6.3. An alternative loglinear model for the ordinal-ordinal table, having fixed scores as well as unknown row and column effects, is log ßij = k + kf + kY + ì,· vj + Ui Vj.

With equally spaced {«,} and {VJ}, Goodman (1979a, 1981a) referred to it as the R + C model because of the additivity in terms of row effects

{ì,} and column effects {v,}. Kateri et al. (1998) generalized this model to include both additive and multiplicative effects.

(a) Show that the L x L model, row effects model, and column effects model are special cases.

(b) Specify constraints to make the model identifiable, and show that residual df = (r - 2)(c - 2), like the RC model.

(c) With equally spaced scores, show that the log local odds ratio has the additive form logÖ^ = y, + 8j. By contrast, for the RC model (6.13), show that log öl· = y,<5y·.

(d) A multiplicative model for the log global odds ratios has form logö^ — YiSj. Is this model invariant to permutations of rows or columns? Why or why not? How does this compare to the RC model?