With the alternative formulation of the twofactor ANOVA mode

With the alternative formulation of the two-factor ANOVA model, considering that the main effects and interaction effects require constraints for identifiability, what is the number of free parameters in the factor effect model?

Solution

Zero-sum Constraints ?. = X i ?i = 0 ?. = X j ?j = 0 (??).j = X i (??)i,j = 0 ?j (for all j) (??).i = X i (??)i,j = 0 ?i (for all i) All of these constraints are satisfied by the above estimates. Notice how these main and interaction effects fit together to give back the treatment means: 45 = 51 ? 7 ? 1+2 43 = 51 ? 7+1 ? 2 65 = 51 + 16 ? 1 ? 1 69 = 51 + 16 + 1 + 1 40 = 51 ? 9 ? 1 ? 1 44 = 51 ? 9+1+1 SAS GLM Constraints As usual, SAS has to do its constraints differently. As in one-way ANOVA, it sets the parameter for the last category equal to zero. ?a = 0 (1 constraint) ?b = 0 (1 constraint) (??)a,j = 0 for all j (b constraints) (??)i,b = 0 for all i (a constraints) The total is 1 + 1 + a + b ? 1 = a + b + 1 constraints (the constraint (??)a,b is counted twice above). Parameters and constraints The cell means model has ab parameters for the means. The factor effects model has (1 + a + b + ab) parameters.