Consider the cell means model, where the errors are independent, N(0,). For testing the hypothesis of equal means, we may use the (p-1) xp matrix H = (I|- J).

a. Determine the matrix H(XX)HT and its inverse (see Appendix A.1.10). Show that the last (p-1) rows of the Helmert matrix defined in Exercise 16.10 are eigenvectors of this matrix and its inverse.

b. Use the data in Example 18.3 to determine the numerator sum of squares for testing this hypothesis. Write the contrasts, a Hp defined by the eigenvectors and determine the sum of squares for each of these contrasts and verify (18.19).