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Abstract The earlier literature on three-factor interactions in a three-way contingency table was concerned mainly with the null hypothesis H 0 that these interactions are all nil. In the present article, we present methods for testing H 0 which are easier to apply than those suggested by earlier writers on this subject, and in addition we develop methods for obtaining simultaneous confidence intervals for estimating the magnitudes of the three-factor interactions in the three-way contingency table, i.e., in the I × J × K contingency table. We also show how to partition a chi-square statistic for testing H 0 in a 2 × 2 × K table into K − 1 asymptotically independent components that can be used to test K − 1 different sub-hypotheses into which the hypothesis H 0 can be partitioned, and we introduce a rough and ready approximate method for testing H 0 in a 2 × J × K table which is much simpler than any other valid method presented heretofore for testing H 0.
Leo A. Goodman (1964) studied this question.