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A Restricted Four-Parameter IRT Model: The Dyad Four-Parameter Normal Ogive (Dyad-4PNO) Model

Author

Listed:
  • Justin L. Kern

    (Department of Educational Psychology, University of Illinois at Urbana-Champaign)

  • Steven Andrew Culpepper

    (Department of Statistics, University of Illinois at Urbana-Champaign)

Abstract

Recently, there has been a renewed interest in the four-parameter item response theory model as a way to capture guessing and slipping behaviors in responses. Research has shown, however, that the nested three-parameter model suffers from issues of unidentifiability (San Martín et al. in Psychometrika 80:450–467, 2015), which places concern on the identifiability of the four-parameter model. Borrowing from recent advances in the identification of cognitive diagnostic models, in particular, the DINA model (Gu and Xu in Stat Sin https://doi.org/10.5705/ss.202018.0420 , 2019), a new model is proposed with restrictions inspired by this new literature to help with the identification issue. Specifically, we show conditions under which the four-parameter model is strictly and generically identified. These conditions inform the presentation of a new exploratory model, which we call the dyad four-parameter normal ogive (Dyad-4PNO) model. This model is developed by placing a hierarchical structure on the DINA model and imposing equality constraints on a priori unknown dyads of items. We present a Bayesian formulation of this model, and show that model parameters can be accurately recovered. Finally, we apply the model to a real dataset.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:psycho:v:85:y:2020:i:3:d:10.1007_s11336-020-09716-3
    DOI: 10.1007/s11336-020-09716-3
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    References listed on IDEAS

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    1. Steven Andrew Culpepper, 2016. "Revisiting the 4-Parameter Item Response Model: Bayesian Estimation and Application," Psychometrika, Springer;The Psychometric Society, vol. 81(4), pages 1142-1163, December.
    2. Ernesto San Martín & Jean-Marie Rolin & Luis Castro, 2013. "Identification of the 1PL Model with Guessing Parameter: Parametric and Semi-parametric Results," Psychometrika, Springer;The Psychometric Society, vol. 78(2), pages 341-379, April.
    3. 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.
    4. Hariharan Swaminathan & Janice Gifford, 1986. "Bayesian estimation in the three-parameter logistic model," Psychometrika, Springer;The Psychometric Society, vol. 51(4), pages 589-601, December.
    5. Gabrielsen, Arne, 1978. "Consistency and identifiability," Journal of Econometrics, Elsevier, vol. 8(2), pages 261-263, October.
    6. E. Maris, 1999. "Estimating multiple classification latent class models," Psychometrika, Springer;The Psychometric Society, vol. 64(2), pages 187-212, June.
    7. David J. Spiegelhalter & Nicola G. Best & Bradley P. Carlin & Angelika Van Der Linde, 2002. "Bayesian measures of model complexity and fit," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(4), pages 583-639, October.
    8. Ernesto Martín & Jorge González & Francis Tuerlinckx, 2015. "On the Unidentifiability of the Fixed-Effects 3PL Model," Psychometrika, Springer;The Psychometric Society, vol. 80(2), pages 450-467, June.
    9. 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.
    10. Edgar C. Merkle & Daniel Furr & Sophia Rabe-Hesketh, 2019. "Bayesian Comparison of Latent Variable Models: Conditional Versus Marginal Likelihoods," Psychometrika, Springer;The Psychometric Society, vol. 84(3), pages 802-829, September.
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    Cited by:

    1. 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.
    2. Justin L. Kern, 2024. "Extending an Identified Four-Parameter IRT Model: The Confirmatory Set-4PNO Model," Journal of Educational and Behavioral Statistics, , vol. 49(3), pages 368-402, June.
    3. Xiangyi Liao & Daniel M. Bolt, 2021. "Item Characteristic Curve Asymmetry: A Better Way to Accommodate Slips and Guesses Than a Four-Parameter Model?," Journal of Educational and Behavioral Statistics, , vol. 46(6), pages 753-775, December.

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