Abstract
Nonparametric tests for testing the validity of polytomous ISOP-models (unidimensional ordinal probabilistic polytomous IRT-models) are presented. Since the ISOP-model is a very general nonparametric unidimensional rating scale model the test statistics apply to a great multitude of latent trait models. A test for the comonotonicity of item sets of two or more items is suggested. Procedures for testing the comonotonicity of two item sets and for item selection are developed. The tests are based on Goodman-Kruskal's gamma index of ordinal association and are generalizations thereof. It is an essential advantage of polytomous ISOP-models within probabilistic IRT-models that the tests of validity of the model can be performed before and without the model being fitted to the data. The new test statistics have the further advantage that no prior order of items or subjects needs to be known.
Similar content being viewed by others
References
Andrich, D. (1978a). A rating formulation for ordered response categories.Psychometrika, 43, 561–573.
Andrich, D. (1978b). Application of a psychometric rating model to ordered categories which are scored with successive integers.Applied Psychological Measurement, 2, 581–594.
Christodoulides, P. (1993).Individuelle politische Profile auf Zypern und der Einfluß der politischen Propaganda auf die politische Ideologie [Individual political profiles in Cyprus and the influence of political propaganda on political ideology]. Unpublished master's thesis, University of Vienna, Vienna, Austria. (In German)
Dabrowska, D., Pleszczynska, E., & Szczesny, (1981). Remarks on multivariate analogues of Kendall's tau.Communications in Statistics—Theory and Methods, Series A,10(23), 2435–2445.
Fischer, G.H., & Ponocny-Seliger, E. (1998).Structural Rasch modeling. Handbook of the usage of LPCM-WIN 1.0. Groningen: ProGAMMA.
Gans, L.P., & Robertson, C.A. (1981). Distributions of Goodman and Kruskal's gamma and Spearman's rho in 2 × 2 tables for small and moderate sample sizes.Journal of the American Statistical Association, 76, 942–946.
Goodman, L.A., & Kruskal, W.H. (1954). Measures of association for cross classifications, Part I.Journal of the American Statistical Association, 49 732–764.
Goodman, L.A., & Kruskal, W.H. (1959). Measures of association for cross classifications, Part II.Journal of the American Statistical Association, 54, 123–163.
Goodman, L.A., & Kruskal, W.H. (1963). Measures of association for cross classifications, Part III.Journal of the American Statistical Association, 58, 310–364.
Goodman, L.A., & Kruskal, W.H. (1972). Measures of association for cross classifications, Part IV.Journal of the American Statistical Association, 67, 415–421.
Guttman, L. (1986). Coefficients of polytonicity and monotonicity. In S. Kotz & N.L. Johnson (Eds.),Encyclopedia of statistical sciences (Vol. 7, pp. 80–87). New York, NY: Wiley.
Hemker, B.T. (1996).Unidimensional IRT models for polytomous items, with results for Mokken scale analysis. Unpublished doctorial dissertation, Universiteit Utrecht, Utrecht Netherlands.
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.
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.
Irtel, H. (1987). On specific objectivity as a concept in measurement. In E.E. Roskam & R. Suck (Eds.),Progress in mathematical psychology (Vol. 1, pp. 35–45). Amsterdam North-Holland: Elsevier.
Irtel, H. (1995). An extension of the concept of specific objectivity.Psychometrika, 60, 115–118.
Irtel, H., & Schmalhofer, F. (1982). Psychodiagnostik auf Ordinalskalenniveau: Meßtheoretische Grundlagen, Modelltest und Parameterschätzung [Psychodiagnostics at the level of ordinal scales: Measurement theoretical foundation, model controls and parameter estimation].Archiv für Psychologie, 134, 197–218.
Joe, H. (2001). Majorization and stochastic orders. In A.A.J. Marley (Ed.),International encyclopedia of the social & behavioral sciences: Mathematics and computer sciences. Oxford, U.K.: Elsevier Science.
Junker, B.W. (1993). Conditional association, essential independence and monotone unidimensional item response models.Annals of Statistics, 21, 1359–1378.
Junker, B.W. (1998). Some remarks on Scheiblechner's treatment of ISOP models.Psychometrika, 63, 73–85.
Junker, B.W., & Ellis, J.L. (1997). A characterization of monotone unidimensional latent variable models.Annals of Statistics, 25, 1327–1343.
Mielke, P.W., Jr. (1983). Goodman-Kruskal tau and gamma. In S. Kotz & N.L. Johnson (Eds.),Encyclopedia of statistical sciences (Vol. 3, pp. 446–449). New York, NY: Wiley.
Mokken, R.J. (1971).A theory and procedure of scale analysis. Paris/Den Haag: Mouton.
Mokken, R.J., & Lewis, C. (1982). A nonparametric approach to the analysis of dichotomous item responses.Applied Psychological Measurement, 6, 417–430.
Molenaar, I.W. (1991). A weighted Loevinger H-coefficient extending Mokken scaling to multicategory items.Kwantitatieve Methoden, 37, 97–117.
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, NY: Springer.
Nandakumar, R., Feng Yu, Hsin-Hung Li & Stout, W. (1998). Assessing Unidimensionality of Polytomous Data.Applied Psychological Measurement, 22, 99–115.
Rasch, G. (1977). On specific objectivity: An attempt at formalizing the request for generality and validity of scientific statements. In M. Blegvad (Ed.),The Danish yearbook of philosophy, 14, (pp. 58–94). Copenhagen: Munksgaard.
Robertson, T., Wright, F.T., & Dykstra, R.L. (1988).Order restricted statistical inference. New York, NY: John Wiley.
Roskam, E.E. (1995). Graded responses and joining categories: A rejoinder to Andrich's “Models for measurement, precision, and the nondichotomization of graded responses”.Psychometrika, 60, 27–35.
Scheiblechner, H. (1995). Isotonic ordinal probabilistic models (ISOP).Psychometrika, 60, 281–304.
Scheiblechner, H. (1998). Corrections of theorems in Scheiblechner's treatment of ISOP models and comments on Junker's remarks.Psychometrika, 63, 87–91.
Scheiblechner, H. (1999) Additive conjoint isotonic probabilistic models (ADISOP).Psychometrika, 64, 295–316.
Shaked, M., & Shantikumar, J. G. (Eds.). (1994).Stochastic orders and their applications. New York, NY: Academic Press.
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.
Sijtsma, K., & Hemker, B.T. (1998). Nonparametric polytomous IRT models for invariant item ordering, with results for parametric models.Psychometrika, 63, 183–200.
Sijtsma, K. (1998). Methodology Review: Nonparametric IRT Approaches to the Analysis of Dichotomous Item Scores.Applied Psychological Measurement, 22, 3–31.
Stouffer, S.A., Guttman, L., & Suchman, E.A. (1950).Measurement and prediction. New York, NY: John Wiley.
Stout, W.F. (1990). A new item response theory modeling approach with application to unidimensionality assessment and ability estimation.Psychometrika, 55, 293–325.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Scheiblechner, H. Nonparametric IRT: Testing the bi-isotonicity of isotonic probabilistic models (ISOP). Psychometrika 68, 79–96 (2003). https://doi.org/10.1007/BF02296654
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02296654