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Measuring theaccuracy ofclinical findings 75

“The wellest of the well:” Healthy volunteers have advantages for measuring test specificity, which requires people

who do not have the target condition. It is often reasonable to assume that healthy volunteers do not have the target

condition because they are in excellent health. Therefore, performing a gold standard test is unnecessary. These advan-

tages are outweighed by the disadvantages. The volunteers are often much younger than patients who undergo the

test in practice, and they are also usually free of diseases that might cause a false-

positive index test. When the test is

evaluated with a broader spectrum of patients, the specificity of the test usually falls.

5.6.2 The second phase oftest evaluation: reluctance toorder thegold standard test because of

over- confidence ina negative index test result

Using the index test result to decide whom to refer for the gold standard test is a special form of spectrum bias:

test- referral bias. Because test- referral bias is a threat to the validity of studies of test performance, it is important to

understand how it affects the measurement of sensitivity and specificity.

Why would a clinician be less likely to obtain the gold standard test on patients with a negative index test? Too often,

a new diagnostic test becomes embedded in clinical practice before high-

quality research has established its rightful

role. In this phase, a clinician may gain confidence in a new test based on few if any high-

mance and a small, and perhaps atypical, personal experience with it. Consequently, a negative index test increases

their belief that the patient does not have the target condition, and so they are less likely to refer the patient for the gold

standard test.

The verified sample and the source population often differ. Would these differences affect patient care? When should

spectrum bias change the interpretation of a test result? These effects of spectrum bias are the subject of the next

twosections.

5.6.3 Effects ofspectrum bias

Effects ofspectrum bias onthe sensitivity ofa test

Spectrum bias causes the verified sample to have a different spectrum of disease than the source population.

To understand the effects of spectrum bias due to selective referral of patients for the gold standard test, imagine

how a patient would become a member of the verified sample in our hypothetical study of the spleen scan to detect

splenomegaly. The index test is the spleen scan and the gold standard test is the weight of the spleen after surgical

removal.

Two forms of spectrum bias affect test sensitivity.

• Disease severity bias: Since the weight of the spleen is the gold standard for a study of the spleen scan, the source

population is enriched with patients whose spleen is large enough to require surgery to remove it. The spleen scan

can detect most of these very large spleens, so that the verification sample– those who undergo splenectomy– is

enriched with patients with large spleens. This form of disease severity bias leads to a study that overestimates the

sensitivity of the scan because larger spleens are easier to detect with a spleen scan.

• Test referral bias: Another reason for referral and enrollment in the verified sample is an enlarged spleen on the

index test, the spleen scan. First, a positive index test is often the reason for referring a patient for the gold standard

test. Second, a positive index test is often sufficient for the clinician to make a firm diagnosis without referring the

patient for the gold standard test. Third, patients with a positive index test, whether referred or not, are not typical

of the entire population who had the index test, many of whom had a negative spleen scan and were not referred.

The net effect of these conflicting biases may be hard to predict.

These biases afflict retrospective studies, in which the physician decides whether to perform the gold standard based

on the clinical need. They should not be a problem with prospective studies, which mandate doing both index test and

gold standard test. However, prospective studies may be subject to referral bias because of failure to adhere to the

study protocol which specifies that all patients who got the index test are to be referred to undergo the gold standard

test. Failure to enroll all eligible patients will lead to a similar bias.

Definition

Test- referral bias: Systematic error in measuring test performance because the results of the index test influence

the decision to refer a patient for the gold standard test.

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76 Medical decision making

To see the effect of test- referral bias, first recall the study of the spleen scan, in which the verified sample were

patients who had been referred for splenectomy because their spleens were very large on the spleen scan and needed

surgical removal (Table5.2).

Now, imagine doing the spleen scan in a hypothetical Source Sample of patients with suspected splenomegaly. This

Source Sample contained 230 patients, of whom only the 50 shown in Table 5.2 were referred for splenectomy because

of their large spleens. Now suppose—hypothetically—that we knew the true state of the spleen in the 180 patients who

had a negative spleen scan and were not referred for splenectomy: 45 of them had an enlarged spleen, and 135 had a

normal size spleen. Table 5.4 shows the results of the study in the Verified Sample (the same results as in Table 5.2) and

as done in the Source Sample. In Table 5.4, the number of Source Sample patients who were not referred for the gold

standard procedure appears in bold face type.

If it were possible to measure the sensitivity and specificity of the spleen scan in the Source Sample, the sensitiv-

ity would be lower (0.25 vs 0.57) and the specificity higher (0.97 vs. 0.67) than as measured in the Verified Sample

(Table 5.5).

Figure5.8 illustrates the effects of test-

referral bias graphically. Focus first on the right half of the figure, which

represents patients with the target condition, those needed to calculate test sensitivity. Source population patients with

a positive index test (T+) all get referred for the gold standard test, while many with a negative test (T-

) do not get

referred (test-

referral bias). In the verified population, the proportion of index test- positive patients is much higher than

in the source population.

In other words, the sensitivity of the test as measured in the verified population is higher than it would be if it were

possible to ascertain the true state of all of the patients in the source population.

As a general rule:

When test- referral bias is present, test sensitivity is higher in the verified sample than in the source population.

Table 5.5 Comparison ofsensitivity andspecicity ofthe spleen scan asmeasured inthe

Veried Sample andthe Source Sample.

Sensitivity Specificity

Verified sample

p TD



|

TD s



|

Source sample

p TD



|

TD





145

150

Table5.4 Hypothetical results ofdoing thespleen scan inthe source sample.

Study population Results of the spleen scan

Number of patients

Spleen weighs >250 g Spleen weighs <250 g Totals

Verified Sample Positive True positive

N=

False positive

N= 5

N= 25

Negative False negative

N= 15

True negative

N= 10

N= 25

Totals N= 35 N= 15 N= 50

Source sample Positive True positive

N= 20

False positive

N= 5

N= 25

Negative False negative

N= 15+ 45

True negative

N= 10+ 135

N= 205

Totals N= 80 N= 150 N= 230

Note: The number of source population patients who were not referred for the gold standard procedure appears in bold face type.

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Measuring theaccuracy ofclinical findings 77

Effect ofspectrum bias onthe specificity ofa test

Spectrum bias due to selective referral also affects the specificity of a test, causing it to be lower in the verified sample

than it would be in the source population. Two forms of spectrum bias are at work:

Disease severity bias: When patients are very sick and the cause has not been identified, clinicians are likely to order

the gold standard test after a negative index test. This practice enriches the verified sample with sick patients, relative

to their proportion in the source population, which includes a broad spectrum of disease severity. Sicker patients have

more severe, more easily detected forms of the target condition and other conditions that lead to referral for the gold

standard test. They are therefore more likely to have positive gold standard test results, both true-

positives but also

false-

positives due to other diseases. More false- positive results mean an index test with a lower measured

specificity.

Test- referral bias: Physicians are less likely to refer a patient with a negative index test to have a gold standard test.

Therefore, the verified sample has relatively few patients with a negative index test, which means relatively few true-

negative test results and lower measured specificity. The left- hand side of Figure5.8 depicts patients who do not have

the target condition, those who are needed to calculate test specificity. Source population patients with a positive index

test all get referred for the gold standard test, while many with a negative test do not get referred (test-

referral bias). In

the verified population, the proportion of index test- negative patients is lower than in the source population, which

means that test- referral bias underestimates specificity, which is the probability of a negative test result in patients who do

not have the target condition.

Thus, as a general rule: test-referral bias causes test specificity to be lower in the verified sample than it would be in the source

population.

Table5.5 uses the spleen scan study to illustrate the effect of test- referral bias on sensitivity and specificity. Sensitivity

is higher and specificity is lower in the verified sample, a specific example of the general rule.

The spleen scan study does not reflect a general principle about the direction of the effect of test- referral bias on the

likelihood ratio, which reflects the joint effects of sensitivity and specificity. Test referral bias affects the numerator and

denominator of the likelihood ratio in the same direction. Therefore, the effect of bias on the ratio depends on the size

of its effect on sensitivity relative to the size of its effect on specificity.

Because of referral biases, the verified sample may contain only a small fraction of the source population. Readers

must scrutinize reports of test performance characteristics for evidence of these biases. One important clue is a

T+

T+ T+

T–

disease

Disease

Source

population

Source

population

Verified population

Specificity = 0.45

Specificity = 0.70

Sensitivity = 0.80

Sensitivity = 0.60

Figure 5.8 This representation of a retrospective study of test performance shows the effect of preferential referral of patients with positive tests

on measurements of sensitivity and specificity. The heights of the T+ rectangles are the same in the source population and the verified population,

reflecting the practice of referring all Test- positive patients to receive the gold standard test. The heights of the T- rectangles in the verified

population are lower than in the source population, reflecting the practice of failing to refer every Test- negative patient to receive the gold standard

test.

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78 Medical decision making

retrospective study design in which the decision to order the gold standard test is up to the clinician rather than a

prospective study, whose study protocol requires doing both the gold standard test and the index test.

The best clues to a trustworthy study are:

• Prospective design

• An accounting of how many patients dropped out before getting the gold standard test and why

• A comparison of the clinical characteristics of the source population and the verified sample.

Spectrum bias is a serious problem for retrospective studies, but remediation is possible. The next section contains

advice about how to adjust sensitivity and specificity to reduce the effect of spectrum bias.

5.6.4 Adjusting forbiased estimates ofsensitivity andspecificity

How should the clinician adjust published estimates of the sensitivity and specificity to improve their application to

the source population? The following discussion, which outlines some general principles, pertains principally to retro-

spective studies of test performance.

General principles

Sensitivity and specificity are conditional probabilities.

Sensitivity is the probability of a positive test given that the patient has the target condition: p[T+|D+].

Specificity is the probability of a negative test given that the patient does not have the target condition: p[T−|D−].

In Chapter3, we defined probability as a statement of opinion about the likelihood that a current state exists or a

future event will occur. To arrive at a probability of the disease in a patient, we anchor on the prevalence of the disease

in the subgroup to which our patient belongs and adjust to take account of the special characteristics of our patient.

We can use the anchoring and adjustment heuristic to adjust a published sensitivity or specificity to take account of

differences between the verified sample and the population to which the patient belongs.

5.6.5 Heuristics foradjusting published reports fordisease severity bias

These heuristics assume that the target population for the test is the average patient in a primary care practice. Test

performance measured in a prospective study in which everyone in the source population received the gold standard

test would apply directly to the source population without any adjustment. Applying the same study to a very sick

inpatient population might require adjustments in the opposite direction to that described here.

Sensitivity: The sicker the verified sample in comparison with the source population, the larger the downward

adjustment of the sensitivity.

Specificity: If the verified sample consists of healthy volunteers, adjust the specificity downward. If the verified

sample is mostly very sick patients, adjust the specificity upward.

These rules of thumb beg the question of how much to adjust the sensitivity and specificity estimates, which is the next topic.

Exact adjustment for spectrum bias

This material is advanced. The reader may want to skip to Section5.7.

Exact correction for spectrum bias is possible in a retrospective study, subject to two related assumptions (Gray

etal.,1984).

• Selection for the gold standard test depends only on the index test result.

• Selection for the gold standard test depends on disease severity only through the correlation between the index

test result and disease severity.

These assumptions are probably valid for screening tests, in which there are no clinical findings to influence selec-

tion for the gold standard test. The assumptions are less likely to be valid when the patient has symptoms and signs

that might influence the clinician to refer the patient for the gold standard test even after a negative index test.

Begg and colleagues showed that, if these two assumptions are valid, the unbiased estimate of p[R|D+] is

where

p[R|D+] is the probability of the index test result conditional upon the presence of the disease. If the result was a posi-

tive test, p[R|D] would be p[T+|D+], which is the sensitivity of the test.

PR PD RS









 







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Measuring theaccuracy ofclinical findings 79

p[D+] is the disease prevalence in the verified sample

p[R] is the probability of the test result in the source population

p[D+|R,S+] is the post- test probability of the disease conditional upon R in the verified sample

Another approach to adjusting for test-referral bias is to focus on estimating the likelihood ratio in the source

population.

If L[R] is the likelihood ratio in the source population, and L*[R] is the likelihood ratio in the verified sample, L[R] =

c × L*R, The correction factor, c, depends only on the odds of the disease in the verified sample and in the source

population,” and it is the same for all results of the test. (Gray et al., 1984).





Odds of target condition verified sample

Odds of target cconditionsource population



This research provides a pathway to measure the likelihood ratio of an index test in the source population of a retro-

spective study of test performance.

1. Measure the likelihood ratio in the verified population.

2. Determine the final diagnosis on all patients in both source and verified samples.

3. Calculate the odds of the target condition in both samples.

4. Calculate c, as noted earlier.

5. Calculate the likelihood ratio (L*[R}) in the source population using the formula listed earlier.

This source population likelihood ratio should give reliable estimates of post- test odds using the odds ratio form of

Bayes’ theorem.

5.7 When tobe concerned about inaccurate measures oftest performance

At this point in the chapter, the reader should be concerned about whether it is possible to measure test performance

so that the results apply to the source population. The answer should be “yes” for well- conducted prospective studies

and “maybe” for retrospective studies. Our focus on accurate measures of test performance should not obscure the

real goal, which is making post- test probability estimates that guide decision making toward choices that maximize

the probability of the outcomes that the patient values most. For this purpose, accurate measurements of test perfor-

mance are sometimes important but not in all cases. Figures 5.9 and 5.10 illustrate this point by showing regions of

pre-

test probability in which the post- test probability changes very little as the sensitivity or specificity of a test

changes quite a lot (see next page). These two figures are examples of sensitivity analysis, a powerful concept in the

analysis of decision making.

When is accurate measurement of sensitivity important? Figure5.9 shows the post-

test probability for any value of

the sensitivity, three values of the pre- test probability, and a specificity of 0.80. Remember that the sensitivity of most

commonly used tests is usually larger than 0.75, which is the shaded area in Figure5.9.

Figure5.9 shows that accurate measurement of test sensitivity is not very important most of the time because the

post- test probability changes very little as the sensitivity changes. Test sensitivity is important if your patient has an

intermediate-

to- high pre- test probability and a negative test result.

Figure5.9 shows that test sensitivity in the range of 0.75–1.0has little effect on the post- test probability for positive

tests regardless of pre- test probability and for negative tests when the pre- test probability is low.

When is accurate measurement of test specificity important? Figure5.10 shows the post-

test probability for various

values of the specificity, three values of the pre- test probability, and a sensitivity of 0.90. Remember that most tests have

a specificity that is greater than 0.8, the shaded region to pay attention to in Figure5.10.

Figure5.10 shows that typical values of test specificity (shaded area) have little or no effect on the post-

test probabil-

ity for negative test results (dashed lines) in any patient. Test specificity strongly affects post-

test probability for patients

with positive test results if the pre- test probability is low and, to a lesser extent, intermediate.

Summary:

• Worry about accurate measurements of test performance when your patient has an unexpected result:

• When the pre- test probability is high and the test is negative

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80 Medical decision making

• When the pre- test probability is low and the test is positive.

• Worry about test performance when your patient has an intermediate probability (i.e., when you are the most

uncertain about the diagnosis).

• Otherwise, do not worry.

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–Specificity

p[D] = 0.1

p[D] = 0.9

p[D] = 0.1

p[D] = 0.5

p[D] = 0.9

Post-testprobability

Figure 5.10 Change in post- test probability because of changes in test specificity. The sensitivity was 0.9 for all calculations. The pre- test

probabilities (denoted by p[D]) were 0.1, 0.5, and 0.9in successive calculations using Bayes’ theorem and varying the 1 − specificity of the test from

0 to 1.0. The post- test probabilities after a positive test are denoted by solid lines. After a negative test, they are denoted by the dashed lines. The

shaded section denotes the range of test specificity typically found in practice (0.8 to 1.0), so that 1 – specificity is low.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

12141618

1101

Sensitivity (%)

p[D] = 0.9

Post-test probability

p[D] = 0.5

p[D] = 0.1

p[D] = 0.5

p[D] = 0.9

Figure 5.9 Change in post- test probability (vertical axis) because of changes in test sensitivity (horizontal axis). The specificity was 0.8 for all

calculations. The pre- test probabilities (denoted by p[D]) were 0.1, 0.5, and 0.9in successive calculations using Bayes’ theorem and varying the

sensitivity of the test from 0 to 1.0. The post- test probabilities after a positive test are denoted by solid lines. After a negative test, they are denoted

by the dashed lines. The sensitivity of most tests in common practice is >0.75.

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Measuring theaccuracy ofclinical findings 81

On reflection, accurate measurements of test performance are most important in all the situations in which a test

result could have a big effect on probability.

5.8 Test results asa continuous variable: theROC curve

In most of this chapter, we have described test results as “positive” and “negative.” Often, we use these terms because

the clinical laboratory provides only the “upper limit of normal” as information to guide test interpretation (by now,

you should be feeling impatient with clinical laboratories that fail to provide sensitivity and specificity, and especially

likelihood ratios, the information that you need to interpret test results!). This part of the chapter describes the advan-

tages of expressing test results as a continuous variable (i.e., a number within a range) rather than as a dichotomous

variable (positive or negative). The principal advantage is the freedom to choose the optimal cut point for the clinical

situation.

5.8.1 The distribution oftest results indiseased andwell individuals

As discussed at the beginning of this chapter, the distribution of test results in patients with the target condition often

overlaps the distribution of results in those who do not have it (Figure5.11). That overlap region is where decision

making requires tradeoffs between the benefits and harms of treating patients with a value on the horizontal axis that

is above the cut point for the test.

When published reports provide information about the distribution of test results in patients who have the target

condition and those who do not have it, we can express the conditional probabilities of test result R (p[R|D+] and

p[R|D-

]) in several ways.

• As the conditional probability of a result anywhere in a range of values of test results, here denoted by the distance

between X

and X

on the horizontal axis (Figure5.12, see next page). From p[R|D+] and p[R|D- ], we can calculate

the likelihood ratio of a test result that falls in this range.

• As the conditional probabilities of all test results above a cutoff value in the overlap region. P[R|D+] would rep-

resent this result (Figure5.13, see next page). From p[R|D+] and p[R|D- ], we can calculate the likelihood ratio of

a test result that falls above the cutoff value.

As discussed at the beginning of this chapter, test results should link to a decision for action. When the clinical labo-

ratory reports a test result as a number (e.g., a serum troponin I value of 50 ng/dl), we must decide if that number

exceeds the cut point for taking action.

The clinician who learns of a test result should always ask the question, “What do I do now?” The options are: do

nothing, get more information, or start treatment. As we shall see in Chapter13, these three options correspond to three

ranges of probabilities: low probability: do nothing; intermediate probability: get more information; and high probabil-

ity: start treatment. According to Bayes’ theorem, the post- test odds of the disease depends on the likelihood ratio of

the test, which in turn depends on the cut point for the test. In this section, we will lay the groundwork for determining

the test result cut point that maximizes the patient’s well- being.

No. of patients

Serum troponin concentration

Individuals

with MI

Individuals

without MI

P[MI] = 0

[MI] = 1.0

Overlap

region

Figure 5.11 Distribution of test results in healthy and diseased individuals.

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82 Medical decision making

= Sensitivity

= False-negative

= True-negative

= Specificity

= Sensitivity

= 1 – specificity

Serum concentration

Figure 5.12 Calculation of p[R | D+] (top panel) and p[R | D- ] (bottom panel) for the range of test results denoted by the distance between X

and X

on the horizontal axis.

= Sensitivity

= Specificity

= False-negatives = Tr ue-positives

= False-positives = Tr ue-negatives

Number

of patients

Number

of patients

disease

Disease

Number of patients

Figure 5.13 Calculation of p[R | D+] (top panel) and p[R+ | D- ] (bottom panel) for all test results above a cutoff value, which is denoted by the vertical

line.

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Measuring theaccuracy ofclinical findings 83

Establishing a criterion for decisive action will require making a tradeoff. To understand the tradeoff, refer to

Figure 5.14 and consider two cases. The only difference between the two panels is the position of the cut point

(verticalline)

• Upper panel: Start at the right- hand end of the distribution of patients with the target condition. Move the cut

point value of the test to the left until over half of the patients with the target condition have a test result above

the cut point (a true- positive). In doing so, some patients who do not have the target condition will have results

that are to the right of the cut point (a false- positive). Thus, to be sure of detecting more of the patients with the

target condition, one must accept a cut point that will lead to treating patients who do not have the target

condition.

• Bottom panel: Move the cut point still further to the left until nearly all patients with the target condition have a test

result that is above the cut-point. False- negative results are few, but now many persons who do not have the target

condition will have results above the cut point (false- positives). Thus, to be sure of detecting and treating more of

the patients with the target condition, one must accept a cut point that will lead to treating more patients who do

not have the target condition.

This example shows that changing the cut point changes the relative proportions of false- negative and false- positive

results. Every cut point defines a different proportion of those that treatment will help relative to those whom treatment

will not help and may harm. Recognizing this outcome as inevitable, we should choose a cut point that maximizes net

benefit across everyone, which is the topic of the next section.

5.8.2 The receiver operating characteristic curve

The receiver operating characteristic (ROC) curve is a graphical method for depicting the tradeoff between the sensi-

tivity and the specificity of a test at various cut points. To obtain an ROC curve, one does the index test and the gold

standard test on patients suspected of having the target condition and calculates the conditional probability of a test

result (p[R|D+] and p[R|D− ]) for each definition of the cut point that defines a positive test result. Figure5.15 is an

ROC curve (see next page). It shows the effect of several cut points for a test that detects different blood levels of a

chemical. The ROC curve is a plot of p[R|D+] on the vertical axis and p[R|D− ] on the horizontal axis. In Figure5.15,

each point corresponds to one of three definitions of a positive test result, as defined by three different cut points

(shown on the inserts).

False positive result

False negative result

Normal

individuals

Normal

individuals

Diseased

individuals

Serum concentration

Number of patientsNumber of patients

Diseased

individuals

Cut-off value

to define

abnormal result

Figure 5.14 Effect of altering the cut point that defines a positive test result. See text for explanation.

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84 Medical decision making

The area of the ROC curve graph has three parts:

• The 45- degree line: Recalling that the likelihood ratio is p[R|D+]/p[R|D− ], the 45- degree line represents test results

for which p[R|D+] equals p[R|D− ], and, therefore, the likelihood ratio is 1.0. Since the post- test odds equals the

pre-

test odds times the likelihood ratio, points on the 45- degree line have no effect on the probability of the

disease.

• The area above the 45- degree line: The likelihood ratio for test results that fall above the 45- degree line is greater than

1.0, indicating that the test result increases the probability of the disease.

• The area below the 45- degree line: The likelihood ratio for points below the 45- degree line is less than 1.0, indicating

that the test result decreases the probability of the disease.

The three inserts in Figure5.15 illustrate the effect of moving the cut point, this time from a higher serum con-

centration of creatine phosphokinase to a lower concentration. This figure reinforces the lesson learned from

Figure5.14.

Figure5.16 displays the ROC curve for ST segment depression on an exercise ECG. One measure of an abnormal

exercise ECG is how far the ST segment falls during exercise as compared with the baseline value prior to exercise. A

cut point corresponding to deeper ST segment depression means a higher specificity but also a lower sensitivity.

This ROC curve illustrates the tradeoffs of choosing different cut points for defining a positive test. Suppose that

the exercise ECG result determines whether a patient receives a coronary arteriogram. Setting the cut point at

≥2 mm of ST segment depression would result in fewer false- positive results (and fewer negative coronary arterio-

grams). This cut point would lead to more false-

negative exercise ECG results (and therefore more coronary artery

disease patients who did not get a coronary arteriogram), and fewer coronary arteriograms (and lower costs). Setting

the cut point at 1.0 mm of ST segment depression would result in fewer false- negative results and more coronary

arteriograms.

False-positive

False-negative

Serum CPK concentration

1-specificity

Sensitivity

No. patients

Disease

False-positive

Note: this slide is a maker

for a preferred version done

in .odg format

False-negative

Serum CPK concentration

No. patients

Disease

False-positive

False-negative

Serum CPK concentration

No. patients

Disease

1. 0

01.0

Figure 5.15 ROC curve for hypothetical test used to detect myocardial infarction.

https://t.me/medicina_free

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