Identifiability of Polychoric Models with Latent Elliptical Distributions
Journal
Psychometrika
Journal Volume
90
Journal Issue
2
Start Page
757
End Page
778
ISSN
0033-3123
1860-0980
Date Issued
2025-01-03
Author(s)
Abstract
The family of polychoric models (PM) categories ordinal data with latent multivariate normal variables. This modeling framework is commonly used to study the association between ordinal variables, often leading to a polychoric correlation model (PCM). Moreover, PM subsumes several well-known psychometric models, such as the structural equation modeling (SEM) with ordinal data. That said, the identifiability of PM has not been addressed in the literature. Meanwhile, in recent years researchers have suggested that the latent variables underlying PM could be generalized to the family of elliptical distributions, such as the multivariate logistic and t distributions. This article concerns the identifiability of PM and PCM with latent elliptical distributions, for which we show that PM is not identifiable and PCM is identifiable. In particular, we prove the identifiability of the polychoric t correlation model based on the copula representation. We then move on to find the set of identifiability constraints of PM through an equivalence-classes approach of identifiability, and demonstrate its use in two applications: one concerns the identifiability of PM on Likert scales and on comparative judgment, and the other concerns the identifiability of ordinal SEM and item factor analysis. Possible implications induced by these identifiability constraints are discussed.
Subjects
elliptical distributions
identifiability constraints
just-identified models
polychoric correlations
polychoric models
SDGs
Publisher
Cambridge University Press (CUP)
Type
journal article