Question

Suppose 1% of the population of Toronto has been infected with SARS-CoV-2 at some point (call the corresponding proportion, 0.01, the prevalence of the virus). There are now antibody tests available for SARS-CoV-2 to detect whether someone has ever been infected. However, like any test, these tests are not perfect and are subject to error at some rate.

Also suppose that, when someone has been infected, the test correctly comes up positive 98% of the time, comes up negative 1.7% of the time, and comes up inconclusive in the remaining 0.3% of cases. Also suppose that when someone has not been infected, the test correctly comes up negative 99% of the time, positive 0.8% of the time, and inconclusive 0.2% of the time.

Let ?D be the event that a randomly selected person in Toronto has truly been infected (?D for disease). Thus, ??Dc is the event that the person has never been infected. Let ++ be the event that a test comes up positive, and – the event it comes up negative. Use ?O for inconclusive.

Draw a probability tree to depict the relationship between these events (see Lecture 7). Use the tree to compute ?(?|+)P(D|+), the probability that someone has been infected conditional on getting a positive test result. What does this result mean in words and why might it be surprising (1-2 sentences)?

Use the same information about testing for SARS-CoV-2 as above. What happens if prevalence changes to 0.10? Redraw the tree, and compute ?(?|+) again using this new prevalence. What do these results suggest about the importance of prevalence in interpreting test results like this (1 sentence)? Suppose a random person in Toronto gets an inconclusive result on an antibody test for SARS-CoV-2. Is the probability that they are infected lower or higher than it was before we knew their test result? Use your tree to explain.

Answer 1