For diagnostic testing, let X true status (1 disease, 2 no disease) and Y diagnosis (1 positive, 2 negative) Let 1 P(Y 1 X 1) and 2 P(Y 1 X 2) Let denote the probability that a subject has the disease a Given that the diagnosis is positive, use Bayes Theorem to show that the probability a subject t...

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For diagnostic testing, let X = true status (1 = disease, 2 = no disease) and Y

Question:

For diagnostic testing, let X = true status (1 = disease, 2 = no disease) and Y = diagnosis (1 = positive, 2 = negative). Let π1 = P(Y = 1 | X = 1) and

π2 = P(Y = 1 | X = 2). Let γ denote the probability that a subject has the disease.

a. Given that the diagnosis is positive, use Bayes’ Theorem to show that the probability a subject truly has the disease is P(X = 1 | Y = 1) = π1γ/[π1γ + π2(1 − γ)].

b. For mammograms for detecting breast cancer, suppose γ = 0.01, sensitivity =

π1 = 0.86, and specificity 1 − π2 = 0.88. Find the positive predictive value.

c. To better understand the answer in (b), find the joint probabilities for the 2 × 2 cross-classification of X and Y . Discuss their relative sizes in the two cells that refer to a positive test result.

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