- PrðX¼þjY¼þÞand PrðX¼jY¼Þare the sensitivity and spe-
cificity, respectively. - PrðY¼þjX¼þÞand PrðY¼jX¼Þare called thepositive pre-
dictivityandnegative predictivity.
With positive predictivity (orpositive predictive value), the question is: Given
that the testXsuggests cancer, what is the probability that, in fact, cancer is
present? Rationales for these predictive values are that a test passes through
several stages. Initially, the original test idea occurs to a researcher. It must
then go through a developmental stage. This may have many aspects (in bio-
chemistry, microbiology, etc.) one of which is in biostatistics: trying the test out
on a pilot population. From this developmental stage, the e‰ciency of the test
is characterized by its sensitivity and specificity. An e‰cient test will then go
through an applicational stage with an actual application ofX to a target
population; and here we are concerned with its predictive values. The simple
example given in Table 3.3 shows that unlike sensitivity and specificity, the
positive and negative predictive values depend not only on the e‰ciency of the
test but also on the disease prevalence of the target population. In both cases,
the test is 90% sensitive and 90% specific. However:
- Population A has a prevalence of 50%, leading to a positive predictive
value of 90%. - Population B has a prevalence of 10%, leading to a positive predictive
value of 50%.
The conclusion is clear: If a test—even a highly sensitive and highly specific
one—is applied to a target population in which the disease prevalence is low
(e.g., population screening for a rare disease), the positive predictive value is
low. (How does this relate to an important public policy: Should we conduct
random testing for AIDS?)
In the actual application of a screening test to a target population (the
applicational stage), data on the disease status of individuals are not available
(otherwise, screening would not be needed). However, disease prevalences are
often available from national agencies and health surveys. Predictive values are
TABLE 3.3
Population A Population B
XXY þY þ
þ 45,000 5,000 þ 9,000 1,000
5,000 45,000 9,000 81,000
116 PROBABILITY AND PROBABILITY MODELS