Suppose that a clinic has devised a differential diagnostic system for two types t = 1, 2 and based on a 1 × c feature vector x on the basis of a fitted binary normal model Pr(t = 1|x, β)=1 − Pr(t = 2|x, β) = Φ(xβT ).

Suppose further that the referred patients have feature vectors which follow a Nc(µ, Ω) distribution. Show that the clinic will diagnose a proportion

Φ

⎧

⎨

⎩

µβT

(1 + βΩβT )

1 2

⎫

⎬

⎭

of referred patients as of type 1.

Suppose that an alternative system based on a binary logistic model with Pr(t = 1|x, γ)=1 − Pr(t = 2|x, γ) = Ψ(xγT )

is used. Find an approximation to the proportion of referred patients diagnosed as of type 1.