Decision theory and Bayesian statistics. More repetition

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Decision theory and Bayesian statistics. More
repetition 

• Tron Anders Moger
• 22.11.2006

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Overview

• Statistical desicion theory
• Bayesian theory and research in health economics
• Review of previous slides

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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 of nature 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 for choosing actions
• The minimax regret criterion for choosing actions
• The expected monetary value criterion for
  choosing actions

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Example
states
actions
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Maximin and minimax

• Maximin Maximize the minimum payoff
• For each row, compute the minimum
• Maximize over the actions
• Minimax regret Minimize the maximum regret
  possible
• Compute the regrets in each column, by finding
  differences to max numbers
• Maximize over the rows
• Find action that minimizes these maxima.

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Example
Find that action C is preferred under the maximin
criterion Regret table
states
actions
Action C is also preferred under the minimax
criterion
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Expected monetary value criterion

• Need probabilities for each state
• Assume P(no outbreak)P195, P(small
  outbreak)P24.5, P(pandemic)P30.5
• EMV(A)P1M11P2M12P3M13
• 00.95-5000.045-1000000.005 -522.5
• EMV(B)-55.45
• EMV(C)-1000
• Should choose action B

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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 (or left to
  right in the book) of tree

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Example
No outbreak (0.95)
0
Action A
Small outbreak (0.045)
-500
Pandemic (0.005)
EMV-522.5
-100000
No outbreak (0.95)
EMV-55.45
-1
Action B
Small outbreak (0.045)
-100
Pandemic (0.005)
-10000
No outbreak (0.95)
EMV-1000
-1000
Small outbreak (0.045)
-1000
Action C
Pandemic (0.005)
-1000
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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

• Prior probabilities P(no outbreak)95, P(small
  outbreak)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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The new information might affect what action we
would take

• But not in this example
• If we find out that birdflu virus has high
  mutation rate, we would still choose action B!
• EMV(A)-5075, EMV(B)-515.8, EMV(C)-1000
• If we find out that birdflu virus has low
  mutation rate, we would still choose action B!
• EMV(A)-104.5, EMV(B)-11.9, EMV(C)-1000

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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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Example

• Found that action B was best using the prior
  probabilities
• However, if there is no outbreak, action A is one
  unit better than B
• Similarily, if there is a pandemic, action C is
  9000 units better than B
• The expected value of perfect information is then
• EVPI0.9510.04500.005900045.95

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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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Repetition What is relevant for the exam

• Probability theory
• Expected values and variance
• Distributions
• Tests, regression, one-way ANOVA and at least an
  understanding of two-way ANOVA are all relevant
  (obviously)
• Interpretation of a time-series regression model
  might also show up
• Do not forget how to interpret SPSS output
  (including graphs and figures)!!
• Also, do not forget the chi-square test!!

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Conditional probability

• If the event B already has occurred, the
  conditional probability of A given B is
• Can be interpreted as follows The knowledge that
  B has occurred, limit the sample space to B. The
  relative probabilities are the same, but they are
  scaled up so that they sum to 1.

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Probability postulates 3

• Multiplication rule For general outcomes A and
  B
• P(A?B)P(AB)P(B)P(BA)P(A)
• Indepedence A and B are statistically
  independent if P(A?B)P(A)P(B)
• Implies that

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The law of total probability - twins

• A Twins have the same gender
• B Twins are monozygotic
• Twins are heterozygotic
• What is P(A)?
• The law of total probability
• P(A)P(AB)P(B)P(A )P( )
• For twins P(B)1/3 P( )2/3
• P(A)1 1/31/2 2/32/3

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Bayes theorem

• Frequently used to estimate the probability that
  a patient is ill on the basis of a diagnostic
• Uncorrect diagnoses are common for rare diseases

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Example Cervical cancer

• BCervical cancer
• APositive test
• P(B)0.0001 P(AB)0.9 P(A )0.001
• Only 8 of women with positive tests are ill

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Probability postulates 4

• Assume that the events
• A1, A2 ,..., An are independent. Then
  P(A1?A2?....?An)P(A1)P(A2) .... P(An)
• This rule is very handy when all P(Ai) are equal
• The complement rule P(A)P( )1

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Example Doping tests

• Lets say a doping test has 0.2 probability of
  being positive when the athlete is not using
  steroids
• The athlete is tested 50 times
• What is the probability that at least one test is
  positive, even though the athlete is clean?
• Define Aat least one test is positive

Complement rule Rule of
independence 50 terms
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Expected values and variance

• Remember the formulas E(aXb) aE(X)b and
• How do you calculate expectation and variance for
  a categorical variable?
• For a continuous variable?
• How do you construct a standard normal variable
  from a general normal variable?
• Finding probabilities for a general normal
  variable?

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Distributions

• Distributions weve talked about in detail
• Binomial
• Poisson
• Normal
• Approximations to normal distributions?
• Other distributions are there just to allow us to
  make test statistics, but you need to know how to
  use them

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Remember this slide? (This was difficult)

• The probabilities for
• A Rain tomorrow
• B Wind tomorrow
• are given in the following table

Some wind
Strong wind
Storm
No wind
No rain
Light rain
Heavy rain
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And this one?

• Marginal probability of no rain
  0.10.20.050.010.36
• Similarily, marg. prob. of light and heavy rain
  0.34 and 0.3. Hence marginal dist. of rain is a
  PDF!
• Conditional probability of no rain given storm
  0.01/(0.010.040.05)0.1
• Similarily, cond. prob. of light and heavy rain
  given storm 0.4 and 0.5. Hence conditional dist.
  of rain given storm is a PDF!
• Are rain and wind independent? Marg. prob. of no
  wind 0.10.050.050.2
• P(no rain,no wind)0.360.20.072?0.1

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Think wheat fields!

• Wheat field was a bivariate distribution of wheat
  and fertilizer
• Only Continuous outcome instead of categorical
• Calculations on previous incomprehensible slide
  is exactly the same as we did for the wheat
  field!
• Mean wheat crop for wheat 1 regardless of
  fertilizer-gtMarginal mean!!
• Mean crop for wheat 1 given that you use
  fertilizer -gtConditional mean!!
• (corresponds to mean for a single cell in our
  field)

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Chi-square test

• Expected cell values Abortion/op.nurses
  1336/706.7
• Abortion/other nurses 1334/706.3
• No abortion/op.nurses 5736/7029.3
• No abortion/other nurses 5734/7027.7
• Can be easily extendend to more groups of nurses
• As long as you have only two possible outcomes,
  this is equal to comparing proportions in more
  than two groups (think one-way ANOVA)

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We get

• This has a chi-square distribution with
  (2-1)(2-1)1 d.f.
• Want to test H0 No association between abortions
  and type of nurse at 5-level
• Find from table 7, p. 869, that the
  95-percentile is 3.84
• This gives you a two-sided test!
• Reject H0 No association
• Same result as the test for different proportions
  in Lecture 4!

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In SPSS
Check Expected under Cells, Chi-square under
statistics, and Display clustered bar charts!
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Next time

• Find some topics you dont understand, and we can
  talk about them