In the two-way layout of Section 7.5 with a = b = 2, denote the first three terms in the partition of 

Xijk − Xi j·

2 by S2 A, S2 B, and S2 AB, corresponding to the A, B, and AB effects (i.e., the α’s, β’s, and γ ’s), and denote by HA, HB, and HAB the hypotheses of these effects being zero. Define a new two-level factor B which is at level 1 when A and B are both at level 1 or both at level 2, and which is at level 2 when A and B are at different levels. Then HB = HAB, SB = SAB, HAB = HB, SAB = SB, so that the B-effect has become an interaction, and the AB-interaction the effect of the factor B

. [Shaffer (1977b).]