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A Class of Multidimensional IRT Models for Testing Unidimensionality and Clustering Items

Published online by Cambridge University Press:  01 January 2025

Francesco Bartolucci
Francesco Bartolucci*
Affiliation:
Università di Perugia
*
Requests for reprints should be sent to Francesco Bartolucci, Dipartimento di Economia, Finanza e Statistica, Università di Perugia, Via Pascoli 20, 06123 Perugia, Italy. E-mail: [email protected]
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Abstract

We illustrate a class of multidimensional item response theory models in which the items are allowed to have different discriminating power and the latent traits are represented through a vector having a discrete distribution. We also show how the hypothesis of unidimensionality may be tested against a specific bidimensional alternative by using a likelihood ratio statistic between two nested models in this class. For this aim, we also derive an asymptotically equivalent Wald test statistic which is faster to compute. Moreover, we propose a hierarchical clustering algorithm which can be used, when the dimensionality of the latent structure is completely unknown, for dividing items into groups referred to different latent traits. The approach is illustrated through a simulation study and an application to a dataset collected within the National Assessment of Educational Progress, 1996.

Keywords

2PL modelEM algorithmlatent class modelNAEP dataRasch model
Type
Original Paper
Information
Psychometrika , Volume 72 , Issue 2 , June 2007 , pp. 141 - 157
Copyright
Copyright © 2007 The Psychometric Society

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Footnotes

The author would like to thank the Editor, an Associate Editor and three anonymous referees for stimulating comments. I also thank L. Scaccia, F. Pennoni and M. Lupparelli for having done part of the simulations.

References

Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Petrov, B.N. & Csaki, F. (eds.), Second international symposium on information theory (pp. 267–281). Budapest: Akademiai Kiado.Google Scholar
Andersen, E.B. (1973). Conditional inference and models for measuring. Copenhagen: Mentalhygiejnisk Forlag.Google Scholar
Bartolucci, F., Forcina, A. (2001). Analysis of capture-recapture data with a Rasch-type model allowing for conditional dependence and multidimensionality. Biometrics, 57, 714–719.CrossRefGoogle ScholarPubMed
Bartolucci, F., Forcina, A. (2005). Likelihood inference on the underlying structure of IRT models. Psychometrika, 70, 31–43.CrossRefGoogle Scholar
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s ability. In Lord, F.M. & Novick, M.R. (eds.), Statistical theories of mental test scores (pp. 395–479). Reading, MA: Addison-Wesley.Google Scholar
Burnham, K.P., & Anderson, D.R. (2002), Model selection and multi-model inference: A practical information-theoretic approach (2nd ed.), New York: Springer-Verlag.Google Scholar
Carstensen, C.H., and Rost, J. (2001). MULTIRA (version 1.63) [Computer software and manual]. Retrived from http://www.multira.de.Google Scholar
Christensen, K.B., & Bjorner, J. B. (2003). SAS macros for Rasch based latent variable modelling (Research Report No. 03/13). Department of Biostatistics, University of Copenhagen.Google Scholar
Christensen, K.B., Bjorner, J.B., Kreiner, S., Petersen, J.H. (2002). Testing unidimensionality in polytomous Rasch models. Psychometrika, 67, 563–574.CrossRefGoogle Scholar
de Leeuw, J., Verhelst, N. (1986). Maximum likelihood estimation in generalized Rasch models. Journal of Educational Statistics, 11, 183–196.CrossRefGoogle Scholar
Dempster, A.P., Laird, N.M., Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, Series B, 39, 1–18.CrossRefGoogle Scholar
Embretson, S.E. (1996). Item response theory models and spurious interaction effects in factorial ANOVA designs. Applied Psychological Measurement, 20, 201–212.CrossRefGoogle Scholar
Forcina, A., Bartolucci, F. (2004). Modelling quality of life variables with non-parametric mixtures. Environmetrics, 15, 519–528.CrossRefGoogle Scholar
Formann, A.K. (1995). Linear logistic latent class analysis and the Rasch model. In Fischer, G.H. & Molenaar, I.W. (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 239–255). New York: Springer-Verlag.CrossRefGoogle Scholar
Glas, C.A.W. (1989). Contributions to estimating and testing Rasch models. Doctoral thesis. Enschede: University of Twente.Google Scholar
Glas, C.A.W., & Verhelst, N.D. (1995). Testing the Rasch model. In Fischer, G.H. & Molenaar, I.W. (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 69–75). New York: Springer-Verlag.CrossRefGoogle Scholar
Goodman, L.A. (1974). Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika, 61, 215–231.CrossRefGoogle Scholar
Hardouin, J.B., Mesbah, M. (2004). Clustering binary variables in subscales using an extended Rasch model and Akaike information criterion. Communications in Statistics. Theory and Methods, 33, 1277–1294.CrossRefGoogle Scholar
Hoijtink, H., Vollema, M. (2003). Contemporary extensions of the Rasch model. Quality & Quantity, 37, 263–276.CrossRefGoogle Scholar
Kelderman, H. (1984). Loglinear Rasch model tests. Psychometrika, 49, 223–245.CrossRefGoogle Scholar
Kelderman, H., Rijkes, C.P.M. (1994). Loglinear multidimensional IRT models for polytomously scored items. Psychometrika, 59, 147–176.CrossRefGoogle Scholar
Kiefer, J., Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many nuisance parameters. Annals of Mathematical Statistics, 27, 887–906.CrossRefGoogle Scholar
Kreiner, S., Christensen, K.B. (2004). Analysis of local dependence and multidimensionality in graphical loglinear Rasch models. Communications in Statistics: Theory and Methods, 33, 1239–1276.CrossRefGoogle Scholar
Lazarsfeld, P.F., & Henry, N.W. (1968). Latent structure analysis. Boston: Houghton Mifflin.Google Scholar
Lindsay, B., Clogg, C., Grego, J. (1991). Semiparametric estimation in the Rasch model and related exponential response models, including a simple latent class model for item analysis. Journal of the American Statistical Association, 86, 96–107.CrossRefGoogle Scholar
Magidson, J., Vermunt, J.K. (2001). Latent class factor and cluster models, bi-plots, and related graphical displays. Sociological Methodology, 31, 223–264.CrossRefGoogle Scholar
Martin-Löf, P. (1973). Statistiska modeller. Anteckningar fr{å}n seminarier las{å}ret 1969–1970, utarbetade av Rolf Sundberg. Obetydligt ändrat nytryck, October 1973. Stockholm: Institütet för Försäkringsmatemetik och Matematisk Statistisk vid Stockholms Universitet.Google Scholar
McKinley, R.L., & Reckase, M.D. (1982). The use of the general Rasch model with multidimensional item response data. Iowa City, IA: American College Testing.Google Scholar
Molenaar, I.W. (1983). Some improved diagnostics for failure of the Rasch model. Psychometrika, 48, 49–72.CrossRefGoogle Scholar
Rasch, G. (1961). On general laws and the meaning of measurement in psychology. Proceedings of the IV Berkeley Symposium on Mathematical Statistics and Probability, 4, 321–333.Google Scholar
Samejima, F. (1996). Evaluation of mathematical models for ordered polychotomous responses. Behaviormetrika, 23, 17–35.CrossRefGoogle Scholar
Stegelmann, W. (1983). Expanding the Rasch model to a general model having more than one dimension. Psychometrika, 48, 259–267.CrossRefGoogle Scholar
Thissen, D. (1982). Marginal maximum likelihood estimation for the one-parameter logistic model. Psychometrika, 47, 175–186.CrossRefGoogle Scholar
Tjur, T. (1982). A connection between Rasch’s item analysis model and a multiplicative Poisson model. Scandinavian Journal of Statistics, 9, 23–30.Google Scholar
van Abswoude, A.A.H., van der Ark, L.A., Sijtsma, K. (2004). A comparative study of test data dimensionality procedures under nonparametric IRT models. Applied Psychological Measurement, 28, 3–24.CrossRefGoogle Scholar
van den Wollenberg, A.L. (1979). The Rasch model and time limit tests. Doctoral thesis. Nijmegen: University of Nijmegen.Google Scholar
van den Wollenberg, A.L. (1982). Two new test statistics for the Rasch model. Psychometrika, 47, 123–140.CrossRefGoogle Scholar
Verhelst, N.D. (2001). Testing the unidimensionality assumption of the Rasch model. Methods of Psychological Research Online, 6, 231–271.Google Scholar
Vermunt, J.K. (2001). The use of restricted latent class models for defining and testing nonparametric and parametric item response theory models. Applied Psychological Measurement, 25, 283–294.CrossRefGoogle Scholar