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On the Unidentifiability of the Fixed-Effects 3PL Model

Published online by Cambridge University Press:  01 January 2025

Ernesto San Martín ,
Jorge González  and
Francis Tuerlinckx
Ernesto San Martín*
Affiliation:
Pontificia Universidad Católica de Chile, Measurement Center Mide UC, Ceppe-UC and Université Catholique de Louvain
Jorge González
Affiliation:
Pontificia Universidad Católica de Chile and Measurement Center Mide UC
Francis Tuerlinckx
Affiliation:
University of Leuven
*
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

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.

Keywords

consistent estimatoridentified parametersparameters of intereststatistical modelunbiased estimator
Type
Original Paper
Information
Psychometrika , Volume 80 , Issue 2 , June 2015 , pp. 450 - 467
Copyright
Copyright © 2014 The Psychometric Society

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References

Baker, F., & Kim, S. (2004). Item response theory: parameter estimation techniques. New York: Marcel Dekker.CrossRefGoogle Scholar
Bamber, D., & Van Santen, J.P.H. (2000). How to asses a model’s testability and identifiability. Journal of Mathematical Psychology, 44, 20–40.CrossRefGoogle Scholar
Basu, D. (1975). Statistical information and likelihood (with discussion). Sankhyā: The Indian Journal of Statistics, 37, 1–71.Google Scholar
Berman, A., & Ben-Israel, A. (1971). More on linear inequalities with applications to matrix theory. Journal of Mathematical Analysis and Its Applications, 27, 367–389.Google Scholar
Berman, A., & Plemmos, R.J. (1994). Nonnegative matrices in the mathematical sciences. Philadelphia: SIAM.CrossRefGoogle Scholar
Birnbaum, A. (1962). On the foundation of statistical inference (with discussion). Journal of the American Statistical Association, 57, 269–326.CrossRefGoogle 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 (pp. 395–479). Reading: Adison-Wesley.Google Scholar
Bunke, H., & Bunke, O. (1974). Identifiability and estimability. Mathematische Operationsforschung und Statistik, 5, 223–233.CrossRefGoogle Scholar
Cao, J., & Stokes, L. (2008). Bayesian IRT guessing models for partial guessing behaviors. Psychometrika, 73, 209–230.CrossRefGoogle Scholar
Castro, L.M., San Martín, E., & Arellano-Valle, R. (2013). A note on the parameterization of multivariate skewed-normal distributions. Brazilian Journal of Probability and Statistics, 27, 110–115.CrossRefGoogle Scholar
Cox, D.R., & Hinkley, D.V. (1974). Theoretical statistics. London: Chapman and Hall.CrossRefGoogle Scholar
De Boeck, P., & Wilson, M. (Eds.) (2004). Explanatory item response models: a generalized linear and nonlinear approach. New York: Springer.CrossRefGoogle 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, 145–151.CrossRefGoogle Scholar
Embretson, S.E., & Reise, S.P. (2000). Item response theory for psychologists. Mahwah: Lawrence Erlbaum Associates.Google Scholar
Engle, R.E., Hendry, D.F., & Richard, J.F. (1983). Exogeneity. Econometrica, 51, 277–304.CrossRefGoogle Scholar
Fisher, R.A. (1922). On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society of London, Series A, 222, 309–368.Google Scholar
Fisher, R.A. (1973). Statistical methods for research workers. New York: Hafner.Google Scholar
Florens, J.P., Mouchart, M., & Rolin, J.-M. (1999). Elements of Bayesian statistics. New York: Marcel Dekker.Google Scholar
Gabrielsen, A. (1978). Consistency and identifiability. Journal of Econometrics, 8, 261–263.CrossRefGoogle Scholar
Haavelmo, T. (1944). The probability approach in econometrics. Econometrica, 12. Supplement iii–vi+1–115.CrossRefGoogle Scholar
Haberman, S. (2004). Joint and conditional maximum likelihood estimation for the Rasch model for binary responses (Technical Report RR-04-20). Princeton, NJ: ETS.Google Scholar
Koopmans, T.C. (1949). Identification problems in economic model construction. Econometrica, 17, 125–144.CrossRefGoogle Scholar
Koopmans, T.C., & Reiersøl, O. (1950). The identification of structural characteristics. The Annals of Mathematical Statistics, 21, 165–181.CrossRefGoogle Scholar
Lancaster, T. (2000). The incidental parameter problem since 1948. Journal of Econometrics, 95, 391–413.CrossRefGoogle Scholar
Le Cam, L., & Schwartz, L. (1960). A necessary and sufficient condition for the existence of consistent estimates. The Annals of Mathematical Statistics, 31, 140–150.CrossRefGoogle Scholar
Linacre, J. (2012). Winsteps (Version 3.75.0) [Computer software]. [Computer software manual].Google Scholar
Lord, F. (1980). Applications of item response theory to practical testing problems. Mahwah: Lawrence Erlbaum Associates.Google Scholar
Manski, C.F. (1995). Identification problems in the social sciences. New York: Harvard University Press.Google Scholar
Maris, G., & Bechger, T. (2009). On interpreting the model parameters for the three parameter logistic model. Measurement: Interdisciplinary Research and Perspective, 7, 75–86.Google Scholar
McCullagh, P. (2002). What is a statistical model? (with discussion). The Annals of Statistics, 30, 1225–1310.CrossRefGoogle Scholar
Molenaar, I.W. (1995). Estimation of item parameters. In Fischer, G.H., & Molenaar, I.W. (Eds.), Rasch models. Foundations, recent developments and applications (pp. 39–51). New York: Springer.Google Scholar
Neyman, J., & Scott, E. (1948). Consistent estimates based on partially consistent observations. Econometrica, 16, 1–32.CrossRefGoogle Scholar
Oulhaj, A., & Mouchart, M. (2003). Partial sufficiency with connection to the identification problem. Metron. International Journal of Statistics, LXI, 267–283.Google Scholar
Paulino, C.D.M., & Pereira, C.A.B. (1994). On identifiability of parametric statistical models. Journal of the Italian Statistical Society, 1, 125–151.CrossRefGoogle Scholar
Raoult, J.P. (1975). Structures statistiques. Paris: Presses Universitaires de France.Google Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. The Danish Institute for Educational Research (Expanded edition, 1980). Chicago: The University Chicago Press.Google Scholar
Revuelta, J. (2010). Estimating difficulty from polytomous categorial data. Psychometrika, 75, 331–350.CrossRefGoogle Scholar
Rothenberg, T.J. (1971). Identification in parametric models. Econometrica, 39, 577–591.CrossRefGoogle Scholar
San Martín, E., & González, J. (2010). Bayesian identifiability: contributions to an inconclusive debate. Chilean Journal of Statistics, 1, 69–91.Google Scholar
San Martín, E., González, J., & Tuerlinckx, F. (2009). Identified parameters, parameters of interest and their relationships. Measurement: Interdisciplinary Research and Perspective, 7, 95–103.Google Scholar
San Martín, E., Jara, A., Rolin, J.-M., & Mouchart, M. (2011). On the Bayesian nonparametric generalization of IRT-type models. Psychometrika, 76, 385–409.CrossRefGoogle Scholar
San Martín, E., & Quintana, F. (2002). Consistency and identifiability revisited. Brazilian Journal of Probability and Statistics, 16, 99–106.Google Scholar
San Martín, E., & Rolin, J. (2013). Identification of parametric Rasch-type models. Journal of Statistical Planning and Inference, 143, 116–130.CrossRefGoogle Scholar
San Martín, E., Rolin, J.M., & Castro, L.M. (2013). Identification of the 1PL model with guessing parameter: parametric and semi-parametric results. Psychometrika, 78, 341–379.CrossRefGoogle ScholarPubMed
Thissen, D., & Wainer, H. (2001). Test scoring. New York: Psychology Press.CrossRefGoogle Scholar
Thurstone, L.L. (1935). The vectors of mind. Multiple-factor analysis for the isolation of primary traits. Chicago: The University Chicago Press.CrossRefGoogle Scholar
Van der Linden, W. (2010). Linking response-time parameters onto a common scale. Journal of Educational Measurement, 47(1), 92–114.CrossRefGoogle Scholar
Van der Linden, W., & Hambleton, R. (1997). Handbook of modern item response theory. New York: Springer.CrossRefGoogle Scholar
Wingersky, M. (1983). LOGIST: a program for computing maximum likelihood procedures for logistic test models. In Hambleton, R.K. (Ed.), Applications of item response theory.Google Scholar
Wood, R.L., Wingersky, M.S., & Lord, F.M. (1976). LOGIST: a computer program for estimating examinee ability and item characteristic curve parameters. Princeton: Educational Testing Service (RM 76-6) [Computer program].Google Scholar
Woods, C. (2008). Ramsay-curve item response theory for the three-parameter logistic item response model. Applied Psychological Measurement, 32, 447–465.CrossRefGoogle Scholar