Analysis of Nonadditive Multiway Classifications

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Abstract
This article considers the problems of testing additivity and estimating σ2 in unreplicated multiway classifications. To model nonadditivity and jointly estimate σ2, the interaction parameter space must be restricted; otherwise the model is saturated. The parameterization we use is a multiway extension of the two-way multiplicative interaction model of Mandel (1971) and Johnson and Graybill (1972a). For example, in a three-way classification, we model interaction as θijk = λδ 1i δ2j δ3k . This structure is a special case of the k-mode principal components model, which has received considerable attention in the psychometric literature (Kapteyn, Neudecker, and Wansbeek 1986). We construct an exact test of λ = 0 and propose an estimator of σ2 that can be used when interaction has been detected. Our test is an approximation to the likelihood ratio test (LRT) of Ho : λ = 0. The proposed test has essentially the same power as the LRT but is easier to compute, and the exact null distribution of the test statistic is known. Selected percentiles of the null distribution are given for three-way classifications. For large |λ/σ|, a transformation of the test statistic is shown to be approximately distributed as a noncentral F and can be used to compute the power of the test. The test and estimator are illustrated on a data set having three rows, three columns, and four layers.