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# Decision theory and Bayesian statistics. Tests and problem solving

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1. Decision theory and Bayesian statistics. Tests and problem solving  Petter Mostad 2005.11.21

2. Overview • Statistical desicion theory • Bayesian theory and research in health economics • Review of tests we have learned about • From problem to statistical test

3. 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.

4. 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.

5. 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

6. Example states actions

7. 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

8. 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?

9. 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:

10. 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

11. 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.

12. 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

13. 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.

14. 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”.

15. 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.

16. 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!

17. One group of normally distributed observations

18. Comparing two groups of observations: matched pairs (D1, …, Dn differences) Large samples:

19. Comparing two groups of observations: unmatched data see book for d.f.

20. Comparing more than two groups of data

21. Studying population proportions (p0 common estimate)

22. Regression tests

23. Model tests

24. Tests for correlation

25. Tests for autocorrelation

26. 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?

27. 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?