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The generalized Logit-Linear Item Response Model for Binary-Designed Items

Published online by Cambridge University Press:  01 January 2025

Javier Revuelta
Javier Revuelta*
Affiliation:
Autonoma University of Madrid
*
Requests for reprints should be sent to Javier Revuelta, Department of Social Psychology and Methodology, Autonoma University of Madrid, 28049 Madrid, Spain. E-mail: javier.revuelta@uam.es
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Abstract

This paper introduces the generalized logit-linear item response model (GLLIRM), which represents the item-solving process as a series of dichotomous operations or steps. The GLLIRM assumes that the probability function of the item response is a logistic function of a linear composite of basic parameters which describe the operations, and the coefficients depend on three design matrices X, Y and Z. The GLLIRM provides a tool for testing hypotheses on the item-solving process and generalizes existing models. An empirical application is included, in which the model is applied to evaluate sources of difficulty and pairwise item interactions in a logical analysis test.

Keywords

generalized logit-linear item response modeldesign matriceslinear restrictionsitem response theory

Information

Type
Theory and Methods
Information
Psychometrika , Volume 73 , Issue 3 , September 2008 , pp. 385 - 405
Copyright
Copyright © 2007 The Psychometric Society

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Footnotes

This research was supported by the Comunidad de Madrid grant CCG06-UAM/ESP-0043.

References

Adams, R.J., Wilson, M., & Wang, W.C. (1997). The multidimensional random coefficients multinomial logit model. Applied Psychological Measurement, 21, 1–23.CrossRefGoogle Scholar
Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Petrov, B.N., & Csaki, F. (Eds.), The second international symposium on information theory, Budapest: Akademiai Kiado.Google Scholar
Albert, J.H. (1992). Bayesian estimation of normal ogive item response curves using Gibbs sampling. Journal of Educational Statistics, 17, 251–269.CrossRefGoogle Scholar
Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561–573.CrossRefGoogle Scholar
Andrich, D. (1985). A latent trait model for items with response dependencies. Implications for test construction and analysis. In Embretson, S.E. (Eds.), Test design. Developments in psychology and psychometrics, New York: Academic Press.Google Scholar
Batchelder, W.H., & Riefer, D.M. (1999). Theoretical and empirical review of multinomial process tree modeling. Psychonomic Bulletin and Review, 6, 57–86.CrossRefGoogle ScholarPubMed
Bechger, T.M., Verstralen, H.H.F.M., & Verhelst, N.D. (2002). Equivalent linear logistic test models. Psychometrika, 67, 123–136.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 tests scores, Reading: Addison–Wesley.Google Scholar
Bock, R.D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37, 29–51.CrossRefGoogle Scholar
Bock, R.D. (1997). The nominal categories model. In van der Linden, W.J., & Hambleton, R.K. (Eds.), Handbook of modern item response theory, New York: Springer.Google Scholar
Bock, R.D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443–459.CrossRefGoogle Scholar
Bock, R.D., Gibbons, R., Schilling, S.G., Muraki, E., Wilson, D.T., & Wood, R. (2003). Testfact 4.0, Lincolnwood: Scientific Software International.Google Scholar
Bradlow, E.T., Wainer, H., & Wang, X. (1999). A Bayesian random effects model for testlets. Psychometrika, 64, 153–168.CrossRefGoogle Scholar
Breusch, T.S., & Pagan, A.R. (1980). The Lagrange multiplier test and its applications to model specification in econometrics. The Review of Economic Studies, 47, 239–253.CrossRefGoogle Scholar
Buse, A. (1982). The likelihood ratio, Wald and Lagrange multiplier tests. An expository note. The American Statistician, 36, 153–157.Google Scholar
Cox, D.R., & Hinkley, D.V. (1974). Theoretical statistics, New York: Chapman and Hall.CrossRefGoogle Scholar
De Boeck, P., & Wilson, M. (2004). Explanatory item response models. A generalized linear and nonlinear approach, New York: Springer.CrossRefGoogle Scholar
Embretson, S. (1984). A general latent trait model for response processes. Psychometrika, 49, 175–186.CrossRefGoogle Scholar
Embretson, S. (1999). Generating items during testing. Psychometric issues and models. Psychometrika, 64, 407–433.CrossRefGoogle Scholar
Fischer, G.H. (1997). Unidimensional linear logistic Rasch models. In van der Linden, W.J., & Hambleton, R.K. (Eds.), Handbook of modern item response theory, New York: Springer.Google Scholar
Fischer, G.H. (2004). Remarks on “Equivalent linear logistic test models” by Bechger, Verstralen, and Verhelst (2002). Psychometrika, 69, 305–315.CrossRefGoogle Scholar
Fischer, G.H., & Molenaar, I.W. (1995). Rasch models. Foundations, recent developments and applications, New York: Springer.Google Scholar
Gelman, A., Carlin, J.B., Stern, H.S., & Rubin, D.B. (1995). Bayesian data analysis, London: Chapman and Hall.CrossRefGoogle Scholar
Glas, C.A.W. (1999). Modification indices for the 2–PL and the nominal response model. Psychometrika, 64, 273–294.CrossRefGoogle Scholar
Haberman, S.J. (2007). The interaction model. In von Davier, M., & Carstensen, C.H. (Eds.), Multivariate and mixture distribution Rasch models, New York: Springer.Google Scholar
Hoijtink, H., & Molenaar, I.W. (1997). A multidimensional item response model. Constrained latent class analysis using the Gibbs sampler and Posterior Predictive Checks. Psychometrika, 62, 171–189.CrossRefGoogle Scholar
Hu, X., & Batchelder, W.H. (1994). The statistical analysis of general processing tree models with the EM algorithm. Psychometrika, 59, 21–47.CrossRefGoogle Scholar
Jannarone, R.J. (1986). Conjunctive item response theory kernels. Psychometrika, 51, 357–373.CrossRefGoogle Scholar
Jannarone, R.J. (1997). Models for locally dependent responses. Conjunctive item response theory. In van der Linden, W.J., & Hambleton, R.K. (Eds.), Handbook of modern item response theory, New York: Springer.Google Scholar
Liu, C.L. (1968). Introduction to combinatorial mathematics, New York: McGraw–Hill.Google Scholar
Maris, E. (1995). Psychometric latent response models. Psychometrika, 60, 523–547.CrossRefGoogle Scholar
Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.CrossRefGoogle Scholar
Mellenbergh, G.J. (1995). Conceptual notes on models for discrete polytomous item responses. Applied Psychological Measurement, 19, 91–100.CrossRefGoogle Scholar
Mislevy, R.J., & Verhelst, N. (1990). Modeling item responses when different subjects employ different solution strategies. Psychometrika, 55, 195–215.CrossRefGoogle Scholar
Muraki, E. (1992). A generalized partial credit model. Application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.CrossRefGoogle Scholar
Muthén, L.K., & Muthén, B.O. (2006). MPLUS. Statistical analysis with latent variables. User’s guide, Los Angeles: Muthén & Muthén.Google Scholar
Patz, R.J., & Junker, B.W. (1999). A straightforward approach to Markov chain Monte Carlo methods. Journal of Educational and Behavioral Statistics, 24, 146–178.CrossRefGoogle Scholar
Rao, C.R. (1973). Linear statistical inference and its applications, (2nd ed.). New York: Wiley.CrossRefGoogle Scholar
Rao, C.R., & Mitra, S.K. (1971). Generalized inverse of matrices and its applications, New York: Wiley.Google Scholar
Reiser, M. (1996). Analysis of residuals for the multinomial item response model. Psychometrika, 61, 509–528.CrossRefGoogle Scholar
Schilling, S., & Bock, R.D. (2005). High-dimensional maximum marginal likelihood item factor analysis by adaptive quadrature. Psychometrika, 70, 533–555.CrossRefGoogle Scholar
Schwarz, G. (1978). Estimating the dimensions of a model. Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
Thissen, D. (2003). Multilog 7.0, Lincolnwood: Scientific Software International.Google Scholar
Tversky, A., & Sattath, S. (1979). Preference trees. Psychological Review, 86, 542–573.CrossRefGoogle Scholar
von Davier, M. (in press). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology.Google Scholar
Yamamoto, K. (1989). Hybrid model of IRT and latent class model (ETS Research Report-89-41). Princeton, NJ: ETS.Google Scholar