Two-Way Crossed Classification with Interaction

Suppose that we relax the requirement of model (3.1.1) that there be exactly one observation in each of the a x b cells of the two-way layout. The model remains the same except that we could now use yijk to designate the k-th observation at the i-th level of A and the j-th level of B, that is, in the (i, j)-th cell. We now suppose that there are n (n ≥ 1) observations in each cell. With n = 1, the model (3.1.1) will be a special case of the model being considered here. With an arbitrary integer value of n, the analysis of variance will be a simple extension of that described in Chapter 3. However, an important and somewhat restrictive implication of the simple additive model discussed in Chapter III is that the value of the difference between the mean responses at two levels of A is the same at each level of B. However, in many cases, this simple additive model may not be appropriate. The failure of the differences between the mean responses at the different levels of A to remain constant over the different levels of B is attributed to interaction between the two factors. Having more than one observation per cell allows a researcher to investigate the main effects of both factors and their interaction. In this chapter, we study the model involving two factors with interaction terms.