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Decision theory and Bayesian statistics. Tests and problem solving

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Presentation Transcript

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

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:

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 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.

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!

Studying population proportions

(p0 common estimate)

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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