2.33 For a 2 x 2 x 2 table, suppose XY (1) = XY (2) = . For a possibly confounding vari- able

2.33 For a 2 x 2 x 2 table, suppose θ_XY(1) = θ_XY(2) = θ. For a possibly confounding vari-

able Z, let θ_c denote the common value of θ_(i)YZ. Let π₁ = P(Z = 1|X = 1, Y = 2)

and π₂ = P(Z = 1|X = 2, Y = 2).

a. Show (Breslow and Day 1980, p. 96) that

θ_XY = θ

θ_cπ₁ + (1 – π₁)

θ_cπ₂ + (1 – π₂)

b. Verify that either odds ratio collapsibility condition in Section 2.3.6 implies that the confounding risk ratio θ_XY/θ equals 1.0.

c. Describe what needs to happen for θ_XY/θ to be far from 1.0. Illustrate with particular values of θ_c > 1 and π₁ > π₂. Describe a study in which such values would be plausible.