Exercise 6.12 (a) Find the model matrix for the orthogonal polynomial model Y = T +e corresponding to the model yi j =0+1xi+2x2 i +3x3

Exercise 6.12

(a) Find the model matrix for the orthogonal polynomial model Y = Tγ +e corresponding to the model yi j =β0+β1xi+β2x2 i +β3x3 i +ei j, i = 1,2,3, 4, j = 1, . . . ,N, where xi = a+(i−1)t.

Hint: First consider the case N = 1.

(b) For the model yi j = μ +αi +ei j, i = 1,2,3, 4, j = 1, . . . ,N, and for k =

1,2, 3, find the contrast Σλikαi such that the test of H0 : Σλikαi = 0 is the same as the test of H0 : γk = 0, i.e., find the polynomial contrasts.