Adjusted means (continued): The notion of an ‘‘adjusted’’ mean was introduced in Exercises 7.2 and 7.3. Consider the main-effects model for the p-way classification: (a) Show that if we constrain each...

Adjusted means (continued): The notion of an ‘‘adjusted’’ mean was introduced in Exercises 7.2 and 7.3. Consider the main-effects model for the p-way classification:

(a) Show that if we constrain each set of effects to sum to 0, then the population marginal mean for category j of factor A is µ_j'...'
= µ + α_A(j).

(b) Let us define the analogous sample quantity, Y_j'...'
[ M + A_A(_j), to be the adjusted mean in category j of factor A. How is this quantity to be interpreted?

(c) Does the definition of the adjusted mean in part (b) depend fundamentally on the constraint that each set of effects sums to 0?

(d) Can the idea of an adjusted mean be extended to ANOVA models that include interactions? (Cf. the discussion of effect displays in this and the preceding chapter.)