a. Assuming that the balanced, one-way classification model is written in over- parameterized form as in (10.55), write the normal equations and observe that the coefficient matrix is singular. Using the methods for determining generalized inverses, described in Appendix A.1.12, determine a solution to the normal equations. Hint: Note that (1) the first p rows and columns of the coefficient matrix form a non-singular matrix, (2) the same is true of the last p rows and columns, and (3) the vector C = (0,n,,n) is linearly independent of the rows of the coefficient matrix.

b. Write the normal equations in matrix form for the two parameterizations given in (10.52) and (10.54). Compare the ease of solving the equations in these two forms and contrast with parameter estimation in the cell means model.10.6 Develop the matrix expression for the numerator sum of squares Ny in (10.20) by using HA, as defined in (10.9). 10.7

a. Verify that the columns of A in (10.44) are eigenvectors of HHT and determine the associated eigenvalues.

b. Let and denote the last two columns of A and verify that +vy and y are also eigenvectors and contrasts that are orthogonal to v, but not to each other. 10.8

a. Verify the relation in (10.48) as follows: Solve the relation as eu for j use (10.47) to write the relation in terms of a,, and then note that J=0.

b. Use the expression for a in terms of a to write M

b. Use the fact that MMI to show that, under the restriction imposed by the requirement in (10.47), the hypothesis matrix must have the form, H (10) J 10.9 The first form of the hypothesis in (10.7) is not of the form (10.50) and hence the relation given in (10.47) and (10.48) does not hold. However, it is reasonable to consider the transformation in (10.46) using H, as defined in (1.8). Determine the expression for j in terms of

a. Hint: To determine the inverse of My we can write H, AH, where A has the form -1 0 A=0 -1 0 Use this matrix to write M, in terms of M, and use the inverse of M, to obtain the inverse of M. 10.10

a. Assuming that the balanced, one-way classification model is written in over- parameterized form as in (10.55), write the normal equations and observe that the coefficient matrix is singular. Using the methods for determining generalized inverses, described in Appendix A.1.12, determine a solution to the normal equations. Hint: Note that (1) the first p rows and columns of the coefficient matrix form a non-singular matrix, (2) the same is true of the last p rows and columns, and (3) the vector C = (0,n,,n) is linearly independent of the rows of the coefficient matrix.

b. Write the normal equations in matrix form for the two parameterizations given in (10.52) and (10.54). Compare the ease of solving the equations in these two forms and contrast with parameter estimation in the cell means model.