Families of Generalized Quasisymmetry Models: A φ-Divergence Approach

The quasisymmetry (QS) model for square contingency tables is revisited, highlighting properties and features on the basis of its alternative definitions. More parsimonious QS-type models, such as the ordinal QS model for ordinal classification variables and models based on association models (AMs) with homogeneous row and column scores, are discussed. All these models are linked to the local odds ratios (LOR). QS-type models and AMs were extended in the literature for generalized odds ratios other than LOR. Furthermore, in an information-theoretic context, they are expressed as distance models from a parsimonious reference model (the complete symmetry for QS and the independence for AMs), while they satisfy closeness properties with respect to Kullback-Leibler (KL) divergence. Replacing the KL by phi divergence, flexible classes of QS-type models for LOR, AMs for LOR, and AMs for generalized odds ratios were generated. However, special QS-type models that are based on homogeneous AMs for LOR have not been extended to phi-divergence-based classes so far, or the QS-type models for generalized odds ratios. In this work, we develop these missing extensions, and discuss QS-type models and their generalizations in depth. These flexible families enrich the modeling options, leading to models of better fit and sound interpretation, as illustrated by representative examples.

Automated summary

This paper revisits the quasisymmetry (QS) model for square contingency tables, discussing more parsimonious QS-type models and models linked to local odds ratios (LOR). It also explores the expression of these models in an information-theoretic context and their closeness properties with respect to Kullback-Leibler (KL) divergence. Additionally, the paper introduces flexible classes of QS-type models and AMs based on different types of divergences, enriching the modeling options for better fit and interpretation.