On the Problem of Interactions in the Analysis of Variance

Abstract

For the two-way analysis of variance without the assumption of zero interactions, consider the following recommendation: Test the zero-interactions hypothesis and, if the hypothesis is not rejected, proceed as if the interactions were zero. This recommendation is unwarranted and may give incorrect decisions with a large probability, as has been pointed out in several articles beginning in the year 1935. Despite such isolated criticism, it is a recommendation suggested in most applied books in the field and used in countless applications.

An apology for the recommendation is that it can be easily improved by considering the power of the test and thus obtaining information on the interactions neglected. We consider this improved recommendation for the goals of (i) an interval estimate of one of the cell expectations, (ii) a simultaneous interval estimate of the cell expectations, and (iii) an estimate of the cell with the largest expectation. We show that the improved recommendation leads to methods inferior to methods based on the replacement of the two-way model by the one-way model in cases (ii) and (iii) and to a method not better in case (i). This refutes the apology for the original unwarranted recommendation: the improved recommendation is not easy to apply and does not always lead to good methods.

Key Words:

• Cell expectations
• Interval estimates
• Statistical tests
• Zero interactions