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Equivalent Linear Logistic Test Models

Published online by Cambridge University Press:  01 January 2025

Timo M. Bechger ,
Huub H. F. M. Verstralen  and
Norman D. Verhelst
Timo M. Bechger*
Affiliation:
CITO, National Institute for Educational Measurement
Huub H. F. M. Verstralen
Affiliation:
CITO, National Institute for Educational Measurement
Norman D. Verhelst
Affiliation:
CITO, National Institute for Educational Measurement
*
Requests for reprints should be send to Timo Bechger, Cito, RO. Box 1034, 6801 MG Arnhem, THE NETHERLANDS. E-Mail: [email protected]
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Abstract

This paper is about the Linear Logistic Test Model (LLTM). We demonstrate that there are infinitely many equivalent ways to specify a model. An implication is that there may well be many ways to change the specification of a given LLTM and achieve the same improvement in model fit. To illustrate this phenomenon, we analyze a real data set using a Lagrange multiplier test for the specification of the model. This Lagrange multiplier test is similar to the modification index used in structural equation modeling.

Keywords

item response theory (IRT)linear logistic test modelLagrange multiplier test
Type
Articles
Information
Psychometrika , Volume 67 , Issue 1 , March 2002 , pp. 123 - 136
Copyright
Copyright © 2002 The Psychometric Society

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References

Aitchison, W., & Silvey, S.D. (1958). Maximum likelihood estimation of parameters subject to restraints. Annals of Mathematical Statistics, 28, 813–828.CrossRefGoogle Scholar
Azzalini, A. (1996). Statistical inference based on the likelihood. London, U.K.: Chapman and Hall.Google Scholar
Baumert, J., Heyn, S., Köller, O., & Lehrke, M. (1992). Naturwissenschaftliche und psychosoziale Bildungsprozesse im Jugendalter (BIJU)—Testdokumantation [Scientific and psychosocial developmental processes in young children], Kiel, Germany: IPN.Google Scholar
Bechger, T.M., Verhelst, N.D., & Verstralen, H.F.M. (2001). Identifiability of nonlinear logistic test models. Psychometrika, 66, 357–372.CrossRefGoogle Scholar
Bekker, P.A., Merckens, A., & Wansbeek, T.J. (1994). Identification, equivalent models, and computer algebra. Boston, MA: Academic Press.Google Scholar
Bollen, K.A. (1909). Structural equations with latent variables. New York, NY: Wiley.Google Scholar
Buse, A. (1982). The likelihood ratio, Wald and Lagrange multiplier test: An expository note. The American Statistician, 36(3), 153–157.Google Scholar
Butter, R.P., De Boeck, P., & Verhelst, N.D. (1998). An item response model with internal restrictions on item difficulty. Psychometrika, 63, 47–63.CrossRefGoogle Scholar
Fischer, G.H. (1974). Einfurung in die Theorie psychologischer Tests: Grundlagen und Anwendungen [Introduction to the theory of psychological tests: Foundations and applications], Bern, Switzerland: Verlag Hans Huber.Google Scholar
Fischer, G.H. (1995). The linear logistic test model. In Fischer, G.H., & Molenaar, I.W. (Eds.), Rasch models: Foundations, recent developments and applications (pp. 131–155). Berlin, Germany: Springer.CrossRefGoogle Scholar
Fisher, F.M. (1966). The identification problem in econometrics. New York, NY: McGraw-Hill.Google Scholar
Glas, C.A.W., & Verhelst, N.D. (1995). Testing the Rasch model. In Fischer, G.H., & Molenaar, I.W. (Eds.), Rasch models: Their foundations, recent developments and applications (pp. 69–95). New York, NY: Springer Verlag.CrossRefGoogle Scholar
Green, S.B., Marquis, J.G., Hersberger, S.L., Thompson, M.S., & McCollam, K.M. (1999). The overparameterized analysis of variance Model. Psychological Methods, 4, 214–233.CrossRefGoogle Scholar
Lang, S. (1987). Linear algebra 3rd ed., New York, NY: Springer Verlag.CrossRefGoogle Scholar
Luijben, T.C.W. (1989). Statistical guidance for model modification in covariance structure analysis. Amsterdam: Sociometric Research Foundation.Google Scholar
Luijben, T.C.W. (1991). Equivalent models in covariance structure analysis. Psychometrika, 56, 653–665.CrossRefGoogle Scholar
MacCallum, R. (1986). Specification searches in covariance structure modeling. Psychological Bulletin, 100, 107–120.CrossRefGoogle Scholar
Penrose, R. (1955). A generalized inverse for matrices. Proceedings of the Cambridge Philosophical Society, 62, 673–677.Google Scholar
Pringle, R.M., & Rayner, A.A. (1971). Generalized inverse matrices with applications to Statistics. London, U.K.: Charles Griffin.Google Scholar
Rao, C.R. (1947). Large sample tests of statistical hypothesis concerning several parameters with applications to the problems of estimation. Proceedings of the Cambridge Philosophical Society, 44, 50–57.Google Scholar
Rost, J. (1996). Testtheory, Testkonstruction [Test theory, test construction], Bern, Germany: Verlag Hans Huber.Google Scholar
Saris, W.E., Satorra, A., & Sörbom, D. (1987). The detection and correction of specification errors in structural equation models. In Glogg, C.C. (Eds.), Sociological Methodology 1987 (pp. 105–129). San Fransisco, CA: Jossey-Bass.Google Scholar
Shapiro, A. (1983). On local identifiability in structural models. Unpublished research report, University of South Africa, Department of Mathematics and Applied Mathematics.Google Scholar
Silvey, S.D. (1959). The Lagrange multiplier test. Annals of Mathematical Statistics, 30, 398–407.CrossRefGoogle Scholar
Sörbom, D. (1989). Model modification. Psychometrika, 54, 371–384.CrossRefGoogle Scholar
Sternberg, R.J. (1985). Human abilities: An information-processing approach. New York, NY: W.H. Freeman.Google Scholar
Tsai, R.-C. (2000). Remarks on the identifiability of Thurstonian ranking models: Case V, Case III, or neither?. Psychometrika, 65, 233–240.CrossRefGoogle Scholar
Verhelst, N.D. (1993). On the standard errors of parameter estimators in the Rasch model. (Measurement and Research department Reports, 93-1) Arnhem, Netherlands: Cito.Google Scholar
Volodin, N.A., & Adams, R.J. (1995). Identifying and estimating a D-dimensional item response model. Berkeley, California: University of California.Google Scholar
Whitely, S.E., & Schneider, L.M. (1981). Information structure of geometric analogies: A test theoretic approach. Applied Psychological Measurement, 5, 383–397.CrossRefGoogle Scholar