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

  • Klaas Sijtsma
  • L. van der Andries Ark
Part of the Lecture Notes in Statistics book series (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.

Keywords

Differential Item Functioning Item Response Theory Latent Trait Item Response Theory Model Item Response Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Agresti, A. (1990). Categorical data analysis. New York: Wiley.MATHGoogle Scholar
  2. 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 Unpublished doctoral dissertation, University of Twente, Enschede, The Netherlands.Google Scholar
  3. Andrich, D. (1978). A rating scale formulation for ordered response categories. Psychometrika, 43, 561–573.MATHCrossRefGoogle Scholar
  4. Barlow, R.E., Bartholomew, D.J., Bremner, J.M., & Brunk, H.D. (1972). Statistical inference under order restrictions. New York: Wiley.MATHGoogle Scholar
  5. 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.MathSciNetMATHCrossRefGoogle Scholar
  6. Debets, P., & Brouwer, E. (1989). MSP: A program for Mokken scale analysis for polytomous items [Software manual]. Groningen: ProGAMMA.Google Scholar
  7. Grayson, D.A. (1988). Two-group classification in latent trait theory: Scores with monotone likelihood ratio. Psychometrika, 53, 383–392.MathSciNetMATHCrossRefGoogle Scholar
  8. 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
  9. 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.CrossRefGoogle Scholar
  10. 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
  11. 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
  12. 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.MATHCrossRefGoogle Scholar
  13. 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.MathSciNetMATHCrossRefGoogle Scholar
  14. Hemker, B.T., Van der Ark, L.A., & Sijtsma, K. (2000). On measurement properties of sequential IRT models. Manuscript submitted for publication.Google Scholar
  15. Junker, B.W. (1993). Conditional association, essential independence and monotone unidimensional item response models. The Annals of Statistics, 21, 1359–1378.MathSciNetMATHCrossRefGoogle Scholar
  16. 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
  17. Junker, B.W., & Sijtsma, K. (2000). Latent and manifest monotonicity in item response models. Applied Psychological Measurement, 24, 65–81.CrossRefGoogle Scholar
  18. Lehmann, E.L. (1959). Testing statistical hypotheses. New York: Wiley.MATHGoogle Scholar
  19. Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.MATHCrossRefGoogle Scholar
  20. Mellenbergh, G.J. (1995). Conceptual notes on models for discrete polytomous item responses. Applied Psychological Measurement, 19, 91–100.CrossRefGoogle Scholar
  21. Mokken, R.J. (1971). A theory and procedure of scale analysis: With applications in political research. The Hague: Mouton.CrossRefGoogle Scholar
  22. Molenaar, I.W. (1982). Mokken scaling revisited. Kwantitatieve Methoden, 8, 145–164.Google Scholar
  23. Molenaar, I.W. (1983). Item steps (Heymans Bulletin HB-83-630-EX). Groningen: University of Groningen, Vakgroep Statistiek en Meettheorie.Google Scholar
  24. 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
  25. Molenaar, I.W. (1991). A weighted Loevinger H-coefficient extending Mokken scaling to multicategory items. Kwantitatieve Methoden, 37, 97–117.Google Scholar
  26. 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
  27. Molenaar, I.W., Debets, P., Sijtsma, K., & Hemker, B.T. (1994). User’s manual MSP [Software manual]. Groningen: ProGAMMA.Google Scholar
  28. Molenaar, I.W., & Sijtsma, K. (1988). Mokken’s approach to reliability estimation extended to multicategory items. Kwantitatieve Methoden, 28, 115–126.Google Scholar
  29. 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
  30. Rosenbaum, P.R. (1987). Probability inequalities for latent scales. British Journal of Mathematical and Statistical Psychology, 40, 157–168.MathSciNetMATHCrossRefGoogle Scholar
  31. Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.Google Scholar
  32. Samejima, F. (1995). Acceleration model in the heterogeneous case of the general graded response model. Psychometrika, 60, 549–572.MathSciNetMATHCrossRefGoogle Scholar
  33. Sijtsma, K. (1988). Contributions to Mokken’s nonparametric item response theory. Amsterdam: Free University Press.Google Scholar
  34. 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.CrossRefGoogle Scholar
  35. Sijtsma, K., & Hemker, B.T. (1998). Nonparametric polytomous IRT models for invariant item ordering, with results for parametric models. Psychometrika, 63, 183–200.MathSciNetMATHCrossRefGoogle Scholar
  36. 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
  37. 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.MathSciNetMATHCrossRefGoogle Scholar
  38. 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.CrossRefGoogle Scholar
  39. Suppes, P., & Zanotti, M. (1981). When are probabilistic explanations possible? Synthese, 48, 191–199.MathSciNetMATHCrossRefGoogle Scholar
  40. Tutz, G. (1990). Sequential item response models with an ordered response. British Journal of Mathematical and Statistical Psychology, 43, 39–55.MathSciNetMATHCrossRefGoogle Scholar
  41. 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
  42. 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
  43. Verhelst, N.D. (1992). Het eenparameter logistisch model (OPLM) [The oneparameter logistic model (OPLM)] (OPD Memorandum 92-3). Arnhem: CITO.Google Scholar
  44. Verhelst, N.D., & Verstralen, H.H.F.M. (1991). The partial credit model with non-sequential solution strategies. Arnhem: CITO.Google Scholar
  45. 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.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Klaas Sijtsma
    • 1
  • L. van der Andries Ark
    • 1
  1. 1.Both authors are at the Department MTOTilburg UniversityTilburgThe Netherlands

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