EXAMPLE 32 The ELISA test for the presence of HIV

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The ELISA test is used to screen blood for the presence of HIV. Like most diagnostic procedures, the test is not foolproof.

  • When a blood sample contains HIV, the ELISA test will give a positive result 99.6% of the time. That is, the false-negative rate, the percentage of tests returning a negative result when the HIV virus is actually present, is .
  • When the blood does not contain HIV, the ELISA test will give a negative result 98% of the time. That is, the false-positive rate, the percentage of tests returning a positive result when the HIV virus is not actually present, is .

A positive result means that the test says that the person has the HIV infection. A negative result means that the test says that the person does not have the virus. The prevalence rate for HIV in the general population is 0.5%. That is, 5 of 1000 persons in the general population have HIV.

Suppose we have samples of blood from 100,000 randomly chosen people.

Problem 1. How many people in the sample of 100,000 have HIV? How many do not?

Solution

The prevalence rate of 0.5% means that people in the sample have HIV. The remainder—99,500—do not.

Problem 2. A positive result is given 99.6% of the time for blood containing HIV. For the 500 people with HIV, how many positive results will the ELISA test return? How many of the 500 people with HIV will receive a negative result?

Solution

The ELISA test will return a positive result for of the 500 people. Thus, two people who actually have HIV will receive a test result indicating that they do not have the virus.

Problem 3. A negative result is given 98% of the time for blood without HIV. For the 99,500 people without HIV, how many negative results will the ELISA test return? Positive results?

Solution

The ELISA test will return a negative result for of the 99,500 people without HIV. The remaining 2%, or 1990 people, will receive positive ELISA test results, even though they do not have the virus.

We can use the counts we found to fill in Table 25.

Table 5.58: Table 25 ELISA test contingency table
In reality
ELISA test results Person has HIV Person does not have HIV Total
Positive 498 1,990 2,488
Negative 2 97,510 97,512
Total 500 99,500 100,000

285

We will use the information in the ELISA test contingency table to solve Problems 4 and 5. If a person is chosen at random from the sample of 100,000, define the following events:

  • : Person has HIV.
  • : Person does not have HIV.
  • Pos: ELISA test returned positive results.
  • Neg: ELISA test returned negative results.

Problem 4. What is the probability that a randomly chosen person actually does have HIV, given that the ELISA results are negative? In other words, find .

Solution

Problem 5. What is the probability that a randomly chosen person actually does not have HIV, given that the ELISA test results are positive? In other words, find .

Solution