The structural model for a nested design with two random factors is given by the formula
where μ is a constant, the αi, βj(i) and εijk are variables and αi is the effect of the ith level in Factor A and βj(i) is the effect of the jth level in Factor B nested within the ith level of Factor A.
We assume that for all i, j and k, the εijk are pairwise independent and the various random factors are independent of the error terms. We also assume that
We can conclude that for all i, j and k, the xijk are independent and normally distributed with mean
and variance
The estimators of μ, αi, βj(i) and εijk are as for fixed factor nested models, as are the values for SSA, SSB(A), etc. This time we have
The null and alternative hypotheses are as follows. There is no interaction between factors A and B.
| Test Desired | Null H0 | Alt Hyp H1 | Statistical Test |
| Effect of factor A | αi = 0 for all i | αi ≠ 0 for some i | MSA/MSB(A) ∼ F(dfB,dfB(A)) |
| Effect of B(A) | βj = 0 for all j | βj ≠ 0 for some j | MSB(A)/MSE ∼ F(dfB(A),dfE) |
Unbiased estimators of 
Charles,
For the expected mean of the two random nested model, I would think that we will need to have µ+αi+βj(i) instead of µ as the factors A and B are random. Please confirm.
Thanks,
-Sun
Hello Sun,
Are you referring to x-bar = E[xijk] = µ? Recall that the mean of the αi is zero and the mean of the βj(i) is also zero.
Charles
Charles,
For the hypotheses we are testing for the random factors, should their variances be tested for the effect of each factor testing?
-Sun
Sun,
If I understand your question correctly, then, yes, you need to test equality of variances for each group.
Charles
Sun,
Are you referring to homogeneity of variances?
Charles