As we have demonstrated, determining the appropriate test statistics in the analysis of variance method depends on finding the expected mean squares. In complex design situations and in the presence of random or mixed models it is tedious to apply the expectation operator. Therefore, it would be helpful to have a formal procedure by which we could derive the expected mean squares for the different terms in the model. Page 523 has listed a set of rules which works for any set of balanced models to derive the expected mean squares. These rules are consistent with the restricted mixed model and can be modified to incorporate the unrestricted model assumptions, as well.
It is worth mentioning that the test statistic is a ratio of two mean squares where the expected value of the numerator mean square differs from the expected value of the denominator mean square by the variance component or the fixed factor in which we are interested. Therefore, under the assumption of the null hypothesis, both the numerator and the denominator of the F ratio have the same EMS.