5.92 Prostate cancer A study of the PSA blood test for diagnosing prostate cancer in men (by R. M. Hoffman et al., BMC Family Practice, vol. 3, p. 19, 2002) used a sample of 2620 men who were 40 years and older. When a positive diagnostic test result was defined as a PSA reading of at least 4, the sensitivity was estimated to be 0.86 but the specificity only 0.33.

a. Suppose that 10% of those who took the PSA test truly had prostate cancer. Given that the PSA was positive, use a tree diagram and/or contingency table to estimate the probability that the man truly had prostate cancer.

b. Illustrate your answer in part a by using a tree diagram or contingency table with frequencies showing what you would expect for a typical sample of 1000 men.

c. Lowering the PSA boundary to 2 for a positive result changed the sensitivity to 0.95 and the specificity to 0.20. Explain why, intuitively, if the cases increase for which a test is positive, the sensitivity will go up but the specificity will go down.