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Interpreting new information: Bayes’ theorem 45
As noted earlier, the probability of a positive test result (p[T+]) is the sum of its probability in diseased patients and
its probability in nondiseased patients:
pT pT DpTD
andand
In the previous section, we rearranged the definition of conditional probability to show that
pT DpDpTD
and|
pT DpDpTD
and|
Since p[T+|D+]=sensitivity and p[T+|D−]=1 − specificity
probability of apositiveresultsensitivity
pD pD111
specificity
probability of anegativeresultsensitivity
pD pD11
specificity
The following is another way to calculate the probability of a negative test.
p
robabilityofanegative test result probabilityofapositive t
1– eestresult
Figure4.6 shows the relationship between the pre- test probability and the probability of a test result when the test has
a sensitivity of 0.8 and a specificity of 0.8. The probability of a negative result is lowest when the pre-
test probability of
the target condition is nearly 1.0. At that probability, the probability that the patient does not have the target condi-
tion is very low, and one should expect few negative test results. Note that the probability of a positive test result is
lowest when the pre-
test probability is low.
4.4 The odds ratio form ofBayes’ theorem
One disadvantage of Bayes’ theorem is that most people need a calculator to do the math. A second disadvantage is
that sensitivity and specificity alone do not convey their combined effect on probability. The solution to these problems
is to rewrite Bayes’ theorem to calculate the post- test odds, which is the same information as the post- test probability
but expressed differently. The odds ratio form of Bayes’ theorem is easy to remember and entails multiplying one
number by another number.
Expressing the pre- test probability of the disease in terms of the pre- test odds of the disease simplifies Bayes’ theorem.
Using the odds ratio format of Bayes’ theorem, anyone can easily update probabilities. Moreover, expressing Bayes’
theorem in its odds ratio format leads directly to a simple, intuitive way to describe the effect of new diagnostic
information on probability. It is the likelihood ratio (LR), one of the most important ideas in this book.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
Pre-test probability
Probability of test result
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.01.0
1.0
0
Positive
test result
Negative
test result
Figure 4.6 Probability of a test result for different pre- test probabilities. The sensitivity of the test is 0.8 and the specificity is 0.8.
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46 Medical decision making
4.4.1 The derivation ofthe odds ratio form ofBayes’ theorem
Derivation:
To derive the odds ratio form of Bayes’ theorem, start with Bayes’ theorem in its familiar form (here we use R to denote
any test result)
pD R
pD pRD
pD pRDpDpRD
|
|
||
Now, convert p[D+], a probability, to odds using the relationship learned in Chapter3:
pD
D
D
odds
odds1
Instead of Odds[D+], we write O[D+]. Substituting
OD
OD
1
where we see p[D+] in Bayes’ theorem, we get this expression:
pD
R
OD pRD
OD
OD pRD
OD
|
|
|
1
1
1
1
OD
OD
pRD|
Using simple algebra, we obtain the odds ratio form of Bayes’ theorem
O
DROD
pRD
pRD
|
|
|
Look carefully at this equation. It’s telling you that the post- test odds of disease D after test result R, (O[D|R]), is
equal to the pre- test odds (O[D]) times a number. That number is the amount that the odds change after the test result,
R. An expression that tells you how much the odds of disease change after new information is so useful that
it...
pRD
pR no D
|
|
...has a name: the likelihood ratio, also referred to as LR.
Definition: Likelihood Ratio=
a number that shows how much the odds of disease change after getting a test result.
Another way to state the odds ratio form of Bayes’ theorem is:
Post-test odds pretestodds likelihood ratio-
The LR is convenient and powerful: a single number that shows how much one’s uncertainty should change after a
test result.
4.4.2 The likelihood ratio: ameasure oftest discrimination
We saw in the previous section that
Likelihood ratio
result in diseased persons
result in nondis
p
p eeasedpersons
Any clinical finding may be characterized by its LR. Different amounts of ST segment depression during an exercise
stress test indicate different likelihoods of coronary artery disease on an arteriogram, as shown in this example (Table4.1).
In contrast to this example of a continuous variable having separate LRs for each of several results, the results can
be combined, a single cut- point chosen, and a dichotomous test result formed, with its own LR. With a dichotomous
result, a result that raises the odds of the target condition is “positive,” and a result that lowers the odds is “nega-
tive.” For example, physicians often treat the amount of ST segment depression on a stress ECG as a dichotomous
variable. Less than 1 mm ST segment depression is a negative test; ST segment depression of ≥1 mm is a positive test,
one that may require additional testing. Table4.2 shows the LR for any amount of ST segment depression that is at
least 1 mm.
Note that the data for Tables4.1 and4.2 are taken from different studies.
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Interpreting new information: Bayes’ theorem 47
Both of the results in Table4.2 have a LR, one corresponding to ≥1 mm ST segment depression (abbreviated LR+)
and another corresponding to <1 mm ST segment depression (abbreviated LR−):
L
R
finding present in persons
finding present i
p
p
diseased
n
np
ersonsnondiseased
From the definitions of sensitivity and specificity, this definition takes the following form:
L
R
sensitivity
specificity
1
L
R
finding absent inpersons
finding absent in
p
p
diseased
noondiseased persons
From the definitions of sensitivity and specificity, this definition takes the following form:
L
R
sensitivity
specificity
1
The LR is especially useful for expressing the discriminatory power of a test result because the clinician needs to
remember only one number (the LR) instead of two numbers (sensitivity and specificity).
• If the LR is 10.0, the odds increase 10- fold after a positive test.
• If the LR is close to 1, the odds change very little.
• If the LR is 0.1, the odds decrease a lot after a negative test.
4.4.3 Using theodds ratio form ofBayes’ theorem
The LR is used in the odds ratio form of Bayes’ theorem. To understand the odds ratio format, recall the following defi-
nitions from Chapter3:
Odds of even
t
the event will occur
the event will not o
p
p
cccur
Odds of even
t
p
p1
Thus, if the probability of an event is 0.33, the odds that the event will occur are:
Odds of even
t
p
p
or
1
033
1033
1
2
12
.
.
:
Table 4.2 Likelihood ratios forexercise ECG results asa dichotomous test.
Test result (mm ST
segment depression) Test result label
Likelihood ratio for
coronary artery disease
<1.0 Negative 0.40
≥1 Positive 7.45
Adapted from Rifkin etal. (1977).
Table 4.1 Likelihood ratios forexercise ECG results.
Test result (mm ST segment
depression)
Likelihood ratio for coronary
artery disease
<1.0 0.40
1.0–1.49 2.09
1.5–1.99 4.50
2.0–2.49 10.80
≥2.5 38.00
Diamond and Forrester (1979).
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48 Medical decision making
If a finding is present, the amount that the pre- test odds increases may be calculated with the odds ratio form of
Bayes’ theorem.
Post-test oddspre-test odds likelihood ratio
The first example of the odds ratio form of Bayes’ theorem shows how to calculate the post- test probability with a test
result that is one point on a continuum of results. The test is a stress ECG, a test whose results can be expressed as a
continuous variable, the amount of ST segment depression (Table 4.1).
Example 1: What is the post-
test probability of coronary artery disease in a middle- aged man with a history of atypical
angina pectoris? His exercise stress ECG showed 2.0
mm ST segment depression.
Step1: Determine the LR for the patient’s stress test result. From Table4.1, we know that ≥2.0
mm ST segment depres-
sion has a LR of 10.8.
Step2: Calculate the pre-
test odds of coronary artery disease. From Table4.1 in Chapter3, we know that a male with a his-
tory of atypical angina pectoris has a pre-
test probability of 0.70. To convert this probability to odds, we do the following:
Odds of even
t
p
p1
07
107
07
03
23
1
.
.
.
.
.:
Step3: The third step is to use the odds ratio form of Bayes’ theorem to calculate the post- test odds:
Post-test odds pre-test odds likelihood ratio
P
ost-test odds
07
03
10 8252
1
.
.
..
:
In this case, the very positive stress test result has increased the probability of coronary artery disease to a virtual
certainty.
In the second example, we repeat an early clinical scenario but now as an example of using the odds ratio form of
Bayes’ theorem. The test result is a dichotomous variable: a lung mass is either present on the chest x-
ray (a positive
test) or it is not (a negative test). In this example, the mass is present.
Example 2: How should a radiologist interpret a lung mass on a chest radiograph taken in a man whose pre-
test prob-
ability of lung cancer is 0.4?
Step1: The first step in answering this question is to calculate the LR for the radiographic finding of a lung mass.
L
R
findingpresent in persons
findingpresent in
p
p
diseased
nonddiseased persons
The probability that a lung mass is present in persons with lung cancer is 0.6, which is the sensitivity of the chest radio-
graph for lung cancer.
The probability that a lung mass is present in persons who do not have lung cancer is 0.04, which is 1 − the specificity
of the chest radiograph. Thus, the specificity of the chest radiograph for lung cancer is 0.96.
Thus, when a finding is present, the LR is:
L
R
sensitivity
specificity
1
The LR+ for a lung mass on the chest x- ray is:
L
R
060
004
15
0
.
.
.
Step2: The second step in obtaining the post- test odds is to convert the pre- test probability to the pre- test odds:
Odds
of event
p
p1
04
104
04
06
06
71
.
.
.
.
.:
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Interpreting new information: Bayes’ theorem 49
Step3: The third step is to use the odds ratio form of Bayes’ theorem to calculate the post- test odds:
Post-test oddspre-test odds likelihood ratio
P
ost-test odds
04
06
15 01
01
.
.
.:
Thus, the post- test odds are higher than the pre- test odds when the chest x- ray shows a mass. The expression “10:1”
is read “10 to one.” In words, “for every ten persons who have lung cancer, 1 person will not have lung cancer.”
Thinking in terms of odds of events and LRs for tests simplifies computation. The odds of an event may be converted
back to the probability of the event, if desired.
P
robability
odds
odds
1
10 1
1101
:
:
P
robability = =
10
11
09
1
.
In the third example of using the odds ratio form of Bayes’ theorem, the test result is a dichotomous variable: a lung
mass is either present (a positive test) or it is not (a negative test). In this example, the mass is absent despite a high
pre-
test probability of lung cancer.
Example 3: How should a radiologist interpret a normal chest radiograph taken because of concern about lung cancer?
The patient had a pre-
test probability of lung cancer equal to 0.4.
Step1: The first step in answering this question is to calculate the LR for lung cancer when the chest radiograph does
not show a lung mass.
L
R
finding absent i
np
ersons
findingabsentin
p
p
diseased
non
ddiseased
persons
The probability that a lung mass is present in persons with lung cancer is 0.6, which means that the probability that a
lung mass is absent in persons with lung cancer is 1 − sensitivity, (Explanation: since a lung mass is either present or
absent, the sum of the two probabilities must add to 1.0).
The probability that a lung mass is absent in persons who do not have lung cancer is 0.96 is the specificity of the
chest x-
ray.
When a finding is absent, the LR is:
L
R
sensitivity
specificity
1
The LR− for a lung mass on the chest x- ray is:
L
R
040
096
04
2
.
.
.
This LR tells us that a negative chest x- ray reduces the odds of lung cancer by a little more than half, which is not likely
to be the end of the story given a pre- test probability of 0.4in this man.
Step2: The second step in obtaining the post- test odds is to convert the pre- test probability to the pre- test odds, as in
the second example:
Odds of even
t
p
p1
04
104
04
06
06
71
.
.
.
.
.:
Step3: The third step is to use the odds ratio form of Bayes’ theorem to calculate the post- test odds:
Post-test oddspre-test odds likelihood ratio
P
ost-test odds
04
06
042028 11
35
.
.
..
::
.
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50 Medical decision making
Thus, the post- test odds are lower than the pre- test odds when the chest x- ray does not show a mass. The expression
“1:3.5” is read, “for every 1 person who has cancer, 3.5 people will not have cancer.”
The post- test probability of lung cancer is quite high despite the normal chest radiograph. We will return to this
point later in the chapter.
While thinking in terms of odds of events and LRs for tests simplifies computation, many people find probability an
easier language of uncertainty. We can convert the odds of an event back to the probability of the event.
P
robability
odds
odds
1
0281
1028 1
022
.:
.:
.
4.5 Lessons tobe learned fromusing Bayes’ theorem
The following is a list of lessons that Bayes’ theorem teaches us. The text lists the lessons in related groups, and the
applicable evidence follows in the form of a plot of pre-
test probability (horizontal axis) and post- test probability
(vertical axis).
Lesson 1: The interpretation of new information depends on what you already knew.
Lesson 2: At low and high pre-
test probabilities, the post- test probability is close to the pre- test probability. The test
has the largest effect for intermediate range probabilities, where diagnostic uncertainty is greatest.
Lesson 3: Close to a probability of zero, the probability rises most steeply after a positive test result. Close to a prob-
ability of 1.0, the probability falls most steeply after a negative test result.
Lesson 4: A positive test is least likely when the pre-
test probability is close to zero. A negative test is least likely
when the pre- test probability is near 1.0.
Lesson 5: A test done to confirm that the probability is close to 0 or 1.0 is highly likely to confirm it and very unlikely
to disconfirm it.
Lesson 6: The bottom-
line: test results are least likely to occur when you need them the most.
4.5.1 Further thoughts
The last three lessons suggest that doing a test to confirm a probability that is close to 0 or to 1.0 is seldom
warranted.
However, disconfirming a strong clinical suspicion is potentially quite useful. An unexpected negative test result
when the pre-
test probability is high, may lower the probability of disease enough to reverse a decision to start treat-
ment. The same holds when the pre- test probability is very low: an unexpected positive result could change
management.
Do these observations mean that one should order a test when the pre-
test probability is very low or very high in the
hope of a test result that could change treatment? The answer is “on occasion, but certainly not as a routine practice,”
for several reasons.
First, the probability of a surprising, disconfirmatory test result is lowest when the pre-
test probability is at the
extreme ends of the probability scale. As shown earlier in Figure4.6, when the pre- test probability is above 0.95, the
probability of a negative test result is as low as 20% for a test whose sensitivity and specificity are 0.8.
Second, a low- probability test result could be a false- positive or a false- negative, due to an error in the laboratory.
Being skeptical of surprising test results is good clinical policy, as is confirming the unexpected finding by repeating
the test.
Third, to change management, the post- test probability would have to cross a treatment threshold probability,
which is the topic of Chapter13. Treatment threshold probabilities are usually less than 0.50 and often considerably
less.
By how much must the probability change to alter the next step in management? Discussions of the role of diagnostic
tests should focus on this question. Before proceeding further, some readers may wish to learn more about how to use
diagnostic tests in clinical practice. The first five pages of Chapter13 provide a brief introduction.
4.5.2 The clinical significance oftest specificity
The considerations when deciding about doing a test are its safety, cost, and suitability for the situation. If the patient’s
probability is relatively low, a test with high specificity (few false- positives) should be the prime consideration. This
section will use Figure4.7 to explain why.
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Interpreting new information: Bayes’ theorem 51
The role of test specificity in choosing between tests
When the pre- test probability is relatively low, the role of testing is to increase the probability of the target condition. High
specificity is far more important than high sensitivity when the pre- test probability is relatively low. Figure4.7 shows that
the probability after a positive test (the upper group of curves) increases as the specificity of the test increases. Specificity
is important when the pre- test probability is low. It is much less important when the pre- test probability is higher.
When the pre-
test probability is high, the role of testing is to lower the probability of the target condition. Figure4.7
shows that the probability after a negative test (the lower group of curves) is the same irrespective of the specificity.
4.5.3 The clinical significance oftest sensitivity
The clinical significance of test sensitivity sharply contrasts with specificity (Figure4.8). Test sensitivity is important
when the pre- test probability is high, the important result is a negative test, and the main effect of a negative test is to
lower the probability.
Figure4.8 illustrates several concepts:
• The sensitivity of a test strongly affects the probability after a negative test result (lower family of curves in Figure4.8).
The higher the sensitivity of a test, the lower the post- test probability of the disease after a negative test result.
• The sensitivity of a test has a small effect on the post- test probability after a positive test result (upper family of
curves in Figure4.8).
FP=0.15
FP=0.05
FP=0.01
FP=0.01, 0.05, 0.15
Pre-test probability
Post-test probability
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.2 0.4 0.6 0.8
1
Figure 4.7 Effect of the specificity of a test on the post- test probability of the disease. The sensitivity of the test was set at 0.90. The post- test
probability was calculated when the specificity was 0.99, 0.95, and 0.85 (in the figure, labeled FP=0.01, FP=0.05, and FP=0.15, respectively,
where FP stands for p[T+|D–]). The upper group of curves represent a positive test result. The lower group of curves represent a negative test result.
Pre-test probability
Post-test probability
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 0.2 0.4 0.6 0.8
1
TP = 0.98
TP = 0.98
TP = 0.75
TP = 0.75
TP = 0.50
TP = 0.50
Figure 4.8 Effect of the sensitivity of a test on the post- test probability of the disease. The specificity of the test was set at 0.95. The post- test
probability was calculated when the sensitivity of a test (where TP stands for p[T+|D+]) was set at 0.98, 0.75, and 0.50. The upper group of curves
represent a positive test result. The lower group of curves represent a negative test result.
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52 Medical decision making
Most tests have good sensitivity or specificity but seldom both. As a guide to choosing a test in specific situations,
Table4.3 summarizes the effects of sensitivity and specificity.
In choosing a test, strategies keyed to one test performance characteristic (sensitivity or specificity) and the patient’s
pre-
test probability can be helpful. However, a diagnostic test has both characteristics, and the interplay between them
is important in determining the post-
test probability. The LR, discussed earlier in this chapter, reflects both sensitivity
and specificity and determines the post-
test odds.
L
R
sensitivity
specificity
LR
sensitivity
specificity
1
1
Post-test odds pre-test odds likelihood ratio
Using the LR to represent test performance has powerful advantages:
• Test performance: In one number the LR reflects two very different but inter- related characteristics of test perfor-
mance: sensitivity (detection) and specificity (discrimination).
• Ease of use: The odds ratio form of Bayes’ theorem requires the multiplication of two numbers. The conditional
probability form of Bayes’ theorem requires 3multiplications, one addition, and one division. With a handy list of
LRs, a clinician can use on-
the- fly Bayesian reasoning to inform the choice of a diagnostic test.
Maintaining a list of commonly used tests and their LRs for common diseases should be on every clinician’s to- do
list. The bibliography for Chapter5 includes a reliable source.
4.6 The assumptions ofBayes’ theorem
Bayes’ theorem is an oversimplification of real life. It requires us to make several assumptions. To understand them,
we shall express Bayes’ theorem in conditional probability notation:
pD
R
pD pRD
pD pRDpDpRD
|
|
||
where:
R=test result
D+=disease present
D−=disease absent
Assumption No. 1: The probability of a test result conditional on the presence of the disease (i.e., sensitivity and specificity) is
independent of the prior probability of the disease.
We assume that this relationship is true whenever we calculate the post- test probability using the same sensitivity
and specificity and different prior probabilities of the disease. The following example shows how this assumption can
be wrong. Imagine a radionuclide scan of the liver that detects metastases from colon cancer. This scan detects all
metastases that are larger than 2 cm. Now consider two patients with recently discovered colon cancer. Unbeknownst
to anyone, both patients have metastases to the liver.
• One has lost weight. Their liver is considerably enlarged and has a stony hard consistency. This patient’s prior
probability of liver metastases is high. If a pathologist could examine this patient’s liver, most of the metastases
Table 4.3 Contrasting effects oftest sensitivity andspecicity.
Sensitivity of test Specificity of test
Preferred pre-
test probability High pre- test probability Low pre- test probability
Test result that matters most for decision- making Negative test Positive test
Direction of the largest effect on probability Lowers probability Increases probability
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Interpreting new information: Bayes’ theorem 53
would be larger than 2 cm, easily detectable by the liver scan. If similar patients were used to measure its sensitiv-
ity for liver metastases the result would be close to 1.0.
• The other patient feels and looks well, and their liver is not enlarged on physical examination. The patient’s prior
probability of liver metastases is low. If similar patients were used to measure the sensitivity of the scan for liver
metastases, the result would be far from 1.0.’
In this hypothetical example, the sensitivity of the test depends on the prior probability of the disease, in violation
of the assumption that the sensitivity and specificity are both constant. Few such examples exist, probably because
few researchers determine the pre- test probability of the target disease when they measure sensitivity and
specificity.
Table4.4 shows a study in which a history of chest pain was obtained in patients along with an exercise ECG and a
coronary arteriogram. In men and women, the sensitivity of the exercise ECG increased as the prior probability of coro-
nary artery disease increased.
Table 4.4 Exercise ECG test performance inpatients withdifferent chest pain syndromes.
Type of chest pain No. patients p[CAD] Exercise ECG sensitivity Exercise ECG specificity
Men Definite angina 487 0.88 0.84 0.71
Probable angina 443 0.67 0.72 0.80
Nonischemic pain 203 0.22 0.46 0.79
Women Typical angina 67 0.58 0.80 0.57
Atypical angina 153 0.35 0.67 0.69
Nonischemic pain 175 0.05 0.22 0.81
Adapted from Weiner etal. (1979).
Assumption No. 2: The probability of a test result conditional on the presence of the disease (i.e., sensitivity and specificity) is
independent of prior test results.
To understand this assumption, imagine that two tests have been done in sequence (first test 1with result R1 and
then test 2with result R2); now calculate the probability of the disease after the result of the second test.
pD
RR
pD pR DR
pD pR DR pD pR D
|
|
||
12
21
21 2
,
,
,
,R1
where:
R1=result of test 1
R2=result of test 2
D+=disease present
D−=disease absent
p[D|R1,R2]=probability of the disease conditional upon the results of test one and test two
p[R2|D+,R1]=probability of result on test 2 conditional upon the patient having the target condition and the result on
test 1
To verify Assumption 2would require the following study. First, divide patients with the target condition into two
groups: one with a normal result on the first test and one with an abnormal result. Second, do the second test in both
groups of patients with the target disease and a gold standard (reference) test on both groups. Then, calculate the sen-
sitivity and specificity of the second test. Assumption 2 is verified (for that test) if they are the same. Since few
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54 Medical decision making
investigators have studied two diagnostic tests done always in the same sequence, Assumption 2 of Bayes’ theorem is
seldom tested. Therefore, clinicians must assume that the sensitivity of a second test in diseased patients is the same regardless
of the results of the first test.
The assumption of conditional independence also applies to test specificity. The specificity of the second test in a
sequence is assumed to be the same regardless of the results of the first test in the sequence.
4.7 Using Bayes’ theorem tointerpret asequence oftests
Diagnostic tests are often used in sequence. An abnormal finding on one test may raise concerns that only can be
resolved by a second test. For example, when a screening test is performed on an asymptomatic person, an abnormal
finding may raise concerns that a second test might resolve. What method should be used to interpret the results of the
second test? The reader who has absorbed the lessons of this chapter will answer as follows:
“Use the post- test probability following the first test as the pre-test probability for the second test. Then, use the sensitivity and
specificity of the second test and Bayes’ theorem to calculate the post-
test probability for the second test.”
We illustrate this concept for an exercise ECG performed on an asymptomatic 45- year- old man who is about to
begin an exercise training program (Figure4.9). The pre- test probability of coronary artery disease is 0.06in asymp-
tomatic men in the fifth decade of life. The sensitivity of the exercise ECG is 0.58 and its specificity is 0.88. The prob-
ability of coronary artery disease if the exercise test provokes ≥1 mm ST segment depression is 0.24, as calculated with
Bayes’ theorem. The odds of CAD corresponding to a probability of 0.24 are 1:3. Thus, following an unexpected
abnormal stress test, the odds are one in four that the patient has significant coronary artery disease. Given the uncer-
tainty about the diagnosis of CAD, the logical next step would be to perform a noninvasive test like coronary CT
angiography.
Before using Bayes’ theorem to calculate the post- test probability of coronary artery disease after coronary CT angi-
ography, the clinician must answer two questions:
1. What should I use for the pre- test probability of coronary artery disease?
This question is easy to answer. The probability of coronary artery disease after the positive exercise ECG is the
probability of the disease prior to the second test.
2. What are the sensitivity and specificity of coronary CT angiography?
The second question seems even more straightforward than the first. The apparently correct answer is its sensitiv-
ity and specificity, and therefore its LR, as measured in a large study of patients who had CT angiography and a
definitive test, such as a coronary arteriogram.
This approach sounds logical, but it has a potential flaw. It assumes that the sensitivity and specificity of the CT
angiogram, and therefore its LR+ and LR-
, are the same in patients with a positive exercise ECG as it is in patients with
a negative exercise ECG.
Recall Assumption 1in the preceding section: The probability of a test result conditional on the presence of the disease (i.e., its sen-
sitivity and specificity) is independent of the prior probability of the disease.
0 0.10 0.20 0.30 0.40 0.90
1.0
Probability of coronary artery disease
Pre-test p[D]
on test #1
Post-test p[D] on test #1 =
Pre-test p[D] on test #2
Positive
Test #1
Figure 4.9 The post- test probability after the first test in a sequence is the pre- test probability for the second test. The pre- test probability for Test
1 is labeled as “Pre- test #1.”
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