Exploratory latent structure analysis using both identifiable and unidentifiable models

TLDR

The paper surveys a broad class of latent structure models that explain relationships among m polytomous manifest variables, encompassing both identifiable and unidentifiable parameterizations. It presents a straightforward procedure for computing maximum‑likelihood frequencies in the m‑way contingency table, assessing parameter identifiability, testing model fit, and substituting unidentifiable models with identifiable alternatives. Illustrative data applications demonstrate the utility of the proposed methods.

Abstract

This paper considers a wide class of latent structure models. These models can serve as possible explanations of the observed relationships among a set of m manifest polytomous variables. The class of models considered here includes both models in which the parameters are identifiable and also models in which the parameters are not. For each of the models considered here, a relatively simple method is presented for calculating the maximum likelihood estimate of the frequencies in the m-way contingency table expected under the model, and for determining whether the parameters in the estimated model are identifiable. In addition, methods are presented for testing whether the model fits the observed data, and for replacing unidentifiable models that fit by identifiable models that fit. Some illustrative applications to data are also included.