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.
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Darroch et al. (1983) studied this question.