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

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Overview

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

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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.

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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.

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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

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Example
states
actions
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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

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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?

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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

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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

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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.

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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

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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.

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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.

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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 ?, the CEAC plots the probability that one
  treatment is more cost-effective than another.

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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!

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One group of normally distributed observations
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Comparing two groups of observations matched
pairs
(D1, , Dn differences)
Large samples
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Comparing two groups of observations unmatched
data
see book for d.f.
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Comparing more than two groups of data
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Studying population proportions
(p0 common estimate)
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Regression tests
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Model tests
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Tests for correlation
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Tests for autocorrelation
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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?

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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?