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On the Proper Interpretation of Residualized Interaction Means in Analysis of Variance: A Reply to Rosnow and Rosenthal

According to Rosnow and Rosenthal the correct interpretation of an interaction in an analysis of variance necessarily involves examining the residual means after removing the lower order effects from the original cell means. They make two errors in their analysis of the residual means. First, they incorrectly interpret the negative residual means as literal negative values. The signs of the residual means actually indicate relative differences between the residual means. Second, since the residual interaction means of a 2 × 2 interaction always form a symmetric, crossed interaction pattern, their interpretation incorrectly suggests that all 2 × 2 interactions are symmetrically crossed. That is not the case. The crossed interaction for the residual means indicates that the change within one level of a factor is relatively less than the change within the other level of that factor. Therefore, the form of the underlying interaction cannot be inferred from the residual interaction means alone. Contrary to Rosnow and Rosenthal's suggestion, the residual interaction means have limited usefulness in interpreting interactions.