Minimal sample size in balanced ANOVA models of crossed, nested, and mixed classifications

We consider balanced one-, two- and three-way ANOVA models to test the hypothesis that the fixed factor A has no effect. The other factors are fixed or random. We determine the noncentrality parameter for the exact F-test, describe its minimal value by a sharp lower bound, and thus we can guarantee the worst case power for the F-test. These results allow us to compute the minimal sample size. We also provide a structural result for the minimum sample size, proving a conjecture on the optimal experimental design.