Refer to the deviance comparison statistic G2(M0 ∣ M1) introduced in Section 4.4.3. For a sequence of s nested binary response models M1,…, Ms, model Ms is the most complex. Let v denote the difference in residual df between M1 and Ms.

a. Explain why for j < k, G2(Mj ∣ Mk) ≤ G2(Mj ∣ Ms).

b. Assume model Mj, so that Mk also holds when k > j. For all k > j, as n → ∞, explain why P[G2(Mj ∣ Mk) > ????2 v (????)] ≤ ????.

c. Gabriel (1966) suggested a simultaneous testing procedure in which, for each pair of models, the critical value for differences between G2 values is ????2 v (????). The final model accepted must be more complex than any model rejected in a pairwise comparison. Since part

(b) is true for all j < k, argue that Gabriel’s procedure has type I error probability no greater than ????.