Hostname: page-component-6898cfcb55-zsp5v Total loading time: 0 Render date: 2026-11-11T20:52:06.464Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonparametric Polytomous IRT Models for Invariant Item Ordering, with Results for Parametric Models

Published online by Cambridge University Press:  01 January 2025

Klaas Sijtsma and
Bas T. Hemker
Klaas Sijtsma*
Affiliation:
Tilburg University
Bas T. Hemker
Affiliation:
CITO National Institute for Educational Measurement
*
Requests for reprints should be sent to Klaas Sijtsma, Department of Research Methodology, Faculty of Social Sciences, Tilburg University, PO Box 90153, 5000 LE Tilburg, THE NETHERLANDS. E-mail: k.sijtsma@kub.nl
Get accessRights & Permissions [Opens in a new window]

Abstract

It is often considered desirable to have the same ordering of the items by difficulty across different levels of the trait or ability. Such an ordering is an invariant item ordering (IIO). An IIO facilitates the interpretation of test results. For dichotomously scored items, earlier research surveyed the theory and methods of an invariant ordering in a nonparametric IRT context. Here the focus is on polytomously scored items, and both nonparametric and parametric IRT models are considered.

The absence of the IIO property in two nonparametric polytomous IRT models is discussed, and two nonparametric models are discussed that imply an IIO. A method is proposed that can be used to investigate whether empirical data imply an IIO. Furthermore, only two parametric polytomous IRT models are found to imply an IIO. These are the rating scale model (Andrich, 1978) and a restricted rating scale version of the graded response model (Muraki, 1990). Well-known models, such as the partial credit model (Masters, 1982) and the graded response model (Samejima, 1969), do no imply an IIO.

Keywords

invariant item orderingitem response theorynonparametric polytomous IRT modelsparametric polytomous IRT models

Information

Type
Original Paper
Information
Psychometrika , Volume 63 , Issue 2 , June 1998, pp. 183 - 200
Copyright
Copyright © 1998 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Andrich, D. (1978). A rating scale formulation for ordered response categories. Psychometrika, 43, 561–573.CrossRefGoogle Scholar
Andrich, D. (1995). Distinctive and incompatible properties of two common classes of IRT models for graded responses. Applied Psychological Measurement, 19, 101–119.CrossRefGoogle Scholar
Cavalini, P. M. (1992). It's an ill wind that brings no good. Studies on odour annoyance and the dispersion of odorant concentrations from industries, The Netherlands: University of Groningen.Google Scholar
Chang, H., & Mazzeo, J. (1994). The unique correspondence of the item response function and item category response functions in polytomously scored item response models. Psychometrika, 59, 391–404.CrossRefGoogle Scholar
Hemker, B. T. (1996). Unidimensional IRT models for polytomous items, with results for Mokken scale analysis, The Netherlands: Utrecht University.Google Scholar
Hemker, B. T., Sijtsma, K., & Molenaar, I. W. (1995). Selection of unidimensional scales from a multidimensional item bank in the polytomous Mokken IRT model. Applied Psychological Measurement, 19, 337–352.CrossRefGoogle Scholar
Hemker, B. T., Sijtsma, K., Molenaar, I. W., & Junker, B. W. (1996). Polytomous IRT models and monotone likelihood ratio of the total score. Psychometrika, 61, 679–693.CrossRefGoogle Scholar
Hemker, B. T., Sijtsma, K., Molenaar, I. W., & Junker, B. W. (1997). Stochastic ordering using the latent trait and the sum score in polytomous IRT models. Psychometrika, 62, 331–347.CrossRefGoogle Scholar
Holland, P. W., & Rosenbaum, P. R. (1986). Conditional association and unidimensionality in monotone latent variable models. The Annals of Statistics, 14, 1523–1543.CrossRefGoogle Scholar
Junker, B. W. (1991). Essential independence and likelihood-based ability estimation for polytomous items. Psychometrika, 56, 255–278.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
Molenaar, W. (1970). Approximations to the Poisson, binomial, and hypergeometric distribution functions, Amsterdam: Mathematical Centre Tracts 31.Google Scholar
Molenaar, I. W. (1997). Nonparametric models for polytomous responses. In van der Linden, W. J., & Hambleton, R. K. (Eds.), Handbook of modern item response theory (pp. 369–380). New York: Springer.CrossRefGoogle Scholar
Molenaar, I. W., Debets, P., Sijtsma, K., & Hemker, B. T. (1994). User's manual MSP, Groningen, The Netherlands: iecProGAMMA.Google Scholar
Muraki, E. (1990). Fitting a polytomous item response model to Likert-type data. Applied Psychological Measurement, 14, 59–71.CrossRefGoogle Scholar
Muraki, E. (1992). A generalized partial credit model: application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.CrossRefGoogle Scholar
Rosenbaum, P. R. (1987). Probability inequalities for latent scales. British Journal of Mathematical and Statistical Psychology, 40, 157–168.CrossRefGoogle Scholar
Rosenbaum, P. R. (1987). Comparing item characteristic curves. Psychometrika, 52, 217–233.CrossRefGoogle Scholar
Samejima, F. (1969). Estimation of latent trait ability using a response pattern of graded scores. Psychometrika Monograph, No. 17.Google Scholar
Scheiblechner, H. (1995). Isotonic ordinal probabilistic models (ISOP). Psychometrika, 60, 281–304.CrossRefGoogle Scholar
Sijtsma, K., Debets, P., & Molenaar, I. W. (1990). Mokken scale analysis for polychotomous items: theory, a computer program and an empirical application. Quality & Quantity, 24, 173–188.CrossRefGoogle Scholar
Sijtsma, K., & Junker, B. W. (1996). A survey of theory and methods of invariant item ordering. British Journal of Mathematical and Statistical Psychology, 49, 79–105.CrossRefGoogle ScholarPubMed
Thissen, D., & Steinberg, L. (1986). A taxonomy of item response models. Psychometrika, 51, 567–577.CrossRefGoogle Scholar
Verhelst, N. D., & Glas, C. A. W. (1995). The one parameter logistic model. In Fischer, G. H., & Molenaar, I. W. (Eds.), Rasch models. Foundations, recent developments, and applications (pp. 215–237). New York: Springer.Google Scholar