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Comparing Latent Structures of the Grade of Membership, Rasch, and Latent Class Models

Published online by Cambridge University Press:  01 January 2025

Elena A. Erosheva
Elena A. Erosheva*
Affiliation:
University of Washington
*
Requests for reprints should be sent to Elena A. Erosheva, University of Washington, Box 354322, Seattle, WA 98195-4322, USA. E-Mail: elena@stat.washington.edu
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Abstract

This paper focuses on model interpretation issues and employs a geometric approach to compare the potential value of using the Grade of Membership (GoM) model in representing population heterogeneity. We consider population heterogeneity manifolds generated by letting subject specific parameters vary over their natural range, while keeping other population parameters fixed, in the marginal space (based on marginal probabilities) and in the full parameter space (based on cell probabilities). The case of a 2 × 2 contingency table is discussed in detail, and a generalization to 2J tables with J ≥ 3 is sketched. Our approach highlights the main distinction between the GoM model and the probabilistic mixture of classes by demonstrating geometrically the difference between the concepts of partial and probabilistic memberships. By using the geometric approach we show that, in special cases, the GoM model can be thought of as being similar to an item response theory (IRT) model in representing population heterogeneity. Finally, we show that the GoM item parameters can provide quantities analogous to more general logistic IRT item parameters. As a latent structure model, the GoM model might be considered a useful alternative for a data analysis when both classes of extreme responses, and additional heterogeneity that cannot be captured by those latent classes, are expected in the population.

Keywords

2×2 contingency tablefuzzy sets modelgeometric approachheterogeneitylocal independence
Type
Theory and Methods
Information
Psychometrika , Volume 70 , Issue 4 , December 2005 , pp. 619 - 628
Copyright
Copyright © 2005 The Psychometric Society

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Footnotes

This work was supported by Award #1R03 AG18986-01 from the National Institute on Aging and NIH grant #1R01 CA94212-01. The presentation of the ideas in this paper owes much to discussions with Stephen Fienberg and Brian Junker, Carnegie Mellon University. The author thanks Jim Ramsay and two anonymous reviewers for their valuable comments on earlier drafts of this paper.

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