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A Note on Weaker Conditions for Identifying Restricted Latent Class Models for Binary Responses

Published online by Cambridge University Press:  01 January 2025

Steven Andrew Culpepper Open the ORCID record for Steven Andrew Culpepper [Opens in a new window]
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, 605 E Springfield Ave, Champaign, IL 61820, USA. Email: sculpepp@illinois.edu
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Abstract

Restricted latent class models (RLCMs) are an important class of methods that provide researchers and practitioners in the educational, psychological, and behavioral sciences with fine-grained diagnostic information to guide interventions. Recent research established sufficient conditions for identifying RLCM parameters. A current challenge that limits widespread application of RLCMs is that existing identifiability conditions may be too restrictive for some practical settings. In this paper we establish a weaker condition for identifying RLCM parameters for multivariate binary data. Although the new results weaken identifiability conditions for general RLCMs, the new results do not relax existing necessary and sufficient conditions for the simpler DINA/DINO models. Theoretically, we introduce a new form of latent structure completeness, referred to as dyad-completeness, and prove identification by applying Kruskal’s Theorem for the uniqueness of three-way arrays. The new condition is more likely satisfied in applied research, and the results provide researchers and test-developers with guidance for designing diagnostic instruments.

Keywords

cognitive diagnosisrestricted latent class modelsidentifiabilityQ-matrix

Information

Type
Theory and Methods
Information
Psychometrika , Volume 88 , Issue 1 , March 2023 , pp. 158 - 174
Copyright
Copyright © 2022 The Author(s) under exclusive licence to The Psychometric Society

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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 10.1214/09-AOS689CrossRefGoogle Scholar
Balamuta, J. J., & Culpepper, S. A. (2021). Exploratory restricted latent class models with monotonicity requirements under pólya-gamma data augmentation. Psychometrika.Google Scholar
Chen, Y., Culpepper, S. A., (2020). A multivariate probit model for learning trajectories: A fine-grained evaluation of an educational intervention Applied Psychological Measurement 44 7–8 515–530 10.1177/0146621620920928 34565932 7495794CrossRefGoogle ScholarPubMed
Chen, Y., Culpepper, S. A., Chen, Y., Douglas, J., (2018). Bayesian estimation of the DINA Q-matrix Psychometrika 83 89–108 10.1007/s11336-017-9579-4 28861685CrossRefGoogle ScholarPubMed
Chen, Y., Culpepper, S., Liang, F., (2020). A sparse latent class model for cognitive diagnosis Psychometrika 85 121–153 10.1007/s11336-019-09693-2 31927684CrossRefGoogle ScholarPubMed
Chen, Y., Culpepper, S. A., Wang, S., Douglas, J., (2018). A hidden Markov model for learning trajectories in cognitive diagnosis with application to spatial rotation skills Applied Psychological Measurement 42(1) 5–23 10.1177/0146621617721250 29881110CrossRefGoogle ScholarPubMed
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 10.1007/s11336-021-09750-9 33751367CrossRefGoogle 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 10.1080/01621459.2014.934827 26294801CrossRefGoogle Scholar
Chiu, C. Y., Douglas, J. A., Li, X., (2009). Cluster analysis for cognitive diagnosis: Theory and applications Psychometrika 74(4) 633–665 10.1007/s11336-009-9125-0CrossRefGoogle Scholar
Culpepper, S. A., (2015). Bayesian estimation of the DINA model with Gibbs sampling Journal of Educational and Behavioral Statistics 40(5) 454–476 10.3102/1076998615595403CrossRefGoogle Scholar
Culpepper, S. A., (2019). Estimating the cognitive diagnosis Q matrix with expert knowledge: Application to the fraction-subtraction dataset Psychometrika 84 333–357 10.1007/s11336-018-9643-8 30456748CrossRefGoogle Scholar
Culpepper, S. A., (2019). An exploratory diagnostic model for ordinal responses with binary attributes: Identifiability and estimation Psychometrika 84(4) 921–940 10.1007/s11336-019-09683-4 31432312CrossRefGoogle ScholarPubMed
de la Torre, J., (2009). DINA model and parameter estimation: A didactic Journal of Educational and Behavioral Statistics 34(1) 115–130 10.3102/1076998607309474CrossRefGoogle Scholar
de la Torre, J., (2011). The generalized DINA model framework Psychometrika 76(2) 179–199 10.1007/s11336-011-9207-7CrossRefGoogle Scholar
de la Torre, J., Douglas, J. A., (2004). Higher-order latent trait models for cognitive diagnosis Psychometrika 69(3) 333–353 10.1007/BF02295640CrossRefGoogle Scholar
DeCarlo, L. T., (2011). On the analysis of fraction subtraction data: The DINA model, classification, latent class sizes, and the Q-matrix Applied Psychological Measurement 35(1) 8–26 10.1177/0146621610377081CrossRefGoogle Scholar
Fang, G., Liu, J., Ying, Z., (2019). On the identifiability of diagnostic classification models Psychometrika 84(1) 19–40 10.1007/s11336-018-09658-x 30673967CrossRefGoogle ScholarPubMed
Gu, Y., Xu, G., (2019). The sufficient and necessary condition for the identifiability and estimability of the DINA model Psychometrika 84(2) 468–483 10.1007/s11336-018-9619-8 29728918CrossRefGoogle ScholarPubMed
Gu, Y., Xu, G., (2020). Partial identifiability of restricted latent class models The Annals of Statistics 48(4) 2082–2107 10.1214/19-AOS1878CrossRefGoogle Scholar
Gu, Y., Xu, G., (2021). Sufficient and necessary conditions for the identifiability of the Q-matrix Statistica Sinica 31 449–472Google Scholar
Haertel, E. H., (1989). Using restricted latent class models to map the skill structure of achievement items Journal of Educational Measurement 26(4) 301–321 10.1111/j.1745-3984.1989.tb00336.xCrossRefGoogle Scholar
Hartz, S. (2002). A Bayesian framework for the unified model for assessing cognitive abilities: Blending theory with practicality (Unpublished doctoral dissertation). University of Illinois at Urbana-Champaign.Google Scholar
Henson, R. A., Templin, J. L., Willse, J. T., (2009). Defining a family of cognitive diagnosis models using log-linear models with latent variables Psychometrika 74(2) 191–210 10.1007/s11336-008-9089-5CrossRefGoogle Scholar
Junker, B. W., Sijtsma, K., (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory Applied Psychological Measurement 25(3) 258–272 10.1177/01466210122032064CrossRefGoogle Scholar
Köhn, H. F., Chiu, C. Y., (2016). A proof of the duality of the DINA model and the DINO model Journal of Classification 33(2) 171–184 10.1007/s00357-016-9202-xCrossRefGoogle 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 10.1007/BF02293554CrossRefGoogle Scholar
Kruskal, J. B., (1977). Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics Linear Algebra and its Applications 18(2) 95–138 10.1016/0024-3795(77)90069-6CrossRefGoogle Scholar
Liu, J., Xu, G., Ying, Z., (2012). Data-driven learning of Q-matrix Applied Psychological Measurement 36(7) 548–564 10.1177/0146621612456591 23926363CrossRefGoogle ScholarPubMed
Madison, M. J., Bradshaw, L. P., (2018). Assessing growth in a diagnostic classification model framework Psychometrika 83 963–990 10.1007/s11336-018-9638-5 30264183CrossRefGoogle Scholar
Masiero, B., Nascimento, V. H., (2017). Revisiting the Kronecker array transform IEEE Signal Processing Letters 24(5) 525–529 10.1109/LSP.2017.2674969CrossRefGoogle Scholar
Shute, V. J., Hansen, E. G., Almond, R. G., (2008). You can’t fatten a hog by weighing it-or can you? Evaluating an assessment for learning system called ACED International Journal of Artificial Intelligence in Education 18(4) 289–316CrossRefGoogle Scholar
Sorrel, M. A., Olea, J., Abad, F. J., de la Torre, J., Aguado, D., Lievens, F., (2016). Validity and reliability of situational judgement test scores: A new approach based on cognitive diagnosis models Organizational Research Methods 19(3) 506–532 10.1177/1094428116630065CrossRefGoogle Scholar
Templin, J. L., Henson, R. A., (2006). Measurement of psychological disorders using cognitive diagnosis models Psychological Methods 11(3) 287 10.1037/1082-989X.11.3.287 16953706CrossRefGoogle ScholarPubMed
von Davier, M., (2008). A general diagnostic model applied to language testing data British Journal of Mathematical and Statistical Psychology 61(2) 287–307 10.1348/000711007X193957CrossRefGoogle ScholarPubMed
Wang, S., Yang, Y., Culpepper, S. A., Douglas, J., (2017). Tracking skill acquisition with cognitive diagnosis models: A higher-order hidden Markov model with covariates Journal of Educational and Behavioral Statistics 43(1) 57–87 10.3102/1076998617719727CrossRefGoogle Scholar
Xu, G., (2017). Identifiability of restricted latent class models with binary responses Annals of Statistics 45(2) 675–707 10.1214/16-AOS1464CrossRefGoogle Scholar
Xu, G., Shang, Z., (2018). Identifying latent structures in restricted latent class models Journal of the American Statistical Association 113(523) 1284–1295 10.1080/01621459.2017.1340889CrossRefGoogle Scholar