The family of polychoric models (PM) considers ordinal data as categorization of latent multivariate normal variables. Such framework is commonly used to study the association between ordinal variables, often leading to the polychoric correlation model (PCM). Moreover, PM subsumes several psychometric models, such as the graded response model (Samejima, 1968; 1997). However, the property of identifiability of PM has not been addressed in the literature. To make the issue more complicated, the normality assumption underlying PM has been challenged recently; researchers have suggested that the latent variables underlying PM could be generalized to elliptical distributions. Two unsolved questions can be posited: (a) Is PM and/or PCM with latent elliptical distributions identifiable? (b) If not, can we find the identifiability constraints of it?
In this research, we investigate the identifiability issue of PM and PCM with latent elliptical distributions by generalizing Rodriguez and Mouchart’s (2003) argument. We first prove the identification of PCM based on the copula representation. We then proceed to find the set of identifiability constraints of PM through the equivalence-classes approach introduced by Tsai (2000). Our results show that the PM of Likert scales (LS) can be identified if we set the first cut-off of items to be zero and the final cut-off of items to be one. Future research should investigate how to equate the scales constructed by LS and by comparative judgment items.