3D ANOVA Assumptions

The 3D ANOVA VI makes the following assumptions:

• Assume that for each p, q, and r, is Normally distributed with mean 0 and variance .
• If a factor, for instance, A, is fixed, assume 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 and C.
• If a factor, for instance, A, is random, assume 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 and C.
• If some of the factors, for instance, A and B, associated with the effect of an interaction are fixed, then assume that the populations of measurements at each level of A and B are Normally distributed with mean
  
  
  
  and variance . For any fixed p, the means sum to zero when summing over all q. Similarly, for any fixed q, sum to zero when summing over all p.
• If any of the factors, for instance, 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, 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 all effects taken to be random variables are mutually independent.