In Question 24 of the extended activities regarding the ELISA test data, you found that the probability that a randomly selected person who tests positive for HIV (the first time) actually has the virus is surprisingly low: P(HIV | + Test) = 0.1175. Now let’s investigate how this probability changes depending on the prevalence rate of HIV— that is, the proportion of people who have HIV. For this problem, the probability of having HIV was believed to be 0.002 (i. e., the prevalence rate was 0.2%). Also, recall that the rate of correct positive test results for infected people (the sensitivity of the test) was 99.7% and the rate of correct negative test results for non-infected people (the specificity of the test) was 98.5%.
a. If the prevalence rate was 0.1%, recalculate P(HIV + Test). Repeat the conditional probability calculation for a prevalence rate of 0.3%. What do you observe about P(HIV | + Test) as the prevalence rate increases? Briefly explain why this makes sense.
b. Create an expression that relates the prevalence rate to P(HIV | + Test). Then produce a plot that displays how P(HIV | + Test) changes as the prevalence rate increases from 0 to 1. Put the prevalence rate on the horizontal axis and P(HIV | + Test) on the vertical axis. Visually inspect the curve to determine an approximate prevalence rate that would be needed for P(HIV | + Test) = 0.95.
c. Now use the equation you found in Part B to find the prevalence rate that would be needed for P(HIV | + Test) = 0.95. Does this prevalence rate seem high? What does it suggest about the population as a whole?
d. To ensure a high probability that a person has HIV when the ELISA test result is positive, is high sensitivity (fixing the specificity level) or high specificity (fixing the sensitivity) more important for a given prevalence rate? Create graphs of P(HIV | + Test) versus values of the sensitivity and P( HIV + Test) versus values of the specificity to help you answer this question.