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Generic Identifiability of the DINA Model and Blessing of Latent Dependence

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  • Yuqi Gu

    (Columbia University)

Abstract

Cognitive diagnostic models are a powerful family of fine-grained discrete latent variable models in psychometrics. Within this family, the DINA model is a fundamental and parsimonious one that has received significant attention. Similar to other complex latent variable models, identifiability is an important issue for CDMs, including the DINA model. Gu and Xu (Psychometrika 84(2):468–483, 2019) established the necessary and sufficient conditions for strict identifiability of the DINA model. Despite being the strongest possible notion of identifiability, strict identifiability may impose overly stringent requirements on designing the cognitive diagnostic tests. This work studies a slightly weaker yet very useful notion, generic identifiability, which means parameters are identifiable almost everywhere in the parameter space, excluding only a negligible subset of measure zero. We propose transparent generic identifiability conditions for the DINA model, relaxing existing conditions in nontrivial ways. Under generic identifiability, we also explicitly characterize the forms of the measure-zero sets where identifiability breaks down. In addition, we reveal an interesting blessing-of-latent-dependence phenomenon under DINA—that is, dependence between the latent attributes can restore identifiability under some otherwise unidentifiable $${\mathbf {Q}}$$ Q -matrix designs. The blessing of latent dependence provides useful practical implications and reassurance for real-world designs of cognitive diagnostic assessments.

Suggested Citation

  • Yuqi Gu, 2023. "Generic Identifiability of the DINA Model and Blessing of Latent Dependence," Psychometrika, Springer;The Psychometric Society, vol. 88(1), pages 117-131, March.
    • Handle: RePEc:spr:psycho:v:88:y:2023:i:1:d:10.1007_s11336-022-09886-2
    DOI: 10.1007/s11336-022-09886-2
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    References listed on IDEAS

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    1. Jonathan Templin & Laine Bradshaw, 2014. "Hierarchical Diagnostic Classification Models: A Family of Models for Estimating and Testing Attribute Hierarchies," Psychometrika, Springer;The Psychometric Society, vol. 79(2), pages 317-339, April.
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    6. Justin L. Kern & Steven Andrew Culpepper, 2020. "A Restricted Four-Parameter IRT Model: The Dyad Four-Parameter Normal Ogive (Dyad-4PNO) Model," Psychometrika, Springer;The Psychometric Society, vol. 85(3), pages 575-599, September.
    7. Kazuhiro Yamaguchi & Kensuke Okada, 2020. "Variational Bayes Inference for the DINA Model," Journal of Educational and Behavioral Statistics, , vol. 45(5), pages 569-597, October.
    8. Jimmy de la Torre, 2011. "The Generalized DINA Model Framework," Psychometrika, Springer;The Psychometric Society, vol. 76(2), pages 179-199, April.
    9. 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.
    10. James Joseph Balamuta & Steven Andrew Culpepper, 2022. "Exploratory Restricted Latent Class Models with Monotonicity Requirements under PÒLYA–GAMMA Data Augmentation," Psychometrika, Springer;The Psychometric Society, vol. 87(3), pages 903-945, September.
    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. Guanhua Fang & Jingchen Liu & Zhiliang Ying, 2019. "On the Identifiability of Diagnostic Classification Models," Psychometrika, Springer;The Psychometric Society, vol. 84(1), pages 19-40, March.
    13. 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.
    14. 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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