Suppose that the model matrix for the two-way ANOVA model Yijk j k

Suppose that the model matrix for the two-way ANOVA model Yijk ¼ µ þ αj þ βk þ γjk þ εijk is reduced to full rank by imposing the following constraints (for r ¼ 2 rows and c ¼ 3 columns):

α2 ¼ 0

β3 ¼ 0

γ21 ¼ γ22 ¼ γ13 ¼ γ23 ¼ 0 These constraints imply dummy-variable (0/1) coding of the full-rank model matrix.75

(a) Write out the row basis of the full-rank model matrix under these constraints.

(b) Solve for the parameters of the constrained model in terms of the cell means. What is the nature of the hypotheses H0: all αj ¼ 0 and H0: all βk ¼ 0 for this parametrization of the model? Are these hypotheses generally sensible?

(c) Let SS*ðα; β; γÞ represent the regression sum of squares for the full model, calculated under the constraints defined above; let SS*ðα; βÞ represent the regression sum of squares for the model that deletes the interaction regressors; and so on. Using the Moore and Krupat data (discussed in Section 8.2), confirm that SS*
ðαjβÞ ¼ SSðαjβÞ
SS*
ðβjαÞ ¼ SSðβjαÞ
SS*
ðγjα; βÞ ¼ SSðγjα; βÞ
but that SS*
ðαjβ; γÞ 6¼ SSðαjβ; γÞ
SS*
ðβjα; γÞ 6¼ SSðβjα; γÞ
where SSð#Þ and SSð#j#Þ give regression and incremental sums of squares under the usual sigma constraints and deviation-coded (1, 0, (1) regressors.

(d) Analyze the Moore and Krupat data using one or more computer programs available to you. How do the programs calculate sums of squares in two-way ANOVA? Does the documentation accompanying the programs clearly explain how the sums of squares are computed?