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Outcome utilities– clinical applications 175
was less than 10 years. The greater their risk aversion, the shorter their acceptable life expectancy for the guaranteed
outcome.
As we saw in the previous section, the answer to this question implies a value for the parameter
used in an expo-
nential utility model. We called
the risk parameter because it characterizes the patient’s risk attitudes. Recall that if
denotes the patient’s answer to the assessment question, the expression we derived for determining the value for
from
is:
05 120
105
.
.
//yearsX
where 0.5 is the probability of surviving the gamble and 20 years is the patient’s resulting life expectancy.
For example, suppose that our 60-
year- old man was risk averse and responded that having a life expectancy of
4 years was equivalent to the gamble. That is X equals 4 years. The value for the patient’s risk parameter then
wouldbe
05 120
105
05 4120
05
015
./ /
.
./ /
.
./
X yearsyears
year
On the other hand, suppose the 60- year- old man was highly risk averse and responded with a life expectancy of only
2 years. The value for the patient’s risk parameter would then be
05 2120
05
040
./ /
.
./
yearsyears
year
Therefore, as discussed in the previous section, the value for the risk parameter γ increases as risk aversion increases.
Figure9.8 shows how the risk parameter value changes with the response to the assessment question we are using
with this patient.
The risk parameter value of 0.15/year we have determined for our patient may seem like just another number
derived from a complicated equation using an answer to a hypothetical question. However, this value provides us
with an understanding of how this patient feels about the two treatments. Recall that the risk parameter value adjusts
an exponential utility model to match this patient’s risk attitudes. We can combine the resulting utility model with
probabilities from the two survival models to determine the patient’s expected utilities for the two treatments. The
optimal treatment for the patient will be the treatment with the greater expected utility.
When we discuss the details of survival models in Chapter11, we will see that survival curves like those shown in
Figure9.8 are not exponential survival models. This means computing the expected utilities for the two treatments
0.0
0.1
0.2
0.3
0.4
0.5
0123456789
10
Exponential utility model parameter (γ)
Life expectancy with disease (X)
Figure9.8 Exponential utility model parameter implied by the choice of life expectancy with disease such that living with the disease is equivalent
to a risky treatment that has a 50% chance of causing immediate death and a 50% of leading to a life expectancy of 20 years.
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176 Medical decision making
requires the longer calculations demonstrated in the previous chapter (see Table8.1). When we do those calculations,
the results are
USurgery
2
3989
.
URadiotherapy
2
5785
.
Since EU(Radiotherapy) is greater than EU(Surgery), it follows that radiotherapy is the preferred treatment for a
60- year- old man with a risk parameter value of 0.15/year.
Figure9.9 uses certainty equivalents to compare the two treatment alternatives based on different risk parameter
values. Recall that a certainty equivalent summarizes a gamble, like a survival model, by determining the guaranteed
outcome that has the same utility as the expected utility for the gamble. The certainty equivalent and the gamble are
equivalent because they have the same utility given the patient’s risk attitudes.
With an exponential utility model, determining a certainty equivalent is a straightforward calculation that depends
on the form of the exponential utility model that has been used.
For example, suppose that u is the expected utility for one of the survival curves. The certainty equivalent for that
survival curve is the value x such that the utility for x equals u. Recall that there are three forms of the exponential
utility model. The expected utilities for the two treatments shown above were computed using the following form of
the exponential utility model:
tility of x
e
x
1
Therefore, the certainty equivalent x for a gamble with expected utility u satisfies the following equation:
e
x
1
Using a little high school algebra, this equation can be rearranged to show that
x
u
ln 1
Similar expressions are used to compute the certainty equivalent for the other two forms of the exponential utility
model.
2.0
3.0
4.0
5.0
6.0
0.00 0.05 0.10 0.15
0.20
Certainty equivalent (years)
Risk parameter
γ
(/year)
Radiotherapy
Surgery
0.15/year
3.26 years
2.97 years
Figure9.9 Comparison of surgery and radiotherapy as treatments for lung cancer in a 60- year- old man based on risk parameter value. Vertical
axis expresses the difference between the two treatments as certainty equivalents.
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Outcome utilities– clinical applications 177
Returning to our 60- year- old patient, we can use this result to determine the certainty equivalents for each of the two
treatment options. Recall that the expected utility we calculated for surgery is 2.3989 (u) and 0.1500/year is this patient’s
risk parameter (γ). Therefore,
for surgery
ln 1
u
ln ./ .
./
101500 2 3989
0 1500
year
year
Similarly,
for radiotherapy
year
year
ln ./ .
./
.
101500 2 5785
0 1500
3226 year
s
In other words, for a 60- year- old man who answered the assessment question with a 4- year life expectancy, the uncer-
tainty they face with surgery has the same utility as facing a guaranteed 2.97 years of life. Similarly, the uncertainty of
facing radiotherapy has the same utility as facing a guaranteed 3.26 years of life. This means for this patient the loss
from undergoing surgery rather than radiotherapy is approximately the difference between living 3.26 or 2.97years.
That difference works out to about 3months.
9.3.4 A simpler assessment question
We have just seen that the answer to an assessment question provides enough information about the patient’s risk
attitudes to identify which treatment they preferred. The process requires that the patient provides a specific value as
the answer to the following question:
Suppose there is a risky treatment for a disease you have. The chances the treatment will cause your immediate death is 50%.
However, if you survive treatment you will live, on average, another 20 years. Your alternative is to take your chances with the
disease, which will mean a shorter average life. How short would your average life with the disease have to be before you would
choosethe risky treatment?
Some patients will be unable to provide an exact value when answering this question. However, that value is needed
to determine the risk parameter. The value for the risk parameter is then used in the expected utility calculations which
completes the comparison of the treatments. What if the patient is unable to fully answer the assessment question?
What if they cannot decide what the exact life expectancy with disease must be so that living with the disease would
be equivalent to the gamble posed in the assessment question?
Result: Certainty equivalent forthe exponential utility model
The following expressions determine the certainty equivalent (CE) for the corresponding exponential utility
model form
1 eu
x
ln 1 u
U
1 e
u
x
ln 1
u
U
ee
ee
u
Ax
AB
ln ee
eu
AAB
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178 Medical decision making
It turns out we do not actually need the patient’s exact answer to the assessment question. Looking at Figure9.10,
notice that the patient in this example would prefer radiotherapy over surgery as long as their risk parameter value is
greater than 0.0533/year. This means that in order to conclude radiotherapy is the best treatment for this patient, we
only need to know that their risk parameter is greater than 0.0533/year.
We might call 0.0533/year the threshold value for the risk parameter since this is where the preferred treatment
switches between surgery and radiotherapy. Figure9.11 is a copy of Figure9.8 and shows that an answer of 6.5 years
to the assessment question implies the risk parameter value of 0.0533/year. In other words, the risk parameter equals
the threshold value if the patient believes that living with a life expectancy of 6.5 years is equivalent to the risky treat-
ment in the assessment question. We will round 6.5 to 6 years to reduce the complexity of the question.
Therefore, in order to decide which treatment is best for this patient, we only need to know that their answer to the
assessment question is less than 6 years. This means we can simplify the assessment question to the following yes-
no
question:
Suppose there is a risky treatment for a disease you have. The chances the treatment will cause your immediate death is 50%.
However, if you survive treatment you will live, on average, another 20 years. Your alternative is to take your chances with the
disease, which would mean the average length of life you can expect is at least 6 years. Would you choose the risky treatment?
2.0
3.0
4.0
5.0
6.0
0.00 0.05 0.10 0.15 0.20
Certain equivalent (years)
Risk parameter γ (/year)
Radiotherapy
Surgery
0.0533/year
Figure9.10 Copy of Figure9.9 showing the risk parameter value γ implying surgery and radiotherapy are equivalent lung cancer treatments for a
60- year- old man.
0.0
0.1
0.2
0.3
0.4
0.5
0123456789
10
Exponential utility model parameter (γ)
Life expectancy with disease (X)
0.0533/year
6.5 years
Figure9.11 Copy of Figure9.8 showing the assessment question answer Ximplying a risk parameter value of 0.0533/year, which is where surgery
and radiotherapy are equivalent lung cancer treatments for a 60- year- old man.
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Outcome utilities– clinical applications 179
If the patient answered “yes”, surgery is their preferred treatment. If the patient answers “no” radiotherapy is the
preferred treatment.
9.3.5 Generalized age- andgender- specific clinical policy
The approach we have described has determined the risk- adjusted clinical policy for a 60- year- old man. The treatment
recommended by the policy uses the patient’s response to a relatively simple question to determine which treatment
best represents how the patient feels about risk. Extending this clinical policy to other demographic groups would
involve the following changes.
The first change adjusts the survival model to reflect how the group’s demographics affect the uncertainty of how
long a patient in the group will live. The discussion of survival models in a later chapter shows how to make age and
gender adjustments to survival curves. For example, Figure9.12 shows the resulting surgery and radiotherapy sur-
vival curves if the demographic group consists of 30- year- old women.
The second change determines how the threshold value for the risk parameter is affected by the survival model
adjustments. Figure9.13 shows the comparison of the treatments for the example of a 30- year- old woman. Notice that
Radiotherapy
Surgery
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
010203040506
070
Probability survival ≥ t years
Length of life, t (years)
Figure9.12 Surgery and radiotherapy survival curves for 30- year- old woman. The survival curves for a 60- year- old man, from Figure9.7, are
shown in grey.
2.0
3.0
4.0
5.0
6.0
0.00 0.05 0.10 0.15
0.20
Certainty equivalent (years)
Risk parameter
γ
(/year)
Radiotherapy
Surgery
0.0809/year
Figure9.13 Comparison of surgery and radiotherapy as treatments for lung cancer in a 30- year- old woman based on risk parameter value. Vertical
axis expresses the difference between the two treatments as certainty equivalents. Comparison of the two treatments for a 60- year- old man, from
Figure9.10, is shown in grey.
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180 Medical decision making
the threshold value for the risk parameter has increased to 0.0809/year for this group. When the risk parameter for this
patient is greater than 0.0809/year, radiotherapy is the preferred treatment. When the risk parameter is less than
0.0809/year surgery is preferred.
The other changes are to the assessment question. First, according to actuarial tables, the life expectancy for a
30-
year- old woman is 51 years. Rounding this life expectancy to 50 years, the hypothetical treatment in the assessment
question will now indicate a 50- year life expectancy if the patient survives the risky treatment. Figure9.14 shows how
answers to the revised assessment question affect the implied risk parameter. Notice that a response of 8.3 years to the
question would imply the threshold value of 0.0809/year for the risk parameter. The second change to the assessment
question incorporates this new value of 8.3 years as the life expectancy for living with the disease. Therefore, rounding
8.3 to 8.0 years, the yes-
no question asked of a 30- year- old woman would be:
Suppose there is a risky treatment for a disease you have. The chances the treatment will cause your immediate death is 50%.
However, if you survive treatment you will live, on average, another 50 years. Your alternative is to take your chances with the
disease, which would mean the average length of life you can expect is at least 8 years. Would you choose the risky treatment?
As before, answering “yes” implies risk attitudes that favor surgery, whereas answering “no” implies risk attitudes
favoring radiotherapy.
Table9.1 generalizes this approach to a range of ages for men and women. Figure9.15 shows a decision tree repre-
sentation of the assessment question that uses the values from the table. The patient’s life expectancy, if they survive
the hypothetical treatment in Option B (X
B
), approximates the patient’s disease- free life expectancy, rounded to the
nearest multiple of 5 years. The life expectancy with Option A (X
A
) is the corresponding life expectancy implying
thethreshold value for the risk parameter with Option A and Option B are equivalent.
Notice that the table contains two additional values p=0.25 and p=0.75. So far the discussion has focused on assess-
ment questions where the survival probabilities were set at 0.50. Using these different survival probabilities would
repeat the assessment question to verify the patient’s response.
Using the notation in Figure9.15, the assessment question can be expressed as follows.
Suppose there is a risky treatment for a disease you have. The chances the treatment will cause your immediate death is p. However,
if you survive treatment you will live another X
B
years on average. Your alternative is to take your chances with the disease, which
would mean the average length of life you can expect is at least X
A
years. Would you choose the risky treatment?
9.3.6 Risk- adjusted clinical policies– what does it all mean?
In this section, we have shown how to personalize a clinical policy by the adjusting treatment recommendations
according to the risk attitudes of the patient. The adjustments are tailored to the patient’s demographics so that (1) the
0.0
0.1
0.2
0.3
0.4
0.5
012345678
910
Exponential utility model parameter (γ)
Life expectancy with disease (X)
0.0809/year
8.3 years
Figure9.14 Exponential utility model parameter implied by the choice of life expectancy with disease such that living with the disease is
equivalent to a risky treatment that has a 50% chance of causing immediate death and a 50% of leading to a life expectancy of 51 years.
Thecorresponding curve used to design the clinical policy for the 60-
year- old man, from Figure9.8, is shown in grey.
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Outcome utilities– clinical applications 181
survival models predicting outcomes reflect that characteristics of the patient and (2) the assessment process is framed
in a clinical context that is meaningful to the patient.
9.4 Helping patients communicate their preferences
This final section shifts the focus away from determining the optimal decision for a patient and considers instead
how treatment decisions should be communicated to patients. In 1979Daniel Kahneman and Amos Tversky pub-
lished a paper titled Prospect Theory: An Analysis of Decision under Risk (Kahneman and Tversky,1979). This paper
described a descriptive model of decision making that ultimately led to the awarding of a Nobel Prize in Economic
Sciences. Prospect Theory provides a general framework for understanding how people actually make decisions.
However, Prospect Theory also provides important insights into how the description of the options in a deci-
sion can distort the patient’s understanding of the decision they face. Kahneman and Tversky called this the
framing problem, which can be seen as a generalization of the cognitive biases discussed in Chapter3. This section
describes how accounting for the framing problem should be considered when discussing treatment options with
patients.
Tversky and Kahneman first described the framing problem in an article published in 1981 (Tversky and
Kahnman,1981). This paper described a study in which subjects were asked to state their preferences when faced with
hypothetical decision problems. The following is an example of one of the decision problems they posed:
Table9.1 Assessment question values used to determine when surgery is preferred to radiotherapy for a patient. Option A in the
question has a given life expectancy X
A
. Option B is a risky treatment with a given survival probability (P) and a given life expectancy
X
B
with survival. Surgery is the preferred treatment if the patient prefers Option B.
Age (years)
X
B
=Life expectancy
with Option B (years)
X
A
=Life expectancy with Option A (years)
p=0.25 p=0.50 p=0.75
Women 30 50 3 8 19
40 40 3 8 17
50 35 3 8 16
60 25 3 7 14
70 15 3 6 10
80 10 3 5 8
90
5 2 3 4
Men 30 45 3 8 18
40 40 3 8 17
50 30 3 7 15
60 20 3 6 12
70 15 3 6 10
80 10 3 6 8
90
5 2 3 4
1–p
Uncertain survival with
X
A
year life expectancy
Uncertain survival with
X
B
year life expectancy
Live 0 years
(D1)
Option A
Option B
p
(D2)
Figure9.15 Decision tree representing general form of the assessment question for a risk- adjusted clinical policy. The value for X
B
represents the
patient’s life expectancy without disease. The value for the survival probability is 0.25, 0.50, or 0.72 and the value for X
B
is selected from Table9.1.
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182 Medical decision making
Problem 1: Given an outbreak that will kill 600 people if untreated, choose between the following two programs:
200 people are saved
Program A
1/3
600 people are saved
No one is saved
2/3
Program B
When this problem was presented to 152 subjects, 72% preferred Program A. This makes sense. All of the outcomes are
expressed as gains in terms of lives saved. Moreover, both programs have the same expected value for the number
saved– 200 people. However, with Program A those 200 saved lives are a certainty, whereas with Program B there is
the possibility that no one will be saved. In Prospect Theory, when outcomes are gains, there is preference for a guar-
anteed outcome over a gamble that awards that same outcome, on average. This preference for certainty is called the
certainty effect for gains.
Tversky and Kahneman also presented a second problem to another 155 subjects:
Problem 2: Given an outbreak that will kill 600 people if untreated, choose between the following two programs:
400 will die
Program C
1/3
No one will die
600 people will die
2/3
Program D
Of course, the two problems describe the same two programs. With Program C, 400members die in the population of
600 meaning that 200 will saved. This is the same as the outcome for Program A in the first problem. Similarly,
Program D is simply a rewording of Program B. However, 78% of the subjects in the second survey preferred Program
D in Problem 2, which reverses the preferences stated for Problem 1.
Notice that Program C and Program D are described as losses rather than gains. Prospect Theory attributes the
switch in preferences to the differences in how people view potential gains and losses. In the terminology of Prospect
Theory, this change in preferences demonstrates the framing effect.
The clinical relevance of the framing effect is that how risks are expressed to patients can affect their choices.
Mathematically, telling a patient that there is a 10% chance they will die from an operation is equivalent to telling the
patient there is a 90% chance they will survive the operation. Both statements contain the same mathematical content,
but they can have a very different cognitive meaning to the patient.
In 1982, a group of researchers, that included one of the coauthors of this book, published the results of a study that
demonstrated the framing effect in a clinical setting (McNeil etal., 1982). This study, once again, used the choice
between surgery or radiotherapy for treatment of lung cancer in a 60- year- old patient. For the purposes of this study,
surgical treatment was summarized by a 10% operative mortality rate with a 6.1- year life expectancy whereas radio-
therapy had a 0% procedure mortality rate with a 4.7-
year life expectancy. The life expectancy for surgery included the
10% mortality rate. The goal of the study was to document how the framing of the choice between the two treatments
would affect preferences.
The study population included 238 patients, as well as 424 radiologists and 491 business school students. This sum-
mary focuses only on the patients, who were outpatients at the Palo Alto VA medical facility. None had lung cancer and
all were male. The average age was 58 years.
Three framing factors were included in the description of how the two treatment options were described. The choice
between the treatment options was described to the study participants using different framing models based on the
following three factors:
Gain vs. loss: This factor expressed the treatment alternatives either in terms of gains (e.g., 90% survive the treatment)
or losses (e.g., 10% die during treatment).
Life expectancy vs. cumulative survival: Treatment alternatives were characterized either by life expectancy (e.g., life
expectancy with surgery is 6.1 years) or survival (e.g., 34% of patients undergoing surgery are alive after 5years).
Treatment identified vs. treatment not identified: The actual treatment was named (e.g., treatment is surgery) or not
(e.g.,Treatment A is used).
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Outcome utilities– clinical applications 183
The third factor was used to determine how past experience (e.g., “my brother had surgery for his lung cancer and
didn’t do well”) affected responses. The subjects were asked to state their preference for a choice between the treat-
ments when described by combinations of the three framing factors.
For example, the following wording was used to describe the treatment decision with the three factors: (1) gain,
(2)cumulative survival, and (3) treatment identified.
Of 100 people having surgery, 90will survive the treatment, 68 will be alive by one year and 34 will have died by five years.
Of100people having radiation therapy, all will survive the treatment, 77will be alive by one year and 22will be alive by five years.
On the other hand, the following wording was used to describe the treatment decision with the three factors: (1) loss,
(2) life expectancy, and (3) treatment not identified.
Of 100 people having Treatment A, 10will die during treatment, 32will have died by one year and 66will have died by five years.
Of 100 people having Treatment B, none will die during treatment, 23will die by one year and 78will die by five years.
Table9.2 shows how preference for radiotherapy changed according to the framing of the treatment decision. In all
cases, the percentage of the patients preferring radiotherapy almost always doubled when the operative mortality rate
for surgery was expressed as a loss rather than a gain. For example, when the long- term survival was expressed as a
life expectancy and the treatments were identified, 31% of the patients preferred radiotherapy when they were told
that the chances of surviving surgery were 90%. The percentage of patients preferring radiotherapy increased to 68%
when the risk for surgery was described as a 10% chance of dying.
The percentages also changed if the treatments were identified in the description of the options. Patients preferred
radiotherapy when they were told the decision was between radiotherapy or surgery. Expressing long- term survival as
a life expectancy rather than cumulative survival probabilities also increased the attractiveness of radiotherapy to the
patients. In short, the framing used to describe the options mattered to the patients when deciding which treatment they
preferred. This means a patient’s preferences for a treatment option can be manipulated by simply changing the word-
ing of how the risks are expressed. When an operative risk is expressed as a probability of death rather than a probability
of survival, the patient is more inclined to choose an alternative that avoids the risk of short- term mortality.
In summary, Kahneman’s and Tversky’s descriptive model of how individuals make decisions reveals the subtle
influence using either gains or losses to describe a decision can have on how that decision is perceived. Clinicians
should be aware of these cognitive biases and try to avoid unconsciously influencing the patient’s choice. Explaining
a choice in both frames, as either a choice between losses or a choice between gains, may help.
Table9.2 Percentage ofpatients preferring radiotherapy over surgery according tothe framing model used todescribe thechoice between
thetreatment options.
Description of long- term
survival
Treatment identified vs. treatment
not identified
Surviving surgery expressed as
gain vs. loss
Percent preferring
radiotherapy
Cumulative survival Treatment identified 10% chance of dying 40%
90% chance of surviving 22%
Treatment not identified 10% chance of dying 35%
90% chance of surviving 19%
Life expectancy Treatment identified 10% chance of dying 68%
90% chance of surviving 31%
Treatment not identified 10% chance of dying 50%
90% chance of surviving 27%
Adapted from McNeil etal. (1982).
Summary
• A parametric utility model determines the values for outcome utilities by using a mathematical expression that
includes terms, called parameters, that can be adjusted to match someone’s risk attitudes.
• The exponential utility uses the following formula to express an individual’s utility for an outcome measured by
the variable x.
x
1
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184 Medical decision making
For example, x might be the length of the individual’s life. The term γ quantifies the individual’s risk attitudes
and is expressed in units that are one over the units of x. The following is an alternate form of the exponential
utility function:
e
x
1
The exponential utility model can be scaled so that U(A)=0 and U(B)=1 by the following form:
ee
ee
Ax
AB
• An individual’s preferences have the delta property for numerically valued outcomes if for any outcomes X
A
, X
B
,
and X
C
and probability p, the following equivalence:
X
A
equivalent to
p
X
B
1−p
X
C
implies the same equivalence if Δ is added to each of the outcomes. That is, for any Δ, it also is true that:
p
X
B
+
∆
X
C
+
∆
1−p
X
A
+ ∆ equivalent to
Someone whose risk attitudes have the delta property is also said to have constant risk attitudes.
• When an individual’s risk attitudes match the delta property their outcome utilities can be determined by an
exponential utility model.
•
Suppose that someone’s risk attitudes mean the following equivalence is true:
Live
X
A
years equivalent to
Live X
B
years
Live 0 years
p
1−p
The value of the exponential utility model parameter γ satisfies the following expression:
1
1
e
e
p
X
X
A
B
The value for γ satisfying this expression can be solved by trial- and- error methods that evaluate the ratio on the
left for different values of the parameter γ until the ratio equals the probability on the right.
• Guaranteed outcome assessment uses questions that ask the individual to compare outcomes that are guaran-
teed lengths of life.
• Uncertain outcome assessment uses questions that ask the individual to compare outcomes that are uncertain,
typically expressed as a life expectancy or a survival model.
• Suppose the following equivalence is determined by an assessment question in which outcomes are expressed
as the life expectancies for exponential survival models, as shown in the following diagram:
p
Life expectancy = X
B
Live 0 years
1−p
Life expectancy =
X
A
equivalent to
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