Rapid Test, Base Rate, and the Cost of a Second Test
A regional clinic uses a rapid antibody test for a condition that affects 3% of the adult population. In validation studies, the test correctly identifies 88% of people who have the condition (sensitivity) and correctly identifies 94% of people who do not (specificity). A patient with no symptoms tests positive. The clinic nurse says, 'The test is 88% accurate, so there is an 88% chance you have the condition.' The patient is alarmed and asks for a second, independent test. The clinic assumes that, conditional on true disease status, the two test results are independent. A positive result triggers a counselling session, and the clinic can afford at most 40 counselling sessions per 1,000 tests administered before changing its protocol. The nurse wants to know the actual probability of disease after one positive result and after two positive results, and whether the '88%' statement is defensible.
1. Using a tree diagram or two-way table, calculate the probability that a person who tests positive actually has the condition. Compare this with the nurse's 88% statement.
Read the worked answer
Let D be having the condition and + be a positive test. We have P(D)=0.03, P(+|D)=0.88, and P(+|D')=1−0.94=0.06.
P(+) = P(+|D)P(D) + P(+|D')P(D') = 0.88×0.03 + 0.06×0.97 = 0.0264 + 0.0582 = 0.0846.
Then
P(D|+) = P(+|D)P(D)/P(+) = 0.0264/0.0846 ≈ 0.3121.
So a positive result gives about a 31.2% chance of having the condition, not 88%. The 88% is P(+|D), the sensitivity, not P(D|+). The nurse has confused the probability of a positive test given disease with the probability of disease given a positive test.
2. A second, independent rapid test is given and is also positive. Calculate the probability that the patient has the condition after two positive results. State any assumption you use.
Read the worked answer
Assume that, conditional on true disease status, the two test results are independent. Then
P(++|D) = 0.88^2 = 0.7744, P(++|D') = 0.06^2 = 0.0036.
So
P(++ ) = 0.7744×0.03 + 0.0036×0.97 = 0.023232 + 0.003492 = 0.026724.
Therefore
P(D|++) = 0.023232/0.026724 ≈ 0.8693.
After two positive tests, the probability of disease is about 86.9%. This is close to 88%, but for a different reason: the second positive result has greatly reduced the false-positive contribution.
3. Per 1,000 tests administered, compare the number of counselling sessions generated by (i) counselling every positive rapid test and (ii) counselling only after two positive tests. The clinic budget allows at most 40 sessions per 1,000 tests. Which policy fits the budget, and what clinical cost does it introduce?
Read the worked answer
Per 1,000 tests, the expected number of positives is 1000×0.0846 = 84.6, so policy (i) generates about 85 counselling sessions. This exceeds the budget of 40.
Policy (ii) uses double positives: 1000×0.026724 = 26.7, so about 27 sessions. This fits the budget.
The clinical cost is that the two-positive rule misses more true cases. There are 1000×0.03 = 30 true cases per 1,000 tests. The two-positive rule detects 30×0.7744 ≈ 23.2 of them, so about 6.8 true cases per 1,000 are missed, compared with 30×0.12 = 3.6 missed under a single-test rule (if every positive were followed up). The clinic must judge whether the budget saving is worth the extra missed or delayed diagnoses.