Assume that for each p, q, and r, is Normally distributed with mean 0 and variance .
If a factor such as A is fixed, assume that the populations of measurements at each level of A are Normally distributed with mean and variance , and that all the populations at each of the levels have the same variance. In addition, assume that sum to zero. Analogous assumptions are made for B.
If a factor such as A is random, assume that the effect of the level of A itself, , is a random variable Normally distributed with mean 0 and variance . Analogous assumptions are made for B.
If all of the factors such as A and B associated with the effect of an interaction are fixed, assume that the populations of measurements at each level are Normally distributed with mean and variance . For any fixed p, sum to zero, when summing over all q. Similarly, for any fixed q, the means sum to zero, when summing over all p.
If any of the factors such as A and B associated with the effect of an interaction are random, assume the effect is a random variable Normally distributed with mean 0 and variance . If A is fixed but B is random, then also assume that for any fixed q the means sum to zero, when summing over all p. Similarly, if B is fixed but A is random, assume that for any fixed p the means sum to zero, when summing over all q.
Assume that all effects taken to be random variables are mutually independent.