The earlier literature on three-factor interactions in a three-way contingency table was concerned mainly with the null hypothesisH0that these interactions are all nil. In the present article, we present methods for testingH0which 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 theI×J×Kcontingency table. We also show how to partition a chi-square statistic for testingH0in a 2 × 2 ×Ktable intoK− 1 asymptotically independent components that can be used to testK− 1 different sub-hypotheses into which the hypothesisH0can be partitioned, and we introduce a rough and ready approximate method for testingH0in a 2 ×J×Ktable which is much simpler than any other valid method presented heretofore for testingH0.