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Measuring theaccuracy ofclinical findings 85
Sensitivity
0.2
0.2
0.4
0.4
0.6
0.6
0.8
0.8
1.0
1.
0
1-specificity
>2.5
>2.0
>1.5
>1.0
>0.5
Figure 5.16 ROC curve for ST segment depression on an exercise electrocardiogram. Each point represents a different amount of ST segment
depression, as indicated on the graph by the number of millimeters of ST depression. Adapted from Diamond and Forrester (1979).
5.8.3 Using theROC curve tocompare tests
The ROC curve is useful for comparing tests. The area under the ROC curve is a single measure of test performance
that takes into account sensitivity and specificity. Of several tests for the same target condition, the test with the great-
est area under its ROC curve best discriminates between people who have the target condition and those who do not
have it. This method has become a widely used approach for comparing tests. A shortcoming is the requirement for
knowing p[R|D+] and p[R|D−
] for each definition of a test result. Two ROC curves could have different shapes and
yet enclose the same area. As seen in the next section, the shape of the ROC curve is often the decisive factor in deciding
when to use a test for a specific indication. Chapter13 discusses another way to decide between two tests: given a pre-
test probability, with which test would the post- test probability cross a treatment threshold probability and thereby
alter the treatment plan?
5.8.4 Setting thecut point fora test
This chapter has prepared us to appreciate the importance of a method for setting the cut point for a test. This method
takes into account two characteristics of the patient: the patient’s pre- test probability of the target condition and the
importance to the patient of avoiding false-
negative and false- positive results. The principle is as follows:
The optimal cut point of a test is the test result corresponding to the point on the ROC curve at which the slope of the tangent
to the ROC curve satisfies the following relationship.
where:
p[D]=the pre-
test probability of the target condition
H=the net harms of treating patients who do not have the target condition
B=the net benefit of treating patients with the target condition
The derivation of this relationship appears in the appendix to this chapter. To find the optimal point on the ROC
curve, the derivation uses the principle that guides expected value decision making: choose the action that maximizes
a person’s expected utility. Thus, to maximize the patient’s well- being, the cut point should be the test result corre-
sponding to the point on the ROC curve where the tangent to the curve has a slope that satisfies this relationship.
Maximizing the patient’s well- being is a powerful foundation for a method to determine the optimal cut point of a test.
Slopeoftangent to ROCcurve
H
B
pD
pD
1
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86 Medical decision making
With a test whose results are expressed as a continuous variable, each point on its ROC curve corresponds to a dif-
ferent test result, each with its own sensitivity and specificity (Figure5.15 illustrates this point) and a different ratio of
Harms to Benefits, as shown in the equation for the slope of the tangent to the ROC curve (above). According to the
equation for defining the optimal cut point:
• A test result corresponding to a point on the ROC curve where its slope is flatter than at the cut point reflects
increased benefit relative to harm. Such results are classified as “positive” and should lead to action consistent
with the target condition being present.
• A test result corresponding to a point on the ROC curve where its slope is steeper than at the cut point reflects
increased harm relative to benefit. Such results are classified as “negative” and should lead to action consistent
with the target condition being absent.
Chapter13 contains a description of the method for determining the harms (H) and benefits (B) of treatment. In this
chapter, we will use subjective judgment to estimate the ratio of H to B. Subjective estimates of this ratio often starts
with a clinician imagining what it would feel like to make a mistake. An error of commission would be treating someone
who does not have the target condition and thereby causing the harms of treatment (H) without corresponding benefit.
An error of omission would be the failure to treat someone with the target condition and depriving the person of the
benefit (B) of treatment.
There are two target populations for this test: (1) those who have symptoms of active disease (test done to confirm
suspected disease) and (2) those who have no symptoms but are worried that they may have it (test done to screen for
disease). The net harms and benefits of treatment in the two populations may differ.
Cut point for disease confirmation: Imagine that a patient is suspected of having the disease that the index test is used
to detect. Based on the patient’s clinical findings, the probability of the target condition is 0.50. Patients with a positive
test undergo a treatment that puts the patient at risk temporarily because of transient bone marrow toxicity. Patients
with the disease derive considerable benefit from disease detection because treatment can prolong survival by several
years. Patients who have similar symptoms but do not have the disease and have a positive test also receive the toxic
treatment. They suffer transient bone marrow depression and no benefit from treatment. The clinician’s ratio of harms
to benefit for disease detection is 1 to 2.5, which reflects a judgment that it is worse to mistakenly withhold the treat-
ment from someone who has the cancer and could benefit from treatment (an error of omission) than it is to give the
treatment to someone with similar symptoms but who does not have the target condition and would suffer only tran-
sient bone marrow depression (an error of commission). In this population, the test result that divides “negative” from
“positive” corresponds to the point on the ROC curve where the tangent has the following slope:
S
lope of ROCcurve
H
B
pD
pD
1
1
25
105
05
04
.
.
.
.
To find the cut point for the test, identify the point on the ROC curve where the tangent to the curve has a slope of
0.4, as shown in Figure5.17. From differential calculus, the slope of the tangent to the ROC curve at any point is
dy
dx
, the
instantaneous change along the vertical axis (sensitivity) for a given change along the horizontal axis (1 − specificity).
With a relatively flat ROC curve at the cut point, Figure5.17 shows that the sensitivity of the test will be high and the
specificity of the test will be low, which means relatively few false- negative results that would lead to failure totreat those
who have the target condition. The likelihood ratio (LR+) for the test at the optimal cut point is 1.51. TheLR- is 0.32.
Cutoff value for screening: Candidates for screening for this cancer face a somewhat different situation. The prevalence of
this disease in healthy people is only 0.001 (1 case in 1000 people). There is considerable benefit to disease detection
because early detection and treatment could prolong life. The clinician’s ratio of harms to benefits for treating the cancer
in a screening population consisting of largely healthy people is 1 to 50, which reflects the judgment that it is worse to
mistakenly give the treatment to an apparently healthy person who does not have the cancer and would suffer temporary
harm from treatment than it is to withhold the treatment from an apparently healthy person who does have the cancer.
Example
Consider a hypothetical antibody test to detect a form of cancer. The scale for the concentration of antibodies is
divided into 10equal- sized intervals. A reputable research group has measured the sensitivity and specificity of
the serum concentration corresponding to each interval.
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Measuring theaccuracy ofclinical findings 87
0.2
0.2
0.4
0.4
0.6
0.6
0.8
0.8
1.0
1.0
1-specificity
Sensitivity
Slope of tangent
to ROC curve = 0.4
LR+ of test at the
tangent to its ROC
curve = 1.51
Figure 5.17 ROC curve for hypothetical test for antibodies to a form of cancer. The dotted lines denote the coordinates of the point on the
curveatwhich the slope of the tangent to the curve is 2.5.
In the screening population, the test should be considered abnormal at the point on the ROC curve where its
slopeis:
S
lope of ROCcurve
H
B
pD
pD
1
1
50
10001
0 001
1
50
0 999
0
.
.
.
..001
20
To find the cutoff value for the test, identify the point on the ROC curve where a tangent to the curve has a slope of
20. A test result corresponding to this point will occur in the lower left-
hand corner of the ROC space plot. A slope of
20means a large ratio of true- positive results to false- positive results and therefore a test result with a high likelihood
ratio. A high likelihood ratio is fitting because the test result must produce a large increase in the probability of the
cancer to justify screening asymptomatic persons. Therefore, the antibody level corresponding to the cut point should
be much higher when testing asymptomatic persons in order to avoid false-
positive results.
This example shows that choosing the definition of an abnormal test result requires attention to all facets of the
clinical situation, including the clinical features of the patient and the consequences of managing a patient with a
positive test.
5.9 Combining data fromstudies oftest performance: thesystematic review
andmeta- analysis
The sensitivity and specificity of commonly used tests have been measured in many studies. The results often differ,
leaving the clinician to decide which study to believe. This problem has largely been solved by a systematic review. A
systematic review summarizes the evidence from studies on the same topic. Systematic reviews have become a power-
ful force in shaping clinical practice, in part because clinical practice guideline panels rely on them to summarize the
evidence that shapes their recommendation, which in turn influence insurance coverage decisions.
The systematic review starts with a systematic literature search to find and evaluate all studies, published and
unpublished. The goal is to identify every study that fits the inclusion and exclusion criteria for the systematic review
and evaluate and grade its study design characteristics. The risk of bias is a key measure of study quality. Factors to be
assessed include (1) patient selection processes (prospective, consecutive patients); (2) blinding to index test results
when assessing gold standard test results (and conversely); (3) adequacy of the gold standard test as a diagnostic
benchmark; and (4) a short time interval between the index test and doing the gold standard test.
The second part of a systematic review is called a meta- analysis, which summarizes the results of high- quality
studies. Studies of test performance are an excellent case example of the need for caution when trying to combine
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88 Medical decision making
results from a group of studies. The same test could perform quite differently in different studies for the following
reasons:
• Differing definitions of the cut point that defines an abnormal result.
• Different criteria for deciding that an image is a true- positive result.
• Different techniques for performing the test.
• The sample of studies includes several generations of the technology.
For these reasons, the first step in a systematic review of a group of studies of a diagnostic test is to do a qualitative
analysis of the body of studies to decide if they are sufficiently alike to consider combining their results.
The product of a meta-
analysis of studies of a diagnostic test is a summary estimate of sensitivity and specificity.
Simply calculating an average sensitivity and, separately, an average specificity is not a valid approach because it
ignores the correlation between sensitivity and specificity. To understand this correlation, imagine the cut point sepa-
rating negative test results from positive test results. As this cut point increases, fewer patients with the target condi-
tion are classified as test-
positive (decreasing sensitivity) while more patients who do not have the target condition are
identified as test-
negative (higher specificity and therefore fewer false- positives). Refer to Figure 5.14 to see the effect
of raising the cut-point that divides positive from negative results. The opposite occurs when the cut point is lowered.
A valid statistical approach to calculating a summary sensitivity and specificity must take account of the correlation
between sensitivity and specificity: as one changes, so does the other. An approach that takes this dependency into
account is the starting point for determining a valid average sensitivity and specificity.
A summary ROC curve is a robust approach to combining the results of diagnostic test accuracy studies. A summary
ROC curve is the line that best fits the points defined by the sensitivity and specificity of individual studies as plotted
in ROC space.
The first step in developing a summary ROC curve is to graph in ROC space the sensitivity (vertical axis) and
1
− specificity (horizontal axis) of the test as measured in each study. Each individual study of the test is one point in
the space defined by the two axes (Figure5.18, left- hand panel).
The next step is to define the line that best fits the points in ROC space. Two valid, equivalent methods are: (1) hier-
archical SROC curve and (2) the bivariate random effects model. Using available statistical software, both methods
give a valid estimate of the ROC curve that represents the individual studies in the ROC space. The methods also give
the average sensitivity and specificity across all the included studies. The right-
hand panel of Figure5.18 represents
these features.
These methods do not work for calculating a summary likelihood ratio. Instead, calculate a summary ROC curve
and the average sensitivity and specificity. Use the latter to calculate a summary likelihood ratio-
positive (sensitivity/
[1–specificity]) and likelihood- negative ([1 − sensitivity]/specificity).
The right-
hand panel shows a summary ROC curve and the point representing the average sensitivity and specific-
ity with a 95% confidence interval. The article by Leeflang etal. (2008) discusses this topic in depth (see Bibliography).
This illustration is a drawing, not a calculated plot. For an example with real- life data, see the article by Leeflang listed
in the Bibliography.
1-specificity
0
Sensitivity
Sensitivity
1.0
1.0
0 0.20 0.40 0.60 0.80
1-specificity
0
1.0
1.00 0.20 0.40 0.60 0.80
Figure 5.18 The left- hand panel shows several hypothetical studies of the accuracy of a diagnostic test plotted in the ROC. Each point represents
a different study of test performance.
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Measuring theaccuracy ofclinical findings 89
Summary
1. The following expressions completely characterize diagnostic information:
• p[R|D+]: Likelihood of observing the finding in patients with the target condition (sensitivity)
• p[R|D−]: Likelihood of the finding occurring in patients who do not have the target condition (1 − specificity).
2. The predictive value- positive is the proportion of positive test results that are true positives in a study of diag-
nostic test performance. The predictive value applies to the population in which it was measured. Applying a
predictive value to other populations will often lead to error.
3. Spectrum bias can occur when test performance is measured in one population (the verified sample) and ap-
plied to another population (the source sample). It occurs mostly in retrospective studies. In the source popula-
tion, sensitivity is typically lower and specificity typically higher than in the verified population.
4. When the results of tests are expressed as a continuous variable, a result is defined by the value of the cut point,
above which the test is positive. Each cut point defines a different test result. Each result has a p[R|D+] and a
p[R|D−]. The ROC curve is a plot of the p[R|D+] (vertical axis) and p[R|D−] (horizontal axis) of each possible
test result.
5. A test has an optimum cut point value for defining a positive test result. Using that p[R|D+] and p[R|D−] to
calculate a post-
test probability will maximize the patient’s expected utility. The optimum cut point is patient-
specific. It depends on the patient’s pre- test probability and the harms and benefits of treatment.
A.5.1 Appendix: derivation ofthe method forusing anROC curve tochoose thedefinition
ofan abnormal test result
The topic of the last section of this chapter is a test that can have many results. The serum concentration of an enzyme
released from damaged heart muscle, such as the serum troponin, can have any biologically reasonable value. When a
test has many possible results, the clinician must pick a result that defines the threshold for acting as if the patient had
the target condition. This appendix contains the derivation of the formula for identifying the point on the ROC curve
that corresponds to the preferred definition of an abnormal result. The decision to give or withhold treatment to a
patient depends on whether the patient’s test result is above or below this cut point.
This derivation depends on concepts that first appear in the next several chapters. The reader should refer to these
chapters as new concepts appear. The author has based this derivation on the one that appears in the Bibliography
(Metz, 1978).
Our goal is to find the definition of an abnormal result that will minimize the net harm of doing the test. Since the
true state of the patient is unknown, we express the net harm as an expected value (expected value decision making is
introduced in Chapter6). We will use the expected utility of the test (U[test]) as our measure of its worth. Refer to
Chapter8 to learn about utility.
UE
UEU
test test no test
(A.5.1)
To calculate the expected utility of doing the test, we use a decision tree (FigureA.5.1, see next page). See Chapter6 for
a description of decision trees. The first chance node on the tree represents the unknown true state of the patient
(D+=has the target condition; D−=does not have the target condition). The second chance node (reading from left to
right) represents the outcome of the test (T+=test positive; T−=test negative). The probability of disease is p(D+). The
probability of a positive test in a diseased patient is p(T+|D+), which is the sensitivity of the test.
Inspection of the tree shows four possible outcomes of the test (true- positive, false- negative, false- positive, and true-
negative). If the patient’s test result is above the optimum cut point that defines an abnormal test result, the patient will
receive treatment (A+). If the result is below this value, the patient will not receive treatment (A−). Thus, depending on
the patient’s true state (D+ or D−) and the outcome of the test, a patient will be in one of four states:
True- positive result: D+A+
False- negative result: D+A−
False- positive result: D−A+
True- negative result: D−A−
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90 Medical decision making
The patient’s utility (or preference) for a true- positive result is U(D+A+) or, in a more compact notation, U
tp
. A true-
positive result implies the treatment of a diseased patient, which presumably is a better outcome than a false- negative
result, which implies not treating a diseased patient. Thus, one would expect U
tp
to be greater than U
fn
.
To obtain the expected utility of the test option, average out at the chance nodes in the tree shown in FigureA.5.1, as
described in Chapter6. To do so, multiply the utility of each outcome times the probability that it will occur, which is
the product of the probabilities along the path to the outcome (the path probability).
Up
DpTD UpDpTD U
pD
test
tp tn
*|**|*
*|**|*pT DU pD pT
DU
fp fn
(A.5.2)
but
pT DpTD andpTD pT D
||
||11
Substituting these relationships and rearranging terms, we get:
UUUpDpTD UU pD pTtest
tp fn fp tn
|( ||D
The ROC curve describes the relationship between the sensitivity and 1 − specificity for different cut points of a diag-
nostic test. Let the function R[ ] denote this relationship, which, plotted in ROC space, would describe the shape of the
ROC curve. The relationship between the sensitivity and 1 − specificity is given by this functional relationship:
pT DRpT D
||
Substituting this relationship in EquationA.5.2,
Utest UU pD RpTD UU pD
tp fn fp tn
| ppT D
|
The next step is from differential calculus. Differentiate U(test) with respect to p(T+|D– ) to find the point on the ROC
curve where U(test), the patient’s utility for testing, is a maximum. This step is an important reason to use this method
for choosing the cut point because it maximizes the patient’s best interests:
dU
dpTD
UU pD
dRpT D
dp
test
tp fn
|
|
TTD
UU pD
|
fp tn
Test
P [D+]
P [D−]
P [T+ |D−]
U(D+A
−)
U(D−A
+)
U(D−A
−)
U(D+A
+)
P [T− |D−]
P [T+ |D+]
T+ → treat
T– → no treat
T+ → treat
T–
→
no treat
P [T− |D+]
Figure A.5.1 Decision tree for calculating the expected utility of doing a test.
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Measuring theaccuracy ofclinical findings 91
Now set the left-hand expression in the preceding equation equal to zero to find the point on the ROC curve where
the utility of doing the test (U[test]) is a maximum.
0
UU pD
dRpT D
dpTD
UU
tp fn fp tn
|
|
pD
Solve this equation for
dR
pT D
dp
TD
|
|
no
no
, the slope of the ROC curve at the cut point that maximizes the value of
testing,
dRpT D
dpTD
pD
pD
UU
UU
|
|
fp tn
tp f
nn
(A.5.3)
In Chapter13, we refer to (U
fp
− U
tn
) as the “harm” of treating a nondiseased person (denoted as H), and (U
tp
− U
fn
) as
the “benefit” of treating a diseased person (denoted as B). Thus, EquationA.5.3 becomes
dR pT D
dpTD
pD
pD
H
B
[|
|
(A.5.4)
The left-
hand side of EquationA.5.4 is the rate of change of the sensitivity with respect to 1 − the specificity of the test,
which is the slope of the ROC curve.
With a test whose results are expressed as a continuous variable, each point on its ROC curve corresponds to a dif-
ferent test result and a different ratio of Harms to Benefits, as shown in EquationA.5.4.
• A test result corresponding to a point on the ROC curve where its slope is flatter than at the cut point reflects
increased benefit relative to harm. Such results are classified as “positive” and should lead to action consistent
with the target condition being present.
• A test result corresponding to a point on the ROC curve where its slope is steeper than at the cut point reflects
increased harm relative to benefit. Such results are classified as “negative” and should lead to action consistent
with the target condition being absent.
EquationA.5.4, which is the same expression that appears in the last section of Chapter5, indicates that the cutoff
value that maximizes the patient’s utility is where the slope of the ROC curve equals the odds that the disease is absent
times the ratio of harms to benefits of treatment.
Bibliography
Begg, C.B. and Greenes, R.A. (1984) Assessment of diagnostic tests when disease verification is subject to selection bias. Biometrics, 39, 207–15.
Information about correcting for disease verification bias in selecting participants in studies of diagnostic test performance.
Bossyut, P.M., Reitsma, J.B., Bruns, D.E. etal. (2003) Standards for reporting of diagnostic accuracy. Towards complete and accurate reporting
of studies of diagnostic accuracy: the STARD Initiative. Annals of Internal Medicine, 138, 40–4.
A brief article describing the key elements of a study of diagnostic test accuracy and how to report them. An accompanying online- only
article goes into much greater depth.
Diamond, G.A. and Forrester, J.S. (1979) Analysis of probability as an aid in the clinical diagnosis of coronary- artery disease. New England
Journal of Medicine, 300, 1350–8.
Gray, R., Begg, C.B., and Greenes, R.A. (1984) Construction of receiver operating characteristic curves when disease verification is subject to
selection bias. Medical Decision Making, 4, 151–64.
If selection for the verified sample depends on the results of the index test, the method described in this article can be used to correct
published sensitivity and specificity and obtain an improved estimate of these data in the source population.
Hanley, J.A. and McNeil, B.J. (1982) The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology, 143, 29–36.
A good article for those who wish to learn more about ROC curves and how they may be used to compare diagnostic tests.
https://t.me/medicina_free
92 Medical decision making
Irwig, L., Tosteson, A.N.A., Gatsonis, C. etal. (1994) Guidelines for meta- analyses evaluating diagnostic tests. Annals of Internal Medicine, 120,
667–76.
An in- depth review of combining the results of different studies of test performance.
Leeflang, M.M.G., Deeks, J.J., Gatsonis, C.G., etal. (2008) Systematic review of diagnostic test accuracy. Annals of Internal Medicine, 149, 889–97.
An up- to- date description of the key elements of a systematic review of diagnostic test accuracy, including reliable statistical methods for
combining the results of studies.
McNeil, B.J., Keeler, E., and Adelstein, S.J. (1975) Primer on certain elements of medical decision making. New England Journal of Medicine, 293,
211–5.
This classic article describes the use of ROC curves, including the method for selecting a definition of an positive result.
Metz, C.E. (1978) Basic principles of ROC Analysis. Seminars in Nuclear Medicine, 8, 283–98.
This article contains a derivation of the method for choosing the optimum cut- off value for a diagnostic test.
Philbrick, J.T., Horwitz, R.I, Feinstein, A.R., etal. (1982) The limited spectrum of patients studied in exercise test research: analyzing the tip of
the iceberg. Journal of the American Medical Association, 248, 2467–70.
The authors show how selection factors limit the number of patients in the source population who can participate in studies of the sensi-
tivity and specificity of a test.
Ransohoff, D.F. and Feinstein, A.R. (1978) Problems of spectrum and bias in evaluating the efficacy of diagnostic tests. New England Journal of
Medicine, 299, 926–30.
A description of the effect of biased patient selection on sensitivity and specificity.
Simel, D.L. and Rennie, D. (2008) The Rational Clinical Examination: Evidence- based Clinical Diagnosis, McGraw- Hill Medical, NewYork.
A source of trustworthy information about the performance of many different diagnostic tests. The following website has the information
in the book as well as subsequent articles in JAMA: https://jamaevidence.mhmedical.com/content.aspx?bookid=845§ionid=61357443.
This site is proprietary. Access is through an institutional subscription or a personal subscription, Short-term access may also be
available.
Weiner, D.A., Ryan, T.J., McCabe, C.H. et al. (1979) Exercise stress testing: correlation among history of angina, ST- segment response and
prevalence of coronary artery disease in the Coronary Artery Surgery Study (CASS). New England Journal of Medicine, 301, 230–5.
This study illustrates how measurement of the sensitivity and specificity in clinical and anatomic subgroups of patients can pay off in new
insights. This study showed that the sensitivity of the exercise electrocardiogram depends on the patient’s history.
Whiting PF,Rutjes AWS,Westwood ME etal. QUADAS- 2: a revised tool for the quality assessment of diagnostic accuracy studies. Annals of
Internal Medicine 2011;155:529–36.
A tool to assess the quality of studies of diagnostic accuracy. It covers patient selection, index test, reference standard, and flow of patients
through the study.
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.
93
CHAPTER6
Decision trees– representing thestructure
ofadecisionproblem
6.1 Introduction
This chapter begins the discussion of decision trees. These diagrams provide a foundation for many of the
topics covered in later chapters. Chapters 3 and 4 discussed how uncertainty complicates medical decision
making.A decisionproblem also involves choices. A test can be performed or skipped. A therapy can be started
ordelayed. The patient can be admitted or sent home. Moreover, choices and uncertainties can interact. Choosing
a test can provide information that reduces uncertainty. That reduced uncertainty can alter the subsequent set
of appropriate treatments. Decision analysis often uses the trees described in this chapter to represent these
interactions.
The focus of this chapter is the structure of decision trees. We will see how decision trees can be organized to repre-
sent the relationships between the decision alternatives and uncertainties in a decision problem. Emphasis will be
placed on how probabilities are used to quantify those uncertainties. We will see how the resulting probabilities can be
combined to quantify the likelihood of the possible outcomes of a decision. Simple examples with a single decision will
illustrate the discussion in this chapter. The simplicity of these examples will mean that a relatively simple calculation
will identify the best decision alternative for the patient.
Ultimately, we will use decision trees in the analysis of complicated problems with multiple interdependent deci-
sions. The analysis of those decision trees will be more complex. How to deal with complexity will be described in the
next chapter.
6.2 Key concepts andterminology
A decision tree is a collection of nodes and branches showing the relationships between the alternatives and uncertainties
in a decision problem. A choice between alternatives is represented by what is called a decision node. The uncertainty
about something like the patient’s condition, the results of tests, or the effect of treatments is represented by what is
called a chance node.
The branches emanating from a decision node correspond to the alternatives available for the choice represented by
the node. Each of the possible alternatives must be represented by exactly one of the branches. For example, suppose
that a decision node represents a choice about the use of two possible drugs. The branches emanating from that
decision node will include separate branches for each of the two drugs. In general, there will be a third branch
6.1
Introduction 93
6.2
Key concepts andterminology 93
6.3
Constructing thedecision tree fora hypothetical decision problem 96
6.4
Constructing thedecision tree fora medical decision problem 103
Epilogue 112
Bibliography 112
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94 Medical decision making
representing the alternative of using neither drug. If medically appropriate, there also will be a fourth branch repre-
senting the
combination of the two drugs.
The uncertainty represented by a chance node is what mathematicians call a random variable. The word “random”
emphasizes what is known about the variable’s actual value. That variable always has a value; however, the value is
not known for certain within the timeframe of the decision problem. The branches emanating from a chance node
correspond to the possible values for the corresponding patient condition, test result, or treatment effect. Each possible
value must be represented by exactly one of the branches. Each of those branches corresponds to a possible chance
node value.
For example, suppose that a chance node represents the uncertainty about two diseases as possible causes for a
patient’s symptoms. There will be separate branches for each of the two diseases representing the possibility that
thecorresponding disease is the sole cause of the symptoms. Often there will be a third branch representing the pos-
sibility that neither disease is causing the symptoms. If medically possible, there also will be a fourth branch rep-
resenting the case where both diseases cause the symptoms. In the terminology we are using, the possible cause
ofthe patient’s symptoms is the random variable and the outcomes associated with the four branches are the
possible values.
6.2.1 Final outcomes
The starting point for a decision tree typically is a decision node. The branches leading from that initial node lead to
other nodes, which in turn have branches leading to yet other nodes in the tree. A given sequence of nodes and
branches, starting at the initial node, constitutes a path that includes the decision node alternatives and chance node
values the patient might experience. Such a path ultimately ends with a final outcome, which summarizes that particu-
lar patient experience.
A key step in the analysis of a decision problem is the assignment of numerical values to the possible final outcomes.
The methods discussed in this book are numerical in nature. In Chapter3, we learned how to represent subjective
beliefs by a number– a probability. The analysis of a decision also uses numerical values to quantify the desirability,
or undesirability, of the possible outcomes.
The numerical values quantifying the desirability of outcomes are called outcome values. How best to determine the
outcome value that is appropriate for the patient is a major challenge in decision analysis. Several later chapters will
focus on this important topic. In the meantime, this chapter, and the one that follows, will use the length of the patient’s
life as the outcome value. Readers who question the use of the length of the patients as the sole measure of an out-
come’s desirability are right to do so. Patients usually are concerned about more than just how long they will live.
However, a fuller discussion of this important topic will have to wait until those later chapters.
Definition: decision trees, decision nodes, andchance nodes
Decision tree: Collection of nodes and branches showing the relationships between the alternatives and
uncertainties in a decision problem.
Decision node: Decision tree node representing a choice between alternatives. Branches emanating from a
decision node correspond to the alternatives for the choice represented by the node.
Chance node: Decision tree node representing uncertainty about something like a patient characteristic, test
result, or treatment effect. The branches emanating from a chance node correspond to the possible
values for the corresponding patient condition, test result, or treatment effect.
Definition: final outcomes andoutcome values
Final outcome: The endpoint for a sequence of nodes and branches in a decision tree, starting at the initial node.
Outcome value: Numerical value quantifying the desirability of a final outcome.
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