A unified treatment is given of the classical additive models for complete factorial experiments and of multiplicative models and Lancaster-additive models for multi-dimensional contingency tables. The models are characterised by properties of being simplest subject to having a prescribed set of marginals. It is shown that, by using averaging operators and the notion of a generalised interaction, the interaction properties of these models can be derived very simply.
"Additive and Multiplicative Models and Interactions." Ann. Statist. 11 (3) 724 - 738, September, 1983. https://doi.org/10.1214/aos/1176346240