For diagnostic testing, let X = true status (1 = disease, 2 = no disease) and Y = diagnosis (1 = positive, 2 = negative). Let π i = P (Y = 1|X = i), i = 1, 2. a. Explain why sensitivity = π 1 and...

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

a. Explain why sensitivity = π₁
and specificity = 1 − π₂.

 b. Let γ denote the probability that a subject has the disease. Given that the diagnosis is positive, use Bayes’s theorem to show that the probability a subject truly has the disease is

c. For mammograms for detecting breast cancer, suppose γ = 0.01, sensitivity = 0.86, and specificity = 0.88. Given a positive test result, find the probability that the woman truly has breast cancer.

d. To better understand the answer in (c), 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.