Hostname: page-component-7f64f4797f-kjzhn Total loading time: 0 Render date: 2025-11-04T06:20:35.753Z Has data issue: false hasContentIssue false
  • English
  • Français

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: esanmart@mat.puc.cl
Get access Rights & Permissions [Opens in a new window]

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

Information

Type
Original Paper
Information
Psychometrika , Volume 78 , Issue 2 , April 2013 , pp. 341 - 379
Copyright
Copyright © 2013 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Adams, R.J., Wilson, M.R., Wang, W. (1997). The multidimensional random coefficients multinomial logit model. Applied Psychological Measurement, 21, 1–23CrossRefGoogle Scholar
Adams, R.J., Wu, M.L. (2007). The mixed-coefficients multinomial logit model: a generalization form of the Rasch model. In von Davier, M., Carstensen, C.H. (Eds.), Multivariate and mixture distribution Rasch models, Berlin: Springer 5775CrossRefGoogle Scholar
Andersen, E.B. (1980). Discrete statistical models with social sciences applications, Amsterdam: North-HollandGoogle Scholar
Béguin, A.A., Glas, C.A.W. (2001). MCMC estimation and some model-fit analysis of multidimensional IRT models. Psychometrika, 66, 541–562CrossRefGoogle Scholar
Berti, P., Pratelli, L., Riggo, P. (2008). Trivial intersection of σ-fields and Gibbs sampling. Annals of Probability, 36, 2215–2234CrossRefGoogle Scholar
Berti, P., Pratelli, L., Riggo, P. (2010). Atomic intersection of σ-fields and some of its consequences. Probability Theory and Related Fields, 148, 269–283CrossRefGoogle Scholar
Birnbaum, A. (1968). Some latent trait models and their use in inferring any examinee’s ability. In Lord, F.M., Novick, M.R. (Eds.), Statistical theories of mental test scores, Reading: Addison-Wesley 395479Google Scholar
Bock, R.D., Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: application of an EM algorithm. Psychometrika, 46, 443–459CrossRefGoogle Scholar
Bock, R.D., Zimowski, M.F. (1997). Multiple group IRT. In van der Linden, W.J., Hambleton, R.K. (Eds.), Handbook of modern item response theory, Berlin: Springer 433448CrossRefGoogle Scholar
Carlin, B.P., Louis, T.A. (2000). Bayes and empirical Bayes methods for data analysis, (2nd ed.). London: Chapman & Hall/CRCGoogle Scholar
De Boeck, P., Wilson, M. (2004). Explanatory item response models. A generalized linear and nonlinear approach, Berlin: SpringerCrossRefGoogle Scholar
Del Pino, G., San Martín, E., González, J., De Boeck, P. (2008). On the relationships between sum score based estimation and joint maximum likelihood estimation. Psychometrika, 73, 145–151CrossRefGoogle Scholar
Eberly, E.E., Carlin, B.P. (2000). Identifiability and convergence issues for Markov chain Monte Carlo fitting of spatial data. Statistics in Medicine, 19, 2279–22943.0.CO;2-R>CrossRefGoogle Scholar
Embretson, S.E., Reise, S.P. (2000). Item response theory for psychologists, Mahwah: Lawrence Erlbaum AssociatesGoogle Scholar
Florens, J.-P., Mouchart, M. (1982). A note on noncausality. Econometrica, 50, 583–591CrossRefGoogle Scholar
Florens, J.-P., Mouchart, M., Rolin, J.-M. (1990). Elements of Bayesian statistics, New York: DekkerGoogle Scholar
Florens, J.-P., Rolin, J.-M. (1984). Asymptotic sufficiency and exact estimability. In Florens, J.-P., Mouchart, M., Raoult, J.-P., Simar, L. (Eds.), Alternative approaches to time series analysis, Bruxelles: Publications des Facultés Universitaires Saint-Louis 121142Google Scholar
Gabrielsen, A. (1978). Consistency and identifiability. Journal of Econometrics, 8, 261–263CrossRefGoogle Scholar
Gelfand, A.E., Sahu, S.K. (1999). Identifiability, improper priors, and Gibbs sampling for generalized linear models. Journal of the American Statistical Association, 94, 247–253CrossRefGoogle Scholar
Gelman, A., Rubin, R. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7, 457–511CrossRefGoogle Scholar
Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to calculating posterior moments, Oxford: Clarendon PressGoogle Scholar
Ghosh, M., Ghosh, A., Chen, M.-H., Agresti, A. (2000). Noninformative priors for one-parameter item response models. Journal of Statistical Planning and Inference, 88, 99–115CrossRefGoogle Scholar
Gustafson, P. (2005). On model expansion, model contraction, identifiability and prior information: two illustrative scenarios involving mismeasured variables (with discussion). Statistical Sciences, 20, 111–140CrossRefGoogle Scholar
Halmos, P. (1951). Introduction to Hilbert space, and the theory of spectral multiplicity, New York: ChelseaGoogle Scholar
Hambleton, R.K., Swaminathan, H., Rogers, H.J. (1991). Fundamentals of item response theory, Thousand Oaks: SageGoogle Scholar
Heidelberg, P., Welch, P. (1992). Simulation run length control in the presence of an initial transient. Operations Research, 31, 1109–1144CrossRefGoogle Scholar
Hutschinson, T.P. (1991). Ability, parameter information, guessing: statistical modelling applied to multiple-choice tests, Rundle Mall: Rumsby Scientific PublishingGoogle Scholar
Karabatsos, G., Walker, S. (2009). Coherent psychometric modelling with Bayesian nonparametrics. British Journal of Mathematical & Statistical Psychology, 62, 1–20CrossRefGoogle ScholarPubMed
Kass, R., Carlin, B., Gelman, A., Neal, R. (1998). Markov chain Monte Carlo in practice: a roundtable discussion. American Statistician, 52, 93–100CrossRefGoogle Scholar
Koopmans, T.C., Reiersøl, O. (1950). The identification of structural characteristics. The Annals of Mathematical Statistics, 21, 165–181CrossRefGoogle Scholar
Lancaster, T. (2000). The incidental parameter problem since 1948. Journal of Econometrics, 95, 391–413CrossRefGoogle Scholar
van der Linden, W., Hambleton, R.K. (1997). Handbook of modern item response theory, Berlin: SpringerCrossRefGoogle Scholar
Lindley, D.V. (1971). Bayesin statistics: a review, Philadelphia: Society for Industrial and Applied MathematicsGoogle Scholar
Maris, G., Bechger, T. (2009). On interpreting the model parameters for the three parameter logistic model. Measurement Interdisciplinary Research & Perspective, 7, 75–86CrossRefGoogle Scholar
McDonald, R.P. (1999). Test theory: a unified treatment, Hillsdale: ErlbaumGoogle Scholar
Millsap, R., Maydeu-Olivares, A. (2009). Quantitative methods in psychology, Thousand Oaks: SageGoogle Scholar
Miyazaki, K., Hoshino, T. (2009). A Bayesian semiparametric item response model with Dirichlet process priors. Psychometrika, 74, 375–393CrossRefGoogle Scholar
Molenaar, I.W. (1995). Estimation of item parameters. In Fischer, G.H., Molenaar, I.W. (Eds.), Rasch models. Foundations, recent developments and applications, New York: Springer (Chapter 3)Google Scholar
Mouchart, M. (1976). A note on Bayes theorem. Statistica, 36, 349–357Google Scholar
Poirier, D.J. (1998). Revising beliefs in nonidentified models. Econometric Theory, 14, 483–509CrossRefGoogle Scholar
R Development Core Team (2006). R: a language and environment for statistical computing [Computer software manual]. Vienna, Austria. http://www.R-project.org (ISBN 3-900051-07-0). Google Scholar
Rao, M.M. (2005). Conditional measures and applications, (2nd ed.). London: Chapman & Hall/CRCCrossRefGoogle Scholar
Rizopoulos, D. (2006). ltm: an R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17(5), 125 http://www.jstatsoft.org/v17/i05/CrossRefGoogle Scholar
Roberts, G.O., Rosenthal, J. (1998). Markov-chain Monte Carlo: some practical implications of theoretical results. Canadian Journal of Statistics, 26(1), 520CrossRefGoogle Scholar
San Martín, E., Del Pino, G., De Boeck, P. (2006). IRT models for ability-based guessing. Applied Psychological Measurement, 30, 183–203CrossRefGoogle Scholar
San Martín, E., González, J. (2010). Bayesian identifiability: contributions to an inconclusive debate. Chilean Journal of Statistics, 1, 69–91Google Scholar
San Martín, E., González, J., Tuerlinckx, F. (2009). Identified parameters, parameters of interest and their relationships. Measurement Interdisciplinary Research & Perspective, 7, 95–103CrossRefGoogle Scholar
San Martín, E., Jara, A., Rolin, J.-M., Mouchart, M. (2011). On the Bayesian nonparametric generalization of IRT-type models. Psychometrika, 76, 385–409CrossRefGoogle Scholar
San Martín, E., Mouchart, M., Rolin, J.M. (2005). Ignorable common information, null sets and Basu’s first theorem. Sankhyā, 67, 674–698Google Scholar
San Martín, E., Quintana, F. (2002). Consistency and identifiability revisited. Brazilian Journal of Probability and Statistics, 16, 99–106Google Scholar
Shiryaev, A.N. (1995). Probability, (2nd ed.). Berlin: SpringerGoogle Scholar
Spivak, M. (1965). Calculus on manifolds: a modern approach to classical theorems of advanced calculus, Cambridge: Perseus Book PublishingGoogle Scholar
Swaminathan, H., Gifford, J.A. (1986). Bayesian estimation in the three-parameter logistic model. Psychometrika, 51, 589–601CrossRefGoogle Scholar
Thissen, D. (2009). On interpreting the parameters for any item response model. Measurement Interdisciplinary Research & Perspective, 7, 104–108CrossRefGoogle Scholar
Thissen, D., Wainer, H. (2001). Item response models for items scored in two categories, Berlin: SpringerGoogle Scholar
Woods, C.M. (2006). Ramsay-curve item response theory (RC-IRT) to detect and correct for nonnormal latent variables. Psychological Methods, 11, 253–270CrossRefGoogle ScholarPubMed
Woods, C.M. (2008). Ramsay-curve item response theory for the three-parameter logistic item response model. Applied Psychological Measurement, 32, 447–465CrossRefGoogle Scholar
Woods, C.M., Thissen, D. (2006). Item response theory with estimation of the latent population distribution using spline-based densities. Psychometrika, 71, 281–301CrossRefGoogle ScholarPubMed
Xie, Y., Carlin, B.P. (2006). Measure of Bayesian learning and identifiability in hierarchical models. Journal of Statistcal Planning and Inference, 136, 3458–3477CrossRefGoogle Scholar