Mixture versus loglinear models in contingency table analysis

We study here two different approaches to model a cluster structure in contingency tables. From a classical independence model, we consider an additional pattern structure given by a family of subsets of cells where we allow a specific mean parameter. We encode such structure through mixture models and loglinear models, and we compare the algebraic equations defining these classes of models. We discuss some examples, and we characterize the models in some special cases.