Two-way classification designs with unequal cell frequencies

Abstract

In a two-way classification design on two factors, say A and B, we apply each factor on varying levels to various experimental units. We assume that this application yields for each unit a quantity which we call the yield of this unit. We denote by ??(/, /) the mean value of the yield obtained when the factor A is applied at level / and the factor B at level /. These levels may be qualitative or quantitative and could assume discrete or continuous values. Usually they are chosen deterministically by the experimenter. In some cases, however, they are selected randomly according to a probability scheme. Even when the levels vary continuously, the experimenter can calibrate or can group them into a finite number of discrete values. We, therefore, assume that / and /can take the values 1, 2, •••, r and 1, 2, •••, s, respectively. The object of a two-way classification design is to make some inferences on the behavior of the mean yield function τ/(I, /). For such purpose, the function ??(/, /) is usually broken up into a general mean /*, a main effect a(I) of the factor A, a main effect /?(/) of the factor B, and an interaction effect γ(I, /) ascribed to the combination of level I of the factor A with level / of the factor B, i.e.,