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A Constrained Metropolis–Hastings Robbins–Monro Algorithm for Q Matrix Estimation in DINA Models

Published online by Cambridge University Press:  01 January 2025

Chen-Wei Liu Open the ORCID record for Chen-Wei Liu [Opens in a new window] ,
Björn Andersson Open the ORCID record for Björn Andersson [Opens in a new window]  and
Anders Skrondal
Chen-Wei Liu*
Affiliation:
National Taiwan Normal University
Björn Andersson
Affiliation:
University of Oslo
Anders Skrondal
Affiliation:
Norwegian Institute of Public Health University of Oslo University of California, Berkeley
*
Correspondence should be made to Chen-Wei Liu, Department of Educational Psychology and Counseling, National Taiwan Normal University, 162, Section 1, Heping E. Road, 10610, Taipei, Taiwan. Email: cwliu@ntnu.edu.tw
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Abstract

In diagnostic classification models (DCMs), the Q matrix encodes in which attributes are required for each item. The Q matrix is usually predetermined by the researcher but may in practice be misspecified which yields incorrect statistical inference. Instead of using a predetermined Q matrix, it is possible to estimate it simultaneously with the item and structural parameters of the DCM. Unfortunately, current methods are computationally intensive when there are many attributes and items. In addition, the identification constraints necessary for DCMs are not always enforced in the estimation algorithms which can lead to non-identified models being considered. We address these problems by simultaneously estimating the item, structural and Q matrix parameters of the Deterministic Input Noisy “And” gate model using a constrained Metropolis–Hastings Robbins–Monro algorithm. Simulations show that the new method is computationally efficient and can outperform previously proposed Bayesian Markov chain Monte-Carlo algorithms in terms of Q matrix recovery, and item and structural parameter estimation. We also illustrate our approach using Tatsuoka’s fraction–subtraction data and Certificate of Proficiency in English data.

Keywords

Diagnostic classification modelsQ matrixstochastic algorithm

Information

Type
Theory and Methods
Information
Psychometrika , Volume 85 , Issue 2 , June 2020 , pp. 322 - 357
Copyright
Copyright © 2020 The Psychometric Society, corrected publication 2024

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References

Agresti, A., & Hitchcock, D. B. (2005). Bayesian inference for categorical data analysis. Statistical Methods & Applications, 14 (3), 297– 330. CrossRefGoogle Scholar
Bock, R. D., & Lieberman, M. (1970). Fitting a response model for n \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} dichotomously scored items. Psychometrika, 35 (2), 179– 197. CrossRefGoogle Scholar
Buck, G., & Tatsuoka, K. Application of the rule-space procedure to language testing: Examining attributes of a free response listening test. Language Testing, (1998). 15 (2), 119– 157. CrossRefGoogle Scholar
Cai, L. (2010). High-dimensional exploratory item factor analysis by a Metropolis-Hastings Robbins-Monro algorithm. Psychometrika, 75 (1), 33– 57. CrossRefGoogle Scholar
Chen, Y., Culpepper, S. A., Chen, Y., & Douglas, J (2017). Bayesian estimation of the DINA Q matrix. Psychometrika, 83 (1), 89– 108. CrossRefGoogle ScholarPubMed
Chen, Y., Culpepper, S., & Liang, F. (2020). A sparse latent class model for cognitive diagnosis. Psychometrika, 85 (1), 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
Culpepper, S. A. (2015). Bayesian estimation of the DINA model with Gibbs sampling. Journal of Educational and Behavioral Statistics, 40 (5), 454– 476. CrossRefGoogle Scholar
Culpepper, S. A. (2019). Estimating the cognitive diagnosis Q \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q$$\end{document} matrix with expert knowledge: Application to the fraction-subtraction dataset Psychometrika, 84 (2), 333– 357. 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., & Chen, Y. (2019). Development and application of an exploratory reduced reparameterized unified model. Journal of Educational and Behavioral Statistics, 44 (1), 3– 24. CrossRefGoogle Scholar
de la Torre, J. (2008). An empirically based method of Q-matrix validation for the DINA model: Development and applications. Journal of Educational Measurement, 45 (4), 343– 362. CrossRefGoogle Scholar
de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76 ((2),), 179– 199. CrossRefGoogle Scholar
de la Torre, J., & Chiu, C. -Y. (2016). A general method of empirical Q-matrix validation. Psychometrika, 81 (2), 253– 273. CrossRefGoogle ScholarPubMed
de la Torre, J., & Douglas, J. A. (2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69 (3), 333– 353. CrossRefGoogle 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. CrossRefGoogle Scholar
DeCarlo, L. T. (2012). Recognizing uncertainty in the Q-matrix via a Bayesian extension of the DINA model. Applied Psychological Measurement, 36 (6), 447– 468. CrossRefGoogle Scholar
Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society: Series B (Methodological), 39 (1), 1– 38. CrossRefGoogle Scholar
Diebolt, J., & Ip, E. H. S. Gilks, W. R., Richardson, S., & Spiegelhalter, D. J. (1996). Stochastic EM: Method and application. Markov chain Monte Carlo in practice, London: Chapman and Hall. 259– 273. Google Scholar
Gelman, A., Hwang, J., & Vehtari, A. (2014). Understanding predictive information criteria for Bayesian models Statistics and Computing 24 (6), 997– 1016. CrossRefGoogle Scholar
George, A. C. Robitzsch, A. (2015). Cognitive diagnosis models in R: A didactic. The Quantitative Methods for Psychology 11, (3), 189– 205. CrossRefGoogle Scholar
George, A. C., & Robitzsch, A. Kiefer, T. Groß, J. Ünlü, A. (2016). The R package CDM for cognitive diagnosis models Journal of Statistical Software 74 (2), 1– 24. CrossRefGoogle Scholar
George, E. I., & McCulloch, R. E. (1997). Approaches for Bayesian variable selection. Statistica Sinica, 7 (2), 339– 373. Google Scholar
Gu, M. G. Kong, F. H.(1998). A stochastic approximation algorithm with Markov chain Monte-Carlo method for incomplete data estimation problems. Proceedings of the National Academy of Sciences, 95 (13), 7270– 7274. CrossRefGoogle ScholarPubMed
Gu, Y., & Xu, G. (2018). Sufficient and necessary conditions for the identifiability of the Q-matrix. Statistica Sinica, Google Scholar
Gu, Y., & Xu, G. (2019). The sufficient and necessary condition for the identifiability and estimability of the DINA model. Psychometrika, 84 (2), 468– 483. CrossRefGoogle ScholarPubMed
Geweke, J. Bernardo, J. M. Berger, J. Dawid, A. P. Smith, JFM (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. Bayesian Statistics 4, Oxford University Press. 169– 193. CrossRefGoogle 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. CrossRefGoogle Scholar
Hartz, S. M. (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. CrossRefGoogle 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. CrossRefGoogle Scholar
Kunina-Habenicht, O. Rupp, A. A. Wilhelm, O.(2009). A practical illustration of multidimensional diagnostic skills profiling: Comparing results from confirmatory factor analysis and diagnostic classification models Studies in Educational Evaluation 35 64– 70. CrossRefGoogle Scholar
Liu, J.(2017). On the consistency of Q-matrix estimation: A commentary Psychometrika 82 (2), 523– 527. CrossRefGoogle ScholarPubMed
Liu, C-W Chalmers, R. P.(2020). A note on computing Louis’ observed information matrix identity for IRT and cognitive diagnostic models British Journal of Mathematical and Statistical Psychology Google Scholar
Liu, J. Xu, G. Ying, Z.(2012). Data-driven learning of Q-matrix Applied Psychological Measurement 36 (7), 548– 564. CrossRefGoogle ScholarPubMed
Liu, J. Xu, G. Ying, Z.(2013). Theory of the self-learning Q-matrix Bernoulli 19 5A 1790– 4011940 CrossRefGoogle ScholarPubMed
Liu, J. S.(1996). Peskun’s theorem and a modified discrete-state Gibbs sampler Biometrika 83 (3), 681– 682. CrossRefGoogle Scholar
Louis, T. A.(1982). Finding the observed information matrix when using the EM algorithm Journal of the Royal Statistical Society: Series B (Methodological) 44 (2), 226– 233. CrossRefGoogle 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– 120. CrossRefGoogle Scholar
Madigan, D. York, J. Allard, D.(1995). Bayesian graphical models for discrete data. International Statistical Review, 63 (2), 215– 232. CrossRefGoogle Scholar
Maydeu-Olivares, A. (2013). Goodness-of-fit assessment of item response theory models. Measurement: Interdisciplinary Research and Perspectives, 11(3), 71–101.Google Scholar
Orlando, M. Thissen, D. (2000). Likelihood-based item-fit indices for dichotomous item response theory models. Applied Psychological Measurement, 24 (1), 50– 64. CrossRefGoogle Scholar
Perks, W.(1947). Some observations on inverse probability including a new indifference rule. Journal of the Institute of Actuaries, 73 (2), 285– 334. CrossRefGoogle Scholar
Philipp, M. Strobl, C. de la Torre, J. Zeileis, A.(2017). On the estimation of standard errors in cognitive diagnosis models. Journal of Educational and Behavioral Statistics, 43 (1), 88– 115. CrossRefGoogle Scholar
Robbins, H. Monro, S.(1951). A stochastic approximation method. The Annals of Mathematical Statistics, 22 (3), 400– 407. CrossRefGoogle Scholar
Rupp, A. A. Templin, J. L.(2008). The effects of Q-matrix misspecification on parameter estimates and classification accuracy in the DINA model. Educational and Psychological Measurement, 68 (1), 78– 96. CrossRefGoogle Scholar
Rupp, A. A. Templin, J. L. Henson, R. A.(2010). Diagnostic measurement: Theory, methods, and applications, New York: Guilford Press. Google Scholar
Schwarz, G.(1978). Estimating the dimension of a model. The Annals of Statistics, 6 (2), 461– 464. CrossRefGoogle Scholar
Sun, Y., Ye, S., Su, G., & Sun, Y. (2016). Q-matrix learning and DINA model parameter estimation. Paper presented at the 2016 International Conference on Behavioral, Economic and Socio-cultural Computing (BESC), Durham, NC, USA. CrossRefGoogle 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– 350. Google Scholar
Tatsuoka, K. K. (1984). Analysis of errors in fraction addition and subtraction problems (Final Report for NIE-G-81-0002). Urbana-Champaign: University of Illinois.Google Scholar
Templin, J. L., & Bradshaw, L. (2014). Hierarchical diagnostic classification models: A family of models for estimating and testing attribute hierarchies. Psychometrika, 79 (2), 317– 339. CrossRefGoogle ScholarPubMed
Templin, J. L., & Hoffman, L. (2013). Obtaining diagnostic classification model estimates using Mplus. Educational Measurement: Issues and Practice, 32 (2), 37– 50. CrossRefGoogle Scholar
Templin, J. L., & Henson, R. A. (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11 (3), 287– CrossRefGoogle 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. CrossRefGoogle ScholarPubMed
von Davier, M. (2014). The DINA model as a constrained general diagnostic model: Two variants of a model equivalency. British Journal of Mathematical and Statistical Psychology, 67 (1), 49– 71. CrossRefGoogle Scholar
von Davier, M., & Sinharay, S. (2010). Stochastic approximation methods for latent regression item response models Journal of Educational and Behavioral Statistics 35 (2), 174– 193. CrossRefGoogle Scholar
Wang, S. (2018). Two-stage maximum likelihood estimation in the misspecified restricted latent class model. British Journal of Mathematical and Statistical Psychology, 71 (2), 300– 333. CrossRefGoogle ScholarPubMed
Wang, W., Song, L., Ding, S., Meng, Y., Cao, C., & Jie, Y. (2018). An EM-based method for Q-matrix validation. Applied Psychological Measurement, 42 (6), 446– 459. CrossRefGoogle ScholarPubMed
Watanabe, S. (2010). Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. Journal of Machine Learning Research, 11(Dec), 3571–3594. Google Scholar
Wei, G. C., & Tanner, M. A. (1990). A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithms. Journal of the American Statistical Association, 85 (411), 699– 704. CrossRefGoogle Scholar
Xu, G. (2017). Identifiability of restricted latent class models with binary responses. The Annals of Statistics, 45 (2), 675– 707. CrossRefGoogle Scholar
Xu, G., & Shang, Z. (2017). Identifying latent structures in restricted latent class models. Journal of the American Statistical Association, 113 (423), 1284– 1295. CrossRefGoogle Scholar
Yang, J. S., & Cai, L. (2014). Estimation of contextual effects through nonlinear multilevel latent variable modeling with a Metropolis-Hastings Robbins-Monro algorithm. Journal of Educational and Behavioral Statistics, 39 (6), 550– 582. CrossRefGoogle Scholar
Zhang, S., Chen, Y., & Liu, Y. (2020). An improved stochastic EM algorithm for large-scale full-information item factor analysis. British Journal of Mathematical and Statistical Psychology, 73 (1), 44– 71. CrossRefGoogle ScholarPubMed