Imani Brown Chapter 5 Diagnostic Testing Consider the mammogram for detecting breast cancer in women. One recent study estimated sensitivity=0.86 and specificity=0.88. Of the women who undergo mammograms at any given time, about 1% are typically estimated to actually have breast cancer; that is, P(S)=0.01. Of the women who receive a positive mammogram result, what proportion actually have breast cancer? Use the following diagram to help you solve this problem: Sensitivity is the probability that the test detects cancer given that it is present. Specificity is the probability that the test does not detect cancer given that it is not present. Both refer to correct results. Show your work: The question asks us to find the proportion of women who actually have cancer given that they received a positive test result. That is, we’re asked to find P(S|POS) where S represents the presence of cancer. The formula for conditional probability gives us P(S|POS) = Now to find P(S and POS) we cannot just multiply P(S) x P(POS) because these events are not independent. So let’s start with P(POS|S) because this is the sensitivity and it is given to us. Don’t forget to use the sensitivity for younger women, which is .001 rather than .01. So using the conditional formula on this quantity, we have P(POS|S) = .86=(P(S and POS))/(.001) = .001*.86 = .00086 Since we are given P(S) = .01, we can solve for P(S and POS) P(S and POS) = .00086 To complete the answer, we still need to find P(POS). From the chart, we can see that there are two ways to get a positive result – a true positive (which is the sensitivity) and a false positive. In the chart, these are represented by the two branches: YES and positive (sensitivity) = (.01)(.86) and the other branch NO and positive = (.99)(.12). Since a positive result can happen either way, we add these two branches to get .0086 + .1188 = .1274 and now we are ready to finish the problem: P(S|POS) = = .999 = (P(S and POS))/(.12) = .999*.12 = .11988 When you return this assignment, you will receive the solution to the above question so that you can check your work. Now answer the question, “Why is the proportion in error likely to be larger for a young population than for an older population?” Because there are more young women in the negative category opposed to the positive so there will be a larger number of false positives than correct positives.