile Method. London: Academic Press; 1992.
- Dilks DW, Canale RP, Meier PG. Development of Bayesian Monte Carlo techniques for water quality model uncertainty. Ecological Modelling. 1992;62:149-162.[CrossRef]
- Patwardhan A, Small MJ. Bayesian methods for model uncertainty analysis with application to future sea level rise. Risk Analysis. 1992;12:513-523.[CrossRef]
- Briggs AH, Ades AE, Price MJ. Probabilistic sensitivity analysis for decision trees with multiple branches: use of the Dirichlet distribution in a Bayesian framework. Med Decis Making. In press.
- WinBUGS Differential Interface (WBDiff). Available from: http://homepages.tesco.net/%7Ecreeping_death/.
- Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A. Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society (B). 2002;64:1-34.
- McCullagh P, Nelder JA. Generalised Linear Models. 2nd ed. London: Chapman and Hall; 1989.
- Naglie G, Detsky AS. Treatment of chronic nonvalvular atrial fibrillation in the elderly: a decision analysis. Med Decis Making. 1992;12:239-249.[Abstract/Free Full Text]
- Gage BF, Cardinalli AB, Albers GW, Owens DK. Cost-effectiveness of Warfarin and aspirin for prophylaxis of stroke in patients with nonvalvular atrial fibrillation. JAMA. 1995;274:1839-1845.[Abstract/Free Full Text]
- Greenberg M, Malenka D, Disch D. Therapeutic strategies for atrial fibrillation: the value of decision analysis. Cardiology Clinics. 1996;14:623-640.[Medline]
[Order article via Infotrieve]
- Fleiss JL. Measures of effect size for categorical data. In: Cooper H, Hedges LV, eds. The Handbook of Research Synthesis. New York: Russell Sage Foundation, 1994. p 245-260.
- Andersen PK, Keiding N. Multi-state models for event history analysis. Stat Methods Med Res. 2002;11:91-115.[Abstract/Free Full Text]
- Keiding N. Event history analysis and inference from observational epidemiology. Stat Med. 1999;18(17-18):2353-2363.[CrossRef][Web of Science][Medline]
[Order article via Infotrieve]
- Mulrow CD. Rationale for systematic reviews. In: Chalmer I, Altman DG, eds. Systenmatic Reviews. London: BMJ Publishing Group, 1995. p 1-8.
- Brand KP, Small MJ. Updating uncertainty in an integrated risk assessment: conceptual framework and methods. Risk Analysis. 1995;15:719-731.[CrossRef][Web of Science]
- Sohn M, Small M, Pantazidou M. Reducing uncertainty in site characterisation using Bayes Monte Carlo methods. Journal of Environmental Engineering. 2000;126:893-902.[CrossRef][Web of Science]
- Raftery AE, Givens GH, Zeh JE. Inference from a deterministic population dynamics model for Bowhead whales (with discussion). Journal of the American Statistical Association. 1995;90:402-430.[CrossRef]
- Brooks SP, Gelman A. Alternative methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics. 1998;7:434-455.[CrossRef][Web of Science]
Medical Decision Making, Vol. 25, No. 6,
633-645 (2005)
DOI: 10.1177/0272989X05282637
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