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Identifiability of Hidden Markov Models for Learning Trajectories in Cognitive Diagnosis

Author

Listed:
  • Ying Liu

    (University of Illinois at Urbana-Champaign)

  • Steven Andrew Culpepper

    (University of Illinois at Urbana-Champaign)

  • Yuguo Chen

    (University of Illinois at Urbana-Champaign)

Abstract

Hidden Markov models (HMMs) have been applied in various domains, which makes the identifiability issue of HMMs popular among researchers. Classical identifiability conditions shown in previous studies are too strong for practical analysis. In this paper, we propose generic identifiability conditions for discrete time HMMs with finite state space. Also, recent studies about cognitive diagnosis models (CDMs) applied first-order HMMs to track changes in attributes related to learning. However, the application of CDMs requires a known $$\varvec{Q}$$ Q matrix to infer the underlying structure between latent attributes and items, and the identifiability constraints of the model parameters should also be specified. We propose generic identifiability constraints for our restricted HMM and then estimate the model parameters, including the $$\varvec{Q}$$ Q matrix, through a Bayesian framework. We present Monte Carlo simulation results to support our conclusion and apply the developed model to a real dataset.

Suggested Citation

  • Ying Liu & Steven Andrew Culpepper & Yuguo Chen, 2023. "Identifiability of Hidden Markov Models for Learning Trajectories in Cognitive Diagnosis," Psychometrika, Springer;The Psychometric Society, vol. 88(2), pages 361-386, June.
    • Handle: RePEc:spr:psycho:v:88:y:2023:i:2:d:10.1007_s11336-023-09904-x
    DOI: 10.1007/s11336-023-09904-x
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    References listed on IDEAS

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    1. Chen, Yunxiao & Liu, Jingchen & Xu, Gongjun & Ying, Zhiliang, 2015. "Statistical analysis of Q-matrix based diagnostic classification models," LSE Research Online Documents on Economics 103183, London School of Economics and Political Science, LSE Library.
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    3. Stéphane Bonhomme & Koen Jochmans & Jean-Marc Robin, 2014. "Estimating Multivariate Latent-Structure Models," SciencePo Working papers Main hal-01097135, HAL.
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    7. 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.
    8. 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.
    9. 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.
    10. Jimmy de la Torre, 2011. "The Generalized DINA Model Framework," Psychometrika, Springer;The Psychometric Society, vol. 76(2), pages 179-199, April.
    11. Jimmy Torre, 2011. "Erratum to: The Generalized DINA Model Framework," Psychometrika, Springer;The Psychometric Society, vol. 76(3), pages 510-510, July.
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    Cited by:

    1. Kazuhiro Yamaguchi, 2023. "Bayesian Analysis Methods for Two-Level Diagnosis Classification Models," Journal of Educational and Behavioral Statistics, , vol. 48(6), pages 773-809, December.

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