threshold probability for acceptance of the diagnosis,
P(acceptance); the threshold probability for accep-
tance being that level of probability at which the
physician and patient agree that the diagnosis is
established with adequate certainty, given the pros
and cons of making the diagnosis and in the knowl-
edge that the diagnosis can subsequently be changed
if the future course of the disease or the response to
therapy are not typical of the diagnosed illness.
Determining a threshold probability explicitly is no
easy task and the most widely applied formal method
for its calculation, clinical decision analysis (Pauker
and Kassirer 1987, Kassirer
et al.
1987), is still
controversial. Still, the notion of a threshold
probability is present, albeit in an informal form, in
most diagnostic reasoning. Believing that an approxi-
mate value of the threshold probability can be identi-
fied, study values that confirm a diagnosis are those
that yield a posterior likelihood of disease at least
equal to the threshold probability. What that means
in terms of result likelihood ratios can be appreciated
by expressing Bayes' formula in the following form,
P[acceptance] =
P
[
pre
]
threshold likelihood ratio
P
[
pre
]
threshold likelihood ratio
+ (
1
+
P
[
pre
])
Rearrangement of this equation yields,
threshold likelihood ratio for acceptance =
(
1
−
P
[
pre
])
P
[
acceptance
]
P
[
pre
] (
1
−
P
[
acceptance
])
Study results with likelihood ratios greater than the
threshold likelihood ratio confirm the diagnosis.
Similar reasoning yields an analogous formula
for the threshold likelihood ratio for rejection of a
diagnosis in which the threshold probability for
rejection of a diagnosis, P[rejection], is that level of
probability at which it is agreed that the diagnosis is
so unlikely that it can be excluded,
threshold likelihood ratio for rejection =
(
1
−
P
[
pre
])
P
[
rejection
]
P
[
pre
] (
1
−
P
[
rejection
])
Study results with likelihood ratios less than the
threshold ratio exclude the diagnosis.
Usually the threshold probability for acceptance
of a diagnosis is different than the threshold proba-
bility for rejecting the diagnosis. That means that
there exist intermediate probabilities that do not
justify acceptance or rejection of the diagnosis.
Patients with these intermediate probabilities require
additional diagnostic workup. Sometimes, however,
the threshold probability for acceptance of a diagno-
sis is equal to the threshold probability for rejecting
the diagnosis; for instance, in situations in which
patients in whom the diagnosis is rejected are to be
seen at some subsequent time, offering another
opportunity to evaluate them for the disorder.
The application of the formulas is illustrated by
again using data of Dallman
et al.
(1981). The prior
probability of iron deficiency in the screen-positive
clinical population is 0.35. The threshold probability
for accepting a diagnosis of iron deficiency and insti-
tuting oral iron therapy might, for example, be
around 0.7, a value twice that of the prior probabil-
ity. Substituting these values into the formula for
the threshold likelihood ratio for acceptance of a
diagnosis yields a ratio of 4.33. Figure 3.12 shows
that a transferrin saturation of 5.5% is associated
with this ratio. Thus, a transferrin saturation of
5.5% or less would be confirmatory of the diagnosis
of iron deficiency. If the threshold probability for
rejecting a diagnosis of iron deficiency were, for
example, 0.1, the threshold likelihood ratio would be
0.21. No value of the transferrin saturation has a
likelihood ratio that low so the measurement of
transferrin saturation alone could not be used as a
tool for the exclusion of a diagnosis of iron
deficiency.
Screening for a disorder
A screening study is one used to detect a
serious, treatable disorder in persons afflicted with
the disorder but who have no clinical findings
suggestive of the condition. Such clinically silent
conditions are sometimes labeled "occult." It seems
reasonable to define the threshold likelihood ratio for
followup of a screening test result as the ratio that
yields a posterior probability of the disorder equal to
the threshold probability for rejecting the diagnosis.
For study values associated with likelihood ratios
greater than the threshold, the disorder cannot be
considered excluded so further evaluation is clearly
justified. The applicable formula is,
threshold likelihood ratio for followup =
(
1
−
prevalence
)
P
[
rejection
]
prevalence
(
1
−
P
[
rejection
])
Notice that, in this usage, the prior probability is the
prevalence of the disorder in the screened
population.
Also notice that the formula reveals
that the larger the threshold probability for rejection
Diagnostic and Prognostic Classification
3-13
