Exploring N-way tables with sums-of-products models

Sums-of-products models can be characterized as a variant of the analysis of variance model, where interaction terms are assumed to be products of single-factor scales and where the assumption that main effects and interaction terms have zero sums is dropped. Because of the ordinal structure inherent in sums of products, these models are appropriate when data satisfy single-factor independence (the direction of the effects of a factor is independent of other factors) but have amplificatory violations of joint independence (the direction of the joint effects of a set of factors may depend on other factors, because a given factor amplifies the effects of some factors more than the effects of others). This paper describes data analysis methods for sums-of-products models that determine which main effect and interaction terms exist. The methods are exploratory in that no error theory is provided. A key concept is that of multi-factor residual with respect to a set of factors, computed by successively "taking out the means" of these factors. The methods involve analysis of multi-factor residuals in terms of their sums of squares and their N -way matrix ranks.