Show that if the effects of the factors in a two-way layout are additive, then there exist unique numbers μ, α1, . . . , αI , and β1, . . . , βJ that satisfy Eqs. (11.7.2) and (11.7.3). Let μ be the average of all θi + ψj values, let αI equal θi minus the average of the θi’s, and similarly for βJ .