The Predictive Value of Clinical Laboratory Test Results
PHILIP G. ARCHER, Sc.D.
Department of Biometrics, University of Colorado
Medical School, Denver, Colorado
Archer, Philip G.: The predictive value of clinical laboratory
test results. Am J Clin Pathol 69: 32-35, 1978. Bayes'
formula is being used with increasing frequency to calculate
predictive values for positive and negative clinical laboratory
test results. Because of its nonlinear form, however, it is
difficult to visualize how changes in the test characteristics
or disease prevalence will effect predictive values. This paper
shows that Bayes' formula is transformed to a linear function through the use of odds, rather than probabilities, thus
facilitating conceptualization, understanding, and memory.
(Key words: Laboratory tests; Predictive value; Bayes'
formula; Probability; Odds ratios.)
where*:
p+
Se
Sp
p
predictive value of a positive test
sensitivity of the test
specificity of the test
prevalence rate (or prior probability) of the
disease
In pragmatic terms, the prevalence rate, p, is an
index (on a scale of 0 to 1.0) of suspicion that the
disease is present before testing, and p + shows the
comparable index of suspicion after having received a
positive test result for a patient who comes from a
population in which the prevalence rate is thought
(believed or known) to be p. The experienced geometer
can illustrate the effects on p + of varying the individual
parameters on the right-hand side of equation 1, but
at best, the graphic illustration is somewhat awkward
to construct and interpret in this nonlinear form. A
similar development in terms of odds in favor of the
presence of disease, rather than the probability of its
presence, reduces the relationship to a linear one in
the odds, thus facilitating both conceptualization and
memory.
THE PREDICTIVE VALUE of "positive" and "negative" laboratory test results has been receiving increased exposure and discussion in the medical literature in recent years, and a recently published book
by Galen and Gambino 2 treats the subject in some
detail from the point of view of the potential users
of clinical laboratory results. Some efforts have been
made to ease the computational aspects of the use of
Bayes' formula, 2 ' 3,6 and to provide geometric illustrations of the effects on predictive values when prevalence, sensitivity, and specificity are individually altered. 3 To date, however, there seems to have been
no elucidation of a unifying concept that will allow the
clinician or research worker easily to grasp the interaction of all of the factors involved. I believe the
following argument will provide such a concept. Although none of the ideas is new, 5 they do not seem
to have been formulated previously in the context of
the interpretation of laboratory results in the practice of clinical medicine.
Odds Ratios
Although odds are commonly used in reference to
betting (racing, poker, football, etc.), with the exception of the literature on relative risk, one doesn't often
see them used in medical applications in this country.
This is similarly true of the British literature, though
there are exceptions in both countries. 1,4 If p is the
prevalence rate of the disease, then the odds in favor of
the disease (relative to 1.0) are:
Bayes' Formula
The "predictive value of a positive test result,"
p + , is defined as the proportion of positive test results
that represent persons who have the disease the test
being used is supposed to detect. In other words, p +
represents the conditional probability of the presence
of disease, given that the patient has had a positive
test result. Bayes' formula gives this as 2 :
PSe
P S e + (1 - S„)(l - p)
=
=
=
=
ci =
l
-
P
_P_
q
(2)
Solving for p in terms of CI, one has:
(1)
CI
1 + CI
Received December 9, 1976; accepted for publication December
27, 1976.
Address reprint requests to Dr. Archer: Department of Biometrics, University of Colorado Medical School, 4200 East Ninth
Ave., Box B-119, Denver, Colorado 80262.
(3)
* Sensitivity is the probability of a positive test, given presence
of the disease. Specificity is the probability of a negative test,
given absence of the disease. Prevalence is the presumed frequency
of the disease in the population tested.
0002-9173-78-0100—0032S00.60 © American Society of Clinical Pathologists
32
33
PREDICTIVE VALUE OF TEST RESULTS
Vol. 69 • No. I
so that the conversion in either direction is a simple
one, which can usually be done in one's head.t Table
1 gives a list of probability values and the corresponding
odds for values of p between 0.05 and 0.95. Figure 1
100.0
shows a graph of p vs.
. For many practical situ1 - p
ations, one can take ft = p whenever p < 0.05, and
ft =
= — when p s 0.95, as can also be seen
1 - p
q
from Figure 1.
If we express the "prior probability," p, in terms
P
of prior odds, ft =
, and the "posterior proba1 -p
bility", p + , in terms of posterior odds, we can derive
y '°:
the following relationship from equation 1:
n+ =
1
1 - Sn
ft = b+n.
(4)
This is a straight line through the origin in the odds
plane, having slope b + =
— .Since both S e and S p
1 — 5»p
will generally be greater than 0.5, b + will generally be
greater than 1.0, showing that the odds favoring presence of the disease increase by a simple multiplicative
factor with the knowledge of a positive test result.
(b + = 1.0 whenever S e + S p = 1, which will generally
not be true of any laboratory test of use to clinicians.)
The "predictive value of a negative test result,"
q", is defined as the proportion of negative test results
that represent persons who do not have the disease the
test being used is supposed to detect. (I have used the
notation q~ here .since we are using the letter p to represent the probability of the presence of the disease.
Similarly, any odds ratio, ft, represents odds favoring
presence of the disease.) Bayes' formula can be used
to express q~ as:
SP(1 - p)
S p (l - p) + (1 - Se)p
1
ft = b-a
(6)
where
ft- =
1 -qq-
1 - p-
2:100
1:100
o.or
FIG. 1. The odds ratio (fi) as a function of prevalence (p) and
two functions that furnish approximations to the odds ratio at
the extremes of the prevalence domain.
straight line through the origin. The slope of the negative result line, b~, will generally be less than 1.0 for
practical values of S e and S p .
Discussion
(5)
In terms of the posterior odds in favor of the presence
of the disease following a negative test result, we have:
ft- =
5:100
(7)
Equation 6, like equation 4, is represented by a
t In case the odds are not given relative to 1.0, say in the form
a>,:oj2, then p is given by p = coj/Ccu, + ai2). For example, odds of
3:2 in favor of the disease being present correspond to a probability
of p = 3/(3 + 2) = 3/5 = 0.60 of its presence.
Both lines (equations 4 and 6) are shown in Figure 2.
For any particular value of the prior odds, say ft,,
Table 1. Relationship between Prevalence (p)
and Odds (ft)
Prevalence
0.05
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
0.95
Odds
0.05
0.11
0.25
0.43
0.67
1.00
1.50
2.33
4.00
9.00
19.00
=
=
=
=
=
=
=
=
=
=
=
5:100
11:100
25:100
43:100
67:100
100:100 = 1:1
1.5:1
2.3:1
4:1
9:1
19.: 1
34
ARCHER
Puteni
OMt
Trior Odds
FIG. 2. Linear relationships between prior and posterior
odds for positive and negative test results.
one can see that the posterior odds will be unchanged
whenever Se + Sp = 1.0. Furthermore, one can easily
visualize both the direction and magnitude of changes
in the posterior odds that would be produced by any
combination of changes in sensitivity, specificity,
and/or prevalence. In addition, the use of equations 4
and 6, together with equations 2, 3, and 7, makes the
calculation of predictive values a very simple matter,
easily carried out by hand or with a pocket calculator.
The following example illustrates the ease of using
Bayes' formula in this form. A prevalence of p = 0.20
would yield (from equation 2) prior odds of D, = 0.25.
If the test to be used is presumed to have a sensitivity
of Se = 0.90 and specificity of Sp = 0.95, then (from
equations 4 and 6) the slopes of the positive and negative result lines would be b + = 18.0 and b" = 0.1053,
which yield (again using equations 4 and 6) posterior
odds of H+ = 4.5 and il" = 0.0263 for positive and
negative test results, respectively. Converting back to
probabilities, the predictive value of a positive test
(from equation 3) would be p + = 0.82, and the predictive value of a negative test (from equations 3 and 7)
would be q- = 1 - p~ = 1 - 0.0256 = 0.97. Thus,
under these circumstances, a positive test result
should cause us to increase our suspicion of the presence of disease from a prior estimate of p = 0.20 to a
posterior estimate of p + = 0.82. On the other hand,
the receipt of a negative test result should cause us
to view the posterior estimate of the absence of disease
as having increased to q" = 0.97 from the prior estimate of q = 1 - p = 0.80.
The graphs shown in Figure 3 may still further
assist in easing calculation, but are given in order to
illustrate the changes in slope of the odds lines pro-
A.J.C.P. • January 1978
duced by changes in the characteristics (sensitivity
and specificity) of the test employed.
From Figure 3A, one can see that substantial changes
in sensitivity have relatively little effect on the positive test slope, b + (and therefore on the posterior
odds, D + , or the posterior probability, p + ), provided
specificity is high. On the other hand, relatively small
changes in specificity can produce very large changes
in b + when Sp is high to begin with.
In a similar vein, Figure 3B illustrates the diminished effect of specificity on changes in the negative
test slope, b~, when sensitivity is high. An increase
in sensitivity, on the other hand, can produce very
large changes in b~ when Se is high to begin with,
particularly when specificity is low.
A further rule of thumb, which can be derived
directly from equation 5, is the fact that whenever
Se and Sp are both greater than 0.5 (which will generally
be the case), q~ never falls below 0.9 until the prevalence, p, is greater than 0.10.
A) b
I-S„
Specificity (S.)
as
0.6
0.7
o.s
Sensitivity ($,)
0.9
B) b --P-
i.o
•" 0
Specificity ( y
O.S
0.6
0.7
O.t
0.9
1.0
Sensitivity (S,)
FIG. 3. Linear relationships between sensitivity and specificity
for fixed values of the slopes of the odds lines. A, the slope of
the positive results line, b + . B, the slope of the negative results
line, b".
PREDICTIVE VALUE OF TEST RESULTS
Vol. 69 • No. 1
While the formulas for the slopes of the odds lines,
b and b _ , given by equations 4 and 6, facilitate computation, it is easier to remember the slopes in terms of
ratios of conditional probabilities. These are given by:
+
b+ =
P(+|D)
P(+|D)
(8)
and
b~ =
P(-|D)
P(-|D)
where:
P(+1D) = probability of a positive test given
presence of the disease
P(+| D) = probability of a positive test given
absence of the disease
P(-1D) = probability of a negative test given
presence of the disease
P(-1D) = probability of a negative test given
absence of the disease.
35
In each case, the slope is given by the ratio of the
probabilities of the test result, conditional on the presence or absence of the disease.
Perhaps many will find it a little awkward to reorient their thinking in terms of odds rather than
(or in addition to) probabilities, but for a real understanding of predictive values, the benefits should be
worth the effort.
References
1. Fleiss JL: Statistical Methods for Rates and Proportions.
New York, John Wiley and Sons, 1973
2. Galen RS, Gambino SR: Beyond normality: The Predictive
Value and Efficiency of Medical Diagnoses. New York,
John Wiley and Sons, 1975
3. Katz MA: A probability graph describing the predictive value
of a highly sensitive diagnostic test. N Engl J Med 291:
1115-1116, 1974
4. Oldham PD: Measurement in Medicine. The Interpretation of
Numerical Data. Philadelphia, J. B. Lippincott, 1968
5. Smith CAB: Biomathematics. Fourth edition. Volume 2, Chapter 19. London, Charles Griffin and Company Limited, 1969
6. Sunderman FW, Van Soestbergen AA: Laboratory suggestion:
Probability computations for clinical interpretations of
screening tests. Am J Clin Pathol 55:105-111, 1971