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Progress in NIRT Analysis of Polytomous Item Scores: Dilemmas and Practical Solutions

  • Chapter
Essays on Item Response Theory

Part of the book series: Lecture Notes in Statistics ((LNS,volume 157))

Abstract

This chapter discusses several open problems in nonparametric polytomous item response theory: (1) theoretically, the latent trait θ is not stochastically ordered by the observed total score X+; (2) the models do not imply an invariant item ordering; and (3) the regression of an item score on the total score X+ or on the restscore R is not a monotone nondecreasing function and, as a result, it cannot be used for investigating the monotonicity of the item step response function. Tentative solutions for these problems are discussed. The computer program MSP for nonparametric IRT analysis is based on models that neither imply the stochastic ordering property nor an invariant item ordering. Also, MSP uses item restscore regression for investigating item step response functions. It is discussed whether computer programs may be based (temporarily) on models that lack desirable properties and use methods that are not (yet) supported by sound psychometric theory.

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References

  • Agresti, A. (1990). Categorical data analysis. New York: Wiley.

    MATH  Google Scholar 

  • Akkermans, L.M.W. (1998). Studies on statistical models for polytomously scored test items. Unpublished doctoral dissertation, University of Twente, Enschede, The Netherlands.

    Google Scholar 

  • Andrich, D. (1978). A rating scale formulation for ordered response categories. Psychometrika, 43, 561–573.

    Article  MATH  Google Scholar 

  • Barlow, R.E., Bartholomew, D.J., Bremner, J.M., & Brunk, H.D. (1972). Statistical inference under order restrictions. New York: Wiley.

    MATH  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.

    Article  MathSciNet  MATH  Google Scholar 

  • Debets, P., & Brouwer, E. (1989). MSP: A program for Mokken scale analysis for polytomous items [Software manual]. Groningen: ProGAMMA.

    Google Scholar 

  • Grayson, D.A. (1988). Two-group classification in latent trait theory: Scores with monotone likelihood ratio. Psychometrika, 53, 383–392.

    Article  MathSciNet  MATH  Google Scholar 

  • Hemker, B.T. (1996). Unidimensional IRT models for polytomous items, with results for Mokken scale analysis. Unpublished doctoral dissertation, Utrecht University, The Netherlands.

    Google Scholar 

  • Hemker, B.T. (2001). Reversibility revisited and other comparisons of three types of polytomous IRT models. In A. Boomsma, M.A.J. van Duijn, & T.A.B. Snijders (Eds.), Essays on item response theory (pp. 277–296). New York: Springer-Verlag.

    Chapter  Google Scholar 

  • Hemker, B.T., & Sijtsma, K. (1998). A comparison of three general types of unidimensional IRT models for polytomous items. Manuscript submitted for publication.

    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.

    Article  Google 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.

    Article  MATH  Google 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Hemker, B.T., Van der Ark, L.A., & Sijtsma, K. (2000). On measurement properties of sequential IRT models. Manuscript submitted for publication.

    Google Scholar 

  • Junker, B.W. (1993). Conditional association, essential independence and monotone unidimensional item response models. The Annals of Statistics, 21, 1359–1378.

    Article  MathSciNet  MATH  Google Scholar 

  • Junker, B.W. (1996). Exploring monotonicity in polytomous item response data. Paper presented at the Annual Meeting of the National Council on Measurement in Education, New York.

    Google Scholar 

  • Junker, B.W., & Sijtsma, K. (2000). Latent and manifest monotonicity in item response models. Applied Psychological Measurement, 24, 65–81.

    Article  Google Scholar 

  • Lehmann, E.L. (1959). Testing statistical hypotheses. New York: Wiley.

    MATH  Google Scholar 

  • Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.

    Article  MATH  Google Scholar 

  • Mellenbergh, G.J. (1995). Conceptual notes on models for discrete polytomous item responses. Applied Psychological Measurement, 19, 91–100.

    Article  Google Scholar 

  • Mokken, R.J. (1971). A theory and procedure of scale analysis: With applications in political research. The Hague: Mouton.

    Book  Google Scholar 

  • Molenaar, I.W. (1982). Mokken scaling revisited. Kwantitatieve Methoden, 8, 145–164.

    Google Scholar 

  • Molenaar, I.W. (1983). Item steps (Heymans Bulletin HB-83-630-EX). Groningen: University of Groningen, Vakgroep Statistiek en Meettheorie.

    Google Scholar 

  • Molenaar, I.W. (1986). Een vingeroefening in item response theorie voor drie geordende antwoordcategorieën [An exercise in item response theory for three ordered answer categories]. In G.F. Pikkemaat & J.J.A. Moors (Eds.), Liber amicorum Jaap Muilwijk (pp. 39–57). Groningen: Econometrisch Instituut.

    Google Scholar 

  • Molenaar, I.W. (1991). A weighted Loevinger H-coefficient extending Mokken scaling to multicategory items. Kwantitatieve Methoden, 37, 97–117.

    Google Scholar 

  • Molenaar, I.W. (1997). Nonparametric models for polytomous responses. In W.J. van der Linden & R.K. Hambleton (Eds.), Handbook of modern item response theory (pp. 369–380). New York: Springer-Verlag.

    Google Scholar 

  • Molenaar, I.W., Debets, P., Sijtsma, K., & Hemker, B.T. (1994). User’s manual MSP [Software manual]. Groningen: ProGAMMA.

    Google Scholar 

  • Molenaar, I.W., & Sijtsma, K. (1988). Mokken’s approach to reliability estimation extended to multicategory items. Kwantitatieve Methoden, 28, 115–126.

    Google Scholar 

  • Molenaar, I.W., & Sijtsma, K. (2000). User’s manual MSP5 for Windows: A program for Mokken scale analysis for polytomous items. Version 5.0 [Software manual]. Groningen: ProGAMMA.

    Google Scholar 

  • Rosenbaum, P.R. (1987). Probability inequalities for latent scales. British Journal of Mathematical and Statistical Psychology, 40, 157–168.

    Article  MathSciNet  MATH  Google Scholar 

  • Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

    Google Scholar 

  • Samejima, F. (1995). Acceleration model in the heterogeneous case of the general graded response model. Psychometrika, 60, 549–572.

    Article  MathSciNet  MATH  Google Scholar 

  • Sijtsma, K. (1988). Contributions to Mokken’s nonparametric item response theory. Amsterdam: Free University Press.

    Google Scholar 

  • Sijtsma, K., Debets, P., & Molenaar, I.W. (1990). Mokken scale analysis for polychotomous items: Theory, a computer program and an application. Quality & Quantity, 24, 173–188.

    Article  Google Scholar 

  • Sijtsma, K., & Hemker, B.T. (1998). Nonparametric polytomous IRT models for invariant item ordering, with results for parametric models. Psychometrika, 63, 183–200.

    Article  MathSciNet  MATH  Google Scholar 

  • Sijtsma, K., & Hemker, B.T. (in press). A taxonomy of IRT models for ordering persons and items using simple sum scores. Journal of Educational and Behavioral Statistics.

    Google 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Sijtsma, K., & Meijer, R.R. (1992). A method for investigating the intersection of item response functions in Mokken’s nonparametric IRT model. Applied Psychological Measurement, 16, 149–157.

    Article  Google Scholar 

  • Suppes, P., & Zanotti, M. (1981). When are probabilistic explanations possible? Synthese, 48, 191–199.

    Article  MathSciNet  MATH  Google Scholar 

  • Tutz, G. (1990). Sequential item response models with an ordered response. British Journal of Mathematical and Statistical Psychology, 43, 39–55.

    Article  MathSciNet  MATH  Google Scholar 

  • Van der Ark, L.A. (2000). Practical consequences of stochastic ordering of the latent trait under various polytomous IRT models. Manuscript in preparation.

    Google Scholar 

  • Van Engelenburg, G. (1997). On psychometric models for polytomous items with ordered categories within the framework of item response theory. Unpublished doctoral dissertation, University of Amsterdam.

    Google Scholar 

  • Verhelst, N.D. (1992). Het eenparameter logistisch model (OPLM) [The oneparameter logistic model (OPLM)] (OPD Memorandum 92-3). Arnhem: CITO.

    Google Scholar 

  • Verhelst, N.D., & Verstralen, H.H.F.M. (1991). The partial credit model with non-sequential solution strategies. Arnhem: CITO.

    Google Scholar 

  • Verweij, A.C., Sijtsma, K., & Koops, W. (1999). An ordinal scale for transitive reasoning by means of a deductive strategy. International Journal of Behavioral Development, 23, 241–264.

    Article  Google Scholar 

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Authors and Affiliations

  1. Both authors are at the Department MTO, Tilburg University, P.O. Box 90153, 5000, LE, Tilburg, The Netherlands

    Klaas Sijtsma & L. van der Andries Ark

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  1. Klaas Sijtsma
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  2. L. van der Andries Ark
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Editors and Affiliations

  1. Department of Statistics and Measurement Theory, University of Groningen, Grote Kruisstraat 2/1, 9712 TS, Groningen, The Netherlands

    Anne Boomsma , Marijtje A. J. van Duijn  & Tom A. B. Snijders ,  & 

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Sijtsma, K., van der Andries Ark, L. (2001). Progress in NIRT Analysis of Polytomous Item Scores: Dilemmas and Practical Solutions. In: Boomsma, A., van Duijn, M.A.J., Snijders, T.A.B. (eds) Essays on Item Response Theory. Lecture Notes in Statistics, vol 157. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0169-1_16

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  • DOI: https://doi.org/10.1007/978-1-4613-0169-1_16

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-95147-8

  • Online ISBN: 978-1-4613-0169-1

  • eBook Packages: Springer Book Archive

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