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

Published online by Cambridge University Press:  01 January 2025

Ernesto San Martín ,
Jean-Marie Rolin  and
Luis M. Castro
Ernesto San Martín*
Affiliation:
Faculty of Mathematics, Pontificia Universidad Católica de Chile Faculty of Education, Pontificia Universidad Católica de Chile Measurement Center MIDE UC CEPPE
Jean-Marie Rolin
Affiliation:
Institut de statistique, biostatistique et sciences actuarielles, Université catholique de Louvain
Luis M. Castro
Affiliation:
Department of Statistics, Universidad de Concepción
*
Requests for reprints should be sent to Ernesto San Martín, Faculty of Mathematics, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Macul, Santiago, Chile. E-mail: [email protected]
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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.

Keywords

2PL model3PL modellocation-scale distributionsfixed effectsrandom effectsidentified parameterparameters of interestHilbert spaceGibbs samplermeasurable separability
Type
Original Paper
Information
Psychometrika , Volume 78 , Issue 2 , April 2013 , pp. 341 - 379
Copyright
Copyright © 2013 The Psychometric Society

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