population is defined by the symptoms elicited by
history taking, the signs revealed by physical exami-
nation, and additional pertinent historic and
demographic data such as age, gender, disease
history, disease exposure, and, in the evaluation of
heritable disorders, family history and geoethnic
lineage. Once diagnostic tests have been performed,
the prior probability is equal to the prevalence of the
disorder in the clinical subpopulation that is charac-
terized by the results of those studies.
Dallman
et al.
(1981) found that the prevalence
of iron deficiency in the apparently healthy 1-year-
olds they studied was approximately 0.09. Thus, at
presentation, the prior probability of iron deficiency
in this population was 0.09. Infants who were
subsequently found to have a low blood hemoglobin
concentration had a number of additional diagnostic
studies performed. The prevalence of iron defi-
ciency in this (low hemoglobin) subpopulation was
found to be 0.35 so, in terms of further testing, these
infants had a prior probability of 0.35.
Posterior probability.
Diagnostic laboratory
studies are ordered with the intent of adjusting the
estimate of the probability of a disorder in a patient
based upon the study results. The revised estimate of
the disorder's probability is called the posterior
probability, P[post]. It is the post-test probability.
The method of adjusting probability estimates to be
discussed here is based upon Bayes' formula for
inverting a conditional probability. The method
possesses a great intuitive appeal, and in addition,
the formulation is provable from the axioms of
probability theory.
Consider the case of a 1-year-old who has a low
blood hemoglobin concentration and who, on subse-
quent testing, is found to have a transferrin satura-
tion of 7.5%. According to the findings of Dallman
et al.
(1981), this child's prior probability of iron
deficiency is 0.35. What is her posterior probability
of iron deficiency given the measured transferrin
concentration? The critical value for transferrin
saturation is 10% so the test is positive for iron
deficiency. There are two ways in which this patient
could have a positive result: she could have a true
positive test result or she could have a false positive
result. The probability of a true positive study
result, P[true positive], equals the product of her
prior probability of iron deficiency times the
probability of registering a positive test result in a
patient with iron deficiency. The latter probability
equals the sensitivity of the study. So,
P[true positive] = P[pre] sens
In this case, the prior probability is 0.35 and the
sensitivity is 0.53 (from Table 3.3), so the probabil-
ity of a true positive result is 0.19. The probability
of a false positive result, P[false positive], equals the
product of the pretest probability that the patient
does not have iron deficiency (1 minus the prior
probability) times the probability of having a
positive result given that she is not iron deficient (1
minus the specificity). Therefore,
P[false positive] = (1-P[pre]) (1-spec)
The prior probability is 0.35 and the specificity is
0.25 (Table 3.3), so the probability of a false
positive result is 0.16.
The patient's posterior probability of iron
deficiency, meaning the probability that she had a
positive result because she is truly iron deficient, is
equal to the probability of a true positive result
divided by the total probability of having a positive
result,
P
[
post
] =
P
[
true positive result
]
P
[
true positive
] +
P
[
false positive
]
=
P
[
pre
]
sens
P
[
pre
]
sens
+ (
1
−
P
[
pre
]) (
1
−
spec
)
This is Bayes' formula. Using it, the posterior
probability of iron deficiency in this patient is 0.54.
This form of Bayes' formula should only be
used when dichotomous interpretation of study
results is obligatory because of the qualitative nature
of the study. When results are quantitative, as for
most laboratory studies, categorizing the result into a
binary classification results in loss of diagnostic
information and unnecessarily restricts the values
that the posterior probability can take.
The use of likelihood ratios based upon the
frequency distributions of results in the pertinent
reference populations allows incorporation of all the
available diagnostic information in the calculation of
the posterior probability of a diagnosis and broadens
the range of values the probability can assume
(Radack 1986). The likelihood ratio is the ratio of
the frequency of a study result in one diagnostic
group to the frequency of the result in another. In
the example being considered here, it is the ratio of
the frequency of a transferrin saturation of 7.5% in
patients with iron deficiency to the frequency of that
result in patients who are iron replete. Examination
of Figure 3.1, the reference frequency histograms
Diagnostic and Prognostic Classification
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