  The paper considers general multiplicative models for complete and incomplete
contingency tables that generalize log-linear and several other models and are
entirely coordinate free. Sufficient conditions of the existence of maximum
likelihood estimates under these models are given, and it is shown that the
usual equivalence between multinomial and Poisson likelihoods holds if and only
if an overall effect is present in the model. If such an effect is not assumed,
the model becomes a curved exponential family and a related mixed
parameterization is given that relies on non-homogeneous odds ratios. Several
examples are presented to illustrate the properties and use of such models.
