Decision theory and bayesian statistics tests and problem solving
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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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Presentation Transcript

Overview
Overview solving 

  • 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
Statistical decision theory solving 

  • 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
Decision theory: Setup solving 

  • 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
Desicion theory: Concepts solving 

  • 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
Example solving 

states

actions


Decision trees
Decision trees solving 

  • 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
Updating probabilities by aquired information solving 

  • 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
Example: Birdflu solving 

  • 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
Expected value of perfect information solving 

  • 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
Expected value of sample information solving 

  • 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
Utility solving 

  • 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
Risk and (health) insurance solving 

  • 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
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
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
Review of tests willingness to pay

  • 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
Comparing two groups of observations: matched pairs willingness to pay

(D1, …, Dn differences)

Large samples:


Comparing two groups of observations unmatched data
Comparing two groups of observations: unmatched data willingness to pay

see book

for d.f.



Studying population proportions
Studying population proportions willingness to pay

(p0 common estimate)


Regression tests
Regression tests willingness to pay


Model tests
Model tests willingness to pay


Tests for correlation
Tests for correlation willingness to pay


Tests for autocorrelation
Tests for autocorrelation willingness to pay


From problem to choice of method
From problem to choice of method willingness to pay

  • 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 method1
From problem to choice of method willingness to pay

  • 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?


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