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A Sparse Latent Class Model for Cognitive Diagnosis

Published online by Cambridge University Press:  01 January 2025

Yinyin Chen ,
Steven Culpepper Open the ORCID record for Steven Culpepper [Opens in a new window]  and
Feng Liang
Yinyin Chen
Affiliation:
University of Illinois at Urbana–Champaign
Steven Culpepper*
Affiliation:
University of Illinois at Urbana–Champaign
Feng Liang
Affiliation:
University of Illinois at Urbana–Champaign
*
Correspondence should be made to Steven Culpepper, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana–Champaign, 725 South Wright Street, Champaign, IL 61820, USA. Email: sculpepp@illinois.edu
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Abstract

Cognitive diagnostic models (CDMs) are latent variable models developed to infer latent skills, knowledge, or personalities that underlie responses to educational, psychological, and social science tests and measures. Recent research focused on theory and methods for using sparse latent class models (SLCMs) in an exploratory fashion to infer the latent processes and structure underlying responses. We report new theoretical results about sufficient conditions for generic identifiability of SLCM parameters. An important contribution for practice is that our new generic identifiability conditions are more likely to be satisfied in empirical applications than existing conditions that ensure strict identifiability. Learning the underlying latent structure can be formulated as a variable selection problem. We develop a new Bayesian variable selection algorithm that explicitly enforces generic identifiability conditions and monotonicity of item response functions to ensure valid posterior inference. We present Monte Carlo simulation results to support accurate inferences and discuss the implications of our findings for future SLCM research and educational testing.

Keywords

sparse latent class modelsBayesian variable selectionidentifiability

Information

Type
Theory and Methods
Information
Psychometrika , Volume 85 , Issue 1 , March 2020 , pp. 121 - 153
Copyright
Copyright © 2020 The Psychometric Society

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References

Albert, J.H., & Chib, S.,(1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88(422) 669–679CrossRefGoogle Scholar
Allman, E.S.,Matias, C.,&Rhodes, J.A., (2009). Identifiability of parameters in latent structure models with many observed variables. The Annals of Statistics, 37, 3099–3132CrossRefGoogle Scholar
Carreira-Perpiñán, M.,Renals, S., (2000). Practical identifiability of finite mixtures of multivariate Bernoulli distributions. Neural Computation, 12, 141–152CrossRefGoogle ScholarPubMed
Chen, Y.,Culpepper, S.A.,Chen, Y.,Douglas, J.,(2018). Bayesian estimation of the DINA Q-matrix. Psychometrika, 83, 89–108CrossRefGoogle 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), 850–866CrossRefGoogle Scholar
Chiu, C-Y.,Douglas, J.A.,Li, X.,(2009). Cluster analysis for cognitive diagnosis: Theory and applications. Psychometrika, 74(4), 633–665CrossRefGoogle Scholar
Cox, D.,Little, J.,O’Shea, D.,Sweedler, M.,(1994). Ideals, varieties, and algorithms. American Mathematical Monthly, 101(6), 582–586Google Scholar
Culpepper, S.A.,(2019). Estimating the cognitive diagnosis Q matrix with expert knowledge: Application to the fraction-subtraction dataset. Psychometrika, 84, 333–357CrossRefGoogle Scholar
Dang, N. .V., (2015). Complex powers of analytic functions and meromorphic renormalization in QFT. arXiv preprint arXiv:1503.00995.Google Scholar
Davier, M., (2005). A general diagnostic model applied to language testing data. ETS Research Report Series, 2005(2)CrossRefGoogle Scholar
de la Torre, J.,(2011). The generalized DINA model framework. Psychometrika, 76(2), 179–199CrossRefGoogle Scholar
de la Torre, J.,Douglas, J.A.,(2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69(3), 333–353CrossRefGoogle Scholar
DiBello, L.V.,Stout, W.F.,Roussos, L.A., Nichols, P.D.,Chipman, S.F.,Brennan, R.L.,(1995). Unified cognitive/psychometric diagnostic assessment likelihood-based classification techniques. Cognitively Diagnostic Assessment, Chapter 15, New York:Routledge 361–389Google Scholar
Fang, G.,Liu, J.,Ying, Z.,(2019). On the identifiability of diagnostic classification models. Psychometrika, 84(1), 19–40CrossRefGoogle ScholarPubMed
Goodman, L.A.,(1974). Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika, 61(2), 215–231CrossRefGoogle Scholar
Gyllenberg, M.,Koski, T.,Reilink, E.,Verlaan, M.,(1994). Non-uniqueness in probabilistic numerical identification of bacteria. Journal of Applied Probability, 31(2), 542–548CrossRefGoogle Scholar
Gyllenberg, M.,Koski, T.,Reilink, E.,Verlaan, M.,(1994). Non-uniqueness in probabilistic numerical identification of bacteria. Journal of Applied Probability, 31(2), 542–548CrossRefGoogle Scholar
Haertel, E.H.,(1989). Using restricted latent class models to map the skill structure of achievement items. Journal of Educational Measurement, 26(4), 301–321CrossRefGoogle Scholar
Hagenaars, J.A., Loglinear models with latent variables 1993 Newbury Park, CA:Sage Publications Inc.CrossRefGoogle Scholar
Hartz, S.M., (2002). A Bayesian framework for the unified model for assessing cognitive abilities: Blending theory with practicality. Ph.D. thesis, University of Illinois at Urbana-Champaign.Google 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), 258–272CrossRefGoogle 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), 281–293CrossRefGoogle 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), 95–138CrossRefGoogle Scholar
Liu, J.,Xu, G.,Ying, Z.,(2013). Theory of the self-learning q-matrix. Bernoulli, 19 5A1790–1817CrossRefGoogle ScholarPubMed
Maris, E.,(1999). Estimating multiple classification latent class models. Psychometrika, 64(2), 187–212CrossRefGoogle Scholar
Mityagin, B., (2015). The zero set of a real analytic function. arXiv preprint arXiv:1512.07276.Google Scholar
Rupp, A.A.,Templin, J.,Henson, R.A., Diagnostic measurement: Theory, methods, and applications 2010 New York:Guilford PressGoogle Scholar
Tatsuoka, C.,(2002). Data analytic methods for latent partially ordered classification models. Journal of the Royal Statistical Society, Series C (Applied Statistics), 51(3), 337–350CrossRefGoogle Scholar
Tatsuoka, K. K., (1984). Analysis of errors in fraction addition and subtraction problems. Final report, Technical report.Google Scholar
Teicher, H., etal (1961). Identifiability of mixtures. The Annals of Mathematical Statistics, 32(1), 244–248CrossRefGoogle Scholar
Templin, J.L.,Henson, R.A.,(2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11(3), 287CrossRefGoogle ScholarPubMed
Xu, G.,(2017). Identifiability of restricted latent class models with binary responses. The Annals of Statistics, 45(2), 675–707CrossRefGoogle Scholar
Xu, G.,Shang, Z.,(2017). Identifying latent structures in restricted latent class models. Journal of the American Statistical Association, 113, 1284–1295CrossRefGoogle Scholar
Yakowitz, S.J.,Spragins, J.D.,(1968). On the identifiability of finite mixtures. The Annals of Mathematical Statistics, 39, 209–214CrossRefGoogle Scholar