# Decision theory and Bayesian statistics

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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.
• a2 is then inadmissible
• The maximin criterion
• The minimax regret criterion
• The expected monetary value criterion
Example
states   No birdflu   Small birdflu    Birdflu
actions        outbreak      outbreak       pandemic
No extra          0            -500        -100000
precautions
Some extra         -1           -100         -10000
precautions
Vaccination      -1000         -1000         -1000
of whole pop.
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:
P(none | high _ mutation)  80%      P(none | low _ mutation)  99%
P( some | high _ mutation)  15%     P( some | low _ mutation)  0.9%
P( pand .| high _ mutation)  5%     P( pand . | low _ mutation)  0.1%
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.
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
– 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!
One group of normally distributed
observations
Goal of test:       Test statistic:     Distribution:

Testing mean of         X  0
standard normal:
normal distribution,      / n                N (0,1)
variance known
t-fordelingen, n-1
Testing mean of          X  0
sx / n                 tn 1
variance unknown
Testing variance of                         Chi-kvadrat, n-1
(n  1) s   2
normal population            02                   n21
Comparing two groups of
observations: matched pairs
Assuming normal                  D  D0
distributions, unknown                                         tn 1
sD / n
variance: Compare
means                    (D1, …, Dn differences)
Sign test: Compare       S = the number of pairs         Bin(n, 0.5)
only which               with positive difference.
Large samples S *  0.5n     Large samples:
observations are
(n>20):           0.5 n           N (0,1)
largest
Wilcoxon signed rank     T=min(T+,T-);               Wilcoxon signed rank
test: Compare ranks      T+ / T- are sum of                statistic
and signs of             positive/negative ranks
differences
Comparing two groups of
observations: unmatched data
Diff. between pop. means:                           x
2        y
2   Standard normal N (0,1)
Known variances                 ( X  Y  D0 ) /    nx       ny

Diff. between pop. means:                          s2        s2
( X  Y  D0 ) /    p
nx        p
ny
tnx ny 2
Unknown but equal variances
Diff. between pop. means:
2        s2
see book
Unknown and unequal             ( X  Y  D0 ) /   sx
nx        y
ny
t            for d.f.
variances
Testing equality of variances             2    2
for two normal populations              s /s
x    y
Fnx 1,ny 1
Assuming identical translated   Based on sum of ranks of Standard normal (n>10)
distributions: test equal       obs. from one group; all
N (0,1)
means: Mann Whitney U test      obs. ranked together
Comparing more than two groups of
data
One-way ANOVA: Testing if          SSG /( K  1)
all groups are equal (norm.)                                   FK 1,n  K
SSW /(n  K )
Kruskal-Wallis test: Testing if Based on sums of ranks
all groups are equal            for each group; all obs.           K 1
2

ranked together
Two-way ANOVA: Testing if           SSG /( K  1)            FK 1,( K 1)( H 1)
all groups are equal, when
you also have blocking           SSE /(( K  1)( H  1))
Two-way ANOVA with
SSI /(( K  1)( H  1))
interaction: Testing if groups                             F( K 1)( H 1), HK ( L 1)
and blocking variable interact    SSE /( HK ( L  1))
Studying population proportions

Test of population                               Standard normal
p 0
proportion in one                                     N (0,1)
group (large             0 (1   0 ) / n
samples)
Comparing the               px  p y             Standard normal
population           p0 (1  p0 ) p0 (1  p0 )

proportions in two        nx          ny              N (0,1)
groups (large
(p0 common estimate)
samples)
Regression tests
Test of regression slope:        b1  *
Is it  * ?
tn  2
sb1
Test on partial regression         b j  *
coefficient: Is it  * ?                               tn  K 1
sb j

Test on sets of partial
regression coefficients:     ( SSE (r )  SSE ) / r
Fr , n  K  r 1
Can they all be set to                 se2
zero (i.e., removed)?
Model tests
Contingency table test: Test if
there is an association             r    c     (Oij  Eij )2
between the two attributes in      
i 1 j 1        Eij
(2r 1)(c1)
a contingency table

Goodness-of-fit test: Counts             K
(Oi  Ei )2
in K categories, compared to             E
i 1
 K 1
2

expected counts, under H0                         i

Tests for normality:
•Bowman-Shelton
•Kolmogorov-Smirnov
*                      *
Tests for correlation
Test for zero population
correlation (normal              r n2
distribution)                                         tn  2
1 r   2

Test for zero correlation   Compute ranks of x-      Special
(nonparametric): Spearman   values, and of y-
rank correlation                                   distribution
values, and compute
correlation of these
ranks
Tests for autocorrelation
n
The Durbin-Watson test
(based on normal
 (et  et 1 ) 2
i2
Special distribution
n
assumption) testing for
autocorrelation in                  et2
i 1
regression data
The runs test              Counting the number Special distribution,
(nonparametric), testing   of ”runs” above and or standard normal
for randomness in time     below the median in
the time series            N (0,1)
for 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?

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