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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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    5. 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.
    6. 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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    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. Christian A. Gregory, 2020. "Are We Underestimating Food Insecurity? Partial Identification with a Bayesian 4-Parameter IRT Model," Journal of Classification, Springer;The Classification Society, vol. 37(3), pages 632-655, October.
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    11. McKee, Douglas & Orlov, George, 2023. "The Economic Statistics Skills Assessment (ESSA)," International Review of Economics Education, Elsevier, vol. 44(C).

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