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Markov versus Medical Markov Modeling – Contrasts and Refinements Gordon Hazen February 2012

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Medical Markov Modeling We think of Markov chain models as the province of operations research analysts However … The number of publications in medical journals – using Markov models – to address medical cost-effectiveness – approaches 300 per year! 2

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Medical Markov Modeling Why the large buy-in from the medical community? – Easy-to-use software that combines decision trees and Markov models (Data, TreeAge) – Simplicity of models Discrete time Transient 3

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Overview of this talk 1.Background on medical Markov modeling 2.Population modeling versus individual-level modeling 3.Product structure in medical Markov models

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Overview 1.Background on medical Markov modeling 2.Population modeling versus individual-level modeling 3.Product structure in medical Markov models.

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Medical Markov Modeling The kind of modeling that is typical 6 IHD = Ischemic heart disease MI = Myocardial infarction (heart attack) A simplification of: Palmer S, Sculpher M, Phillips Z, Robinson M, Ginnelly L, Bakhai A et al. Management of non-ST elevation acute coronary syndrome: how cost-effective are glycoprotein IIb/IIIa antagonists in the U.K. National Health Service?. International J Cardiology 100 (2005) 229-40.

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The kind of modeling that is typical Cohort analysis 7

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Our preference: Continuous-time Cohort analysis in continuous time 8 p MI = t p 0 = t p 1 = t

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Our preference – continuous time Discounted expected quality-adjusted life years: 9

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Cohort analysis in continuous time Intervention: Post-MI mortality rate 1 = 0.1/yr is decreased by 75% and the MI incidence rate = 0.12/yr is decreased by 50%. 10 6.62 QALY/patient10.28 QALY/patient

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Continuous-time version of cohort analysis Let dt 0 to obtain 11 … the Kolmogorov differential equations. The cohort analysis procedure is merely the Euler method for solving the Kolmogorov equations.

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Overview 1.Background on medical Markov modeling 2.Population modeling versus individual-level modeling 3.Product structure in medical Markov models.

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Question: How to incorporate population issues? An intuitive approach: Restart following death 13 Then compute steady-state probabilities in the resulting irreducible chain. Open routing process Closed routing process

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Question: How to incorporate population issues? Balance equations for steady-state probabilities: 14 Intervention assumptions: – Post-MI mortality rate 1 = 0.1/yr is decreased by 75% – MI incidence rate = 0.12/yr is decreased by 50%. Results: PostMI increases from 23.0% to 38.5% The population is less healthy! So what is wrong here?

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Population issues: A more rigorous approach Observation: A population of non-interacting individuals is equivalent to a Jackson network of infinite-server queues. 15

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Equilibrium results Jackson network balance equations 16 Theorem (e.g. Serfozo 1999): The counts n j of individuals in health state j are, at equilibrium, independent Poisson variables with means j given by the solution to the balance equations.

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Equilibrium results Solve balance equations with entrance rate = 1000/yr More survivors under intervention!

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The closed routing process again Convert the open routing process to a closed one in the following way 18 OpenClosed

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Open versus closed routing Theorem (Hazen and Huang 2011): One may obtain equilibrium means from steady state probabilities, and vice versa: 19 Equilibrium means j Steady-state probabilities j.

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Open versus closed results 20 Equilibrium means j Steady-state probabilities j.

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Example: Re-analysis of the preventive use of tamoxifen Original analysis: Col et al 2002 Tamoxifen – an estrogen agonist/antagonist – an effective therapy against established breast cancer Evidence that it can reduce breast cancer incidence But life-threatening side effects – endometrial cancer – vascular events. Would the benefit of its prophylactic use in healthy women be worth the associated risks? 21

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Preventive use of tamoxifen: Our model Cure rate models for breast and endometrial cancer treatment – Mortality decreases in time survived after cancer diagnosis. – This cannot be directly modeled as a Markov model – it is semi-Markov. – Cure rate model with unobserved states Cured/ Not Cured allows implicit mortality to decrease over time survived. 22 Breast cancer incidence and treatment Endometrial cancer incidence and treatment

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Preventive use of tamoxifen: Our model Overall model is the Cartesian product of the two factors below and a third Background Mortality factor. More on this later … 23 Breast cancer incidence and treatment Endometrial cancer incidence and treatment

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Preventive use of tamoxifen: Our model Estimated parameters (max likelihood estimates) 24

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Preventive use of tamoxifen: Our model Product structure for quality of life – Q jk = Q bj Q ek – more on this later 25 Model entry rate 0 = 110,000/yr – 2.3 M women reaching age 50 each year x 4.8% at high risk for breast cancer

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Preventive use of tamoxifen: Results A more nuanced picture of the effects of this intervention than just incremental QALYs. 26 Incremental QALYs/woman Incremental equilibrium means Incremental equilibrium probabilities

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Overview 1.Background on medical Markov modeling 2.Population modeling versus individual-level modeling 3.Product structure in medical Markov models.

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Markov models with product structure Product structure is relatively common in medical Markov models 28 Roach PJ, Fleming C, Hagen MD, Pauker SG. Prostatic cancer in a patient with asymptomatic HIV infection: are some lives more equal than others? Med Decis Making. 1988 Apr- Jun;8(2):132-44.

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Markov models with product structure Much simpler depiction of model structure: Independent factors 29

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Product structure is relatively common Schousboe et al. considered 5 different types of fractures: – hip fracture – clinical vertebral (Cv) fracture – radiographic vertebral (Rv) fracture – distal forearm (Df) fracture – other fracture In principle this should allow 2 5 = 32 state combinations corresponding to 5 factors each at 2 possible levels. What their model actually did: 6 states – 5 states corresponding to a single fracture type – 1 other state corresponding to the combination of the worst two possible fracture types – Disadvantage: Such a model “forgets” past fractures when a new fracture occurs, which the 32-state model would not do. 30 Schousboe JT, Nyman JA, Kane RL, Ensrud KE. Cost-effectiveness of alendronate therapy for osteopenic postmenopausal women. Ann Intern Med. 2005 May 3;142(9):734-41.

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Advantages of explicitly accounting for product structure Model formulation: Simpler to merely consider one factor at a time Model presentation: Simple factors easier to understand and critique. – Model is less likely to be perceived as a “black box” Computational advantages as well when factors are independent. 31

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Computation of QALYs under product structure 32

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Computation of QALYs under product structure 33

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Computation of QALYs under product structure 34

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Computation of expected cost under product structure 35

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Advantages of factored computation Computational work in cohort analysis is proportional to the number of state transitions Suppose the number of transitions in a factor with s non- death states is roughly s also. Then assuming s states in each factor and f factors, – s f transitions in the overall model under naïve cohort analysis – s f transitions in cohort decomposition – Big advantage for large f Caveat: s and f are not usually large. 36

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Cohort decomposition issues How often are factors independent? – Ans: More often not probabilistically independent. – But one factor is almost always probabilistically independent: Background mortality. How reasonable is the product form for the quality coefficient v(x)? – Empirical support for product form in HUI literature – additive decomposition is not supported. – Often only one factor carries quality adjustments, in which case product form holds by default. 37

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Summary These are just the basics – Population modeling Population model Jackson network One can get at equilibrium population issues by solving the usual balance equations for steady-state probabilities and scaling them up appropriately. – Product structure Common feature of medical Markov models Recognizing it can assist in model formulation and presentation, as well as computation. – Drawbacks for continuous-time models Medical researchers don’t “get” the models Software not widely available There is more to do here …

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