Log-linear model is often applied in the analysis of contingency tables, especially when more than one of the classification factors can be regarded as response variables. Consider a 2 × 2 × K contingency table induced by classification factors (X, Y, V ). Namely, both X and Y have two levels and V has K levels. The cell frequencies can be modeled by Poisson models. Consider the following model log µ = β0 + β1 ZX + β2 ZY + β31 ZV 1 + β32 ZV 2 + ··· + β3,K−1 ZV K−1 + β4 ZX · ZY . (8.34) Let µijk denote the expected frequency in the ijk-th cell for i = 1, 2, j = 1, 2, and k = 1, . . ., K. Express the conditional odds ratio θXY (k) between X and Y , conditioning on V = k, log θXY (k) = log µ00k µ11k µ01k µ10k in terms of the regression parameters β’s. Then argue that, under model (8.34), X and Y are independent when conditioning on V.