Multivariate positivity rules.
The positivity
rules for combination testing just discussed rely upon
critical values derived from the univariate (one test)
result frequency distributions for the reference
populations. In the setting of combination testing it
is also possible, and often desirable, to define
positivity rules which arise from a consideration of
the multivariate (multiple test) result frequency
distributions that arise from the application of the
test combination to the respective reference popula-
tions. These are called multivariate positivity rules.
Discriminant functions.
Positivity rules based
on discriminant functions separate diagnostically
positive test result combinations from diagnostically
negative combinations by defining a curve (two
tests) or surface (multiple tests) that divides the
space of test result combinations into the two
diagnostic regions (Figure 3.7). When a linear
discriminant function is used, the diagnostic regions
are separated by a straight line or a plane. The slope
of the line, or the orientation of the plane, is selected
by statistical rules to yield maximum separation of
the result frequency distributions of the diagnostic
classes and, therefore, maximum diagnostic
discrimination (Solberg 1978, Strike 1996). The
location of the line or plane, which is specified by
the value of an axis intercept, establishes the separa-
tion of the diagnostic categories and thereby serves
as the critical value determining the performance
characteristics of the study combination. For combi-
nations of two tests, the diagnostic classification of
individuals can be accomplished graphically, by
plotting their test results, or algebraically, by calcu-
lating a discriminant score,
discriminant score = b1 result 1 + b2 result 2
and comparing the score to the stipulated critical
score value. For more than two test results, the
algebraic approach is used.
Although linear discriminant function positivity
rules are common in the medical literature, the valid
application of this technique is limited by its statisti-
cal constraints. In particular, it is necessary that, in
the clinical population of interest, the variances of
the individual test results as well as the covariances
of all test pairs must be the same for individuals with
the disease and those who are disease-free. This is a
criterion that is rarely satisfied. Fortunately,
quadratic discriminant analysis can be used when the
individual test variances and the test pair covariances
are unequal; however, the test result distributions
must be multivariate normal. As shown in Figure
3.7, quadratic discriminant functions yield maximum
separation of the result frequency distributions of the
diagnostic classes using a curved line or a curved
surface.
Diagnostic ratios.
Diagnostic ratios are a
multivariate approach to result interpretation when
only two laboratory studies are concerned. Positiv-
ity rules based on diagnostic ratios separate the
diagnostic space into two regions using a straight
line that passes through the origin (Figure 3.8).
Patients are classified by plotting their test results or
by calculating the value of the diagnostic ratio,
diagnostic ratio
=
result
1
result
2
and comparing it to the critical value for the ratio.
Ratios have proved most useful when the values
of the analytes change in opposite directions in
response to disease. The ratio of the two magnifies
the changes and thereby increases the diagnostic
Diagnostic and Prognostic Classification
3-7
Figure 3.7
Discriminant function positivity rules for the interpretation of a two-test combination. Left graph, diagnostic
spaces defined by a linear discriminant function; right graph, diagnostic spaces defined by a quadratic discriminant function.
Study 1 result
Study 2 result
positive
negative
Study 1 result
Study 2 result
positive
negative
