Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-08T16:16:51.174Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonparametric Goodness-of-Fit Tests for the Rasch Model

Published online by Cambridge University Press:  01 January 2025

Ivo Ponocny
Ivo Ponocny*
Affiliation:
University of Vienna
*
Requests for reprints should be sent to Ivo Ponocny, Institut für Psychologie, Universität Wien, Liebiggasse 5, A-1010 Wien, AUSTRIA. E-Mail: ivo.ponocny@univie.ac.at
Get access Rights & Permissions [Opens in a new window]

Abstract

A Monte Carlo algorithm realizing a family of nonparametric tests for the Rasch model is introduced which are conditional on the item and subject marginals. The algorithm is based on random changes of elements of data matrices without changing the marginals; most powerful tests against all alternative hypotheses are given for which a monotone characteristic may be computed from the data matrix; alternatives may also be composed. Computation times are long, but exactp-values are approximated with the quality of approximation only depending on calculation time, but not on the number of persons. The power and the flexibility of the procedure is demonstrated by means of an empirical example where, among others, indicators for increased item similarities, the existence of subscales, violations of sufficiency of the raw score as well as learning processes were found. Many of the features described are implemented in the program T-Rasch 1.0 by Ponocny and Ponocny-Seliger (1999).

Keywords

nonparametric testsIRTgoodness-of-fit testsRasch modelexact tests
Type
Articles
Information
Psychometrika , Volume 66 , Issue 3 , September 2001 , pp. 437 - 459
Copyright
Copyright © 2001 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.)

Footnotes

The author wishes to thank Alexander Kaba, Birgit Bukasa, and Ulrike Wenninger of Österreichisches Kuratorium für Verkehrssicherheit (Austrian Traffic Safety Board) for allowing a data set to be used for the empirical example, and Elisabeth Ponocny-Seliger and the reviewers for many helpful comments. The menu-driven program T-Rasch 1.0 by Ponocny and Ponocny-Seliger (1999) can be obtained from Assessment Systems Corporation (http: //www.assess.com) or from the authors. (Note that it also performs exact person fit tests.)

References

Andersen, E.B. (1973). A goodness of fit test for the Rasch model. Psychometrika, 38, 123–140.CrossRefGoogle Scholar
Andersen, E.B. (1980). Discrete statistical models with social science applications. Amsterdam: North-Holland Publishing.Google Scholar
Békéssy, A., Békéssy, P., & Komlós, J. (1972). Asymptotic enumeration of regular matrices. Studia Scientiarum Mathematicarum Hungarica, 7, 343–353.Google Scholar
Bents, R., & Blank, R. (1991). Der Myers-Briggs Typenindikator (MBTI) [The Myers-Briggs type indicator], Weinheim: Beltz..Google Scholar
Bukasa, B., & Wenninger, U. (1998). MAT. Nonverbaler Intelligenztest Testmanual [MAT. Nonverbal test of intelligence. Test manual], Wien: Kuratorium für Verkehrssicherheit, Institut für Verkehrspsychologie.Google Scholar
Cressie, N., & Holland, P.W. (1983). Characterizing the manifest probabilities of latent trait models. Psychometrika, 48, 129–141.CrossRefGoogle Scholar
Fischer, G.H. (1974). Einführung in die Theorie psychologischer Tests [Introduction to mental test theory], Berne: Huber.Google Scholar
Fischer, G.H. (1981). On the existence and uniqueness of maximum likelihood estimates in the Rasch model. Psychometrika, 46, 59–77.CrossRefGoogle Scholar
Fischer, G.H. (1993). Notes on the Mantel-Haenszel procedure and another chi-squared test for the assessment of DIF. Methodika, 7, 88–100.Google Scholar
Fischer, G.H. (1995). Some neglected problems in IRT. Psychometrika, 60, 449–487.CrossRefGoogle Scholar
Fischer, G.H., & Molenaar, I.W. (1995). Rasch models. Foundations, recent developments, and applications. New York, NY: Springer.Google Scholar
Folks, J.L. (1984). Combination of independent tests. In Krishnaiah, P.R., & Sen, P.K. (Eds.), Handbooks of statistics, Vol. 4 (pp. 113–121). Amsterdam: North Holland.Google Scholar
Formann, A.K. (1979). WMT. Wiener Matrizen-Test. Ein Rasch-skalierter sprachfreier Intelligenztest [WMT. Viennese matrices test. A Rasch-scaled non-verbal test for intelligence], Weinheim: Beltz.Google Scholar
Glas, C.A.W. (1988). The derivation of some tests for the Rasch model from the multinomial distribution. Psychometrika, 53, 525–546.CrossRefGoogle Scholar
Glas, C.A.W. (1999). Modification indices for the 2-PL and the nominal response model. Psychometrika, 64, 273–294.CrossRefGoogle Scholar
Glas, C.A.W., & Verhelst, N.D. (1995). Testing the Rasch model. In Fischer, G.H., & Molenaar, I.W. (Eds.), Rasch models. Foundations, recent developments, and applications (pp. 37–51). New York, NY: Springer-Verlag.Google Scholar
Jannarone, R.J. (1986). Conjunctive item response theory kernels. Psychometrika, 51, 357–373.CrossRefGoogle Scholar
Kelderman, H. (1984). Loglinear Rasch model tests. Psychometrika, 49, 223–245.CrossRefGoogle Scholar
Klauer, K.C. (1991). An exact and optimal standardized person test for assessing consistency with the Rasch model. Psychometrika, 56, 213–228.CrossRefGoogle Scholar
Klauer, K.C. (1995). The assessment of person fit. In Fischer, G.H., & Molenaar, I.W. (Eds.), Rasch models. Foundations, recent developments, and applications (pp. 37–51). New York, NY: Springer-Verlag.Google Scholar
Ledermann, W., & Vajda, S. (1980). Handbook of applicable mathematics, Vol. I: Algebra. Chichester, U.K.: Wiley.Google Scholar
Lehmann, E.L. (1986). Testing statistical hypotheses. New York, NY: Wiley.CrossRefGoogle Scholar
Li, M.F., & Olejnik, S. (1997). The power of Rasch person-fit statistics in detecting unusual response patterns. Applied Psychological Measurement, 21, 215–231.CrossRefGoogle Scholar
Molenaar, I.W. (1983). Some improved diagnostics for failure in the Rasch model. Psychometrika, 48, 49–72.CrossRefGoogle Scholar
Molenaar, I.W. (1995). Estimation of item parameters. In Fischer, G.H., & Molenaar, I.W. (Eds.), Rasch models. Foundations, recent developments, and applications (pp. 37–51). New York, NY: Springer-Verlag.Google Scholar
Neyman, J., & Pearson, E.S. (1933). On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society, Series A, 231, 289–337.Google Scholar
Ponocny, I. (1996). Kombinatorische Modelltests für das Rasch-Modell [Combinatorial goodness-of-fit tests for the Rasch model], Austria: University of Vienna.Google Scholar
Ponocny, I. (2000). Exact person fit-indexes for the Rasch model. Psychometrika, 65, 29–42.CrossRefGoogle Scholar
Ponocny, I., & Ponocny-Seliger, E. (1999). T-Rasch 1.0. Groningen: ProGAMMA.Google Scholar
Prabhu, N.U. (1965). Stochastic processes. Basic theory and its applications. New York, NY: Macmillan.Google Scholar
Rao, C.R. (1973). Linear statistical inference and its applications 2nd ed., New York, NY: Wiley.CrossRefGoogle Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Kopenhagen: Danish Institute for Educational Research.Google Scholar
Raven, J.C. (1976). Standard progressive matrices. Oxford: Oxford Psychologists Press.Google Scholar
Stephens, M.A. (1986). Tests for the uniform distribution. In D'Agostino, R.B., & Stephens, M.A. (Eds.), Goodness-of-fit techniques. New York, NY: Dekker.Google Scholar
Verbeek, A., & Kroonenberg, P.M. (1985). A survey of algorithms for exact distributions of test statistics inr ×c contingency tables with fixed margins. Computational Statistics & Data Analysis, 3, 159–185.CrossRefGoogle Scholar