4.14 In the two-way layout (see Example 3.4.11), the EM algorithm can be very helpful in computing ML estimators in the unbalanced case. Suppose that we observe Yijk : N(ξij , σ2

), i = 1,... ,I, j = 1,... ,J, k = 1,...,nij , where ξij = µ+ αi + βj + γij . The data will be augmented so that the complete data have n observations per cell.

(a) Show how to compute both the E-step and the M-step of the EM algorithm.

(b) Under the restriction

i αi =

j βj =

i γij =

j γij = 0, show that the EM sequence converges to the ML estimators corresponding to an unweighted means analysis.

(c) Under the restriction

i ni·αi =

j n·jβj =

i ni·γij =

j ·j γij = 0, show that the EM sequence converges to the ML estimators corresponding to a weighted means analysis.