Towards an optimum test for non-additivity in Tukey's model

In a two-way classification model with one observation per cell, the existence of an optimum test for non-additivity is investigated under Tukey's model. An expression for the local conditional (given the complete sufficient statistic under the null hypothesis) power of any invariant similar test is provided. It turns out that no optimum test exists in the class of invariant similar tests. It is further shown that there exist two exact F tests, different from Tukey's test, which are locally more powerful than Tukey's test under suitable restrictions on the main effects. Local unbiasedness of Tukey's test as well as the proposed tests is also established.