Exercise 7.7.3 Consider the model yi jk = +i+ j +i j +ei jk, i =

Exercise 7.7.3 Consider the model yi jk = μ +αi+η j +γi j +ei jk, i = 1,2,3, 4, j = 1,2, 3, k = 1, . . . ,Ni j, where for i = 1 = j, Ni j = N, and N11 = 2N.

This model could arise from an experimental design having α treatments of No Treatment (NT), a1, a2, a3 and η treatments of NT, b1, b2. This gives a total of 12 treatments: NT, a1, a2, a3, b1, a1b1, a2b1, a3b1 b2, a1b2, a2b2, and a3b2. Since NT is a control, it might be of interest to compare all of the treatments to NT. If NT is to play such an important role in the analysis, it is reasonable to take more observations on NT than on the other treatments. Find sums of squares for testing

(a) no differences between a1, a2, a3,

(b) no differences between b1, b2,

(c) no {a1,a2,a3}×{b1,b2} interaction,

(d) no differences between NT and the averages of a1, a2, and a3 when there is interaction,

(e) no differences between NT and the average of a1, a2, and a3 when there is no interaction present,

(f) no differences between NT and the average of b1 and b2 when there is interaction, (g) no differences between NT and the average of b1 and b2 when there is no interaction present.
Discuss the orthogonality relationships among the sums of squares. For parts

(e) and (g), use the assumption of no interaction. Do not just repeat parts

(d) and (f)!