Consider two medical tests, A and B, for a virus. Test A is 95% effective at rec
ID: 3624901 • Letter: C
Question
Consider two medical tests, A and B, for a virus. Test A is 95% effective at recognizing the virus when it is present, but has a 10% false positive rate (indicating that the virus is present, when it is not). Test B is 90% effective at recognizing the virus, but has a 5% false positive rate. The two tests use independent methods of identifying the virus. The virus is carried by 1% of all people. Say that a person is tested for the virus using only one if the tests, and that test comes back positive for carrying the virus. which test returning positive is more indicative of someone really carrying the virus? Justify your answer mathematically.Explanation / Answer
Let V be the statement that the patient has the virus, and A and B the statements that the medical tests A and B returned positive, respectively. The problem statement gives: P(V ) = 0.01 P(A|V ) = 0.95 P(A|¬V ) = 0.10 P(B|V ) = 0.90 P(B|¬V ) = 0.05 The test whose positive result is more indicative of the virus being present is the one whose posterior probability, P(V |A) or P(V |B) is largest. One can compute these probabilities directly from the information given, finding that P(V |A) = 0.0876 and P(V |B) = 0.1538, so B is more indicative. Equivalently, the questions is asking which test has the highest posterior odds ratio P(V |A)/P(¬V |A). From the odd form of Bayes theorem: P(V |A)/P(¬V |A)=P(A|V )/P(A|¬V ) P(V )/P(¬V ) we see that the ordering is independent of the probability of V , and that we just need to compare the likelihood ratios P(A|V )/P(A|¬V ) = 9.5 and P(B|V )/P(V |¬V ) = 18 to find the answer.