MEDICAL DIAGNOSTIC OR SCREENING TESTS
These are tests to detect if a person in a certain target population has a certain disease or
condition. The test gives either a positive or a negative test result.
 A positive test result indicates or suggests that a person has the disease or condition. A
person who gets a positive result may then be clinically examined, treated, or given a more
definitive test.
 A negative test result indicates that the person does not have the disease or condition.
But the test is not perfect because there can be false positive and false negative results.
 A false positive occurs when a person without the disease gets a positive test result.
 A false negative occurs when a person with the disease gets a negative test result.
Symbols for events:


+ = The person gets a positive test result.
 = The person gets a negative test result.
D = The person has the disease.
DC = The person does not have the disease.
The probability of a false positive = P(+ | DC) = the probability that a person who does
not have the disease will test positive = the proportion of people without the disease who
would get a positive test result if they were tested.
The probability of a false negative = P( | D) = the probability that a person who has the
disease will test negative = the proportion of people with the disease who would get a
negative test result if they were tested.
The false positive and false negative probabilities are related to the following two parameters
of the test.
 Sensitivity of the test = P(+ | D) = 1 - P( | D) = 1 - false negative probability.
 Specificity of the test = P( | DC) = 1 - P(+ | DC) = 1 - false positive probability.
Sensitivity measures the ability of the test to detect the presence of the disease.
Specificity measures the ability of the test to detect the absence of the disease.
A test with high sensitivity and high specificity is able to discriminate between the absence
and the presence of the disease. Unfortunately, there is a trade-off between sensitivity and
specificity; for example, when a test’s sensitivity is increased, the specificity may suffer.
The clinically important probabilities are the following:
 Prevalence of the disease = P(D) = the proportion of people in the target population
with the disease.
 Predictive value of a positive test result = P(D | +).
 Predictive value of a negative test result = P(DC | ).
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If we know the sensitivity, specificity, and prevalence, we can calculate these two predictive
values using Bayes’ formula. We first calculate, using the total probability law, the
probability that a randomly selected person from the target population would test positive for
the disease or condition, if tested. The formula works by breaking the population into those
with the disease and those without the disease.
P(+) = P(+ | D)P(D) + P(+ | DC)P(DC)
= P(+ | D)P(D) + [1 - P( | DC)][1 - P(D)]
= sensitivity  prevalence + [1 - specificity][1 - prevalence].
We now use Bayes’ theorem to get the predictive values:
P(D | +)


P( | D)  P(D)
P(  )
sensitivit y  prevalence
.
P(  )
P(  | D C )  P( D C )
P()
P( | D C )  [1  P(D)]

1  P(  )
specificit y  [1 - prevalence ]

.
1  P(  )
P(DC | ) 
If the prevalence is extremely low, then the predictive value of a positive test result will be
much lower than what you would expect for a good test. In fact, it can be shown that P(D | +)
approaches zero as the prevalence approaches zero. On the other hand, the predictive value of
a negative test result, P(DC | ), approaches one as the prevalence approaches zero.
ESTIMATING THE PREVALENCE, P(D)
Assume that the sensitivity and specificity of a test for disease D are known. These depend
only on the test and are not related to the prevalence of the disease in the population. Earlier,
we used the total probability law to express P(+), the probability of a positive test result, in
terms of the prevalence, sensitivity, and specificity. Using algebra, we can solve for the
prevalence in terms of P(+), sensitivity, and specificity. The resulting equation is:
P ( D) 
P()  (1  specificit y)
.
sensitivit y  (1  specificit y)
To estimate P(D), we obtain an estimate of P(+) and plug it into the above formula. To
estimate P(+), we test a large sample of n people from the target population and determine n+,
the number of people who test positive. Then n+/n, the proportion of people in the sample
who test positive for the disease, serves as an estimate of P(+).
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