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Exploratory Restricted Latent Class Models with Monotonicity Requirements under Pòlya—gamma Data Augmentation

Published online by Cambridge University Press:  01 January 2025

James Joseph Balamuta Open the ORCID record for James Joseph Balamuta [Opens in a new window]  and
Steven Andrew Culpepper
James Joseph Balamuta*
Affiliation:
University of Illinois Urbana-Champaign
Steven Andrew Culpepper
Affiliation:
University of Illinois Urbana-Champaign
*
Correspondence should be made to James Joseph Balamuta, Departments of Informatics and Statistics, University of Illinois Urbana-Champaign, 725 South Wright Street, Champaign, IL61820, USA. Email: balamut2@illinois.edu
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Abstract

Restricted latent class models (RLCMs) provide an important framework for supporting diagnostic research in education and psychology. Recent research proposed fully exploratory methods for inferring the latent structure. However, prior research is limited by the use of restrictive monotonicity condition or prior formulations that are unable to incorporate prior information about the latent structure to validate expert knowledge. We develop new methods that relax existing monotonicity restrictions and provide greater insight about the latent structure. Furthermore, existing Bayesian methods only use a probit link function and we provide a new formulation for using the exploratory RLCM with a logit link function that has an additional advantage of being computationally more efficient for larger sample sizes. We present four new Bayesian formulations that employ different link functions (i.e., the logit using the Pòlya—gamma data augmentation versus the probit) and priors for inducing sparsity in the latent structure. We report Monte Carlo simulation studies to demonstrate accurate parameter recovery. Furthermore, we report results from an application to the Last Series of the Standard Progressive Matrices to illustrate our new methods.

Keywords

restricted latent class modelsBayesianPòlya—gamma data augmentation

Information

Type
Theory & Methods
Information
Psychometrika , Volume 87 , Issue 3 , September 2022 , pp. 903 - 945
Copyright
Copyright © 2021 The Psychometric Society

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References

Chen, Y.,Culpepper, S., Liang, F., (2020). A sparse latent class model for cognitive diagnosis Psychometrika 85 121–153CrossRefGoogle ScholarPubMed
Chen, Y., & Culpepper, S. A. (2020). A multivariate probit model for learning trajectories: A fine-grained evaluation of an educational intervention. Applied Psychological Measurement.CrossRefGoogle Scholar
Chen, Y.,Culpepper, S. A., Chen, Y., & Douglas, J., (2018). Bayesian estimation of the DINA Q Psychometrika 83(1) 89–108CrossRefGoogle 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) 5–23CrossRefGoogle 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
Chen, Y., Liu, Y., Culpepper, S. A. & Chen, Y. (2020). Estimation of K and Q matrix in restricted latent class models. In International Meeting of the Psychometric Society, Virtual.Google Scholar
Chen, Y.,Liu, Y., Culpepper, S. A., & Chen, Y., (2021). Inferring the number of attributes for the exploratory Dina model Psychometrika 10.1007/s11336-021-09750-9CrossRefGoogle ScholarPubMed
Chiu, C. Y.,Douglas, J. A., Li, X., (2009). Cluster analysis for cognitive diagnosis: Theory and applications Psychometrika 74(4) 633–665CrossRefGoogle Scholar
Culpepper, S. A., (2015). Bayesian estimation of the DINA model with Gibbs sampling Journal of Educational and Behavioral Statistics 40(5) 454–476CrossRefGoogle Scholar
Culpepper, S. A., (2019). Estimating the cognitive diagnosis \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec {Q}$$\end{document} matrix with expert knowledge: Application to the fraction—subtraction dataset Psychometrika 84(2) 333–357CrossRefGoogle Scholar
Culpepper, S. A., (2019). An exploratory diagnostic model for ordinal responses with binary attributes: Identifiability and estimation Psychometrika 84(4) 921–940CrossRefGoogle ScholarPubMed
Culpepper, S. A., & Chen, Y., (2019). Development and application of an exploratory reduced reparameterized unified model Journal of Educational and Behavioral Statistics 44(1) 3–2410.3102/1076998618791306CrossRefGoogle Scholar
Culpepper, S. A., & Hudson, A., (2018). An improved strategy for Bayesian estimation of the reduced reparameterized unified model Applied Psychological MeasurementCrossRefGoogle ScholarPubMed
Culpepper, S. A., & Park, T., (2017). Bayesian estimation of multivariate latent regression models: Gauss versus Laplace Journal of Educational and Behavioral Statistics 42(5) 591–616CrossRefGoogle 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
Eddelbuettel, D. (2013). Seamless R and C++ integration with Rcpp. Springer. ISBN 978-1-4614-6867-7. https://10.1007/978-1-4614-6868-4.CrossRefGoogle Scholar
Eddelbuettel, D., & Balamuta, J. J., (2018). Extending R with C++: A brief introduction to Rcpp The American Statistician 72(1) 28–36CrossRefGoogle Scholar
Eddelbuettel, D., & François, R. (2011). Rcpp: Seamless R and C++ integration. Journal of Statistical Software, 40(8), 1—18. https://10.18637/jss.v040.i08.10.18637/jss.v040.i08CrossRefGoogle Scholar
Eddelbuettel, D., & Sanderson, C., (2014). RcppArmadillo: Accelerating R with high-performance C++ linear algebra Computational Statistics and Data Analysis 71 1054–106310.1016/j.csda.2013.02.005CrossRefGoogle Scholar
Fang, G., Liu, J., & Ying, Z. (2019). On the identifiability of diagnostic classification models. Psychometrika.CrossRefGoogle Scholar
Gu, Y., & Xu, G., (2020). Partial identifiability of restricted latent class models Annals of Statistics 48(4) 2082–2107CrossRefGoogle 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) 301–321CrossRefGoogle 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
Heller, J., & Wickelmaier, F., (2013). Minimum discrepancy estimation in probabilistic knowledge structures Electronic Notes in Discrete Mathematics 42 49–56CrossRefGoogle 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) 191–210CrossRefGoogle 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) 358–374CrossRefGoogle ScholarPubMed
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
Macready, G. B., & Dayton, C. M., (1977). The use of probabilistic models in the assessment of mastery Journal of Educational Statistics 2(2) 99–120CrossRefGoogle Scholar
Myszkowski, N., & Storme, M., (2018). A snapshot of g? Binary and polytomous item-response theory investigations of the last series of the standard progressive matrices (SPM-LS) Intelligence 68 109–11610.1016/j.intell.2018.03.010CrossRefGoogle 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) 1339–1349CrossRefGoogle Scholar
Polson, N. G., Scott, J. G., & Windle, J. (2013b). Bayesian inference for logistic models using P, ólya-gamma latent variables. Retrieved from arXiv:1205.0310 (Most recent version: Feb. 2013).CrossRefGoogle Scholar
R Core Team. (2020). R: A language and environment for statistical computing, [Computer software manual]. Vienna, Austria. Retrieved from https://www.R-project.org/.Google Scholar
Raven, J. & Raven, J. (2003). Raven progressive matrices. In: McCallum, R. S. (Ed.), Handbook of nonverbal assessment (pp. 223—237). Springer US. https://10.1007/978-1-4615-0153-4_11.CrossRefGoogle Scholar
Raven, J. C., (1941). Standardization of progressive matrices, 1938 British Journal of Medical Psychology 19(1) 137–150CrossRefGoogle Scholar
Sanderson, C., & Curtin, R., (2016). Armadillo: A template-based C++ library for linear algebra Journal of Open Source Software 1(2) 26CrossRefGoogle 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—316.CrossRefGoogle 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) 506–532CrossRefGoogle Scholar
Tatsuoka, K. K. (1984). Analysis of errors in fraction addition and subtraction problems. Computer-Based Education Research Laboratory: University of Illinois at Urbana-Champaign.Google Scholar
Templin, J. L., & Henson, R. A., (2006). Measurement of psychological disorders using cognitive diagnosis models Psychological Methods 11(3) 287CrossRefGoogle ScholarPubMed
Templin, J. L.,Henson, R. A., Templin, S. E., & Roussos, L., (2008). Robustness of hierarchical modeling of skill association in cognitive diagnosis models Applied Psychological Measurement 32 559–574CrossRefGoogle Scholar
Tjoe, H., & de la Torre, J., (2013). Designing cognitively-based proportional reasoning problems as an application of modern psychological measurement models Journal of Mathematics Education 6(2) 17–26Google Scholar
von Davier, M., (2008). A general diagnostic model applied to language testing data British Journal of Mathematical and Statistical Psychology 61(2) 287–307CrossRefGoogle ScholarPubMed
von Davier, M., (2009). Some notes on the reinvention of latent structure models as diagnostic classification models Measurement: Interdisciplinary Research and Perspectives 7 67–74Google Scholar
Wang, S.,Yang, Y., Culpepper, S. A., & Douglas, J. A., (2018). Tracking skill acquisition with cognitive diagnosis models: a higher-order, hidden Markov model with covariates Journal of Educational and Behavioral Statistics 43(1) 57–87CrossRefGoogle Scholar
Windle, J., Polson, N. G., & Scott, J. G. (2014). Sampling pólya-gamma random variates: Alternate and approximate techniques. arXiv preprint arXiv:1405.0506.Google Scholar
Xu, G., (2017). Identifiability of restricted latent class models with binary responses Annals of Statistics 45(2) 675–707CrossRefGoogle Scholar
Xu, G., & Shang, Z., (2018). Identifying latent structures in restricted latent class models Journal of the American Statistical Association 113(523) 1284–1295CrossRefGoogle Scholar
Zhang, Z.,Zhang, J., Lu, J., & Tao, J., (2020). Bayesian estimation of the DINA model with P, ólya-gamma Gibbs sampling Frontiers in Psychology 11 384CrossRefGoogle Scholar