Exercise 7.7.4 Consider the linear model yi j = + i + j + ei j

Exercise 7.7.4 Consider the linear model yi j = μ + αi + ηj + ei j , i = 1, . . . , a, j = 1, . . . ,

b. As in Section 1, write X = [X0, X1, . . . , Xa, Xa+1, . . . , Xa+b]. If we write the observations in the usual order, we can use Kronecker products to write the model matrix. Write X = [J, X∗, X∗∗], where X∗ = [X1, . . . , Xa], and X∗∗ =

[Xa+1, . . . , Xa+b].Using Kronecker products, X∗ = [Ia ⊗ Jb], and X∗∗ = [Ja ⊗ Ib].

In fact, with n = ab, J = Jn = [Ja ⊗ Jb]. Use Kronecker products to show that X

∗(I − [1/n]J n n )X∗∗ = 0. In terms of Section 1, this is the same as showing that C(Z1, . . . , Za) ⊥ C(Za+1, . . . , Za+b). Also show that [(1/a)J a a

⊗ Ib] is the perpendicular projection operator ontoC(X∗∗) and that Mη = [(1/a)J a a

⊗ (Ib − (1/b)J b b )].