A Discussion of the Two-Way Mixed Model

In his text on the analysis of variance, Sheffe (1959) presents a detailed development of the mixed model for two factors. The model, as initially described, appears to be quite general and is capable of accomodating a wide variety of experimental situations in which one factor is assumed fixed and the other random. Other authors have proposed different models which appear to differ from Scheff6's model in various ways. These models have been the subject of considerable discussion, with much of the controversy focusing on the expressions for the expected value of the mean square associated with the random factor. The purpose of this paper is to relate some of the existing models to the Scheffe model in an attempt to provide the user with some guidance in the choice of an appropriate model. The question of the correct expressions for the expected mean squares is also resolved. Although primary consideration is given to the relations between contending, infinite population models, one relation to the limiting form of the finite model is also noted. For ease of reference, the models are formally stated in Section 2. These statements also contain the basic relations between the models. These relations are then discussed in Section 3, and in Section 4, the correct expressions for the expected means squares are given.