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Decision tree analysis 115
Clearly, Test is the best alternative because 15.0 years is longer than 14.4 years. As long as the expected values are
unchanged by the operation, the obvious choice made for this reduced decision tree is the correct choice for the original
decision tree.
Therefore, the key to the folding-
back operation is how to replace a chance node or a decision node by a single
outcome with an equivalent expected value. How this is done depends on the type of node.
Consider the following generic chance node:
Branch probability = p
1
Outcome value = x
1
Branch probability = p
2
Outcome value = x
2
Branch probability = p
3
Outcome value = x
3
Branch probability = p
n
Outcome value = x
n
This generic chance node represents an uncertainty that has n possibilities. Each possibility corresponds to a
branchprobability denoted by p
1
, ..., p
n
. In turn, each of the branches leads to outcomes that have the values denoted
by x
1
,..., x
n
.
We are only interested in expected values so facing the uncertainty represented by this node would be equivalent to
facing a single outcome that has the same expected value. As we saw in the previous chapter, the expected value for a
chance node is computed by multiplying each branch probability by the corresponding outcome value and summing
the results. That is
Expected value px px
px
nn
11 22
Of course, the expected value for a single outcome is just the value for that outcome. Therefore, we can replace the
generic chance node shown earlier by an outcome that has a value equal to the chance node’s expected value. Moreover,
the replacement will not change the expected value for any decision alternative that leads to this chance node.
P [Test
+
]
P [Test
–
]
0.48
0.52
No treatment
Treatment
No treatment
Treatment
No treatment
Treatment
D
+
and no treatment (4 years)
D
+
and treatment (12 years)
D
+
and no treatment and test (3 year)
D
–
and no treatment and test (19 years)
D
–
and treatment and test (15 years)
D
+
and no treatment and test (3 year)
D
+
and treatment and test (11 years)
D
–
and no treatment and test (19 years)
D
–
and treatment and test (15 years)
D
+
and treatment and test (11 years)
D
–
and treatment (16 years)
D
–
and no treatment (20 years)
Test
No test
(A1)
(A2)
(A3)
(A4)
(A6)
(A7)
(A8)
(A9)
(A10)
(A11)
(A5)
P [D
+
]
0.40
P [D
–
]
0.60
P [D
+
]
0.40
P [D
–
]
0.60
P [D
+
| Test
+
]
0.75
P [D
–
| Test
+
]
0.25
P [D
+
| Test
+
]
0.75
P [D
–
| Test
+
]
0.25
P [D
+
| Test
–
]
0.08
P [D
–
| Test
–
]
0.92
P [D
+
| Test
–
]
0.08
P [D
–
| Test
–
]
0.92
Figure7.1 Decision tree Problem 1, showing branch probabilities and life expectancies for final outcomes. Copy of decision tree in Figure6.8.
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116 Medical decision making
Decision nodes are replaced using a similar operation. Consider the following generic decision node:
Outcome value = x
1
Outcome value = x
2
Outcome value = x
3
Outcome value = x
n
Decision alternative 1
Decision alternative 2
Decision alternative 3
Decision alternative n
This generic decision node represents a choice between n alternatives. Each alternative is represented by a branch
leading to an outcome. The values for those outcomes are denoted by x
1
, ..., x
n
.
The goal is to make choices that maximize expected value. Therefore, if faced with the decision represented by this
node, the choice would be the outcome with the greatest value. This means a decision node can be replaced the pos-
sible outcomes for the node that has the greatest value. As with the chance node, this replacement will not change the
expected value computed for any path that leads to this decision node.
The following demonstrates how the folding- back operation uses these two replacement steps to reduce the decision
tree shown in Figure7.1 for Problem 1. Keep in mind that the goal is to determine the life expectancies that would
result from choosing either Test or No Test. Those life expectancies will determine the best alternative for the decision
represented by node A1.
The folding-
back operation starts with the nodes encountered just before the final outcomes. In the case of the deci-
sion tree shown in Figure7.1, these are the chance nodes corresponding to the uncertainty about the disease (chance
nodes A3, A4, A7, A8, A10, and A11).
Consider chance node A3. This node represents an uncertainty that can result in a life expectancy of 4.0 or 20.0 years. The
corresponding branch probabilities are 0.40 and 0.60, respectively. Therefore, the life expectancy for chance nodeA3 is:
LE
04040060 20 0136.. .. .yearsyears years
Using the logic described earlier, chance node A3 can be replaced by an outcome that has a life expectancy of
13.6years without changing the life expectancies that will be calculated for decision node A1.
Figure7.2 shows the result of repeating this replacement operation for each of the chance nodes representing the
disease uncertainty. The result is a simpler decision tree with six fewer nodes. However, this simpler decision tree will
still lead to the same life expectancy calculations for decision node A1. Therefore, this first step in the folding-
back
operation has reduced the original decision tree to an equivalent decision tree with six fewer nodes.
At first glance, it may appear that the folding- back operation has eliminated the uncertainty about the disease. That
uncertainty originally complicated the treatment decision represented by the tree. In reality, the folding- back operation
has incorporated the disease uncertainty into the revised life expectancies for the new final outcomes.
Continuing the folding-
back operation, now consider the nodes encountered just before the final outcomes in the
reduced decision tree shown in Figure7.2. These are the decision nodes representing the choice between Treatment or
Notreatment (decision nodes A2, A6 and A9).
Consider decision node A2. This node represents the choice between a life expectancy of 13.6 years and a life expec-
tancy of 14.4 years. Once again, the goal is to maximize life expectancy. Therefore, the decision at node A2 would be to
choose 14.4 years, which corresponds to the Treatment alternative. This means that facing the decision represented by
P [Test
+
]
0.48
P [Test
–
]
0.52
No treatment
Treatment
No treatment
Treatment
No treatment
Treatment
Test
No test
LE = 13.6 years = (0.40 × 4.0 years) + (0.60 × 20.0 years)
LE = 14.4 years = (0.40 × 12.0 years) + (0.60 ×
16.0 years)
LE = 7.0 years = (0.75 × 3.0 years) + (0.25 × 19.0 years)
LE = 12.0 years = (0.75 × 11.0 years) + (0.25 ×
15.0 years)
LE = 17.8 years = (0.08 × 3.0 years) + (0.92 × 19.0 years)
LE = 14.7 years = (0.08 × 11.0 years) + (0.92 × 15.0 years)
(A1)
(A2)
(A6)
(A9)
(A5)
Figure7.2 Reduced decision tree after step1 of the folding- back operation is applied to the decision tree in Figure7.1. The chance nodes for
disease uncertainty in Figure7.1 have been replaced with life expectancies computed for the eliminated chance nodes.
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Decision tree analysis 117
node A2 would be equivalent to facing an outcome that has a life expectancy of 14.4 years. Therefore, node A2 can
bereplaced with an outcome with a life expectancy of 14.4 years without changing the life expectancies that will be
calculated for decision node A1.
The decision tree shown in Figure7.3 has used this logic to replace each of the treatment decision nodes in Figure7.2.
The resulting decision tree is equivalent to the original decision tree but with nine fewer nodes.
Finally, consider chance node A5 in Figure7.3. This node represents an uncertainty that can result in a life expectancy
of 12.0 or 17.8 years. The corresponding branch probabilities are 0.48 and 0.52, respectively. This means the life expec-
tancy for chance node A5 is:
LE
048120 052178 15 0.. .. .yearsyears years
Therefore, as shown in Figure7.4, the third step in the folding- back operation results in an equivalent decision tree
with a single decision node (A1). One of the alternatives for node A1 is to forgo the test, which has a life expectancy
of14.4 years. The other alternative for node A1 is to perform the test, which has a life expectancy of 15.0 years. Because
15.0 years is greater than 14.4 years, it follows that the life-
expectancy maximizing decision is to have the test.
In summary, the folding- back operation reduces the original decision tree for the problem to an equivalent decision
tree with a single decision node. The reduction follows a step- by- step process that replaces each node based on one of
the following two operations:
Chance node: Replaced by an outcome with life expectancy equal to the life expectancy computed for the possible
outcomes that can follow the node.
Decision node: Replaced by an outcome with life expectancy equal to the maximum life expectancy for the possible
alternatives that can be chosen at the node.
In effect, the folding-
back operation determines the life expectancy faced at each of the nodes in the decision tree.
0.48
0.52
Test
No test
LE
= 14.4 years = Maximum of 13.6 years and 14.4 years
LE = 12.0 years = Maximum of 7.0 years and 12.0 years
LE
= 17.8 years = Maximum of 17.8 years and 14.7 years
(A1)
(A5)
P [Test
+
]
P [Test
–
]
Figure7.3 Reduced decision tree after step2 of the folding- back operation is applied to the decision tree in Figure7.2. The decision nodes for the
choice of treatment in Figure7.2 have been replaced with life expectancies for the best alternative for the eliminated decision nodes.
Test
No Test
LE = 14.4 years
LE = 15.0 years = (0.48×12.0 years) + (0.52 ×
17.8 years)
(A1)
Figure7.4 Reduced decision tree after step3 of the folding- back operation is applied to the decision tree in Figure7.3. The chance nodes for the
test result uncertainty in Figure7.3 have been replaced with life expectancies computed for that chance node.
Folding- back operation
1. Locate the nodes encountered just prior to a final outcomes in the decision tree. Replace each of those
penultimate nodes as follows:
• In the case of a chance node, replace the node by an outcome with an outcome value that equals the expected
value for that node, computed by multiplying the probabilities for the branches emanating from the node by
the values for the corresponding final outcomes.
• In the case of a decision node, replace the node with an outcome that has a value equal to maximum outcome
value for all of the final outcomes reachable from the node.
2. If the resulting decision tree consists of a single decision node, the best decision alternative is the alternative
corresponding to the greatest outcome value.
3. Otherwise, repeat starting at Step1.
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118 Medical decision making
7.2.2 Chance node ordering revisited
Before applying the folding- back operation to a clinical problem, we will return to the topic of how nodes are ordered
in a decision tree. The ordering of nodes in a decision tree often matches the temporal order in which the correspond-
ing uncertainty would be resolved or a decision would be made. Section6.4.4 described an alternate ordering of the
nodes in the arteriography decision tree discussed in Chapter6. Determining some of the branch probabilities was
simpler if the chance node representing the uncertainty about the presence of coronary artery disease was repositioned
in the tree. How much flexibility is there in the positioning of a chance node in a decision problem like Problem 1,
which involves more than one decision?
For example, Figure7.1 shows a decision tree for Problem 1 that positions the disease state chance nodes just before
the final outcomes. This placement makes sense conceptually because the patient’s actual disease state often does not
become apparent until the effects of treatment are observed. However, this ordering of the nodes required calculations,
both for the test result branch probabilities as well as the branch probabilities for the possible disease state after the test
result is known.
Figure7.5 shows an alternate decision tree for Problem 1. This revised decision tree places the chance node repre-
senting the disease state uncertainty chance nodes that would appear to avoid those branch probability calculations.
However, the resulting decision tree is called “problematic” for reasons that will become apparent.
Figure7.6 shows the so-
called problematic decision tree after the first step of the folding back operation. For this
decision tree, the process starts with the treatment decision nodes since these are positioned just before the final out-
comes (decision nodes D3, D4, D7, D8, D10, and D11). In Figure7.6, each of these decision nodes has been replaced by
outcomes with the corresponding maximum life expectancies. For example, decision node D3 is a choice between a life
expectancy of 4.0 and 12.0 years. Therefore, this decision node has been replaced by an outcome with a life expectancy
of 12.0 years. Similarly, replacements were made for the other five decision nodes.
Figure7.7 shows the continuation of the folding back operation. As shown in Figure7.6, the first step concluded with
a decision tree that placed the chance nodes for the test result as the nodes leading to the final outcomes (chance nodes
D6 and D9). These two chance nodes are replaced by outcomes with the corresponding life expectancies.
The remaining chance nodes in resulting decision tree (see Figure7.8) represent the uncertainty about the disease
(chance nodes D2 and D5). Once again, these chance nodes are replaced by outcomes with the life expectancies com-
puted for the corresponding node.
No treatment
Treatment
D
–
and treatment and test (15 years)
Test
No test
(D1)
(D3)
P [D
+
]
P [D
+
]
P [D
–
]
P [D
–
]
0.40
0.60
(D2)
No treatment
Treatment
(D11)
No treatment
Treatment
(D10)
No treatment
Treatment
(D8)
No treatment
Treatment
(D7)
No treatment
Treatment
(D4)
P [Test
+
| D
+
]
P [Test
–
| D
+
]
P [Test
+
| D
–
]
P [Test
–
| D
–
]
0.90
0.10
(D6)
0.20
0.80
(D9)
0.40
0.60
(D5)
D
+
and no treatment (4 years)
D
+
and treatment (12 years)
D
–
and treatment (16 years)
D
–
and no treatment (20 years)
D
+
and no treatment and test (3 year)
D
+
and treatment and test (11 years)
D
+
and no treatment and test (3 years)
D
+
and treatment and test (11 year)
D
–
and no treatment and test (19 years)
D
–
and treatment and test (15 years)
D
–
and no treatment and test (19 years)
Figure7.5 Problematic decision tree for Problem 1. The chance nodes for the disease state have been moved to before the treatment decision node.
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Decision tree analysis 119
Figure7.8 shows the problematic decision tree after the third and final step in the folding- back operation. The life
expectancies for the chance nodes representing the uncertainty about the disease (nodes D2 and D5) have been com-
puted. The result is a simple choice decision between an option that would lead to a life expectancy of 16.8 years and
an option that would lead to a life expectancy of 15.8 years.
The preferred option for the decision posed in Figure7.8 is clear– the No test alternative has a higher life expectancy
than the Test alternative. However, this result is the opposite of the result determined by folding back the original deci-
sion tree for Problem 1. That is why the problematic decision tree is problematic. Moving the disease state chance node
to a position before the treatment decision node resulted in a tree that no longer accurately represents the actual deci-
sion problem.
In hindsight, the mistake with the problematic decision tree was apparent back in Figure7.6. This figure showed the
decision tree after the first step of the folding back operation. The reduction step determined the optimal treatment
decision based on the test result. However, for a given disease state, the same treatment decision is optimal for either
test result. In other words, the test result no longer matters. Of course, this makes no sense. Why perform the test if the
disease is already known, as implied by the revised decision tree?
The example provided by the problematic decision tree may now seem obvious; however, it illustrates the one rule
that must be followed when ordering the nodes in a decision tree. A chance node represents an uncertainty.
P [D
+
]
P [D
+
]
P [D
–
]
P [D
–
]
P [Test
+
| D
+
]
P [Test
–
| D
+
]
P [Test
+
| D
–
]
P [Test
–
| D
–
]
LE = 12.0 years = Maximum of 4.0 years and 12.0 years
LE = 20.0 years = Maximum of 20.0 years and 16.0 year
s
LE = 11.0 years = Maximum of 3.0 years and 11.0 years
LE = 11.0 years = Maximum of 3.0 years and 11.0 years
LE = 19.0 years = Maximum of 19.0 years and 15.0 year
s
LE = 19.0 years = Maximum of 19.0 years and 15.0 year
s
Test
No test
(D1)
0.40
0.60
(D2)
(D6)
0.40
(D5)
(D9)
0.60
0.90
0.10
0.20
0.80
Figure7.6 Reduced decision tree after step1 of the folding- back operation is applied to the problematic decision tree shown in Figure7.5.
Thedecision nodes for the choice of treatment in Figure7.5 have been replaced with life expectancies for the best alternative for the eliminated
decision nodes.
P [D
+
]
P [D
–
]
P [D
+
]
P [D
–
]
LE = 12.0 years
LE = 20.0 years
LE = 11.0 years = (0.90×11.0 years) + (0.10×
11.0 years)
LE = 19.0 years = (0.20×19.0 years) + (0.80×
19.0 years)
Test
No test
(D1)
0.40
0.60
0.60
(D2)
0.40
(D5)
Figure7.7 Reduced decision tree after step2 of the folding- back operation is applied to the decision tree in Figure7.6. The chance nodes for the
test result uncertainty in Figure7.6 have been replaced with life expectancies computed for these chance nodes.
LE = 16.8 years = (0.40×12.0 years) + (0.60×
20.0 years)
LE = 15.8 years = (0.40×11.0 years) + (0.60×19.0 years)
Test
No test
(D1)
Figure7.8 Reduced decision tree after step3 of the folding- back operation is applied to the decision tree in Figure7.7. The chance nodes for the
disease uncertainty in Figure7.7 have been replaced with life expectancies computed for those chance nodes.
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120 Medical decision making
Thebranches originating from the node represent how that uncertainty might be resolved. Therefore, a chance node
cannot be placed before a decision node representing a choice that would not be made before that uncertainty would
be resolved in the actual decision problem.
7.2.3 Two- stage decision inthe management ofcoronary artery disease
We now will use the folding- back operation to analyze a more detailed example involving the management of coro-
nary artery disease. The problem we will analyze extends the arteriography decision example (Problem 2) discussed
in the previous chapter.
The extension to Problem 2 analyzed in this chapter adds the optional use of an exercise stress test (EST). We will
consider only a small part of what can be learned from this multifaceted test. The results of an exercise stress test
willbe classified as either “positive” or “negative.” A positive result supports a diagnosis of coronary artery disease
whereas a negative result contradicts a diagnosis of coronary artery disease. The example also will include the pos-
sibility of an indeterminate exercise stress test result. This result occurs when the patient is unable to achieve the
necessary exercise level. Table7.1 shows the probabilities for positive, negative, and indeterminate exercise stress test
Node- ordering rule
When ordering the chance nodes in a decision tree, the relative position of one chance node relative to another
chance node is arbitrary. However, the position of chance node A relative to a decision node B must be such that
if node A is placed before node B then, in the actual decision problem, the uncertainty represented by node A
must be fully resolved before the decision represented by node B would be made.
Definition: Problem 2
Decide whether to order coronary arteriography for a 65- year- old woman with stable angina and who is consid-
ering PCI or CABG because of dissatisfaction with drug- based therapy. Assume:
NOCAD CAD
Probability of disease state 0.3000 0.7000
Probability of death during arteriography 0.0001* 0.0010
Probability of death during PCI/CABG N.A. 0.0295
Life expectancy with no treatment 19.2 years* 5.7 years*
Life expectancy with drug- based therapy 19.2 years* 11.4 years
Life expectancy with PCI/CABG N.A. 15.2 years
* Low- confidence parameter values.
Table7.1 Probability ofpositive, negative, andindeterminate exercise stress test results conditioned
onthe patient’s disease state.
Assumption
Disease state
NOCAD CAD
Probability exercise stress test result is indeterminate 0.3317 0.2720
Probability exercise stress test result is negative 0.6154 0.3523
Probability exercise stress test result is positive 0.0529 0.3757
Adapted from Bartel etal. (1974).
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Decision tree analysis 121
results, conditioned on the patient’s disease state from one of the many published studies of exercise stress testing
(Barteletal.,1974). These measures of the test’s diagnostic accuracy will be used in our example.
There is a small chance an exercise stress test will cause the patient’s death. In 1980 a national survey of facilities
conducting exercise stress tests reported approximately one death per 10 000 tests (Stuart and Ellestad,1980). As was
the case with coronary arteriography, all of the fatal complications with exercise stress testing occurred in patients with
confirmed coronary artery disease.
In summary, we will consider the case of the 65-
year- old woman, we called Ms. Maple in the preceding chapter. She
is thought to have coronary artery disease, although the diagnosis is not certain. This patient’s condition currently is
managed by a drug- based therapy. Because of poor compliance with this therapy, more aggressive treatment is being
considered. Depending on further assessment of Ms. Maple’s heart, the more aggressive treatment will either be per-
cutaneous coronary intervention or coronary artery bypass graft surgery. These more aggressive treatments require
coronary arteriography. Before proceeding with coronary arteriography, Ms. Maple could first undergo an exercise
stress test. This less invasive test could provide information that affects the probability of coronary artery disease for
Ms. Maple. We will call this example Problem 3.
7.2.4 Decision tree fortwo- stage coronary artery disease management decision
Figure7.9a and b shows a decision tree for Problem 3. This problem requires a more complicated decision tree that no
longer fits on a single page. However, portions of the tree have the same structure as the decision tree for Problem 2
discussed in the previous chapter.
Figure7.10 shows the alternate decision tree that was developed for Problem 2in the previous chapter. Recall that
this version of the tree for Problem 2 placed the chance node representing the uncertainty about Ms. Maple’s disease
immediately after the arteriography decision node. Referring to Figures7.9a and 7.9b, which show a portion of the
decision tree for Problem 3, notice that after the decision to forgo the exercise stress test, the portion of the tree starting
with decision node E2 is identical to the decision tree for Problem 2 shown in Figure7.10.
The changes to the decision tree, required by the addition of the exercise stress test decision, start with chance node
E8 in Figure7.9a. This node represents the uncertainty about Ms. Maple’s survival if she undergoes the exercise stress
test. Problem 3 assumes this test is not life- threatening if Ms. Maple does not have coronary artery disease. However,
there is a small probability (0.0001) that Ms. Maple will die during an exercise stress test if she does have coronary
artery disease. Since the probability of coronary artery disease is 0.70 for Ms. Maple, it follows that
P
ESTPCADP ESTCADDieDie
,...07000001 0 00007 0.
.0001
Definition: Problem 3
Decide whether to order coronary arteriography for a 65- year- old woman with stable angina and who is
considering PCI or CABG because of dissatisfaction with drug- based therapy. An exercise stress test also can be
ordered and the result used to make the arteriography decision. Assume:
NOCAD CAD
Probability of disease state 0.3000 0.7000
Probability of death during arteriography 0.0001* 0.0010
Probability of death during PCI/CABG N.A. 0.0295
Life expectancy with no treatment 19.17 years* 5.70 years*
Life expectancy with drug- based treatment 19.17 years* 11.39 years
Life expectancy with PCI/CABG treatment N.A. 15.21 years
Probability for indeterminate EST result 0.3317 0.2720
Probability negative EST result 0.6154 0.3523
Probability positive EST result 0.0529 0.3757
Probability of death during EST 0.0000 0.0001*
* Low- confidence parameter values.
.
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122 Medical decision making
Figure7.9 (a) Decision tree for Problem 3. The portions of the tree in the grey regions are expanded in Figure7.9b. (b) Portions of the decision tree
for Problem 3 are not expanded in Figure7.9a. These expansions are for the portions of the decision tree shown in the grey regions in Figure7.9a.
(E6) in Figure 7.9b
(E5) in Figure 7.9b
Death
LE = 0 years
NO EST
EST
P [Die | EST]
0.0001
(E1)
(E8)
(E9)
Rx = MED & NO CA
D
LE = 19.2 years
Rx = MED & CAD
LE = 11.4 years
(E11)
(E12)
NO ART
ART
(E6) in Figure 7.9b
(E5) in Figure 7.9b
Rx = MED & NO CA
D
LE = 19.2 years
Rx = MED & CAD
LE = 11.4 years
P [NO CAD]
P [NO CAD]
0.3000
(E3)
(E4)
NO ART
ART
(E2)
Rx = MED & NO CA
D
LE = 19.2 years
Rx = MED & CAD
LE = 11.4years
(E14)
(E15)
NO ART
ART
(E13)
Rx = MED & NO CA
D
LE = 19.2 years
Rx = MED & CAD
LE = 11.4 years
(E17)
(E18)
NO ART
ART
(E16)
P [EST
±
]
0.2842
P [EST
−
]
0.3797
P [EST
+
]
0.3361
(E6) in Figure 7.9b
(E5) in Figure 7.9b
(E6) in Figure 7.9b
(E5) in Figure 7.9b
P [Live | EST]
0.9999
P [CAD]
P [CAD]
0.7000
0.3000
0.7000
(a)
P [NO CAD | EST
±
]
P [NO CAD | EST
±
]
0.3501
P [CAD | EST
±
]
0.6499
0.3501
P [CAD | EST
±
]
0.6499
P [NO CAD | EST
−
]
P [CAD | EST
−
]
P [NO CAD | EST
−
]
P [CAD | EST
−
]
P [NO CAD | EST
+
]
P [CAD | EST
+
]
P [NO CAD | EST
+
]
P [CAD | EST
+
]
0.4862
0.5138
0.4862
0.5138
0.0472
0.9528
0.0472
0.9528
EST±
: Indeterminate EST result
EST−
: Negative EST result
EST+: Positive EST result
(E10)
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Decision tree analysis 123
Assuming Ms. Maple survives the exercise stress test, chance node E9 represents the uncertainty about the
test result. Recall there are three possibilities: (1) indeterminate result, (2) negative result, and (3) positive result.
The
following abbreviations are used to conserve space in Figures7.9a and7.9b:
EST
+
−
: denotes indeterminate exercise stress test result
EST
−
: denotes negative exercise stress test result
EST
+
: denotes positive exercise stress test result
The corresponding branch probabilities are computed using the conditional probabilities provided in the definition
for Problem 3. For example,
P
ESTPESTNOCAD PNOCAD PEST CADPCAD
0 3317 0 3000 0 2639 0 7000 0 2842.. .. .
Similarly,
PEST
0 6154 0 3000 0 2787 0 7000 0 3797.. .. .
PEST
0 0529 0 3000 0 4575 0 7000 0 3361.. .. .
Rx = None & NO CAD
LE = 19.2 years
0.9999
Death
LE = 0 years
P [Die | ART, NO CAD]
P [Die | ART, CAD]
P [Live | ART, CAD]
P [Live | ART, NO CAD]
0.0001
(E5)
Figure 7.9a
0.9705
Rx = PCI/CABG &
CAD
LE = 15.2 years
Death
LE = 0 years
0.9990
P [Die | PCI/CABG]
P [Live | PCI/CABG]
0.0295
Death
LE = 0 years
0.0010
(E7)
(E6)
Figure 7.9a
(b)
Figure7.9 (Continued )
NO ART
ART
Rx = MED & NO CAD
(19.2 years)
Rx = MED & CAD
(11.4 years)
P [NO CAD]
P [NO CAD]
P [CAD]
P [CAD]
0.3000
0.7000
(C1)
(C2)
0.3000
0.7000
(C3)
Rx = None & NO CAD
(19.2 years)
0.9999
Death
(0 years)
0.0001
(C4)
Rx = PCI/CABG & CA
D
(15.2 years)
Death
(0 years)
0.9990
P [Die | PCI/CABG, CAD]
P [Live | PCI/CABG, CAD]
0.0295
0.9705
Death
(0 years)
0.0010
(C5)
(C6)
P [Die | ART, NO CAD]
P [Live | ART, NO CAD]
P [Die | ART, CAD]
P [Live | ART, CAD]
Figure7.10 Copy of Figure6.11 showing the alternate decision tree for Problem 2.
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124 Medical decision making
The portion of the tree following chance node E9 represents the three different arteriography decisions that would
be faced after the exercise stress test result is known. These three decisions (decision nodes E10, E13, and E16) each
have the same two alternatives ART and NO ART. What differs between these three decisions is the probability of
coronary artery disease given the corresponding exercise stress test result.
For example, decision node E10 represents the arteriography decision if the exercise stress test result is indetermi-
nate. That test result determines the probabilities for the branches originating from the subsequent disease chance
nodes (E11 and E12). We use Bayes’ formula from Chapter4 to compute those branch probabilities:
P
NO CADEST
PNOCAD PEST NO CAD
PNOCAD PEST NO C
AAD PCAD PEST CAD
0 3000 0 3317
0 3000 0 3317 0
..
.. .77000 0 2720
0 3501
.
.
which means
P
CADEST PNOCAD EST
11
0 3501 0 6499
..
Similar calculations determine the branch probabilities for the disease chance nodes (E14, E15, E17, and E18)
following the other two test results.
The remainder of the decision tree for Problem 3 represents the uncertainty about surviving arteriography and the
resulting treatment. For example, if Ms. Maple does not undergo coronary arteriography, she will remain on her cur-
rent drug-
based therapy, denoted by Rx=MEDin the decision tree. Her actual disease state is denoted by CAD and
NOCAD in the decision tree. The corresponding life expectancies in the definition for Problem 3 determine what
Ms.Maple’s life expectancy will be without arteriography.
On the other hand, if Ms. Maple does undergo arteriography, she faces the small risk to her survival from the proce-
dure. Chance nodes E5 and E6 in Figure7.9b represent this uncertainty. Chance node E5 represents Ms. Maple’s risk of
death from the procedure if she does not have coronary artery disease (0.0001). Chance node E6 represents Ms. Maple’s
risk of death if she does have coronary artery disease (0.0010).
Arteriography could discover that Ms. Maple does not have coronary artery disease. This would mean she is not a
candidate for more aggressive treatment. Moreover, her medical therapy would be discontinued. As discussed in the
previous chapter, this would mean she faces the life expectancy that is roughly of a 65-
year- old woman in the general
population.
If arteriography discovers significant narrowing in her coronary arteries Ms. Maple will undergo either a percutane-
ous coronary intervention (PCI) or coronary artery bypass graph surgery (CABG), depending on additional assess-
ments of her heart. Chance node E7 represents the uncertainty about surviving the subsequent treatment. Problem 3
assumes the probability of a perioperative death is 0.0295 for Ms. Maple.
The decision tree for Problem 3has more nodes and branches than the decision tree for Problem 2. However, more
importantly, the life expectancies for the decision options in this more complicated tree can no longer be determined by
simply multiplying the branch probabilities on each of the paths. Calculating the outcome probabilities must account
for how the exercise stress test affects the subsequent arteriography decision. This requires the folding- back operation.
7.2.5 Folding- back operation applied totwo- stage coronary artery disease decision problem
The folding- back operation starts with part 2 of this decision tree, which is reproduced in Figure7.11. The two
fragments of the decision tree shown in Figure7.11 represent the uncertainty about the consequences of arteriography
for Ms. Maple. If Ms. Maple does not have coronary artery disease, the only uncertainty with arteriography is the small
probability that she will die during the procedure. Otherwise, arteriography would reveal the patency of Ms. Maple’s
coronary arteries. This result would indicate that further treatment is unnecessary. Problem 3 assumes Ms. Maple’s life
expectancy without treatment and with disease would be 19.2 years. Therefore, the life expectancy for node E5 in
Figure7.11 is determined by the expression.
2
LE
0 0001 009999 19 219 1981....yearsyears years
2
The unrealistic number of significant figures are required to show how undergoing arteriography reduces her life expectancy.
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