Summary This paper presents a method of partitioning a χ2 statistic for the I × J × K contingency table (viz. the χ2 statistic that is based upon the likelihood-ratio criterion for testing the null hypothesis that the three variables pertaining to the three-way table are mutually independent) into additive components that can be used to test (1) the null hypothesis of zero three-factor interaction, (2) the null hypothesis that the partial association between two of the variables in the three-way table is zero, and (3) some null hypotheses concerning the two-way marginal distributions. This paper also discusses the estimation of the degree of partial association between two of the variables in the I × J × K table both in the case where the observed row and column marginals in each of the K different I × J tables (which form the I × J × K table) are considered given, and also in the case where these marginals are not fixed. Both the conditional maximum-likelihood estimator (given the observed row and column marginals in the K different I × J tables) and the unconditional maximum-likelihood estimator are presented, and for the special case where I = J = 2 various approximations to these estimators are compared.
Leo A. Goodman (1969) studied this question.