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Outcome utilities– clinical applications 185
The patient’s risk parameter then is given by:
pX X
p
AB
//1
1
• The parameter γ for the exponential utility model is called the risk parameter. When γ is positive, the individual
is risk averse. When γ is negative, the individual is risk tolerant. When γ is zero, the individual is risk
indifferent.
•
A risk- adjusted clinical policy includes the patient’s risk attitudes in the actions recommended for the care of the
patient.
•
The following expressions determine the certainty equivalent (CE) for the corresponding exponential utility
model form:
EU=1 − e
−γx
=u
ln 1 u
U
1 e
u
x
ln 1
u
U
ee
ee
u
Ax
AB
ln ee
eu
AAB
• How the descriptions of options are worded can greatly change a patient’s preferences for their alternatives in a
decision problem. Options that are described as gains, such as the probability of surviving a procedure, are more
attractive than options that are described as losses, such as the probability of dying from a procedure. Patients
also tend to place more importance on outcomes that are certain when compared to options that include uncer-
tainty. Collectively, these distortions to how possibilities are perceived, based on how they are described, are
known as the framing effect.
Epilogue
The concept of utility devised by von Neumann and Morgenstern provides a method for incorporating a patient’s risk
preferences into the decisions affecting their lives. However, the clinical application of utility requires an assessment
process that captures the patient’s preferences in a mathematical expression. Utility assessment involves questions that
many patients will find to be difficult to grasp. Utility assessment also can be time consuming. This chapter has
described an approach to utility modeling that focuses on these two challenges.
The foundation of the approach described in this chapter is a mathematical expression called a parametric utility
model. Several parametric utility models have been developed for the use of decision analysis in other domains.
Thischapter focused on use of the exponential utility model, which has a single parameter we called the risk param-
eter. An exponential utility model can be matched to the preferences of a patient by adjusting the value of the risk
parameter. The input from the patient required to make this adjustment can be captured by the patient’s answer to a
single assessment question. Techniques for wording this question were explored that reduce the cognitive challenges
in the assessment process.
This chapter demonstrated the use of parametric utility model by showing how a simple utility assessment process
could be used to personalize the recommendations of a clinical policy that targets a patient population. A clinical pol-
icy can be seen as identifying patient groups for whom a particular clinical action is appropriate. When risk attitudes
are used to define the boundaries for those groups, the assessment process simplifies to a relatively simple yes-
no
question. The structure of that question can be tailored to the patient’s circumstances, further reducing the cognitive
challenges of participating in the utility assessment process.
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186 Medical decision making
Therefore, an approach to representing patient preferences with regard to risk has been demonstrated. However,
what is missing from this approach is any consideration of how decisions can affect the quality of life. How to address
that consideration is the focus of the next chapter.
A.9.1 Exponential utility model parameter nomogram
This appendix presents a nomogram that simplifies determination of the exponential utility model parameter from the
answer to an assessment question based on deterministic outcomes such as live 5 years or live 25 years. The following
shows the structure of the assessment question:
Live
X
A
years equivalent to
Live X
B
years
Live 0 years
p
1−p
where
A
is less than
B
. Recalling the discussion of utility assessment in Chapter8, note that this question can be
posed either as lifetime-
tradeoff assessment (
A
is the unknown) or survival- tradeoff assessment (
p
is the unknown).
That is, the patient can be asked to state their certain equivalence for a risky treatment that results in a length of
life equal to
B
with a known probability of survival
p
. The patient’s answer to the lifetime- tradeoff assessment
question is
A
.
The following equality must hold when the two options represented in the assessment question are equivalent:
AB B
yearsyears yearsyears
10
Or, since we are using the exponential utility model
1
epe
XX
AB
which can be rearranged
1
1
e
e
p
X
X
A
B
Once again, our goal is to determine the value for γ that satisfies this expression for the values of X
A
, X
B
, and p in the
assessment question. We can simplify our task by rescaling the lifetimes to be expressed as proportions of X
B
, the life-
time if the patient survives the risky treatment in the assessment question. In effect, this eliminates one of the terms in
the expression. That is let
X
X
A
B
The assessment question would then be expressed in terms of a new time unit that lasts
B
years.
p
Live 1 time units
Live 0 time units
1−p
Live
ϕ time units equivalent to
Repeating the same steps as before leads to the expression:
1
1
e
e
p
The nomogram in FigureA.9.1 shows the relationship between the exponential utility model parameter
and the
survival probability p, for given values of
.
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Outcome utilities– clinical applications 187
For example, the assessment question used to illustrate the discussion in the main part of this chapter considered
Patient A. This individual believed that a guaranteed lifetime of 5 years is equivalent to a risky treatment that results
in a lifetime of 25 years with probability 0.5. In other words,
A
= 5years
B
= 25years
so
X
X
A
B
5
25
02
years
years
.
This means that we are interested in the curve labelled “0.200” in the nomogram. The value for p in the assessment
question is 0.5. Starting at this value on the vertical axis, we see that the corresponding value on the horizontal axis
is3.3. In other words, the exponential utility model parameter is 3.3when lifetime is measured in 25-
year time units.
Inorder to convert time back to 1-
year time units, we divide 3.3 by 25 years to show that
33
25
13
.
./
years
year
Which is the same value for γ determined by the iterative approach described in the main part of the chapter.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.975
X
A
/X
B
= 0.990
0.950
0.900
0.800
0.700
0.600
0.500
0.400
0.300
0.200
0.100
0.050
0.025
0.010
–5 –4 –3 –2 –1
012345
Survival probability for risky treatment in assessment (p)
Exponential utility model parameter (
γ
)
3.3
0.5
Figure A.9.1 Nomogram for determining the exponential utility model parameter γ from the answer to an assessment question that compared a
guaranteed life of length X
A
with a risk treatment that results in a life of length X
B
with probability p. Final value for γ is determined by dividing the
corresponding value on the horizontal axis by X
B
.
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188 Medical decision making
Bibliography
Kahnman, D. and Tversky, A. (1979) Prospect theory: an analysis of decision under risk. Econometrica, 47, 263–91.
McNeil, B.J., Weichselbaum, R., and Pauker, S.G. (1978) Fallacy of the five- year survival in lung cancer. New England Journal of Medicine, 299,
1397–1401.
McNeil, B.J., Pauker, S.G., Sox, H., and Tversky, A. (1982) On the elicitation of preferences for alternative therapies. New England Journal of
Medicine, 306, 1259–62.
Nease, R.F., Jr, Kneeland, T., O’Connor, G.T. etal. (1995) Variation in patient utilities for outcomes of the management of chronic stable
Angina. JAMA, 723, 1185–90
Raffia, H. (1968) Decision Analysis Introductory Lectures on Choices under Uncertainty, Addison- Wesley, Reading, MA.
Tversky, A. and Kahnman, D. (1981) The framing of decisions and the psychology of choice. Science, 211, 453–8.
https://t.me/medicina_free
Medical Decision Making, Third Edition. Harold C. Sox, Michael C. Higgins, Douglas K. Owens, and Gillian Sanders Schmidler.
© 2024 John Wiley & Sons Ltd. Published 2024 by John Wiley & Sons Ltd.
189
CHAPTER10
Outcome utilities– adjusting forthe quality oflife
10.1 Introduction
This chapter shows how to modify outcome utilities to reflect a patient’s concerns about the quality of their life. The
previous chapters treated the length of life as the patient’s only concern. In Chapter8, utility was introduced as a
method for valuing the possible lengths of life so that a patient’s risk attitudes are represented. The concepts presented
in this chapter will show how to broaden the view of outcomes beyond simply on how long life will last. Life is more
than just the passing of time. Life provides us with opportunities to experience what we enjoy and value as humans.
The pain and disabilities caused by diseases and their treatments can reduce our access to and enjoyment of those
experiences. That reduction lessens what we call the quality of life.
Anyone familiar with medicine knows that the quality of life matters to most patients. But does quality matter
enough to change preferences for options that also affect how long the patient will live? Are patients willing to make
tradeoffs between the length of life and its quality? A study that examined this fundamental question provides the
starting point for the chapter. Not surprisingly, the study showed that patients will accept a reduction in the probabil-
ity of a longer life in exchange for a better quality of life. Perhaps, more importantly, the study demonstrated a method
for measuring how patients feel about the quality of the life they will live. That method is what we will call the
quality- lifetime tradeoff model.
As the name suggests, this approach to measuring quality preferences is closely related to the lifetime- tradeoff
assessment method discussed in Chapter8. The corresponding assessment method measures how a patient feels about
a symptom or disability that reduces the quality of their life by the length of life they would give up to avoid that
symptom or disability. For example, a patient might believe that a life lasting 10more years with a painful symptom
would be equivalent to living only 8more years, but without that symptom. The reduction from 10 to 8 years quantifies
the importance of the painful symptom to that patient.
The previous chapter showed how mathematical expressions called parametric models can reduce the complexity of
analyzing clinical decisions. In this chapter, we will see that the quality- lifetime tradeoff model can be approximated
by a parametric model that extents the utility models for the length of life to reflect quality preferences. The result is a
simple expression for determining outcome utilities that quantifies both risk attitudes and quality preferences.
This
useful expression also requires a minimal assessment process. Using this parametric model requires assumptions
about the patient’s preferences that will be carefully considered in this chapter.
This chapter also describes an approach to capturing quality preferences that will be called the quality- survival trade-
off model. This second model measures how a patient feels about a symptom or disability by the risk to survival they
10.1
Introduction 189
10.2
Example– why thequality oflife matters 190
10.3
Quality- lifetime tradeoff models 193
10.4
Quality- survival tradeoff models 203
10.5
What does it all mean?– anextended example 209
Epilogue 217
Bibliography 217
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190 Medical decision making
would accept to avoid that symptom or disability. For example, suppose that in order to avoid a painful symptom, a
patient would be willing to undergo a risky treatment that had a 5% chance of causing immediate death. If the patient
survives the treatment, they will be free of that symptom for their remaining 10 years of life. In this case, the 5% risk
the patient would accept quantifies the importance of the symptom to that patient.
Like the quality-
lifetime tradeoff model, the quality- survival tradeoff model has a parametric form that uses a
simple assessment process. However, we will see that the assumptions required for this second model are fewer
innumber. Both models provide an effective representation of quality preferences; however, the fewer assump-
tions required by the quality-
survival tradeoff model suggest it is a superior approach to quality preference
representation.
This chapter concludes the discussion of outcome utilities that started in Chapter8. The discussion in these three
chapters has developed a general approach to quantifying a patient’s preferences for the possible outcomes of clinical
decisions. The final section of this chapter concludes this discussion of outcome utilities by determining the utilities for
the example used in a later chapter that describes the selection and interpretation of diagnostic tests.
10.2 Example– why thequality oflife matters
Why should the analysis of medical decisions consider the importance patients place on the quality of their lives? Most
patients want to avoid the unpleasant symptoms of their diseases and side effects of their treatments. However, is the
desire to avoid a reduction in the quality of life important enough to risk the possibility of shortening life? McNeil, and
her same two colleagues from the earlier lung cancer study, conducted a second study that explored the answer to this
question.
The previous two chapters discussed the study of preferences for lung cancer treatments by McNeil and her
colleagues (McNeil et al.,1978). That study demonstrated the importance of risk aversion when determining the
treatment a patient prefers. In their second study, McNeil and her colleagues demonstrated the importance of quality
preferences in treatment decisions (McNeil etal.,1981). They addressed the question: “Is the desire to avoid a reduction
in the quality of life ever important enough to take a chance on shortening the length of life?”
This second study involved the treatment of laryngeal cancer localized to the vocal cords (Stage III). The 3-
year
survival rates for the three treatment alternatives included in the study are listed in Table10.1.
Surgical removal of the larynx– laryngectomy– leads to better survival for patients with Stage III laryngeal cancer;
however, this treatment results in the loss of normal speech. After removal of the larynx, many patients can learn an
esophageal speech or adopt the use of a mechanical voice synthesizer– what we will call artificial speech. Depending on
the level of proficiency achieved with these substitutes, verbal communication often is restored to near pre-
surgery
levels. However, some patients live the rest of their lives with a diminished ability to communicate. The surgery itself
has little risk of fatal complications. Therefore, the survival models used by McNeil and her colleagues excluded the
possibility of immediate death.
Unlike the lung cancer study, this second study interviewed healthy volunteers rather than patients with laryngeal
cancer. The average age of the volunteers was 40 years. McNeil and her colleagues used a two- step process to deter-
mine their volunteers’ outcome utilities. These outcome utilities quantified both the volunteers’ risk attitudes as well
as their quality preferences for retaining normal speech.
The first step assessed the utility for the length of life with normal speech. The same assessment questions developed
for the lung cancer study were used. Recall how this was done. Each volunteer was asked the following question:
Suppose you have a disease that will cause your death in 5 years. Also suppose that if this disease were cured you could expect to live
another 25 years. If there is a risky treatment that will either cure the disease or cause your immediate death, how small would the
risk of death have to be for you to choose the treatment?
Table10.1 Laryngeal cancer treatment alternative and3- year survival rates from.
Treatment 3- year survival rate
Primary surgery 60%
Radiotherapy 30% to 40%
Radiotherapy with salvage surgery 42% to 48%
Adapted from McNeil etal. (1981).
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Outcome utilities– adjusting forthe quality oflife 191
The answer to this question characterized the volunteer’s risk attitudes for the length of their life. Using the methods
described in the previous chapter, the parameter for an exponential utility model– what we called the risk parameter–
can be fitted to an answer to this question. Figure10.1 shows an exponential utility model fitted to the answers pro-
vided by one of the volunteers. The risk parameter for this volunteer was 0.0815/year. We will refer to this volunteer
as Patient B.
The second step in the assessment process determined how utility would change if normal speech were no longer
possible. This was done by asking each volunteer to state the reduction in the length of their lives they would accept if
it meant avoiding the loss of normal speech.
For example, a volunteer would be asked to suppose that life would continue for another 25 years but speech would
be limited to the substitutes for normal speech available after laryngectomy. In order to fully understand this option, a
volunteer was given a written description of the artificial or synthesized speech possible after surgery. This description
of a patient’s post-
treatment quality state included the other effects of surgery, such as the artificial opening to the
trachea necessary for breathing after laryngectomy. The volunteer also would hear recorded examples of the possible
artificial or synthesized speech. After reading the description and hearing the recordings, the volunteer was asked how
many years they would be willing to subtract from those 25 years with artificial speech if doing so would mean retain-
ing normal speech. This question was repeated for several lengths of life with artificial speech.
Figure10.2 shows the responses for the volunteer we are calling Patient B. When asked to imagine facing 25 years
with artificial speech, this volunteer believed that 12.5 years with normal speech would be an equivalent outcome. The
same volunteer believed that 10 years with artificial speech would be equivalent to 7 years with normal speech. Below
5 years, this volunteer would not accept any shortening of life to avoid the loss of normal speech.
By the definition of utility, believing that two outcomes are equivalent means that those two outcomes have the same
utility. This means, for Patient B, the volunteer with the preferences shown in Figure10.2, the following must be true:
eeech
years without normal speechyears with n
10
07
0
..
oormal speech
years without normal speechyea
25
01
25
..
rrs with normal speech
We already stated that Patient B’s utilities for life with normal speech also match the outcome utilities shown in
Figure10.1. That is
50 038
70
..
.
years with normal speech
years with normal s
ppeech
years with normal speech
05
0
12
5073
.
..U
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0510 15 20
25
Utility
Length of life with normal speech (years)
0.3839
0.5000
0.7338
7.0 12.55.0
Figure10.1 Utility for life (u(t)) with normal speech assessed for a laryngeal cancer study volunteer referred to as Patient B. Utility has been scaled
so that the utility for 25 years of life with normal speech is 1.0.
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192 Medical decision making
For Patient B, it follows that:
Using the same reasoning, Patient B’s utility for 10 years and 25 years of life without normal speech would be:
A smooth curve fitted to these outcome utilities is shown in Figure10.3, together with the utility for life with normal
speech first shown in Figure10.1. The two utility curves in Figure10.3 provide a complete set of utilities for the pos-
sible outcomes faced by Patient B when choosing between the laryngeal cancer treatment options.
McNeil and her colleagues used this two-
step process to determine the outcome utilities for each of their volun-
teers. They then computed each volunteer’s expected utility for the three treatment options. Recall that computing an
expected utility requires a survival model that provides the probabilities for each of the possible lengths of life. In
order to represent the range of institutional differences in the effectiveness of radiotherapy, two different survival
models for radiotherapy were used. One survival model, called the average result for radiotherapy, was based on the
lesser of the 3- year survival rates shown in Table10.1 (30% for simple radiotherapy and 42% for radiotherapy with
salvage surgery). The other survival model, called the best result for radiotherapy, was based on the greater of the
3-
year survival rate shown in Table10.1 (40% for simple radiotherapy and 48% for radiotherapy with salvage sur-
gery). A single survival model for surgical treatment was used, based on the 3-
year survival rate of 60% shown in
Table10.1. All three of the survival models were adjusted to reflect the normal survival model for someone who is 40
years old.
Expected utility accounts for an individual’s risk attitudes when faced with an uncertain outcome. Adjusting the
length of life to account for the loss of normal speech means that the expected utilities calculated by McNeil and her
colleagues determined their volunteers’ preferences for the resulting quality of their lives as well as the length of their
lives. This means choosing the treatment with the greatest expected utility computed for a volunteer would be consist-
ent with that volunteer’s preferences.
Table 10.2 summarizes the results of the expected utility calculations for the 37 volunteers. Not surprisingly, the
percentages in Table10.2 show how preference for surgical treatment depends on the survival achieved by the alterna-
tives based on radiotherapy. When radiotherapy would achieve what McNeill and her colleagues labelled average
0
5
10
15
0510 15 20 25
Lifetime with normal speech (years)
Lifetime with artificial speech (years)
5 years
5 years
10 years
7 years
25 year
s
12.5 years
Assessed values
Life with artificial speech w
2
(t)
Life with normal speech w
1
(t)
Fitted curve
Figure10.2 Quality- lifetime tradeoff functions for life with normal speech (w
1
(t)) and life with artificial speech (w
2
(t)). The curve for w
2
(t) is based on
quality preference values assessed for volunteer referred to as Patient B.
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Outcome utilities– adjusting forthe quality oflife 193
results, surgery was preferred by 92% of the volunteers. When radiotherapy would achieve what was labelled best
results, surgery was preferred by 57% of the volunteers.
This section started with the question– does quality of life matter to patients? Table10.2 shows that the answer
is “yes,” at least for some of the participants in the laryngeal cancer study. Notice that none of the treatments were
preferred by every participant. How these individuals weighed the importance of quality of life and the length
of life affected which treatment they preferred. McNeal and her colleagues worked with healthy volunteers
ratherthan patients actually facing the psychological trauma of a cancer diagnosis. Also, the average age of their
volunteers was 40 years, which is less than the average age of patients with laryngeal cancer. Nevertheless, the
assessment technique used in this study, and the subsequent analysis, show that feelings about the quality of
lifeshould influence a treatment decision even when significant reductions in survival are possible. In short, the
quality of life matters.
10.3 Quality- lifetime tradeoff models
McNeil and her colleagues used what we call a quality- lifetime tradeoff model in their study of laryngeal cancer. The
quality- lifetime tradeoff model is one of the commonly used approaches for capturing how quality- of- life preferences
can affect the treatment preferred by a patient. Stated formally, a quality- lifetime tradeoff model represents the utility
for an outcome that has a given length of life with a quality- reducing symptom, by determining an equivalent length
of life without the symptom. The utility for the original symptomatic lifetime equals the utility for that equivalent
shorter symptom- free lifetime.
Table10.2 Laryngeal cancer treatment preferences determined by comparison ofexpected utility
calculations based onutility assessments for37 healthy volunteers.
Radiotherapy result
Treatment
Primary surgery Radiotherapy Radiotherapy with salvage surgery
Average 92% 3% 5%
Best 57% 19% 24%
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1. 0
0510 15 20
25
Utility
Length of life (years)
Utility with artificial speech
Utility with normal speech
u(12.5 years)
u(7 years)
u(T)
u(5 years)
Figure10.3 Utility for length of life with artificial speech. Utility for length of life with normal speech is shown, for comparison, as a dotted curve.
The former is derived from the latter assuming the quality- lifetime tradeoff function shown in Figure10.2. All utilities have been scaled so that the
utility for 25 years of life with normal speech is 1.0.
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194 Medical decision making
This section describes the quality- lifetime tradeoff model and shows a simplification based on a parametric model
similar to the exponential utility model described in the previous chapter.
Using the following notation will simplify our discussion of the quality- lifetime tradeoff model. Denote a patient’s
utility for symptom-
free life as follows:
utility for life lastingyearssymptom-free
For example, Figure10.1 showed u(t) for one of the volunteers in the laryngeal cancer study. We refer to this volun-
teer as Patient B. The previous chapter described how to determine u(t) for someone. Those methods apply in this
chapter as well.
A second key relationship we will use quantifies the length-
of- life tradeoffs that would be acceptable to avoid a
quality- reducing symptom. This relationship will be denoted as follows:
i
ymptom-free lifetime equivalent to a lifetime of iin quality statei
th
The subscript i is added to this function to denote the possibility of more than one combination of symptoms and
disabilities. Living with one of those combinations will be referred to as living in a quality state. The function w
i
(t) will
be called the quality- lifetime tradeoff function for the i
th
quality state.
For example, the study reported by McNeil and her colleagues involved two quality states: life with normal speech
and life with artificial speech. Figure10.2 showed Patient B’s quality- lifetime tradeoff functions for these two quality
states.
If life with normal speech is thought of as quality state 1, note that by definition
wt t
1
The more interesting case in the laryngeal cancer decision is w
2
(t), the quality- lifetime tradeoff function for life
with artificial speech. Remember that Patient B had the following equivalences for life with and without normal
speech:
• 5 years with normal artificial equivalent to 5 years with normal speech
• 10 years with normal artificial equivalent to 7 years with normal speech
• 25 years with normal artificial equivalent to 12.5 years with normal speech
Figure10.2 showed the values for w
2
(t) implied by these equivalences.
Writing the mathematical expression for w
2
(t) is more complicated.
tt
tt
tt
2
5
5050 55
10
10 0371
01
0
if
if
if
.
.
However, w
2
(t) is key to how the quality- lifetime tradeoff model adjusts outcome utilities to account for quality pref-
erences. By definition, living t years with artificial speech is equivalent to living w
2
(t) years with normal speech.
Expressed as outcome utilities this means:
t
2
years with normal speechyears with artificial sppeech
Typically, the next step is to assume that
22
years with normal speech
Definition: quality- lifetime tradeoff model
A quality- lifetime tradeoff model represents the utility for an outcome that has a given length of life with a quality-
reducing symptom by determining an equivalent outcome with a shorter length of life without the symptom. The
utility for the original outcome is the utility for that equivalent shorter symptom-
free lifetime.
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