Ordered alternatives for interaction effects

The problem of testing hypotheses against ordered alternatives for treatment effects in a one-way layout is discussed by many authors; see, for example, Bartholomew (1959a, b; 1961). For interaction effects in a two-way layout, it is required to specify the ordered alternatives appropriate to practical situations and then to establish the test procedures. James (1961) defined an ordered alternative for a simple case. In the present paper several types of ordered alternatives are defined for the case of the two-factor interaction with one factor A indicating the treatments and the other factor B expressing the stages or degrees of the effects of A. For these ordered alternatives, appropriate methods for testing the null hypothesis are proposed. The powers of these test procedures are examined numerically for the seven different patterns of the alternatives. One real example from a drug experiment is shown as an illustration.