Hostname: page-component-74656f9884-rdgjk Total loading time: 0 Render date: 2026-11-09T04:09:15.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Restricted Latent Class Models for Nominal Response Data: Identifiability and Estimation

Published online by Cambridge University Press:  27 December 2024

Ying Liu  and
Steven Andrew Culpepper Open the ORCID record for Steven Andrew Culpepper [Opens in a new window]
Ying Liu
Affiliation:
University of Illinois at Urbana-Champaign
Steven Andrew Culpepper*
Affiliation:
University of Illinois at Urbana-Champaign
*
Correspondence should be made to Steven Andrew Culpepper, Department of Statistics, University of Illinois at Urbana-Champaign, Computing Applications Building, Room 152, 605 E. Springfield Ave., Champaign, IL61820, USA. Email: sculpepp@illinois.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Restricted latent class models (RLCMs) provide an important framework for diagnosing and classifying respondents on a collection of multivariate binary responses. Recent research made significant advances in theory for establishing identifiability conditions for RLCMs with binary and polytomous response data. Multiclass data, which are unordered nominal response data, are also widely collected in the social sciences and psychometrics via forced-choice inventories and multiple choice tests. We establish new identifiability conditions for parameters of RLCMs for multiclass data and discuss the implications for substantive applications. The new identifiability conditions are applicable to a wealth of RLCMs for polytomous and nominal response data. We propose a Bayesian framework for inferring model parameters, assess parameter recovery in a Monte Carlo simulation study, and present an application of the model to a real dataset.

Keywords

restricted latent class modelsnominal response datacognitive diagnosis modelidentifiabilityBayesian

Information

Type
Theory and Methods
Information
Psychometrika , Volume 89 , Issue 2 , June 2024 , pp. 592 - 625
Copyright
Copyright © 2023 The Author(s), under exclusive licence to The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Allman, E. S., Matias, C., Rhodes, J. A.. (2009). Identifiability of parameters in latent structure models with many observed variables. Annals of Statistics, 37, 3099–3132.CrossRefGoogle Scholar
Bacci, S., Bartolucci, F., Gnaldi, M.. (2014). A class of multidimensional latent class IRT models for ordinal polytomous item responses. Communications in Statistics-Theory and Methods, 43(4), 787–800.CrossRefGoogle Scholar
Balamuta, J. J., & Culpepper, S. A. (2022). Exploratory restricted latent class models with monotonicity requirements under Polya–gamma data augmentation. Psychometrika 1–43.CrossRefGoogle Scholar
Bartolucci, F.. (2007). A class of multidimensional IRT models for testing unidimensionality and clustering items. Psychometrika, 72, 141–157.CrossRefGoogle Scholar
Bradshaw, L., Templin, J.. (2014). Combining item response theory and diagnostic classification models: A psychometric model for scaling ability and diagnosing misconceptions. Psychometrika, 79(3), 403–425.CrossRefGoogle ScholarPubMed
Brooks, S. P., Gelman, A.. (1998). General methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics, 7(4), 434–455.CrossRefGoogle Scholar
Chen, J., de la Torre, J.. (2013). A general cognitive diagnosis model for expert-defined polytomous attributes. Applied Psychological Measurement, 37(6), 419–437.CrossRefGoogle Scholar
Chen, J., Zhou, H.. (2017). Test designs and modeling under the general nominal diagnosis model framework. PLoS ONE, 12(6.Google ScholarPubMed
Chen, Y., Culpepper, S., Liang, F.. (2020). A sparse latent class model for cognitive diagnosis. Psychometrika, 85, 121–153.CrossRefGoogle ScholarPubMed
Chen, Y., Liu, J., Xu, G., Ying, Z.. (2015). Statistical analysis of Q-matrix based diagnostic classification models. Journal of the American Statistical Association, 110(510), 850–866.CrossRefGoogle Scholar
Chen, Y., Liu, Y., Culpepper, S. A., Chen, Y.. (2021). Inferring the number of attributes for the exploratory DINA model. Psychometrika, 86(1), 30–64.CrossRefGoogle ScholarPubMed
Chiu, C.-Y., Douglas, J., Li, X.. (2009). Cluster analysis for cognitive diagnosis: Theory and applications. Psychometrika, 74, 633–665.CrossRefGoogle Scholar
Culpepper, S. A.. (2019). An exploratory diagnostic model for ordinal responses with binary attributes: Identifiability and estimation. Psychometrika, 84(4), 921–940.CrossRefGoogle ScholarPubMed
Culpepper, S. A., & Balamuta, J. J. (2021). Inferring latent structure in polytomous data with a higher-order diagnostic model. Multivariate Behavioral Research 1–19.Google Scholar
Dang, N. V. (2015). Complex powers of analytic functions and meromorphic renormalization in QFT. arXiv preprint arXiv:1503.00995 .Google Scholar
De La Torre, J.. (2009). A cognitive diagnosis model for cognitively based multiple-choice options. Applied Psychological Measurement, 33(3), 163–183.CrossRefGoogle Scholar
de la Torre, J.. (2011). The generalized DINA model framework. Psychometrika, 76(2), 179–199.CrossRefGoogle Scholar
DeYoreo, M., Reiter, J. P., Hillygus, D. S.. (2017). Bayesian mixture models with focused clustering for mixed ordinal and nominal data. Bayesian Analysis, 12(3), 679–703.CrossRefGoogle Scholar
DiBello, L. V., Henson, R. A., Stout, W. F.. (2015). A family of generalized diagnostic classification models for multiple choice option-based scoring. Applied Psychological Measurement, 39(1), 62–79.CrossRefGoogle ScholarPubMed
Dunson, D. B., Xing, C.. (2009). Nonparametric Bayes modeling of multivariate categorical data. Journal of the American Statistical Association, 104(487), 1042–1051.CrossRefGoogle Scholar
Fang, G., Liu, J., Ying, Z.. (2019). On the identifiability of diagnostic classification models. Psychometrika, 84(1), 19–40.CrossRefGoogle ScholarPubMed
George, E. I., McCulloch, R. E.. (1993). Variable selection via Gibbs sampling. Journal of the American Statistical Association, 88(423), 881–889.CrossRefGoogle Scholar
Gnaldi, M., Bacci, S., Kunze, T., Greiff, S.. (2020). Students’ complex problem solving profiles. Psychometrika, 85, 469–501.CrossRefGoogle ScholarPubMed
Goodman, L. A.. (1974). Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika, 61(2), 215–231.CrossRefGoogle Scholar
Gu, Y., & Dunson, D. B. (2021). Bayesian pyramids: Identifiable multilayer discrete latent structure models for discrete data. arXiv preprint arXiv:2101.10373 .Google Scholar
Holmes, C. C., Held, L.. (2006). Bayesian auxiliary variable models for binary and multinomial regression. Bayesian Analysis, 1(1), 145–168.Google Scholar
Huang, G.-H., Bandeen-Roche, K.. (2004). Building an identifiable latent class model with covariate effects on underlying and measured variables. Psychometrika, 69(1), 5–32.CrossRefGoogle Scholar
Jiang, Z., Templin, J.. (2019). Gibbs samplers for logistic item response models via the Pólya-Gamma distribution: A computationally efficient data-augmentation strategy. Psychometrika, 84(2), 358–374.CrossRefGoogle ScholarPubMed
Jimenez, A., Balamuta, J. J., & Culpepper, S. A. (2023). A sequential exploratory diagnostic model using a Pólya-gamma data augmentation strategy. British Journal of Mathematical and Statistical Psychology.CrossRefGoogle Scholar
Kruskal, J.. (1977). Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Algebra and its Applications, 18, 95–138.CrossRefGoogle Scholar
Kruskal, J. B.. (1976). More factors than subjects, tests and treatments: An indeterminacy theorem for canonical decomposition and individual differences scaling. Psychometrika, 41(3), 281–293.CrossRefGoogle Scholar
Kuo, B.-C., Chen, C.-H., Yang, C.-W., Mok, M. M. C.. (2016). Cognitive diagnostic models for tests with multiple-choice and constructed-response items. Educational Psychology, 36(6), 1115–1133.CrossRefGoogle Scholar
Ma, W., de la Torre, J.. (2016). A sequential cognitive diagnosis model for polytomous responses. British Journal of Mathematical and Statistical Psychology, 69(3), 253–275.CrossRefGoogle ScholarPubMed
Ma, W., de la Torre, J.. (2019). Category-level model selection for the sequential G-DINA model. Journal of Educational and Behavioral Statistics, 44(1), 45–77.CrossRefGoogle Scholar
MacQueen, J. (1967). Some methods for classification and analysis of multivariate observations. In Proceedings of the fifth berkeley symposium on mathematical statistics and probability (Vol. 1, pp. 281–297).Google Scholar
Matsaglia, G., Styan, G. P. H.. (1974). Equalities and inequalities for ranks of matrices. Linear and Multilinear Algebra, 2(3), 269–292.CrossRefGoogle Scholar
McHugh, R. B.. (1956). Efficient estimation and local identification in latent class analysis. Psychometrika, 21(4), 331–347.CrossRefGoogle Scholar
Mityagin, B. S.. (2020). The zero set of a real analytic function. Mathematical Notes, 107, 529–530.CrossRefGoogle Scholar
Murray, J. S., Reiter, J. P.. (2016). Multiple imputation of missing categorical and continuous values via Bayesian mixture models with local dependence. Journal of the American Statistical Association, 111(516), 1466–1479.CrossRefGoogle Scholar
Polson, N. G., Scott, J. G., Windle, J.. (2013). Bayesian inference for logistic models using Pólya-gamma latent variables. Journal of the American Statistical Association, 108(504), 1339–1349.CrossRefGoogle Scholar
Royle, J. A., Link, W. A.. (2005). A general class of multinomial mixture models for anuran calling survey data. Ecology, 86(9), 2505–2512.CrossRefGoogle Scholar
Shear, B. R., & Roussos, L. A. (2017). Validating a distractor-driven geometry test using a generalized diagnostic classification model. In Understanding and investigating response processes in validation research (pp. 277–304). Springer.CrossRefGoogle Scholar
Si, Y., Reiter, J. P.. (2013). Nonparametric Bayesian multiple imputation for incomplete categorical variables in large-scale assessment surveys. Journal of Educational and Behavioral Statistics, 38(5), 499–521.CrossRefGoogle Scholar
Templin, J., Henson, R. A., Rupp, E., Jang, A. A., & Ahmed, M. (2008). Cognitive diagnosis models for nominal response data. New York, NY: In Annual Meeting of the National Council on Measurement in Education.Google Scholar
Vermunt, J. K., Van Ginkel, J. R., Van der Ark, L. A., Sijtsma, K.. (2008). Multiple imputation of incomplete categorical data using latent class analysis. Sociological Methodology, 38(1), 369–397.CrossRefGoogle Scholar
Wagner, R. F., Wells, K. A.. (1985). A refined neurobehavioral inventory of hemispheric preference. Journal of Clinical Psychology, 41(5), 671–676.3.0.CO;2-1>CrossRefGoogle ScholarPubMed
Xu, G.. (2017). Identifiability of restricted latent class models with binary responses. Annals of Statistics, 45(2), 675–707.CrossRefGoogle Scholar
Xu, G., Shang, Z.. (2018). Identifying latent structures in restricted latent class models. Journal of the American Statistical Association, 113(523), 1284–1295.CrossRefGoogle Scholar
Yigit, H. D., Sorrel, M. A., de la Torre, J.. (2019). Computerized adaptive testing for cognitively based multiple-choice data. Applied Psychological Measurement, 43(5), 388–401.CrossRefGoogle ScholarPubMed