Introduction to decision analysisJouni Tuomisto

THL

Decision analysis is done for purpose: to inform and thus improve action

QRA

Decisions by an individual vs. in a society

• In theory, decision analysis is straightforward with a single decision-maker: she just has to assess her subjective probabilities and utilities and maximize expected utility.

• In practice, there are severe problems: assessing probabilities and utilities is difficult.

• However, in a society things become even more complicated:– There are several participants in decision-making.– There is disagreement about probabilities and utilities.– The decision models used are different.– The knowledge bases are different. NOTE! In this course,

"knowledge" means both scientific (what is?) and ethical (what should be?) knowledge.

Probability of an event x

• If you are indifferent between decisions 1 and 2, then your probability of x is p=R/N.

p

1-p

Red

x does not happen

x happens

White ballDecision 1

Red ball

Decision 2

Prize

100 €

0 €

100 €

0 €

Outcome measures in decision analysis

Outcome measures in decision analysis

– DALY: disability-adjusted life year

– QALY: quality-adjusted life year

– WTP: willingness to pay

– Utility

Disability-adjusted life year

– The disability-adjusted life year (DALY) is a measure of overall disease burden, expressed as the number of years lost due to ill-health, disability or early death. (Wikipedia)

– Originates from WHO to measure burden of disease in several countries in the world.

DALYs in the world 2004

– Source: Wikipedia

How to calculate DALYs

– DALY= YLL+YLD– YLL=Years of life lost

– YLD=Years lived with disability

– YLD = #cases*severity weight*duration of disase

– More DALYs is worse.

Weighting of DALYs

– Discounting

– present value Wt = Wt+n*(r+1)-n

– Where W is weight, r is discount rate, and n is number of years into the future and t is current time

– Typically, r is something like 3 %/year.

– Age weighting– W = 0.1658 Y e-0.04 Y

– where W is weight and Y is age in years

Discounting Wt = Wt+n (1+r)-n

Present value of a future outcome at different discount rates

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60

Years into the future

Ne

t p

rese

nt

valu

e

0

0.01

0.03

0.05

Age weighting with DALYs W = 0.1658 Y e-0.04 Y

Age weighting in DALY

00.20.40.60.8

11.21.41.61.8

0 20 40 60 80 100

Age (years)

Estimating QALY weights

• Time-trade-off (TTO): Choose between:– remaining in a state of ill health for a period of time, – being restored to perfect health but having a shorter life

expectancy.

• Standard gamble (SG): – Choose between:– remaining in a state of ill health for a period of time, – a medical intervention which has a chance of either

restoring them to perfect health, or killing them.

• Visual analogue scale (VAS): Rate a state of ill health on a scale from 0 to 100, with 0 representing death and 100 representing perfect health.

QALY weight of disease x (standard gamble)

• Adjust u in such a way that you are indifferent between decisions 1 and 2.

• Then, your QALY weight is u(x).

u

1-u Dead

Healthy

Live with disease

Disease

Treatment

Utility

?

0

1

Standard descriptions for QALYs

• E.g. as the EuroQol Group's EQ5D questionnaire

• Categorises health states according to the following dimensions:

– mobility, – self-care, – usual activities (e.g. work, study, homework or

leisure activities), – pain/discomfort – anxiety/depression.

Measuring utilities

• Adjust u in such a way that you are indifferent between the two options.

• Then, your utility for option x is u(x).

u

1-u Worst outcome

Best outcome

Choose option xOption

Choose gamble

Utility

?

0

1

Utility of money is not linear

Utility of money

CAFE clean air for Europe

Value of statistical life VSL

• Measure the willingness to accept slightly higher mortality risk.

– E.g. a worker wants 50 € higher salary per month as a compensation for a work which has 0.005 chance of fatal injury in 10 years.

– 50 €/mo*12 mo/a*10 a / 0.005 = 1200000 € / fatality

• VSL is the marginal value of a small increment in risk. Of course, it does NOT imply that a person’s life is worth VSL.

• A similar measure: VOLY = value of life year.

The ultimate decision criterion: expected utility

• Max(E(u(dj)))=Maxj (∑i u(dj,θi) p(θi) )

• Calculate the expected utility for each decision d option j.

• Pick the one with highest expected utility.

Which option is the best?0.03

0.15

Healthy

Swine flu

ReactionVaccination

Swine flu

Do nothing

Utility

0.3

1

0.3

0

Healthy

0.003

1

Which option is the best?

• u(Vaccination)=0.976

• u(Do nothing)=0.895 Choose vaccination

0.03

0.15

Healthy

Swine flu

ReactionVaccination

Swine flu

Do nothing

u; E(u)

0.3;0.009

1;0.85

0.3;0.045

0;0

Healthy

0.003

1;0.967

Limitations of decision trees

• A decision tree becomes quickly increasingly complex. This only contains two uncertain variables and max three outcomes of a variable.

0.03

0.15

Healthy

Vaccination

Swine flu

Do nothing

Complications

0.003Swine flu

Complications

Swine flu

Causal diagrams: a powerful tool for description

Vaccination Outcome

Complications

Bayesian belief networks

• Arrows describe causal connections by conditional probabilities.

• P(swine flu|vaccination)• P(complications|vaccination, swine flu)• P(outcome|swine flu, complicatons)These probabilities describe the whole model.

Estimating societal costs of health impacts

• In theory, all costs should be estimated.

• In practice, the main types considered include– Health case costs (medicine, treatment…).– Loss of productivity (absence from work, school).– WTP of the person to avoid the disease.– The societal cost of disease to other people (relatives

etc) is NOT considered.

St Petersburg paradox

• Consider the following game of chance: you pay a fixed fee to enter and then a fair coin is tossed repeatedly until a tail appears, ending the game. The pot starts at 1 dollar and is doubled every time a head appears. You win whatever is in the pot after the game ends. Thus you win 1 dollar if a tail appears on the first toss, 2 dollars if a head appears on the first toss and a tail on the second, 4 dollars if a head appears on the first two tosses and a tail on the third, 8 dollars if a head appears on the first three tosses and a tail on the fourth, etc. In short, you win 2k−1 dollars if the coin is tossed k times until the first tail appears.

• What would be a fair price to pay for entering the game?

• Solved by Daniel Bernoulli, 1738

St Petersburg paradox (2)

• To answer this we need to consider what would be the average payout: With probability 1/2, you win 1 dollar; with probability 1/4 you win 2 dollars; with probability 1/8 you win 4 dollars etc. The expected value is thus

Example of a model with causal diagram

• Dampness and asthma