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Markov Models: Overview Gerald F. Kominski, Ph.D. Professor, Department of Health Services

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Markov Models: Why Are They Necessary? n Conventional decision analysis models assume: -Chance events -Limited time horizon -Events that do not recur n What happens if we have a problem with: -An extended time horizon, say, over a lifetime -Events can reoccur throughout a lifetime

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Decision Tree for Atrial Fibrillation

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State-Transition Diagram for Atrial Fibrillation Well Post-StrokeDead p 12 =0.2 p 22 =0.9 p 33 =1.0 p 11 =0.7 p 23 =0.1 p 13 =0.1 The probabilities for all paths out of a state must sum to 1.0. Death is known as an absorbing state, because individuals who enter that state cannot transition out of it.

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Transition Probabilities Well Post- Stroke Dead Well0.70.20.1 0.00.90.1 Dead0.00.01.0 State of Current Cycle State of Next Cycle Transition probabilities that remain constant over time are characteristic of stationary Markov models, aka Markov chains

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Markov Model Definitions n Any process evolving over time with uncertainty is a stochastic process, and models based on such processes are stochastic or probabilistic models n If the process is both stochastic and the behavior of the model in one time period (i.e., cycle) does not depend on the previous time period, the process is Markovian -The process has “lack of memory” -Even processes where the previous state does matter can be made Markovian through definition of temporary states know as tunnel states

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Tunnel States Well Post-Stoke 1 Post-Stroke 2Post-Stroke 3 Post-Stroke Dead

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Defining a Markov Model n Define the initial states n Determine the cycle length n Consider possible transitions among states n Determine transition probabilities n Determine utilities, and costs (if cost-effectiveness analysis), for each state

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Evaluating Markov Models: Cohort Simulation State CycleWell Post- Stroke Dead Sum of Years Lived Survival 010,00000 17,0002,0001,0009,0000.9000 24,9003,2001,9008,1000.8100 33,4303,8602,7107,2900.7290 42,4014,1603,4396,5610.6561 51,6814,2244,0955,9050.5905 61,1764,1384,6865,3140.5314 78243,9595,2174,7830.4783 93019,99910.0001 940010,00000.0000 The data in the last column is used to produce a survival curve, aka a Markov trace.

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Estimating Markov Models: Monte Carlo Simulation n Instead of processing an entire cohort and applying probabilities to the cohort, simulate a large number (e.g., 10,000) cases proceeding through the transition matrix -Monte Carlo simulation -TreeAge will do this for you quickly, without programming n The advantage of this approach is that it provides estimates of variation around the mean n Monte Carlo simulation is most valuable because it permits efficient modeling of complex prior history -Such variables are known as tracker variables

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Example of a 5-State Markov Source: Kominski GF, Varon SF, Morisky DE, Malotte CK, Ebin VJ, Coly A, Chiao C. Costs and cost- effectiveness of adolescent compliance with treatment for latent tuberculosis infection: results from a randomized trial. Journal of Adolescent Health 2007;40(1):61-68.

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Key Assumptions of the Markov Model VariableValue (Range)Reference Efficacy of IPT0.85 (0.75-0.98)19 Cost of treating active TB$22,500 ($17,000-$30,000)17 Cost of IPTVaries by study group and whether 6-month IPT is completed Current study TB cases per 100,000250 (120-560)20 TB case fatality rate0.0045-0.16 (varies with age)17 All-cause mortality rate per 100,000 19-15,476 (varies with age)National Center for Health Statistics, 1999 mortality tables Hepatotoxicity of IPT0.0008 (age<35, started IPT) 0.0012 (age<35, completed IPT) 21 Hepatitis fatality rate0.00221 Cost of treating IPT-induced hepatitis $11,250 ($8,500-$15,000)Authors’ assumption QALY – Healthy1.00 (0.95-1.00)Authors’ assumption QALY – Positive Skin Test, but Incomplete IPT0.90 (0.80-0.95)Authors’ assumption QALY – Active TB0.50 (0.20-0.90)Harvard Center for Risk Analysis QALY – IPT-induced hepatitis0.75 (75-0.90)Harvard Center for Risk Analysis Discount rate0.03 (0.00-0.07)Panel on Cost-Effectiveness

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