Interactions, partial interactions, and interaction contrasts in the analysis of variance.

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Abstract
A two-step framework for the interpretation of significant two-treatment inter- actions is proposed. First, a contrast between the means of one treatment is estimated separately at each level of the second treatment. A partial interaction test tests the hypothesis that the variabiliy among these contrasts is zero. The second step consists of computing a difference between the separately estimated contrasts. An interaction contrast test tests the hypothesis that this difference is zero. The familywise Type I error rate can be controlled at alpha by employ- ing Gabriel's simultaneous test procedure for partial interaction tests and Scheffe's method for interaction contrast tests. Recently, an issue has arisen concerning a posteriori tests after detection of a significant interaction in the analysis of variance. Mara- scuilo and Levin (1970) and Levin and Mara- scuilo (1972) have criticized the use of simple effects tests in which the means within a single row or column of the data matrix are compared. Their basic criticism is that the null hypotheses tested by simple effects tests are not coherent with the null hypothesis tested by the a priori omnibus interaction test. To maintain a coherent analysis, Marascuilo and Levin recommended testing interaction contrasts after detection of a significant interaction. In commenting on the approach of Mara- scuilo and Levin (1970), Games (1973) argued that although interaction contrasts are co- herent with the omnibus interaction test, these contrasts do not easily lend themselves to meaningful behavioral interpretations. Con- sequently, Games recommended performing simple effects tests. Levin and Marascuilo (1973) replied by emphasizing the flexibility of the interaction contrast approach. Thus far, there is little evidence that the issue has been

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