Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-08T03:07:56.500Z Has data issue: false hasContentIssue false
  • English
  • Français

Constant Latent Odds-Ratios Models and the Mantel-Haenszel Null Hypothesis

Published online by Cambridge University Press:  01 January 2025

David J. Hessen
David J. Hessen*
Affiliation:
University of Amsterdam
*
Requests for reprints should be sent to David J. Hessen, Department of Psychology, University of Amsterdam, Roetersstraat 15, 1018 WB, Amsterdam, THE NETHERLANDS. E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In the present paper, a new family of item response theory (IRT) models for dichotomous item scores is proposed. Two basic assumptions define the most general model of this family. The first assumption is local independence of the item scores given a unidimensional latent trait. The second assumption is that the odds-ratios for all item-pairs are constant functions of the latent trait. Since the latter assumption is characteristic of the whole family, the models are called constant latent odds-ratios (CLORs) models. One nonparametric special case and three parametric special cases of the general CLORs model are shown to be generalizations of the one-parameter logistic Rasch model. For all CLORs models, the total score (the unweighted sum of the item scores) is shown to be a sufficient statistic for the latent trait. In addition, conditions under the general CLORs model are studied for the investigation of differential item functioning (DIF) by means of the Mantel-Haenszel procedure.

Keywords

IRTodds-ratiosufficient statisticweak item independencemanifest monotonicityRasch modelDIFMantel-Haenszel procedure
Type
Original Paper
Information
Psychometrika , Volume 70 , Issue 3 , September 2005 , pp. 497 - 516
Copyright
Copyright © 2005 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

This research was supported by the Dutch Organization for Scientific Research (NWO), grant number 400-20-026.

References

Barton, M.A., Lord, F.M. (1981) An upper asymptote for the three-parameter logistic item-response model. Research Bulletin 81-20. Educational Testing Service, Princeton NJCrossRefGoogle Scholar
Birnbaum, A. (1968). Some latent trait models and their uses in inferring an examinee’s ability. In Lord, F.M., Novick, M.R. (Eds.), Statistical Theories of Mental Test Scores (pp. 397–479). Reading, MA: Addison-Wesley.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. Bern: Huber. (Introduction to the theory of psychological tests)Google 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). Derivations of the Rasch model. In Fischer, G.H., Molenaar, I.W. (Eds.), Rasch Models: Foundations, Recent Developments and Applications (pp. 15–38). Berlin Heidelberg, New York: Springer.CrossRefGoogle Scholar
Fischer, G.H. (1995). Some neglected problems in IRT. Psychometrika, 60, 459–487.CrossRefGoogle Scholar
Grayson, D.A. (1988). Two-group classification in latent trait theory: Scores with monotone likelihood ratio. Psychometrika, 53, 383–392.CrossRefGoogle Scholar
Hanson, B.A. (1998). Uniform DIF and DIF defined by differences in item response functions. Journal of Educational and Behavioral Statistics, 23, 244–253.CrossRefGoogle 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.CrossRefGoogle Scholar
Hessen, D.J. (2004). A new class of parametric IRT models for dichotomous item scores. Journal of Applied Measurement, 5, 385–397.Google ScholarPubMed
Holland, P.W., Thayer, D.T. (1988). Differential item performance and the Mantel–Haenszel procedure. In Wainer, H., Braun, H.I. (Eds.), Test Validity (pp. 129–145). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Huynh, H. (1994). A new proof for monotone likelihood ratio for the sum of independent Bernoulli random variables. Psychometrika, 59, 77–79.CrossRefGoogle Scholar
Junker, B.W. (1993). Conditional association, essential independence and monotone unidimensional item response models. The Annals of Statistics, 21, 1359–1378.CrossRefGoogle Scholar
Junker, B.W., Sijtsma, K. (2000). Latent and manifest monotonicity in item response models. Applied Psychological Measurement, 24, 65–81.CrossRefGoogle Scholar
Lehmann, E.L. (1994). Testing statistical hypotheses. New York: Wiley/Chapman Hall.Google Scholar
Lord, F.M., Novick, M.R. (1968). Statistical theories of mental test scores. Reading, MA: Addison-Wesley.Google Scholar
Mantel, N., Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute, 22, 719–748.Google ScholarPubMed
Meredith, W., Millsap, R.E. (1992). On the misuse of manifest variables in the detection of measurement bias. Psychometrika, 57, 289–311.CrossRefGoogle Scholar
Millsap, R.E., Everson, H.T. (1993). Methodology review: Statistical approaches for assessing measurement bias. Applied Psychological Measurement, 17, 297–334.CrossRefGoogle Scholar
Millsap, R.E., Meredith, W. (1992). Inferential conditions in the statistical detection of measurement bias. Applied Psychological Measurement, 16, 389–402.CrossRefGoogle Scholar
Mokken, R.J. (1971). A theory and procedure of scale analysis. Berlin: De Gruyter.CrossRefGoogle Scholar
Mokken, R.J., Lewis, C. (1982). A nonparametric approach to the analysis of dichotomous item responses. Applied Psychological Measurement, 6, 417–430.CrossRefGoogle Scholar
Mokken, R.J., Lewis, C., Sijtsma, K. (1986). Rejoinder to “The Mokken scale: A critical discussion”. Applied Psychological Measurement, 10, 279–285.CrossRefGoogle Scholar
Raju, N.S. (1988). The area between two item characteristic curves. Psychometrika, 53, 495–502.CrossRefGoogle Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Nielsen and Lydiche.Google Scholar
Rosenbaum, P.R. (1984). Testing the conditional independence and monotonicity assumptions of item response theory. Psychometrika, 49, 425–435.CrossRefGoogle Scholar
Rosenbaum, P.R. (1987). Comparing item characteristic curves. Psychometrika, 52, 217–233.CrossRefGoogle Scholar
Scheiblechner, H. (1995). Isotonic ordinal probabilistic models (ISOP). Psychometrika, 60, 281–304.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
Weitzman, R.A. (1996). The Rasch model plus guessing. Educational and Psychological Measurement, 56, 779–790.CrossRefGoogle Scholar
Zwick, R. (1990). When do item response function and Mantel–Haenszel definitions of differential item functioning coincide. Journal of Educational Statistics, 15, 185–197.CrossRefGoogle Scholar