Hartley (1967) proposed the following method for computing the expected value of mean squares for the mixed model with unbalanced data. For a given method of computing the 40V table, as if the model were fixed, let Q(y) = y By a mean square associated with a random effect. be

a. Assuming that tr(B) 1 and BX-0, show that the expected value of this mean square is given by EQ)] +XBX).

b. Show that the coefficient of o, in this expression can be written as k = (XBX)=(x) where, is the /th column of X. Note that the quantities in the sum can be evaluated by recomputing the AOV table using each column of X, as the response vector in the model.

e. Show that the expected mean squares can be computed by evaluating = -- AOV tables. Herec is the number of columns in X,

d. Use this method to obtain a formula for the expected value of the treatment mean square in the one-way classification random model with both balanced and unbalanced data. 15.5 Assuming the unbalanced, one-way classification random model:

a. Write the covariance structure in matrix form, and determine the inverse of the covariance matrix.

b. Show that the last N-1 rows of the Helmert matrix H, defined in Exercise 16.10, can be used as the matrix BT, required for the REML method. Section 15.3 15.6 Verify the REML likelihood equations in (15.51) Hint: Apply the relations in (15.6) to the likelihood By N(0, BVB), noting the relation in (15.33). Section 15.4