Decision theory and Bayesian statistics. Tests and problem solving Petter Mostad 2005.11.21
Overview • Statistical desicion theory • Bayesian theory and research in health economics • Review of tests we have learned about • From problem to statistical test
Statistical decision theory • Statistics in this course often focus on estimating parameters and testing hypotheses. • The real issue is often how to choose between actions, so that the outcome is likely to be as good as possible, in situations with uncertainty • In such situations, the interpretation of probability as describing uncertain knowledge (i.e., Bayesian probability) is central.
Decision theory: Setup • The unknown future is classified into H possible states: s1, s2, …, sH. • We can choose one of K actions: a1, a2, …, aK. • For each combination of action i and state j, we get a ”payoff” (or opposite: ”loss”) Mij. • To get the (simple) theory to work, all ”payoffs” must be measured on the same (monetary) scale. • We would like to choose an action so to maximize the payoff. • Each state si has an associated probability pi.
Desicion theory: Concepts • If action a1 never can give a worse payoff, but may give a better payoff, than action a2, then a1 dominates a2. • a2 is then inadmissible • The maximin criterion • The minimax regret criterion • The expected monetary value criterion
Example states actions
Decision trees • Contains node (square junction) for each choice of action • Contains node (circular junction) for each selection of states • Generally contains several layers of choices and outcomes • Can be used to illustrate decision theoretic computations • Computations go from bottom to top of tree
Updating probabilities by aquired information • To improve the predictions about the true states of the future, new information may be aquired, and used to update the probabilities, using Bayes theorem. • If the resulting posterior probabilities give a different optimal action than the prior probabilities, then the value of that particular information equals the change in the expected monetary value • But what is the expected value of new information, before we get it?
Example: Birdflu • Prior probabilities: P(none)=95%, P(some)=4.5%, P(pandemic)=0.5%. • Assume the probabilities are based on whether the virus has a low or high mutation rate. • A scientific study can update the probabilities of the virus mutation rate. • As a result, the probabilities for no birdflu, some birdflu, or a pandemic, are updated to posterior probabilities: We might get, for example:
Expected value of perfect information • If we know the true (or future) state of nature, it is easy to choose optimal action, it will give a certain payoff • For each state, find the difference between this payoff and the payoff under the action found using the expected value criterion • The expectation of this difference, under the prior probabilities, is the expected value of perfect information
Expected value of sample information • What is the expected value of obtaining updated probabilities using a sample? • Find the probability for each possible sample • For each possible sample, find the posterior probabilities for the states, the optimal action, and the difference in payoff compared to original optimal action • Find the expectation of this difference, using the probabilities of obtaining the different samples.
Utility • When all outcomes are measured in monetary value, computations like those above are easy to implement and use • Central problem: Translating all ”values” to the same scale • In health economics: How do we translate different health outcomes, and different costs, to same scale? • General concept: Utility • Utility may be non-linear function of money value
Risk and (health) insurance • When utility is rising slower than monetary value, we talk about risk aversion • When utility is rising faster than monetary value, we talk about risk preference • If you buy any insurance policy, you should expect to lose money in the long run • But the negative utility of, say, an accident, more than outweigh the small negative utility of a policy payment.
Desicion theory and Bayesian theory in health economics research • As health economics is often about making optimal desicions under uncertainty, decision theory is increasingly used. • The central problem is to translate both costs and health results to the same scale: • All health results are translated into ”quality adjusted life years” • The ”price” for one ”quality adjusted life year” is a parameter called ”willingness to pay”.
Curves for probability of cost effectiveness given willingness to pay • One widely used way of presenting a cost-effectiveness analysis is through the Cost-Effectiveness Acceptability Curve (CEAC) • Introduced by van Hout et al (1994). • For each value of the threshold willingness to pay λ, the CEAC plots the probability that one treatment is more cost-effective than another.
Review of tests • Below is a listing of most of the statistical tests encountered in Newbold. • It gives a grouping of the tests by application area • For details, consult the book or previous notes!
Comparing two groups of observations: matched pairs (D1, …, Dn differences) Large samples:
Comparing two groups of observations: unmatched data see book for d.f.
Studying population proportions (p0 common estimate)
From problem to choice of method • Example: You have the grades of a class of studends from this years statistics course, and from last years statistics course. How to analyze? • You have measured the blood pressure, working habits, eating habits, and exercise level for 200 middleaged men. How to analyze?
From problem to choice of method • Example: You have asked 100 married women how long they have been married, and how happy they are (on a specific scale) with their marriage. How to analyze? • Example: You have data for how satisfied (on some scale) 50 patients are with their primary health care, from each of 5 regions of Norway. How to analyze?