  The extended class of multiplicative row-column (RC) association models,
introduced in this paper for two-way contingency tables, allows users to select
both the type of logit (local, global, continuation, reverse continuation)
suitable for the row and column classification variables and the scale on which
interactions are measured. As in \cite{Kateri95} for the case of local logits,
our extended class of bivariate interactions is linked to divergence measures
and, by means of a representation theorem, we provide reconstruction formulas
for the joint probabilities depending on pairs of logit types. These results
are the key to show that, given marginal logits, our extended interactions
determine uniquely the bivariate distribution. We also determine the kind of
positive association which is implied by our extended interactions being non
negative. Quick model selection within this wide class can be performed by an
efficient algorithm for computing maximum likelihood estimates which exploits
the properties of a reduced rank constraint imposed on the matrix of extended
interactions and allows for additional linear constraint on marginal logits. An
application to social mobility data is presented and discussed.
