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Markov Models II

Markov Models II. HS 249T Spring 2008. Brennan Spiegel, MD, MSHS. VA Greater Los Angeles Healthcare System David Geffen School of Medicine at UCLA UCLA School of Public Health CURE Digestive Diseases Research Center UCLA/VA Center for Outcomes Research and Education (CORE ). Topics.

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Markov Models II

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  1. Markov Models II HS 249T Spring 2008 Brennan Spiegel, MD, MSHS VA Greater Los Angeles Healthcare System David Geffen School of Medicine at UCLA UCLA School of Public Health CURE Digestive Diseases Research Center UCLA/VA Center for Outcomes Research and Education (CORE)

  2. Topics • More on Markov models versus decision trees • More examples of Markov models • Calculating annual transition probabilities • Time independent (Markov chains) • Time dependent (Markov processes) • Temporary and tunnel states • Half-cycle corrections

  3. Disadvantages of Traditional Decision Trees • Limited to one-way progression without opportunity to “go back” • Can become unwieldy in short order • Difficult to capture the dynamic path of moving between health states over time • Often fails to accurately reflect clinical reality

  4. Markov Models • Allow dynamic movement between relevant health states • Allow enhanced flexibility to better emulate clinical reality • Acknowledge that different people follow different paths through health and disease

  5. Example Markov Model Inadomi et al. Ann Int Med 2003

  6. Alive Barrett Alive No Barrett Year 0 Dead No Barrett Dead Barrett Markov Model

  7. Markov Model Alive Barrett Alive No Barrett Dead No Barrett Dead Barrett

  8. Markov Model Alive Barrett Alive No Barrett Dead No Barrett Dead Barrett

  9. Markov Model Year 1 Alive Barrett Alive No Barrett Dead No Barrett Dead Barrett

  10. Markov Model Alive Barrett Alive No Barrett Dead No Barrett Dead Barrett

  11. Markov Model Alive Barrett Alive No Barrett Dead No Barrett Dead Barrett

  12. Markov Model End Alive Barrett Alive No Barrett Dead No Barrett Dead Barrett

  13. Decision Trees and Markov Models may Co-Exist • Both provide different types of information • Information from both is not mutually exclusive • Markov model can be “tacked” onto end of a traditional decision tree

  14. No Cirrhosis Normal Lifespan Cirrhosis Markov Model Virological Response Normal Lifespan No Cirrhosis Normal Lifespan No Response Cirrhosis Markov Model Response Start Adefovir Resistance No Response No Resistance Con’t Lamivudine No Therapy Inteferon Chronic HBV Lamivudine Adefovir Adefovir Salvage

  15. To Cirrhosis Markov Model Markov Model #1 Uncomplicated Cirrhosis Chronic HBV Virological Resistance Chronic HBV on Treatment Virological Response Virological Relapse

  16. Markov Model #2 Uncomplicated Cirrhosis Complicated Cirrhosis Hepatocellular Carcinoma Liver Transplant Death

  17. No GI or CV Complications Dyspepsia Myocardial Infarction GI Bleed Post Myocardial Infarction Post GI Bleed Death

  18. START Sub-Clinical HE Overt HE Clinical Response Hepatocellular Cancer Non-HE Complication Liver Transplantation Death

  19. Annual Probability Estimates Annual Probability Estimate Cirrhosis in HBeAg(-) 4.0% Cirrhosis in HBeAg(+) 2.2% Chronic HBV  liver cancer 1.0% Cirrhosis  liver cancer 2.1% Compensated cirrhosis  decompensated 3.3% Decompensated cirrhosis  liver transplant 25% Liver cancer  liver transplant 30% Death in compensated cirrhosis 4.4% Death in decompensated cirrhosis 30% Death in liver cancer 43%

  20. 40 = 8% 5 Converting Data Into Annual Probability Estimates Cannot simply divide long-term data by number of years Example: If 5-year risk of an event is 40%, then annual risk does not amount to:

  21. Converting Data Into Annual Probability Estimates General rule for converting long-term data into annual probabilities: 1-(1-x)Y = Probability at Y Years

  22. Example of Converting Long Term Data into Annual Probability If probability of bleed at 5 years = 0.40, then the annual probability = x, as follows:  1- (1-x)5 = 0.40 (1-x)5 = 1 – 0.40   (1-x)5 = 0.60  (1-x)= 0.902 x = 0.097 … or 9.7%

  23. Example of Converting Long Term Data into Annual Probability Check for errors by back calculating using the inverse equation: 1-(1-annual probability)Y = probability at Y years  1-(1- 0.097)5 = 0.40  1-(0.903)5 = 0.40 0.40 = 0.40

  24. Steps to Combining Time-Independent Transition Probabilities Step 1  Collect and abstract relevant studies Step 2  Select common cycle length Step 3  Convert all studies to common cycle length units Step 4  Calculate common cycle transition probabilities Step 5  Combine common cycle probabilities

  25. Example Mean = 21.3% / 12-month cycle

  26. Many Probabilities are Time Dependent • Time independence is usually a simplifying assumption • Progress though many systems in health care (biological, organizational, psychosocial, etc) are erratic and non-linear • May need to account for time-dependent transitional probabilities using: • Tables • Tunnels

  27. Cycle Probability Cycle Probability 1 0.05 1 0.1 2 0.05 2 0.08 3 0.05 3 0.07 4 0.05 4 0.06 5 0.05 5 0.05 6 0.05 6 0.04 7 0.05 7 0.03 8 0.05 8 0.02 Time Independent Time Dependent Using Tables for Time-Dependent Probabilities • Tables allow transition probabilities to vary cycle-by-cycle • Allow greater precision for processes that are non-linear

  28. Time Independent: Linear Curve Time Dependent: Non-linear Diminishing Returns Time Dependent: Non-linear Accelerating Returns Probability Cycle

  29. Using Tunnels States • Some events can interfere with otherwise orderly Markov chains • Can get “stuck in a rut” that removes subjects from the usual flow of events • e.g. developing cancer • Tunnel states add flexibility to Markov models: • Model getting “stuck in the rut” • Compartmentalize processes into component states • Can model various “recovery states” from the “rut” • Can incorporate time-dependent transitions

  30. Example Prior to Tunnel State

  31. Example With Tunnel State

  32. Half-Cycle Corrections • In “real life,” events can occur anytime during a given cycle – it is usually a random event • The default setting for Markov models is for events to occur at the exact end of each cycle • Yet the default setting can lead to errors in the calculation of average values • Will tend to overestimate benefits (e.g. life expectancy) by about half of a cycle

  33. Rationale for Half-Cycle Corrections “In whatever cycle a ‘member’ of the cohort analysis dies, they have already received a full cycle’s worth of state reward, at the beginning of the cycle. In reality, however, deaths will occur halfway through a cycle on average. So, someone that dies during a cycle should lose half of the reward they received at the beginning of the cycle.” - TreeAge Pro Manual, p476

  34. 1.0 0.8 0.6 0.4 0.2 0.0 Proportion Alive AUC=2 0 1 2 3 4 Cycle

  35. 1.0 0.8 0.6 0.4 0.2 0.0 Proportion Alive AUC=2.5 0 1 2 3 4 Cycle

  36. 1.0 0.8 0.6 0.4 0.2 0.0 Proportion Alive AUC=2.0ish 0 1 2 3 4 Cycle

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