For diagnostic testing, let
X = true status (1 = disease, 2 = no disease)
and Y = diagnosis (1 = positive, 2 = negative). Let πi = 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
π
1γ/[π1γ + π2(1 − γ )]
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.