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Markov Chain Population Models in Medical Decision Making

Markov Chain Population Models in Medical Decision Making. Gordon Hazen Min Huang Northwestern University. Markov models (individual-level) in medical decision making. Intervention that reduces disease mortality rate. Conventional outcome measure— QALYs for an individual (or a cohort).

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Markov Chain Population Models in Medical Decision Making

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  1. Markov Chain Population Models in Medical Decision Making Gordon Hazen Min Huang Northwestern University M. Huang Northwestern Univ.

  2. Markov models (individual-level) in medical decision making Intervention that reduces disease mortality rate M. Huang Northwestern Univ.

  3. Conventional outcome measure—QALYs for an individual (or a cohort) M. Huang Northwestern Univ.

  4. From individual to population Motivation: To study a whole population • Equilibrium distribution of a population • 2. Equilibrium measure of effectiveness of • an intervention 1. no equilibrium 2. no births Individual-level models — M. Huang Northwestern Univ.

  5. Augment model by allowing “births” Intervention that reduces disease mortality rate M. Huang Northwestern Univ.

  6. Population model and its routing Population model Routing process M. Huang Northwestern Univ.

  7. Population no longer dies out—reaches new equilibrium after intervention M. Huang Northwestern Univ.

  8. Time-homogeneous individual-level Markov models Individual Markov model State space {0,1,2,..,J,-1}, where ‘-1’ representing ‘Death’ is an absorbing state Transition rates M. Huang Northwestern Univ.

  9. Population models Population Markov model State space: Transition rates: — Open Jackson processes Serfozo Serfozo R. Introduction to Stochastic Networks. Springer 1999. M. Huang Northwestern Univ.

  10. Routing processes Individual-level model State space: {0,1,2,..,J,-1}, where ‘-1’ is a source/sink node Transition rates M. Huang Northwestern Univ.

  11. Properties is irreducible, then at equilibrium: If • are independent, ~Poisson( ) equilibrium population means • Conditional on total population size |n|, n is • multinomial equilibrium population proportions M. Huang Northwestern Univ.

  12. Equilibrium population means is the unique collection of positive numbers that satisfy balance equations of routing process i.e. . Here Q is a submatrix of the rate matrix of the routing process, and also a submatrix of the rate matrix of the underlying individual model, corresponding to all nonabsorbing states, i.e., health states {0,1,…,J}. M. Huang Northwestern Univ.

  13. What measures of quality are possible at the population level? Measures of health Individual QALYs : QALYs for an individual starting in state j Equilibrium population measures M. Huang Northwestern Univ.

  14. Average Lifetime QALY ALQ Mean QALY of randomly selected individual from equilibrium population M. Huang Northwestern Univ.

  15. Total Lifetime QALY TLQ Mean total QALYs of all individuals in equilibrium population M. Huang Northwestern Univ.

  16. Average QALYs per Year AQ/yr One-year QALY of randomly selected individual from equilibrium population M. Huang Northwestern Univ.

  17. Total QALYs per Year TQ/yr One-year QALY of all individuals in equilibrium population M. Huang Northwestern Univ.

  18. Discount rate = 3% Discounted Total QALYs DTQ Mean total discounted QALYs for this and all subsequent generations of population. M. Huang Northwestern Univ.

  19. Relationships between measures DTQ TQ/yr if the population is in equilibrium from t=0. TLQ ALQ TQ/yr AQ/yr TQ/yr AQ/yr M. Huang Northwestern Univ.

  20. The simple illustrative example— differences among measures Intervention that reduces disease mortality rate M. Huang Northwestern Univ.

  21. Evaluating interventions using these measures: M. Huang Northwestern Univ.

  22. Insight • Problem: average measures do not account for population size increase due to better survival. • Caution in choosing population measures M. Huang Northwestern Univ.

  23. Example: tamoxifen use to prevent breast cancerCol Col N.F., Orr R.K., Fortin J.M. Survival impact of tamoxifen use for breast cancer risk reduction: projections from a patient-specific Markov model, Med Decis Making 2002; 22: 386-393. M. Huang Northwestern Univ.

  24. Non-homogeneous individual-level Markov models 1. Human background survival Background mortality rate (Gompertz) 2. The other factor: a homogeneous Markov process M. Huang Northwestern Univ.

  25. Population models Mean density with respect to age a of the population in state j at time t: Theorem: M. Huang Northwestern Univ.

  26. Notations: equilibrium mean density with respect to age a of the population in state j, equilibrium expected total population count in state j. Conclusions: M. Huang Northwestern Univ.

  27. Measures of health Individual QALYs : QALYs for an individual starting from age a0 in state j Equilibrium population measures ALQ TLQ AQ/yr TQ/yr TLQ M. Huang Northwestern Univ.

  28. Example: tamoxifen use to prevent breast cancerCol M. Huang Northwestern Univ.

  29. Summary • Population Markov models for medical decision making. • Population measures of interventions • Age-dependency. M. Huang Northwestern Univ.

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