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On the Unidentifiability of the Fixed-Effects 3PL Model

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  • Ernesto Martín
  • Jorge González
  • Francis Tuerlinckx

Abstract

The paper offers a general review of the basic concepts of both statistical model and parameter identification, and revisits the conceptual relationships between parameter identification and both parameter interpretability and properties of parameter estimates. All these issues are then exemplified for the 1PL, 2PL, and 1PL-G fixed-effects models. For the 3PL model, however, we provide a theorem proving that the item parameters are not identified, do not have an empirical interpretation and that it is not possible to obtain consistent and unbiased estimates of them. Copyright The Psychometric Society 2015

Suggested Citation

  • 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.
    • Handle: RePEc:spr:psycho:v:80:y:2015:i:2:p:450-467
    DOI: 10.1007/s11336-014-9404-2
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    References listed on IDEAS

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    1. San Martin, Ernesto & Rolin, Jean-Marie, 2013. "Identification of parametric Rasch-type models," LIDAM Reprints ISBA 2013002, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Jing Cao & S. Stokes, 2008. "Bayesian IRT Guessing Models for Partial Guessing Behaviors," Psychometrika, Springer;The Psychometric Society, vol. 73(2), pages 209-230, June.
    3. Rothenberg, Thomas J, 1971. "Identification in Parametric Models," Econometrica, Econometric Society, vol. 39(3), pages 577-591, May.
    4. Guido Pino & Ernesto Martín & Jorge González & Paul Boeck, 2008. "On the Relationships between Sum Score Based Estimation and Joint Maximum Likelihood Estimation," Psychometrika, Springer;The Psychometric Society, vol. 73(1), pages 145-151, March.
    5. Gabrielsen, Arne, 1978. "Consistency and identifiability," Journal of Econometrics, Elsevier, vol. 8(2), pages 261-263, October.
    6. San Martin, Ernesto & Jara, Alejandro & Rolin, Jean-Marie & Mouchart, Michel, 2011. "On the Bayesian nonparametric generalization of IRT-type models," LIDAM Reprints ISBA 2011012, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    7. 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.
    8. Abderrahim Oulhaj & Michel Mouchart, 2003. "Partial sufficiency with connection to the identification problem," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(2), pages 267-283.
    9. Javier Revuelta, 2010. "Estimating Difficulty from Polytomous Categorical Data," Psychometrika, Springer;The Psychometric Society, vol. 75(2), pages 331-350, June.
    10. Ernesto San Martín & Alejandro Jara & Jean-Marie Rolin & Michel Mouchart, 2011. "On the Bayesian Nonparametric Generalization of IRT-Type Models," Psychometrika, Springer;The Psychometric Society, vol. 76(3), pages 385-409, July.
    11. Lancaster, Tony, 2000. "The incidental parameter problem since 1948," Journal of Econometrics, Elsevier, vol. 95(2), pages 391-413, April.
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    Cited by:

    1. Hao Wu, 2016. "A Note on the Identifiability of Fixed-Effect 3PL Models," Psychometrika, Springer;The Psychometric Society, vol. 81(4), pages 1093-1097, December.
    2. 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.
    3. Sandip Sinharay & Jens Ledet Jensen, 2019. "Higher-Order Asymptotics and Its Application to Testing the Equality of the Examinee Ability Over Two Sets of Items," Psychometrika, Springer;The Psychometric Society, vol. 84(2), pages 484-510, June.
    4. Paula Fariña & Jorge González & Ernesto San Martín, 2019. "The Use of an Identifiability-Based Strategy for the Interpretation of Parameters in the 1PL-G and Rasch Models," Psychometrika, Springer;The Psychometric Society, vol. 84(2), pages 511-528, June.
    5. Page, Garritt L. & San Martin, Ernesto & Torres Irribarra, David & Van Bellegem, Sébastien, 2024. "Temporally Dynamic, Cohort-Varying Value-Added Models," LIDAM Discussion Papers CORE 2024009, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Stefano Noventa & Sangbeak Ye & Augustin Kelava & Andrea Spoto, 2024. "On the Identifiability of 3- and 4-Parameter Item Response Theory Models From the Perspective of Knowledge Space Theory," Psychometrika, Springer;The Psychometric Society, vol. 89(2), pages 486-516, June.
    7. Garritt L. Page & Ernesto San Martín & David Torres Irribarra & Sébastien Van Bellegem, 2024. "Temporally Dynamic, Cohort-Varying Value-Added Models," Psychometrika, Springer;The Psychometric Society, vol. 89(3), pages 1074-1103, September.

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