Two - Way Analysis of Variance. Suppose that two different sets of treatments are of interest. Let y ijk be the k th observation level i of the fi rst treatment type and level j of the second treatment type. The two - way analysis - of -

variance model is

where τ1 is the effect of level i of the fi rst treatment type, γj is the effect of level j of the second treatment type, ( τγ ) ij is an interaction effect between the two treatment types, and εijk is an NID(0, σ2 ) random - error component.

a. For the case a = b = n = 2, write down a regression model that corresponds to the two - way analysis of variance.

b. What are the y vector and X matrix for this regression model?

c. Discnss how the regression model could be used to test the hypotheses H0 : τ1 = τ2 = 0 (treatment type 1 means are equal), H0 : γ1 = γ2 = 0 (treatment type 2 means are equal), and H0 : ( τγ ) 11 = ( τγ ) 12 = ( τγ ) 22 = 0 (no interaction between treatment types).

Yijk +++ (TY)j + Eijk i=1, 2,..., a, j=1,2,..., b, k = 1, 2,...,n