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Introduction to decision analysis
Jouni Tuomisto
THL
Decision analysis is done for purpose:
to inform and thus improve action
Q
R
A
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:
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–
–
–
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
p
Decision 1
1-p
Red ball
White ball
Prize
100 €
0€
x happens 100 €
Decision 2
x does not
happen
0€
• If you are indifferent between decisions 1 and 2,
then your probability of x is p=R/N.
Outcome measures in decision analysis
Outcome measures in decision
analysis
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–
–
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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.
Disability weights
• http://en.opasnet.org/w/Disability_weights
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
Net present value
1.2
1
0.8
0
0.01
0.03
0.05
0.6
0.4
0.2
0
0
20
40
Years into the future
60
Age weighting with DALYs
W = 0.1658 Y e-0.04 Y
Age weighting in DALY
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
20
40
60
Age (years)
80
100
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)
Utility
Disease
?
Healthy
1
Dead
0
Live with
disease
u
Treatment
1-u
• Adjust u in such a way that you are
indifferent between decisions 1 and 2.
• Then, your QALY weight is u(x).
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
Utility
Option
?
Choose option x
u
Choose
gamble
1-u
Best outcome 1
Worst
outcome
• Adjust u in such a way that you are
indifferent between the two options.
• Then, your utility for option x is u(x).
0
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
0.3
Healthy
1
0.003
Side effect
0
0.15
Swine flu
Vaccination
Do nothing
Swine flu
Utility
Healthy
0.3
1
Which option is the best?
0.03
Vaccination
Do nothing
u; E(u)
Swine flu 0.3;0.009
Healthy 1;0.967
0.003
Side effect
0.15
Swine flu 0.3;0.045
Healthy
• u(Vaccination)=0.976
• u(Do nothing)=0.895 Choose vaccination
0;0
1;0.85
Limitations of decision trees
0.03
Vaccination
0.003
0.15
Do nothing
Swine flu
Complicat
ions
Swine flu
Complicat
ions
Healthy
• A decision tree becomes quickly increasingly
complex. This only contains two uncertain variables
and max three outcomes of a variable.
Causal diagrams: a powerful tool for
describing decision analysis models
Swine flu
Outcome
Vaccination
Complicat
ions
Bayesian belief networks
• Arrows are causal dependencies described by
conditional probabilities.
• P(swine flu | vaccination)
• P(complications | swine flu)
• P(outcome | swine flu, complicatons)
• These probabilities describe the whole model.
Functional models
• Arrows are causal dependencies described by
(deterministic) functions.
• swine flu = f1(vaccination)
• complications = f2(swine flu)
• outcome = f3(swine flu, complicatons)
• These functions describe the whole model.
Functional vs. probabilistic
dependency
• Va1=2.54*Ch1^2
Va2=normal(2.54*Ch1^2,2)
Estimating societal costs of health
impacts
•
•
In theory, all costs should be estimated.
In practice, the main types considered include
1. Health case costs (medicine, treatment…).
2. Loss of productivity (absence from work, school).
3. 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