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

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  • Chen-Wei Liu

    (National Taiwan Normal University)

  • Björn Andersson

    (University of Oslo)

  • Anders Skrondal

    (Norwegian Institute of Public Health
    University of Oslo
    University of California, Berkeley)

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.

Suggested Citation

  • Chen-Wei Liu & Björn Andersson & Anders Skrondal, 2020. "A Constrained Metropolis–Hastings Robbins–Monro Algorithm for Q Matrix Estimation in DINA Models," Psychometrika, Springer;The Psychometric Society, vol. 85(2), pages 322-357, June.
  • Handle: RePEc:spr:psycho:v:85:y:2020:i:2:d:10.1007_s11336-020-09707-4
    DOI: 10.1007/s11336-020-09707-4
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    References listed on IDEAS

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    3. Yunxiao Chen & Jingchen Liu & Gongjun Xu & Zhiliang Ying, 2015. "Statistical Analysis of Q -Matrix Based Diagnostic Classification Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 850-866, June.
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    6. Jimmy Torre & Jeffrey Douglas, 2004. "Higher-order latent trait models for cognitive diagnosis," Psychometrika, Springer;The Psychometric Society, vol. 69(3), pages 333-353, September.
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    8. Li Cai, 2010. "High-dimensional Exploratory Item Factor Analysis by A Metropolis–Hastings Robbins–Monro Algorithm," Psychometrika, Springer;The Psychometric Society, vol. 75(1), pages 33-57, March.
    9. Jingchen Liu, 2017. "On the Consistency of Q-Matrix Estimation: A Commentary," Psychometrika, Springer;The Psychometric Society, vol. 82(2), pages 523-527, June.
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    11. Yinyin Chen & Steven Culpepper & Feng Liang, 2020. "A Sparse Latent Class Model for Cognitive Diagnosis," Psychometrika, Springer;The Psychometric Society, vol. 85(1), pages 121-153, March.
    12. Yuqi Gu & Gongjun Xu, 2019. "The Sufficient and Necessary Condition for the Identifiability and Estimability of the DINA Model," Psychometrika, Springer;The Psychometric Society, vol. 84(2), pages 468-483, June.
    13. Steven Andrew Culpepper, 2019. "An Exploratory Diagnostic Model for Ordinal Responses with Binary Attributes: Identifiability and Estimation," Psychometrika, Springer;The Psychometric Society, vol. 84(4), pages 921-940, December.
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    Cited by:

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    3. Pablo Nájera & Francisco J. Abad & Chia-Yi Chiu & Miguel A. Sorrel, 2023. "The Restricted DINA Model: A Comprehensive Cognitive Diagnostic Model for Classroom-Level Assessments," Journal of Educational and Behavioral Statistics, , vol. 48(6), pages 719-749, December.
    4. Motonori Oka & Kensuke Okada, 2023. "Scalable Bayesian Approach for the Dina Q-Matrix Estimation Combining Stochastic Optimization and Variational Inference," Psychometrika, Springer;The Psychometric Society, vol. 88(1), pages 302-331, March.
    5. Meng-Ta Chung & Shui-Lien Chen, 2021. "A Deterministic Learning Algorithm Estimating the Q-Matrix for Cognitive Diagnosis Models," Mathematics, MDPI, vol. 9(23), pages 1-11, November.
    6. Yinghan Chen & Ying Liu & Steven Andrew Culpepper & Yuguo Chen, 2021. "Inferring the Number of Attributes for the Exploratory DINA Model," Psychometrika, Springer;The Psychometric Society, vol. 86(1), pages 30-64, March.

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