computed from lognormal modeling of the data in
Figure 3.14. The posterior probability of a good
outcome as a function of acetaminophen half-life, as
calculated for a prevalence of 0.6, is also shown
(right graph). At this prevalence, which was chosen
because it is the prevalence in the study of Saunders
et al.
(1980), the study value corresponding to a
probability of 0.5 is 8.3 h.
Regrettably, it is not infrequent for the prognos-
tic performance of a laboratory study to be reported
in such a way that it is impossible to calculate likeli-
hood ratios for the study. Instead, one is forced to
calculate posterior probabilities solely from the
reported values of the fractions correctly classified
for a limited set of critical values. For two prognos-
tic categories, the form of Bayes' formula that must
then be used is,
P[post]=
prevalence
$
FCC
1
prevalence
$
FCC
1
+ (
1
−
prevalence
)(
1
−
FCC
2
)
where
FCC
1
and
FCC
2
are the fractions correctly
identified as reported for the critical value closest to
the study result. Note the similarity of this formula
and that for the calculation of diagnostic probability
using sensitivity and specificity.
Logistic functions
The probability curve shown in Figure 3.20
(right graph) was computed by modeling the
frequency data of each of the prognostic groups and
then using the model parameters to calculate, by
Bayes' formula, the probability of the indicated
outcome. An alternative computational approach is
to estimate the parameters of a model that directly
describes the sigmoidal relationship between the
value of the study result and the probability of
membership in the prognostic group. Such models
are called probability models (Liao 1994). By far
the most commonly used probability model is the
logistic model,
P
[
post
] =
e
b
0
+
b
1
result
e
b
0
+
b
1
result
+
1
where
b0
and
b1
are the model parameters.
Probability modeling using logistic regression
has a number of very desirable features that explain
its great appeal: it can be used to model posterior
probabilities for test result combinations, in which
case it has the form,
p
[
post
] =
e
b
0
+
bi resulti
e
b
0
+
bi resulti
+
1
it can use qualitative and semiquantitative study
results and categorical variables either alone or in
combination with quantitative study results, it enjoys
considerable robustness to departures from the statis-
tical constraints regarding data normality and
variance/covariance structure, and it can be used
when there are multiple prognostic groups (Strike
1996). Because logistic modeling allows the inclu-
sion of other pertinent demographic and clinical
data, logistic functions are the most common way to
calculate posterior probabilities from prognostic
study results. Care must be taken in their use,
however, because logistic functions include the
effect of prognostic group prevalence; it is
Diagnostic and Prognostic Classification
3-18
0
5
10
15
20
25
Acetaminophen half-life (h)
0.01
0.1
1
10
100
Likelihood ratio of a good outcome
0
5
10
15
20
25
Acetaminophen half-life (h)
0
0.2
0.4
0.6
0.8
1
Probability of a good outcome
Figure 3.18
Bayesian calculation of posterior probabilities of a good outcome in acetaminophen poisoning. Left graph, the
likelihood ratio of a good outcome as a function of acetaminophen half-life as derived from the lognormal frequency distribu-
tion models of the data in Figure 3.14. Right graph, the probability of a good outcome as a function of acetaminophen half-
life given a value of 0.6 for the prevalence of a good outcome.
