2. Consider the loglinear model of independence for a two-way contingency table. This has equation for expected frequencies {µij} in an I × J contingency table

log µij = λ + λ X i + λ Y j

(a) Motivate this model, by showing how the definition of statistical independence of two categorical variables implies that a loglinear model of this form holds.

(b) To allow for association between X and Y, this model is extended to 

log µij = λ + λ X i + λ Y j + λ XY ij

For a 2 × 2 contingency table, express the log odds ratio in terms of expected frequencies, and use it to show that the odds ratio for this model equalsexp λ XY 11 + λ XY 22 − λ XY 12 − λ XY 21
. (Hence the two-factor interaction parameters provide information about the XY association.)