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Decision theory and Bayesian statistics. More repetition . Tron Anders Moger 22.11.2006. Overview. Statistical desicion theory Bayesian theory and research in health economics Review of previous slides. Statistical decision theory.

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Decision theory and Bayesian statistics. More repetition

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## Decision theory and Bayesian statistics. More repetition

Tron Anders Moger

22.11.2006

### Overview

• Statistical desicion theory

• Bayesian theory and research in health economics

• Review of previous slides

### 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 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.

### 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 for choosing actions

• The minimax regret criterion for choosing actions

• The expected monetary value criterion for choosing actions

states

actions

### 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.

### Example

Find that action C is preferred under the maximin criterion

Regret table:

states

actions

Action C is also preferred under the minimax criterion

### Expected monetary value criterion

• Need probabilities for each state

• Assume P(no outbreak)=P1=95%, P(small outbreak)=P2=4.5%, P(pandemic)=P3=0.5%

• EMV(A)=P1*M11+P2*M12+P3*M13=

0*0.95-500*0.045-100000*0.005= -522.5

• EMV(B)=-55.45

• EMV(C)=-1000

• Should choose action B

### 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

### 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

### 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:

• 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:

### 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

### 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

### 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

• EVPI=0.95*1+0.045*0+0.005*9000=45.95

### 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 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.

### 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!!

### 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.

### Probability postulates 3

• Multiplication rule: For general outcomes A and B:

P(AB)=P(A|B)P(B)=P(B|A)P(A)

• Indepedence: A and B are statistically independent if P(AB)=P(A)P(B)

• Implies that

### 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(A|B)P(B)+P(A| )P( )

For twins: P(B)=1/3 P( )=2/3

P(A)=1 · 1/3+1/2 · 2/3=2/3

### 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

### Example: Cervical cancer

• B=Cervical cancer

• A=Positive test

• P(B)=0.0001P(A|B)=0.9 P(A| )=0.001

• Only 8% of women with positive tests are ill

### 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

### Example: Doping tests

• Let’s 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 A=at least one test is positive

Complement rule Rule of independence 50 terms

### Expected values and variance

• Remember the formulas E(aX+b) = 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?

### Distributions

• Distributions we’ve 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

### 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

### And this one?

• Marginal probability of no rain: 0.1+0.2+0.05+0.01=0.36

• 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/(0.01+0.04+0.05)=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: 0.1+0.05+0.05=0.2

P(no rain,no wind)=0.36*0.2=0.072≠0.1

### 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->Marginal mean!!

• Mean crop for wheat 1 given that you use fertilizer ->Conditional mean!!

(corresponds to mean for a single cell in our field)

### Chi-square test:

• Expected cell values: Abortion/op.nurses: 13*36/70=6.7

Abortion/other nurses: 13*34/70=6.3

No abortion/op.nurses: 57*36/70=29.3

No abortion/other nurses: 57*34/70=27.7

• 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)

### 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!

### In SPSS:

Check Expected under Cells, Chi-square under statistics, and

Display clustered bar charts!

### Next time:

• Find some topics you don’t understand, and we can talk about them