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SUMMARY Two new tests are described in connexion with analysis of variance. The first is for the presence of qualitative interaction between a treatment and an intrinsic factor, both with qualitatively defined levels. Such an interaction is said to be present if there are two levels of the treatment factor and two levels of the intrinsic factor such that the treatment difference has opposite signs at the two levels of the intrinsic factor. The second test is for the presence of a treatment effect which can be tested via any of a hierarchical sequence U 1, U 1 + U 2, …, U 1 + … + Ur of sums of squares. An allowance for selection is calculated.
Azzalini et al. (Sun,) studied this question.