Asymmetric Scaling Models for Square Contingency Tables: Points, Circles, Arrows and Odds Ratios

Abstract

We study two asymmetric scaling methods within the context of loglinear modelling of square contingency tables: the distance-radius model and the slide-vector model. The usual association parameters of a loglinear model are replaced by a distance term. We are specifically interested in whether and how the asymmetry in these methods translates to odds ratio structures, as the latter is a primary measure of association for contingency tables. We define models in terms of distances and squared distances. We show that the distance-radius model with distances and the slide-vector model with squared distances do not represent asymmetry in the odds ratios. Finally, we also study models without main effects, where the distances directly represent the observed frequencies. We show that in that case the distance-radius model perfectly represents marginal heterogeneity.