Hostname: page-component-77c78cf97d-v4t4b Total loading time: 0 Render date: 2026-04-25T04:14:37.826Z Has data issue: false hasContentIssue false
  • English
  • Français

Nested Logit Models for Multiple-Choice Item Response Data

Published online by Cambridge University Press:  01 January 2025

Youngsuk Suh  and
Daniel M. Bolt
Youngsuk Suh*
Affiliation:
University of Texas at Austin
Daniel M. Bolt
Affiliation:
University of Wisconsin-Madison
*
Requests for reprints should be sent to Youngsuk Suh, Department of Educational Psychology, University of Texas at Austin, 1 University Station D5800, Austin, TX 78712, USA. E-mail: yssuh327@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Nested logit item response models for multiple-choice data are presented. Relative to previous models, the new models are suggested to provide a better approximation to multiple-choice items where the application of a solution strategy precedes consideration of response options. In practice, the models also accommodate collapsibility across all distractor categories, making it easier to allow decisions about including distractor information to occur on an item-by-item or application-by-application basis without altering the statistical form of the correct response curves. Marginal maximum likelihood estimation algorithms for the models are presented along with simulation and real data analyses.

Keywords

Multiple-choice itemsmultiple-choice modelsnested logit modelsnominal response modelmarginal maximum likelihood estimationitem informationdistractor selection informationdistractor category collapsibility

Information

Type
Original Paper
Information
Psychometrika , Volume 75 , Issue 3 , September 2010 , pp. 454 - 473
Copyright
Copyright © 2010 The Psychometric Society

References

Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19(6), 716–723.CrossRefGoogle Scholar
Baker, F.B., Kim, S.-H. (2004). Item response theory: Parameter estimation techniques, (2nd ed.). New York: Marcel Dekker.CrossRefGoogle Scholar
Bechger, T.M., Maris, G., Verstralen, H.H.F.M., Verhelst, N.D. (2005). The Nedelsky model for multiple-choice items. In van der Ark, L.A., Croon, M.A., Sijtsma, K. (Eds.), New developments in categorical data analysis for the social and behavioral sciences, Mahwah: Lawrence Erlbaum Associates.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., Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443–459.CrossRefGoogle Scholar
Bozdogan, H. (1987). Model selection Akaike’s information criterion (AIC): The general theory and its analytical extensions. Psychometrika, 52, 345–370.CrossRefGoogle Scholar
Center for Placement Testing (1998). Mathematics placement test form 98-X. University of Wisconsin-Madison.Google Scholar
Digital Equipment Corporation (1997). DIGITAL Visual Fortran Version 6.0.A (Computer program). Maynard, MA.Google Scholar
Drasgow, F., Levine, M.V., Williams, E.A. (1985). Appropriateness measurement with polychotomous item response models and standardized indices. British Journal of Mathematical and Statistical Psychology, 38, 67–86.CrossRefGoogle Scholar
Drasgow, F., Levine, M.V., Tsien, S., Williams, B., Mead, A.D. (1995). Fitting polytomous item response theory models to multiple-choice tests. Applied Psychological Measurement, 19, 143–166.CrossRefGoogle Scholar
Hutchinson, T.P. (1991). Ability, partial information, and guessing: Statistical modeling applied to multiple-choice tests, Rundle Mall: Rumsby Scientific.Google Scholar
Kendall, M.G., Staurt, A. (1967). The advanced theory of statistics, New York: Hafner.Google Scholar
Love, T.E. (1997). Distractor selection ratios. Psychometrika, 62, 51–62.CrossRefGoogle Scholar
McFadden, D. (1981). Econometric models of probabilistic choice. In Manski, C.F., McFadden, D. (Eds.), Structural analysis of discrete data with econometric application (pp. 198–272). Cambridge: MIT Press.Google Scholar
McFadden, D. (1982). Qualitative response models. In Hildebrand, W. (Eds.), Advances in econometrics (pp. 1–37). Cambridge: Cambridge University Press.Google Scholar
Revuelta, J. (2004). Analysis of distractor difficulty in multiple-choice items. Psychometrika, 69, 217–234.CrossRefGoogle Scholar
Revuelta, J. (2005). An item response model for nominal data based on the rising selection ratios criterion. Psychometrika, 70, 305–324.CrossRefGoogle Scholar
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph Supplement, 17.Google Scholar
Samejima, F. (1972). A general model for free-response data. Psychometrika Monograph Supplement, 18.Google Scholar
Samejima, F. (1977). Weakly parallel tests in latent trait theory with some criticisms of classical test theory. Psychometrika, 42, 193–198.CrossRefGoogle Scholar
Samejima, F. (1979). A new family of models for the multiple choice item (Research Report No. 79-4). Knoxville: University of Tennessee, Department of Psychology.CrossRefGoogle Scholar
San Martin, E., del Pino, G., De Boeck, P. (2006). IRT models for ability-based guessing. Applied Psychological Measurement, 30, 183–203.Google Scholar
Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
Thissen, D., Steinberg, L. (1984). A response model for multiple choice items. Psychometrika, 49, 501–519.CrossRefGoogle Scholar
Thissen, D., Steinberg, L. (1986). A taxonomy of item response models. Psychometrika, 51, 567–577.CrossRefGoogle Scholar
Train, K. (2003). Discrete choice methods with simulation, Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Wollack, J.A. (1997). A nominal response model approach for detecting answer copying. Applied Psychological Measurement, 21, 307–320.CrossRefGoogle Scholar
Wollack, J.A., Bolt, D.M., Cohen, A.S., Lee, Y.-S. (2002). Recovery of item parameter in the nominal response model: A comparison of marginal maximum likelihood estimation and Markov chain Monte Carlo estimation. Applied Psychological Measurement, 26, 339–352.CrossRefGoogle Scholar
Zimowski, M., Muraki, E., Mislevy, R., Bock, R.D. (2003). BILOG-MG 3, Chicago: Scientific Software (Computer program)Google Scholar