4.11 (a) In Example 4.11, show that (Xijk i j ij )...

4.11

(a) In Example 4.11, show that

(Xijk − µ − αi − βj − γij )

2 = S2 + S2

µ + S2

α + S2

β + S2

γ

where S2 = (Xijk−Xij ·)

2, S2

µ = IJm(X···−µ)

2, S2

α = J m(X1··−X···−αi)

2, and S2

β , S2γ are defined analogously.

(b) Use the decomposition of

(a) to show that the least squares estimators of µ, αi,...

are given by (4.32) and (4.33).

(c) Show that the error sum of squares S2 is equal to (Xijk − ξˆ

ij )

2 and hence in the canonical form to n j=s+1Y 2 j .