Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-08T09:22:12.209Z Has data issue: false hasContentIssue false
  • English
  • Français

Sample Size Determination for Rasch Model Tests

Published online by Cambridge University Press:  01 January 2025

Clemens Draxler
Clemens Draxler*
Affiliation:
Ludwig-Maximilians-University Munich
*
Requests for reprints should be sent to Clemens Draxler, Department Psychology, Ludwig-Maximilians-Universität München, Leopoldstraße 13, 80802 Munich, Germany. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with supplementing statistical tests for the Rasch model so that additionally to the probability of the error of the first kind (Type I probability) the probability of the error of the second kind (Type II probability) can be controlled at a predetermined level by basing the test on the appropriate number of observations. An approach to determining a practically meaningful extent of model deviation is proposed, and the approximate distribution of the Wald test is derived under the extent of model deviation of interest.

Keywords

Rasch modelWald testsample sizeerror of the second kind (Type II error)
Type
Original Paper
Information
Psychometrika , Volume 75 , Issue 4 , December 2010 , pp. 708 - 724
Copyright
Copyright © 2010 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.)

References

Agresti, A. (2002). Categorical data analyses, New York: Wiley.CrossRefGoogle Scholar
Aitchison, J., Silvey, S.D. (1958). Maximum likelihood estimation of parameters subject to. restraints. Annals of Mathematical Statistics, 29, 813–828.CrossRefGoogle Scholar
Andersen, E.B. (1970). Asymptotic properties of conditional maximum likelihood estimators. Journal of the Royal Statistical Society, Series B, 32, 283–301.CrossRefGoogle Scholar
Andersen, E.B. (1973). A goodness of fit test for the Rasch model. Psychometrika, 38, 123–140.CrossRefGoogle Scholar
Andersen, E.B. (1977). Sufficient statistics and latent trait models. Psychometrika, 42, 69–81.CrossRefGoogle Scholar
Andersen, E.B. (1980). Discrete statistical models with social science applications, Amsterdam: North-Holland.Google Scholar
Andersen, E.B. (1995). Polytomous Rasch models and their estimation. In Fischer, G.H., Molenaar, I.W. (Eds.), Rasch models—foundations, recent developments and applications (pp. 271–291). New York: Springer.Google Scholar
Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561–573.CrossRefGoogle Scholar
Andrich, D. (1978). A binomial latent trait model for the study of Likert-style attitude questionnaires. British Journal of Mathematical and Statistical Psychology, 31, 84–98.CrossRefGoogle Scholar
Chen, Y., Small, D. (2005). Exact tests for the Rasch model via sequential importance sampling. Psychometrika, 70, 11–30.CrossRefGoogle Scholar
Cohen, J. (1988). Statistical power analyses for the behavioral sciences, New York: Erlbaum.Google Scholar
Fischer, G.H. (1974). Einführung in die Theorie psychologischer Tests. (Introduction to the theory of psychological tests), Bern: Huber.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., Verhelst, N.D. (1989). Extensions of the partial credit model. Psychometrika, 54, 635–659.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. 69–95). New York: Springer.Google Scholar
Glas, C.A.W., Verhelst, N.D. (1995). Tests of fit for polytomous Rasch models. In Fischer, G.H., Molenaar, I.W. (Eds.), Rasch models—foundations, recent developments and applications (pp. 325–352). New York: Springer.Google Scholar
Kelderman, H. (1984). Log linear Rasch model tests. Psychometrika, 49, 223–245.CrossRefGoogle Scholar
Kelderman, H. (1989). Item bias detection using log linear IRT. Psychometrika, 54, 681–697.CrossRefGoogle Scholar
Martin-Löf, P. (1973). Statistiska modeller. (Statistical models. Notes from seminars 1969–1970 by Rolf Sundberg.) Stockholm.Google Scholar
Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.CrossRefGoogle Scholar
Molenaar, I.W. (1983). Some improved diagnostics for failure of the Rasch model. Psychometrika, 48, 49–72.CrossRefGoogle Scholar
Ponocny, I. (2001). Nonparametric goodness-of-fit tests for the Rasch model. Psychometrika, 66, 437–460.CrossRefGoogle Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests, Copenhagen: The Danish Institute of Education Research (Expanded Edition, 1980. Chicago: University of Chicago Press).Google Scholar
Rasch, G. (1961). On general laws and the meaning of measurement in psychology, Berkeley: University of California Press.Google Scholar
Satorra, A., Saris, W.E. (1985). The power of the likelihood ratio test in covariance structure analysis. Psychometrika, 50, 83–90.CrossRefGoogle Scholar
Snijders, T.A.B. (1991). Enumeration and simulation methods for 0-1 matrices with given marginals. Psychometrika, 56, 397–417.CrossRefGoogle Scholar
Stroud, T.W.F. (1972). Fixed alternatives and Wald’s formulation of the noncentral asymptotic behavior of the likelihood ratio statistic. Annals of Mathematical Statistics, 43, 447–454.CrossRefGoogle Scholar
van den Wollenberg, A. (1982). Two new test statistics for the Rasch model. Psychometrika, 47, 123–140.CrossRefGoogle Scholar
Verhelst, N.D. (2008). An efficient MCMC algorithm to sample binary matrices with fixed marginals. Psychometrika, 73, 705–728.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
Verhelst, N.D., Glas, C.A.W., Verstralen, H.H.F.M. (1994). OPLM: Computer program and manual, Arnhem: CITO.Google Scholar
Wald, A. (1943). Tests of statistical hypothesis concerning several parameters when the number of observations is large. Transactions of the American Mathematical Society, 54, 426–482.CrossRefGoogle Scholar
Wilson, M., Masters, G.N. (1993). The partial credit model and null categories. Psychometrika, 58, 87–99.CrossRefGoogle Scholar