Decision theory and Bayesian statistics. Tests and problem solving

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# Decision theory and Bayesian statistics. Tests and problem solving - PowerPoint PPT Presentation

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.

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### Decision theory and Bayesian statistics. Tests and problem solving

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.
• 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.
• 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:

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?