In a diagnostic test for a disease, let D denote the event of having the disease, and let + (−) denote a positive (negative) diagnosis by the test. Let

????1 = P(+ ∣ D) (the sensitivity), ????2 = P(+ ∣ Dc) (the false positive rate), and

???? = P(D) (the prevalence). More relevant to a patient who has received a positive diagnosis is P(D ∣ +), the positive predictive value.

a. Show that P(D ∣ +) = ????1????∕[????1???? + ????2(1 − ????)].

b. Suppose niyi ∼ bin(ni

, ????i), i = 1, 2. When ???? is known, explain how to simulate in a simple manner to obtain a 95% posterior interval for P(D ∣ +) based on independent uniform priors for ????1 and ????2. Illustrate using n1 = n2 = 100, y1 = y2 = 0.95, and ???? = 0.005. Explain the influence of ???? on why P(D ∣ +) seems to be so small.