An extended class of RC association models: estimation and main properties

TL;DR

This paper introduces an extended class of RC association models for contingency tables, allowing flexible logit types and interaction scales, with theoretical properties, estimation methods, and an application to social mobility data.

Contribution

It extends RC association models by incorporating various logit types and scales, providing reconstruction formulas, and an efficient estimation algorithm with practical application.

Findings

Extended models determine unique joint distributions given marginals.

Non-negative extended interactions imply positive association.

Efficient maximum likelihood estimation with linear constraints.

Abstract

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.