Statistical Methods in Medical Research 4ed

The proportion of true positives among the apparent positives is sometimes
called the predictive value of a positive test; the proportion of true negatives
among the apparent negatives is the predictive value of a negative test. In case
(a), the true prevalence is 500/1000 ˆ 0 5 and the predictive value of a positive
test is high (0 9). In case (b), however, where the true prevalence is 100/1000 ˆ
0 1, the same predictive value is only 0 5. The predictive values are conditional
probabilities of the test results and may be calculated using Bayes' theorem
(§3.3), with the prevalences of disease and non-disease as the prior probabilities
and the likelihoods of the test results obtained from the sensitivity and specificity.
Case (b) illustrates the position in many presymptomatic screening procedures
where the true prevalence is low. Of the subjects found positive by the
screening test, a rather high proportion may be false positives. To avoid this
situation the test may sometimes be modified to reduce a, but such a step often
results in an increased value of b and hence a reduced value of 1 b; the number
of false positives among the apparent positives will have been reduced, but so
will the number of true positives detected.
The sort of modification referred to in the last sentence is particularly
relevant when the test, although dichotomous, is based on a continuous measurement.
Examples are the diagnosis of diabetes by blood-sugar level or of
glaucoma by intraocular pressure. Any change in the critical level of the measurement
will affect a and b. One very simple model for this situation would be
to assume that the variable, x, on which the test is based is normally distributed
with the same variance s 2 for the normal and diseased populations, but with
different means, m N and m D (Fig. 19.1). For any given a, the value of b depends
solely on the standardized distance between the means,
D ˆ mD :
s
If the critical value for the test is the mid-point between the means, 1
2 …mN ‡ mD), a and b will both be equal to the single-tail area of the normal distribution
beyond a standardized deviate of 1
2 D. To compare the merits of different tests one
could, therefore, compare their values of D; tests with high values of D will
differentiate between normal and diseased groups better than those with low
values. There is a clear analogy here with the generalized distance as a measure of
the effectiveness of a discriminant function (p. 467); the discrimination is performed
here by the single variable x.
Instead of an all-or-none classification as healthy or diseased, it may sometimes
be useful to express the strength of the evidence for any individual falling
into each of the two groups. For the model described above, the logarithm of the
likelihood ratio is linearly related to x, as shown in the lower part of Fig. 19.1.
(This is a particular case of the more general result for discriminant functions
m N
19.9 Diagnostic tests 695