QUANTITATIVE RISK STRATIFICATION IN MARKOV CHAINS WITH QUASI-STATIONARY DISTRIBUTIONS

1. L2 L1 L3 LN · · · D QUANTITATIVE RISK STRATIFICATION IN MARKOV CHAINS WITH QUASI-STATIONARY DISTRIBUTIONS • David C Chan, MD, MSc1; Philip K Pollett, PhD2; Milton C Weinstein, PhD3 • 1 Division of General Internal Medicine, Brigham and Women’s Hospital, Boston, MA; • 2 University of Queensland, Queensland, Australia; 3 Harvard School of Public Health, Boston, MA. DISCUSSION RESULTS PURPOSE • Quasi-stationary distributions exist in Markov chains with properties that are natural for many disease models. • Quasi-stationary distributions allow for quantitative definitions of risk within an established framework of Markov chain modeling. • Patient risk in trials can be quantitatively inferred from observed outcomes. Outcomes can then be extrapolated to populations of different risk. • Limitation: Quasi-stationary distributions do not account for population influx. • In clinical practice, treatment decisions often rely on the concept of risk stratification. • The decision in these cases is not whether all patients should be treated as a group, but how many patients should be treated when the group is risk-stratified. • We considered a general Markov chain with living states L1, …, LN and an absorbing death state D. • A unique positive quasi-stationary distribution exists for the living states, if and only if patients in L1 are at lowest risk for progression or death. • As an example, heart failure patients with greater previous hospitalizations (H) are more likely to die or be hospitalized. A positive quasi-stationary distribution therefore exists for surviving patients. • This quasi-stationary distribution allows for quantitative risk stratification of living patients. • Outcomes for quantitatively defined populations at risk (e.g. “highest quintile”) can be evaluated. • Risk can be inferred from observed outcomes, and expected outcomes can be extrapolated from theoretical risk. BACKGROUND • Markov chains are frequently used to model disease processes in decision analysis. • A Markov chain in which all living patients eventually die is called a transient chain with an absorbing state (death). • Quasi-stationary distributions represent the limiting probability conditional on no absorption (the eventual probability of a surviving patient being in a certain state). CONCLUSIONS • Risk stratification is a necessary part of many clinical decisions. • Quantitative risk stratification is made possible by quasi-stationary distributions intrinsic in many Markov chains. • This allows for communication about decisions for precisely how many patients, rather than simplistic ones for all or none. METHODS • We considered a general Markov chain with N living states and an absorbing death state D. • Using Markov chain theory, we solved for the conditions under which a positive quasi-stationary distribution exists. • We considered the specific example of targeting risky heart failure patients, and we explored the use of quasi-stationary distributions to quantify risk. Contact Information: David Chan, MD, MSc Brigham & Women’s Hospital 75 Francis Street, Boston, MA 02115 (617) 732-6660 E-mail: dcchan@partners.org