In the log-linear parametrization all the interactions are contrasts of logarithms of joint probabilities and this is the main reason why this parametrization is not convenient to express hypotheses on marginal distributions or to model ordered categorical data. On the contrary Hierarchical Multinomial Marginal models (HMM) (Bartolucci et al. 2007) are based on parameters, called generalized marginal interactions, which are contrasts of logarithms of sums of probabilities. HMM models allow great flexibility in choosing the marginal distributions, within which the interactions are defined, and they are a useful tool for modeling marginal distributions and for taking into proper account the presence of ordinal categorical variables. … Hierarchical Multinomial Marginal Models (HMM)