Inadiagnostictestforadisease,asin Section 2.1.4 let D denote theeventofhavingthe disease, andlet + denote apositivediagnosisbythetest.Letthe sensitivity 1 = P(+ S

Inadiagnostictestforadisease,asin Section 2.1.4 let D denote theeventofhavingthe disease, andlet + denote apositivediagnosisbythetest.Letthe sensitivity π1 = P(+ S D), the false positiverate π2 = P(+ S Dc), andthe prevalence ρ = P(D). Relevanttoapatientwhohas receivedapositivediagnosisis P(D S +), the positivepredictivevalue.

(a) Showthat P(D S +) = π1ρ~[π1ρ + π2(1 − ρ)].

(b) Assumingindependent Yi ∼ binom(ni, πi), i = 1, 2, with n1 = n2 = 100, suppose y1 = 95 and y2 = 5. Assumingthat ρ = 0.005, simulatetoobtaina95%posteriorintervalfor P(D S +)
based onindependentuniformpriorsfor π1 and π2. Dothisbyrandomlygenerating10,000 π1 and π2 from theirposteriordistributionsandthenfindingtherelevantpercentilesof the 10,000correspondinggeneratedvaluesof P(D S +). (Classicalconfidenceintervalsfor P(D S +) are considerablymorecomplex.)

(c) Repeat(b)for ρ =