Abstract
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 ϕ 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 ϕ-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.
Highlights
We focus on the QS model, penetrating its features by considering the alternative equivalent definitions of QS in terms of cell probabilities, local odds ratios (LOR), and as a model measuring departure from the S model
QS is mostly expressed in terms of cell probabilities, while it can alternatively be expressed in terms of local odds ratios Its definition as a departure model from the more parsimonious model of complete symmetry provides additional interpretation features
We considered the ordinal QS (OQS) model, a more parsimonious QS-type model, applicable if the classification scale is ordinal, which imposes a special structure on the main effects of the model
Summary
A special type of square contingency table that occurs often in studies of correlated or repeated categorical measurements, e.g., in panels or social mobility studies, is a table with row and column classification variables measured on the same scale, which can be nominal or ordinal. AMs for generalized odds ratios ([16]) were extended to a broader family through the φ-divergence ([17]) Combining these two families of models discussed in [15,17], we introduce here new flexible classes of QS models for generalized odds ratios that are based on φ-divergence. Aiming at an indepth discussion of its nature and properties, as consolidated by the alternative possible definitions of QS, and consideration of special QS-type models, with an emphasis on QS-type models that are based on homogeneous AMs. Reviewing extensions of QS-type models towards two directions: (a) in an informationtheoretic setup by replacing the role of KL divergence by φ-divergence, and (b) considering them for generalized odds ratios other than LOR.
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