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Identification of the 1PL Model with Guessing Parameter: Parametric and Semi-parametric Results

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  • Ernesto San Martín
  • Jean-Marie Rolin
  • Luis Castro

Abstract

In this paper, we study the identification of a particular case of the 3PL model, namely when the discrimination parameters are all constant and equal to 1. We term this model, 1PL-G model. The identification analysis is performed under three different specifications. The first specification considers the abilities as unknown parameters. It is proved that the item parameters and the abilities are identified if a difficulty parameter and a guessing parameter are fixed at zero. The second specification assumes that the abilities are mutually independent and identically distributed according to a distribution known up to the scale parameter. It is shown that the item parameters and the scale parameter are identified if a guessing parameter is fixed at zero. The third specification corresponds to a semi-parametric 1PL-G model, where the distribution G generating the abilities is a parameter of interest. It is not only shown that, after fixing a difficulty parameter and a guessing parameter at zero, the item parameters are identified, but also that under those restrictions the distribution G is not identified. It is finally shown that, after introducing two identification restrictions, either on the distribution G or on the item parameters, the distribution G and the item parameters are identified provided an infinite quantity of items is available. Copyright The Psychometric Society 2013

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  • 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.
    • Handle: RePEc:spr:psycho:v:78:y:2013:i:2:p:341-379
    DOI: 10.1007/s11336-013-9322-8
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    1. R. Bock & Murray Aitkin, 1981. "Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm," Psychometrika, Springer;The Psychometric Society, vol. 46(4), pages 443-459, December.
    2. 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.
    3. Lancaster, Tony, 2000. "The incidental parameter problem since 1948," Journal of Econometrics, Elsevier, vol. 95(2), pages 391-413, April.
    4. A. Béguin & C. Glas, 2001. "MCMC estimation and some model-fit analysis of multidimensional IRT models," Psychometrika, Springer;The Psychometric Society, vol. 66(4), pages 541-561, December.
    5. MOUCHART, Michel, 1976. "A note on Bayes theorem," LIDAM Reprints CORE 253, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. 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.
    7. Poirier, Dale J., 1998. "Revising Beliefs In Nonidentified Models," Econometric Theory, Cambridge University Press, vol. 14(4), pages 483-509, August.
    8. Carol Woods & David Thissen, 2006. "Item Response Theory with Estimation of the Latent Population Distribution Using Spline-Based Densities," Psychometrika, Springer;The Psychometric Society, vol. 71(2), pages 281-301, June.
    9. Florens, J P & Mouchart, M, 1982. "A Note on Noncausality," Econometrica, Econometric Society, vol. 50(3), pages 583-591, May.
    10. Carol M. Woods & David Thissen, 2006. "Item Response Theory with Estimation of the Latent Population Distribution Using Spline-Based Densities," Psychometrika, Springer;The Psychometric Society, vol. 71(2), pages 281-301, June.
    11. 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.
    12. Gabrielsen, Arne, 1978. "Consistency and identifiability," Journal of Econometrics, Elsevier, vol. 8(2), pages 261-263, October.
    13. 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).
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    Cited by:

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    3. Yu-Wei Chang & Nan-Jung Hsu & Rung-Ching Tsai, 2017. "Unifying Differential Item Functioning in Factor Analysis for Categorical Data Under a Discretization of a Normal Variant," Psychometrika, Springer;The Psychometric Society, vol. 82(2), pages 382-406, June.
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    6. 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.
    7. 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.
    8. Elizabeth Ooi, 2020. "Give mind to the gap: Measuring gender differences in financial knowledge," Journal of Consumer Affairs, Wiley Blackwell, vol. 54(3), pages 931-950, September.
    9. 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.
    10. Wim J. Linden & Michelle D. Barrett, 2016. "Linking Item Response Model Parameters," Psychometrika, Springer;The Psychometric Society, vol. 81(3), pages 650-673, September.

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