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Decision trees– representing thestructure ofadecisionproblem 105

to establish a clinical policy that will be applied to any 65- year- old woman with stable angina. In this case, the

lifeexpectancy of the patient with a normal arteriography finding probably should be estimated by 19.2 years, the life

expectancy derived from studies of patient survival with treatments for coronary artery disease. The life expectancy of

19.2 years is a value that could be defended by objective evidence from these studies.

On the other hand, suppose the goal of the analysis is to determine the best decision for the specific individual rep-

resented by Ms. Maple in this example. Ms. Maple might differ significantly from that “typical” stable angina patient

being considered for an invasive and expensive diagnostic procedure. Her physician might consider Ms. Maple to be

truly disease free, except for the possibility of blockages in her coronary arteries. In this case, her physician might

choose to assume Ms. Maple’s life expectancy will be close to 24.2 years if the arteriography findings are negative. The

discussion of sensitivity analysis in the next chapter will show how to address these discrepancies.

In either case, recall that the focus of this chapter, and the next chapter, is to find the choice that maximizes her life

expectancy. Table6.1 shows the values for Ms. Maple’s life expectancies that will be used in this discussion. These life

expectancies are sex and age- adjusted from the life expectancies published by CASS.

Notice that most of the life expectancies in Table6.1 can only be determined to be within a range. For example, as we

just saw, there is uncertainty about how long Ms. Maple will live if she does not have coronary artery disease and has

no treatment. Her life expectancy if she has coronary artery disease and is untreated is even less certain. Based on pub-

lished data, all we can say is that, almost certainly, her life expectancy will be less than the 11.4 years reported by studies

of patients with coronary artery disease who are treated with drug- based therapy (Mock etal.,1982). Similarly, unneces-

sary drug therapy, if Ms. Maple does not have coronary artery disease, will not increase her life expectancy over what

she could expect without treatment. Therefore, Ms. Maple’s life expectancy with unnecessary drug therapy can only be

bounded above by the 19.2 years found in published data for untreated patients without disease. Once again, the discus-

sion of sensitivity analysis later in the next chapter shows how to manage these problematic parameter values.

The entry in Table6.1 for unnecessary aggressive treatment is “Not Applicable” (NA). This entry follows from one

of the assumptions that will be made. Arteriography surely is not a perfect test of coronary artery disease. There is

some evidence that this test can produce both false- positive and false- negative results. However, most likely those

misleading results are extremely rare. Therefore, this analysis will assume that arteriography is a perfect test. The pro-

cedure also is the necessary first step for the more aggressive treatments. Therefore, assuming that arteriography is a

perfect test is equivalent to assuming that Ms. Maple has PCI or CABG if and only if she has coronary artery disease.

We also must consider the risks to Ms. Maple’s survival because of the procedures she might undergo. Since we

assume that arteriography is a perfect diagnostic test and PCI or CABG will not be considered without a positive arte-

riography finding, the risk of these treatments for nondiseased patients is not a consideration. On the other hand, fatal

complications do result from the more aggressive therapies. The risk from PCI is very low, probably causing approxi-

mately one death per 1000 cases (Peterson etal.,2010). However, some patients subjected to more aggressive treatment

for coronary artery disease must undergo CABG because of the location of the coronary obstructions and other consid-

erations. Therefore, this example will assume a risk of death equal to 0.0295 for patients undergoing PCI/CABG.

Large studies of arteriography have reported fatal complications (Bourassa and Nobel, 1976). However, those

deathsinvariably occur in patients with coronary artery disease. This means the possibility of fatal arrhythmias or sig-

nificant vascular trauma during catheterization does not appear to be a significant risk for patients without coronary

artery disease. Therefore, the risk of death because of arteriography will be set at 0.0010 for patients with coronary artery

disease. This risk of death will be set at 0.0001 for patients without coronary artery disease who undergo arteriography.

6.4.2 Simple decision inthe management ofcoronary artery disease

The next chapter will analyze a coronary artery disease management decision that includes an additional diagnostic

test. However, the coronary artery disease management decision– called Problem 2 – discussed in this chapter will

focus only on the decision of whether Ms. Maple should have coronary arteriography.

Table6.1 Life expectancy forMs. Maple derived fromthe CASS data.

Treatment

Disease state

NOCAD CAD

No treatment (NONE) 19.2–24.2 years <11.4 years

Drug-

based therapy (MED) <19.2 years 11.4 years

Aggressive treatment (PCI/CABG) N.A. 15.2 years

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106 Medical decision making

This decision assumes that Ms. Maple faces a perioperative risk of 0.0295 if she has coronary artery disease. As stated

earlier, we also will assume that she will not undergo one of the more aggressive treatments if she does not have coro-

nary artery disease. Ms. Maple’s risk of death from arteriography is assumed to be 0.0010 or 0.0001 depending on

whether she does or does not have coronary artery disease. We also assume that her life expectancies, depending on

her treatment and actual disease state, are what are shown in Table6.1. Finally, we assume that this decision is made

with the opinion that the probability Ms. Maple actually has coronary disease is 0.7000. This probability matches the

prevalence of positive arteriograms in the CASS study.

Figure6.9 shows a decision tree for Problem 2. The branch labeled NOART represents the decision to forgo the

procedure. That branch leads to a chance node, labeled B2, which represents the uncertainty about Ms. Maple’s disease

state. Node B2 has two branches, corresponding to the two possible disease states. Those two branches lead to the

outcomes representing the continuation of drug- based therapy for the corresponding disease states. Notice that the

outcomes are shown in Figure6.2 with the corresponding values assumed for Ms. Maple’s life expectancy. For example,

Problem 2 assumes the life expectancies reported by studies of survival for medically treated coronary artery disease,

adjusted to a 65- year- old woman (11.4 years).

The other branch from node B1, labeled ART, leads to the portion of the decision tree representing how undergoing

arteriography will affect Ms. Maple’s life expectancy. This begins with the chance node labeled B3, which represents

the uncertainty about surviving the procedure.

Definition: Problem 2

Decide whether to order coronary arteriography for a 65- year- old woman with stable angina and who is consider-

ing PCI or CABG because of dissatisfaction with drug-

based therapy. Assume:

NOCAD 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.

Death

(0 years)

NO ART

ART

Die from ART

Survive ART

ART ⇒ NO CAD

Death

(0 years)

Die from PCI/CABG

Survive PCI/CABG

ART

⇒

CAD

Rx = MED & NO CAD

(19.2 years)

Rx = MED & CAD

(11.4 years)

NO CAD

CAD

(B1)

(B2)

(B3)

(B4)

(B5)

Rx = None & NO CAD

(19.2 years)

Rx = PCI/CABG & CA

(15.2 years)

Figure6.9 Decision tree for Problem 2. Chance nodes are ordered according to how the corresponding uncertainties in the problem typically

resolve over time.

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Decision trees– representing thestructure ofadecisionproblem 107

Two branches originate from node B3. The branch labeled “Die from ART” represents the possibility of a fatal

complication from arteriography. This branch leads to the outcome labeled “Death,” which has a life expectancy of

0years. The other branch, labeled “SurviveART,” represents what happens if Ms. Maple survives the procedure.

Thisbranch leads to node A4, representing the uncertainty about what arteriography will reveal if it does not cause

Ms. Maple’s death.

Problem 2 divides the range of arteriography findings into two groups represented by the abbreviations:

NOCAD

No significant coronary artery narrowing was found

CAD

Significant narrowing was found

The decision tree in Figure6.9 labels the corresponding branches ART

⇒ NOCAD and ART ⇒ CAD, respectively.

Because Problem 2 assumes arteriography to be a perfect test for coronary artery disease, the findings on the arterio-

gram will define Ms. Maple’s final treatment. A negative finding means that she does not have coronary artery disease.

So the drug-

based therapy would be discontinued. A positive finding means that Ms. Maple faces the risk involved

with the more aggressive treatment indicated by the arteriogram, as presented by the chance node B5.

So far, the Problem 2 decision tree has ordered the nodes according to how the corresponding uncertainties typically

would resolve over time. For example, once the arteriography procedure has been ordered, the uncertainty about the

patient’s survival would be resolved first. Next, the uncertainty about the procedure findings would be resolved.

Typically, the last uncertainty resolved would be Ms. Maple’s actual disease state.

This temporal ordering of the chance nodes seems natural. Often this ordering helps assure that the tree represents

all considerations in a decision problem. However, in a problem with only one decision node, there are no firm rules

about the ordering of the nodes except that the primary decision is placed first and the final outcomes are placed last.

The math used to analyze the tree does not care about the order of the nodes. On the other hand, other orderings of the

nodes can reduce the work in determining the branch probabilities. These alternate orders are discussed later in this

section.

6.4.3 Determining thebranch probabilities

Figure6.10 labels the branch probabilities for the decision tree in Figure6.9. Note that some of these probabilities can

be determined directly from the problem definition. For example, the probabilities on the branches starting at node B2

are the probabilities that Ms. Maple has, or does not have, coronary artery disease. Therefore, from the definition of

Problem 2:

P NO CAD



 0 3000.

P CAD



 0 7000.

Death

(0 years)

NO ART

ART

P[Die|ART]

0.0007

P[Live|ART]

0.9993

P[ART ⇒ NO CAD]

Death

(0 years)

P[Die|PCI/CABG, CAD]

0.0295

P[Live|PCI/CABG, CAD]

0.9705

Rx = MED & NO CAD

(19.2 years)

Rx = MED & CAD

(11.4 years)

P[NO CAD]

0.3000

P[CAD]

0.7000

(B1)

(B2)

(B3)

(B4)

0.3000

P[ART ⇒ CAD]

0.7000

(B5)

Rx = None & NO CAD

(19.2 years)

Rx = PCI/CABG & CA

(15.2 years)

Figure6.10 Decision tree for Problem 2with branch probabilities.

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108 Medical decision making

Notice that because arteriography is assumed to be a perfect test for coronary artery disease, the branch probabilities

for the possible arteriography findings equal the probabilities for the corresponding disease states. That is

PART NOCADPNO CAD







 030.

and

PART CADPCAD











070.

Other branch probabilities require more thought. For example, what is the probability that Ms. Maple will survive

coronary arteriography? This uncertainty corresponds to the branch probability labeled “P[Live

∣ ART]” in Figure6.10.

The same method used to determine the probability of a positive test result in Problem 1 is used.

The probability of surviving arteriography depends on Ms. Maple’s disease state. If her disease state is NOCAD,

Ms.Maple’s chance of surviving is 0.9999. Using the notation of conditional probability, this can be written:

P ARTNOCADLive|and



 0 9999.

P ARTNOCADDie| and



 0 0001.

This mathematical expression reads as “the probability of surviving conditioned on undergoing arteriography and

not having coronary artery disease equals 0.0001.” Similarly, if her disease state is CAD, her chance of surviving arte-

riography is

P ARTCADLive|and



 0 9990.

ARTCADDie| and





0010

Before arteriography, the presence or absence of coronary artery disease is unknown. Therefore, combining the prob-

abilities for each disease state with the corresponding probabilities of dying during arteriography yields:

P ARTLive|











0 3000 0 9999 0 7000 0 9990 0 9993.. .. .

In short, P[Live ∣ ART] is the sum of the probabilities of living for each of the disease states, weighted by the corre-

sponding probabilities for those health states.

The complication requiring this calculation is that Ms. Maple’s disease state affects her chances of surviving the

procedure. In the terminology of probability theory, Ms. Maple’s disease state and her survival are not independent

events. Knowing one event changes the probability of the other event. Put another way, not knowing the disease state

means that the value for the arteriography risk must account for each possible value for the disease state random

variable.

6.4.4 Alternate chance node ordering

Let us consider one of the alternate orderings of the nodes mentioned earlier. Figure6.11 shows an alternate decision

tree that simplifies how the branch probabilities are determined for Problem 2. The important change is the placement

of the nodes representing the uncertainty about Ms. Maple’s disease state. The first decision tree placed those nodes

just before the final outcomes. These are nodes B2 and B4 in Figure6.10. Notice that these nodes are located late in the

tree. The decision tree shown in Figure6.11 places the chance nodes for the disease state uncertainty just after the pri-

mary decision node. These chance nodes are C2 and C3 in Figure6.11. Otherwise, the trees shown in Figures6.10

and6.11 are the same.

The significance of changing the order of the nodes becomes apparent when determining the branch probabilities.

Consider the probability of the branch from node C4 to Death. This branch now is on a path that includes knowing that

Ms. Maple does not have coronary artery disease. Therefore, the branch probability measures the likelihood that

Ms.Maple will die from the arteriography if there are no significant blockages in her coronary arteries. According to

the definition for Problem 2, this probability equals 0.0001. Similarly, the probability for the branch from node C5 to

Death is 0.0010. Because the disease state chance node has been moved closer to the decision node, values for these

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Decision trees– representing thestructure ofadecisionproblem 109

branch probabilities can be taken directly from the problem definition. If Ms. Maple survives arteriography and has

coronary artery disease, the same simplifications hold for the probabilities on the branches originating from node C5

in Figure6.11.

In general, for the purposes of determining the branch probabilities, the optimal ordering of the nodes depends on

how the uncertainty in the problem is defined. In the case of Problem 2, the uncertainties about the outcome of the

arteriography and surviving the procedures are conditioned on Ms. Maple’s actual disease state. Therefore, placing the

chance node for her disease state first in the tree means that many of the branch probabilities can be taken directly from

the problem statement.

6.4.5 Computing thelife expectancy forthe decision alternatives

Recall that our goal is to determine the life expectancies for the two treatment alternatives. Since our focus in this

example is the patient’s life expectancy, the alternative with the greater life expectancy will be the preferred

alternative.

Once the branch probabilities are known, straightforward arithmetic determines the life expectancies. Recall that life

expectancy for an alternative is calculated by multiplying the outcome probability by the outcome life expectancy for

each outcome. This calculation is simple for the NOART decision because there are only two possible outcomes:

LE NO ART













0 3000 19 207000 11 4

13 7

.. ..

yearsyears

years

Calculating the life expectancy for the ART decision involves more terms but still is just arithmetic:

LE ART











0 3000 0 0001 003000 0 9999 19 20.. .. .yearsyears .. .

... .

7000 0 0010 0

0 7000 0 9990 0 0295 00700











years

years0009990 0 9705 15 2

16 1







.. .

years

Note in both calculations that each of the branch probabilities on a path from the initial decision to a final outcome

is conditioned on the branch probabilities that have been encountered earlier in the path.

For example, consider the path that starts with the decision ART, passes through chance node C5 and chance node

C6, and ends with the outcome labeled CADRx=PCI/CABG. Figure6.12 focuses on this single path and the corre-

sponding terms in the summation for LE[ART]. The first branch probability on this path is P[CAD], which equals

0.7000. Nothing else is known at this point on the path so the probability is not conditioned by any prior information.

The next branch probability is P[Live

∣ ART, CAD], the probability of surviving arteriography if Ms. Maple has coronary

Death

(0 years)

NO ART

ART

P [NO CAD]

0.3000

P [CAD]

0.7000

Death

(0 years)

Death

(0 years)

P[Die | PCI/CABG, CAD]

0.0295

P[Live | PCI/CABG, CAD]

0.9705

Rx = MED & NO CAD

(19.2 years)

Rx = MED & CAD

(11.4 years)

P [NO CAD]

0.3000

P [CAD]

0.7000

(C1)

(C2)

(C3)

(C5)

(C6)

Rx = None & NO CAD

(19.2 years)

Rx = PCI/CABG & CA

(15.2 years)

P [Die | ART, NO CAD]

(C4)

0.0001

P [Die | ART, CAD]

0.0010

P [Live | ART, NO CAD]

0.9999

P [Live | ART, CAD]

0.9990

Figure6.11 Alternate Decision Tree for Problem 2, with branch probabilities. The chance node for the uncertainty about the disease state has

been moved forward.

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110 Medical decision making

artery disease. This term comes after setting the disease state to NOCAD. Therefore, this branch probability is the

probability Ms. Maple survives arteriography if she has coronary artery disease. From the definition of Problem 2,

P ARTCADLive  ,.



 0 9990

The third branch probability on the selected path measures the likelihood that Ms. Maple survives the treatment

if she actually has coronary artery disease. According to the definition of Problem 2, this third and final branch

probability is

P PCICABGCADLive / ,.



 0 9705

The value for the last branch probability in the selected path is Ms. Maple’s life expectancy with coronary artery

disease if she has the aggressive therapy indicated for her disease. From the definition of Problem 2,

LE Rx PCICABGCAD



/years&.15 2

Note that moving the chance node for the disease state forward only delays the arithmetic needed to compute some

of the branch probabilities. With Problem 2, moving the node for the uncertainty about Ms. Maple’s disease state sim-

plified computing the probability that she would survive arteriography. With the original decision tree for Problem 2

(see Figure6.10) the probability that Ms. Maple survives arteriography had to be computed.

The alternate design for the tree was drawn with the chance node for Ms. Maple’s disease state placed before the

other chance nodes in Problem 2. This allowed for the probability of surviving arteriography to be taken directly from

the problem statement.

Figure6.13 focuses on the two paths in the alternate decision tree that lead to the outcomes in which Ms. Maple

survives arteriography. Figure6.13 also highlights the corresponding terms in the summation for LE[ART]. Notice that

these two terms contain the same multiplication steps that were used to compute the branch probability P[Live

∣ ART]

in the expression:

P ARTLive 











0 3000 0 9999 0 7000 0 9990 0 9993.. .. .

In other words, the alternate design for the decision tree may have simplified the determination of some of

the branch probabilities; however, the computation that is avoided is simply moved to the life expectancy

calculation.

P [Die | PCI/CABG, CAD]

0.0295

0.0010

P [Die | ART, NO CAD]

P [Die | ART, CAD]

0.0001

NO ART

ART

(C1)

P [NO CAD]

P [CAD]

0.3000

0.7000

(C3)

Rx = None & NO CAD

(19.2 years)

0.9999

Death

(0 years)

(C4)

Rx = PCI/CABG & CA

(15.2 years)

Death

(0 years)

0.9990

0.9705

Death

(0 years)

(C5)

(C6)

LE ART = (0.3000 × 0.0001 ×

(0.3000 × 0.9999 ×

0 years)

19.2 years)

+ (0.7000 × 0.0010 ×0 years)

+ (0.7000 × 0.9990 × 0.0295 ×

(0.7000 × 0.9990 × 0.9705 ×

0 years)

+ 15.2 years)

= 15.7 years

P [Live | ART, NO CAD]

P [Live | ART, CAD]

P [Live | PCI/CABG, CAD]

Figure6.12 Alternate decision tree for Problem 2 focusing on one possible outcome resulting from the decision to order arteriography and the

corresponding term in the expression for Ms. Maple’s life expectancy with arteriography.

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Decision trees– representing thestructure ofadecisionproblem 111

In theory, calculating the life expectancies for the two decision alternatives completes the analysis. The life expec-

tancy for ordering the arteriography is 16.1years. The life expectancy for forgoing arteriography is 13.7years. Therefore,

the decision tree analysis concludes with the recommendation that Ms. Maple has arteriography because this decision

would maximize her life expectancy.

In reality, this first calculation of the life expectancies would be where the real work begins in the analysis of a

decision tree. The definition of Problem 2 assumes several problematic values that should be examined more carefully.

This is done with sensitivity analysis, a topic discussed in the next chapter.

P [Die | PCI/CABG, CAD]

0.0295

P [Die | ART, NO CAD]

P [Die | ART, CAD]

0.0001

NO ART

ART

(C1)

P [NO CAD]

P [CAD]

0.3000

0.7000

(C3)

Rx = None & NO CAD

(19.2 years)

0.9999

Death

(0 years)

(C4)

Rx = PCI/CABG & CAD

(15.2 years)

Death

(0 years)

0.9990

0.9705

Death

(0 years)

(C5)

(C6)

LE ART = (0.3000 × 0.0001 ×

(0.3000 × 0.9999 ×

0 years)

19.2 years)

+ (0.7000 × 0.0010 ×0 years)

+ (0.7000 × 0.9990 × 0.0295 ×

(0.7000 × 0.9990 × 0.9705 ×

0 years)

+ 15.2 years)

= 15.7 years

P [Live | ART, NO CAD]

P [Live | ART, CAD]

P [Live | PCI/CABG, CAD]

0.0010

Figure6.13 Alternate decision tree for Problem 2 focusing on the two outcomes with Ms. Maple surviving arteriography. The corresponding terms

in the expression for Ms. Maple’s life expectancy with arteriography also are highlighted.

Summary

• A decision tree is used in decision analysis to depict the structure of a decision problem.

• The decision tree diagram is drawn using square- shaped nodes to represent choices or decisions and circular-

shaped nodes to represent uncertainties or chances. Typically, an initial decision node provides the root for a

decision tree. Sequences of decision node and chance node branches ultimately lead to each of the possible

outcomes faced by the patient.

•

Each branch originating from a decision node represents the options available for the corresponding decision in

the problem. The options represented by the decision node branches are organized so that exactly one option can

be chosen. Collectively, the options represented by the branches from a decision node represent all possible

choices for that decision.

• Each branch originating from a chance node represents the possibilities for the corresponding uncertainty in the

problem. The possibilities represented by the chance node branches are organized so that exactly one possibility

can be true. Collectively, the possibilities represented by the branches from a chance node represent all possibili-

ties for that uncertainty.

• The definition of a chance node branch includes a branch probability that measures the likelihood of the corre-

sponding possibility, conditioned on the choices and possibilities that must occur to reach that branch.

• Multiplying the branch probabilities encountered on the path to a final outcome determines the probability that

the outcome will occur.

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112 Medical decision making

Epilogue

This chapter described the organization of tree diagrams that represent the interactions between the decisions and

uncertainties in a decision problem. The examples used to illustrate the process were simple. This meant we could

easily compute life expectancies for the decision alternatives. Our focus was finding the decision alternative that

wouldresult in the greatest life expectancy for the patient. Therefore, a simple calculation found a solution to the

decision problem.

For example, with our hypothetical patient named Ms. Maple, we calculated that her life expectancy would be

16.1years if she underwent arteriography. We also calculated that her life expectancy would be 13.7 years if she did

notundergo arteriography. Therefore, we concluded that undergoing arteriography would be the best decision for

Ms.Maple because it would maximize her life expectancy.

This conclusion was based on straightforward calculations that determined the outcome probabilities by simply

multiplying the branch probabilities encounter on each path to an outcome. That simple calculation was possible

because of the simple structure of the decision tree.

The next chapter will expand on the concepts described in this chapter to show the analysis of more complicated

decision problems.

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Cardiology/American Heart Association task force on practice guidelines (Committee on Coronary Angiography). Journal of the American

College of Cardiology, 33, 1756–824.

Taylor, H.A., Deumite, N.J., Chaitman, B.R. etal. (1989) Asymptomatic left main coronary artery disease in the coronary artery surgery study

(CASS) registry. Circulation, 79, 1171–9.

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Medical Decision Making, Third Edition. Harold C. Sox, Michael C. Higgins, Douglas K. Owens, and Gillian Sanders Schmidler.

113

CHAPTER7

Decision tree analysis

7.1 Introduction

This chapter demonstrates how to analyze a decision tree. In particular, this chapter describes the analysis of compli-

cated decision trees that include multiple decision nodes. The previous chapter used a decision tree example that

had only a single decision node. The final step in analysis of that simple decision tree was the comparison of the

expected values calculated for the decision alternatives. Recall that expected value was calculated by combining the

outcome probabilities with outcome values that quantify how much one outcome is preferred relative to another.

The fundamental assumption of decision tree analysis is that the alternative with the greatest expected value is the

best choice.

This basic approach of comparing alternatives based on their expected values is common to all of the methods

discussed in this book. However, the method described in this chapter– based on what is called the folding- back

operation

– calculates an expected value using an approach that differs from what was demonstrated for the simple

example in the previous chapter.

That example was simple because there was only one decision node. This meant the expected value for an alterna-

tive could be determined directly by calculating the probabilities for each of the possible outcomes. Those were called

the outcome probabilities. Because the initial node represented the only decision in the tree, the probability for an out-

come could be calculated by simply multiplying the branch probabilities encountered on the path leading to that

outcome. Branch probabilities are expressed as conditional probabilities that accounted for how one chance node

value depended on another chance node value. This meant multiplying the branch probabilities produced a valid

outcome probability.

This direct approach to calculating the outcome probabilities becomes difficult to use when a decision tree

represents multiple interdependent decisions. The multiplication of the branch probabilities on the path to an

outcome must account for each of the decisions on the path that leads to those branches. With a medical decision

involving multiple tests and multiple treatment options, tracking how those decisions interact can be compli-

cated.The folding back operation manages that complexity by shifting the focus from outcome probabilities to

the expected values computed from those outcome probabilities. The folding back operation that will be

describedstill computes the outcome probabilities; however, it does so in the background while computing the

expected value.

The previous chapter stressed the importance of how values are determined for outcomes. These two chapters use

the patient’s length of life as the value for an outcome. If Outcome A means the patient will live twice as long as the

The folding- back operation also is called “averaging out” the decision alternatives.

7.1 Introduction 113

7.2

Folding- back operation 114

7.3

Sensitivity analysis 126

Epilogue 133

Bibliography 133

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114 Medical decision making

patient would live with Outcome B, then Outcome A is preferred twice as much as Outcome B. We called this

approach life expectancy analysis since it identifies the alternative that maximizes how long the patient will live

onaverage.

Later chapters will discuss alternate outcome values that provide a more complete representation of what

is important in a decision. Those preference measures will have different numerical values; however, the same

folding back operation will still determine an expected value that can be used to compare the alternatives in a

decision.

Finally, this chapter also will demonstrate a process called sensitivity analysis. Recall the medical example called

Problem 2in the previous chapter. Constructing the decision tree for this problem depended on several problematic

decision tree parameters. These parameters were problematic because there was limited or conflicting empirical evi-

dence for their values. The sensitivity analysis described in this chapter will demonstrate a systematic examination of

these low-

confidence values.

7.2 Folding- back operation

The decision tree for Problem 2, first described in Chapter6, is easy to evaluate because there is only one decision–

whether to order coronary arteriography. The branch probabilities on the paths to the various outcomes fully describe

the outcome uncertainties for this problem. Simple arithmetic computes the corresponding life expectancies for the

patient. However, what if additional decision nodes are encountered on those paths to the final outcome? How do

wecompute life expectancy for a more complicated decision problem involving multiple decisions?

One answer is what we call the folding-

back operation.

7.2.1 Folding- back operation applied tohypothetical problem

Problem 1, first described in Chapter6, will be used to introduce the folding- back operation. The decision tree in

Figure7.1 is the same decision tree shown in Figure6.8 for Problem 1. Recall that this decision problem concerns the

management of a hypothetical disease that may be present (D

) or absent (D

−

). Initial assessment establishes a prob-

ability of 0.40 for the disease. Management of the disease requires a choice between two alternatives: Treatment and

Notreatment, which will lead to different life expectancies, depending on the disease state. In Problem 1, there is a test

that can be used to change the probability of the disease. The treatment decision can be based on the result of this test.

However, undergoing the test also affects the patient’s length of life. Figure7.1 shows a decision tree for this problem,

including the various branch probabilities that were derived in the previous chapter. The outcome values are expressed

as lengths of life for the patient.

The folding- back operation involves a series of steps that sequentially eliminate nodes without changing the expected

values for alternatives represented by the initial decision node. The process ends when that initial decision node is

reached. The resulting tree is equivalent to the original tree, only reduced to a simple choice between alternatives with

known expected values. For example, suppose that the decision tree shown in Figure7.1 could be reduced to the

following simple decision tree:

Test

No Test

LE = 14.4 year

LE = 15.0 years

Definition: Problem 1

Decide whether to test and treat a patient who may have a disease

• Probability of disease is 0.40.

• With disease, treatment increases survival from 4 to 12 years.

• Without disease, treatment decreases survival from 20 to 16 years.

• Test has a true- positive rate of 0.90 and a false- positive rate of 0.20.

• Undergoing the test reduces survival by 1 year.

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