Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-07T18:21:16.877Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic Order in Dichotomous Item Response Models for Fixed, Adaptive, and Multidimensional Tests

Published online by Cambridge University Press:  01 January 2025

Wim J. van der Linden
Wim J. van der Linden*
Affiliation:
University of Twente
*
Requests for reprints should be sent to W. J. van der Linden, Department of Educational Measurement and Data Analysis, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Dichotomous IRT models can be viewed as families of stochastically ordered distributions of responses to test items. This paper explores several properties of such distributions. In particular, it is examined under what conditions stochastic order in families of conditional distributions is transferred to their inverse distributions, from two families of related distributions to a third family, or from multivariate conditional distributions to a marginal distribution. The main results are formulated as a series of theorems and corollaries which apply to dichotomous IRT models. One part of the results holds for unidimensional models with fixed item parameters. The other part holds for models with random item parameters as used, for example, in adaptive testing or for tests with multidimensional abilities.

Keywords

item response theory (IRT)stochastic ordercomputerize adaptive testing (CAT)multidimensional testsclassical test theory
Type
Original Paper
Information
Psychometrika , Volume 63 , Issue 3 , September 1998 , pp. 211 - 226
Copyright
Copyright © 1998 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

Portions of this paper were presented at the 9th European Meeting of the Psychometric Society, Leiden, The Netherlands, July 4–7, 1995. The author is indebted to the referees for their comments on the previous version of the manuscript, as well as to the Greek fisherman who picked up the only copy of the set of handwritten notes for this paper from the harbor of Karpathos.

References

Ahmed, A. N., León, R., & Proschan, F. (1981). Generalized association, with applications in multivariate statistics. Annals of Statistics, 9, 168–176.CrossRefGoogle Scholar
Apostol, T. M. (1967). One-variable calculus, with an introduction to linear algebra, New York, NY: Wiley.Google Scholar
Casella, G., & Berger, R. L. (1990). Statistical inference, Pacific Grove, CA: Wadsworth & Brooks/Cole.Google Scholar
Chuan, D. T., Chen, J. J., & Novick, M. R. (1981). Theory and practice for the use of cut-scores for personnel decisions. Journal of Educational Statistics, 6, 129–152.CrossRefGoogle Scholar
Ellis, J. L. (1993). Subpopulation invariance of patterns in covariance matrices. British Journal of Mathematical and Statistical Psychology, 46, 231–254.CrossRefGoogle Scholar
Ellis, J. L., & van den Wollenberg, A. L. (1993). Local homogeneity in latent trait models: A characterization of the homogenous monotone IRT model. Psychometrika, 58, 417–429.CrossRefGoogle Scholar
Esary, J. D., Proschan, F., & Walkup, D. W. (1967). Association of random variables, with applications. Annals of Mathematical Statistics, 38, 1466–1474.CrossRefGoogle Scholar
Fischer, G. H., & Molenaar, I. W. (1995). Rasch models: Foundations, recent developments, and applications, New York City, NY: Springer-Verlag.CrossRefGoogle Scholar
Glas, C. A. W. (1992). A Rasch model with a multivariate distribution of ability. In Wilson, M. (Eds.), Objective measurement: Theory into practice, Norwood, NJ: Ablex.Google Scholar
Grayson, D. A. (1988). Two-group classification in latent trait theory: Scores with monotone likelihood ratio. Psychometrika, 53, 283–292.CrossRefGoogle Scholar
Hambleton, R. K., & Swaminathan, H. (1985). Item response theory: Principles and applications, Boston, MA: Kluwer-Nijhoff Publishing.CrossRefGoogle Scholar
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.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
Holland, P. W. (1981). When are item response models consistent with observed data?. Psychometrika, 46, 79–92.CrossRefGoogle Scholar
Holland, P. W. (1990). On the sampling theory foundations of item response theory models. Psychometrika, 55, 577–601.CrossRefGoogle Scholar
Holland, P. W., & Rosenbaum, P. R. (1986). Conditional association and unidimensionality in monotone latent variable models. Annals of Statistics, 14, 1523–1543.CrossRefGoogle 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. (1991). Essential independence and likelihood-based ability estimations for polytomous items. Psychometrika, 56, 255–278.CrossRefGoogle Scholar
Junker, B. W. (1993). Conditional association, essential independence, and monotone unidimensional item response models. Annals of Statistics, 21, 1359–1378.CrossRefGoogle Scholar
Kelley, T. L. (1939). The selection of upper and lower groups for the validation of test items. Journal of Educational Psychology, 30, 17–24.CrossRefGoogle Scholar
Lehmann, E. L. (1986). Testing statistical hypotheses 2nd ed.,, New York, NY: Wiley.CrossRefGoogle Scholar
Lord, F. M. (1980). Applications of item response theory to practical testing problems, Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Mislevy, R. J., & Wu, P.-K. (1988). Inferring examinee ability when some item responses are missing, Princeton, NJ: Educational Testing Service.CrossRefGoogle Scholar
Mokken, R. J. (1971). A theory and procedure of scale analysis, Den Haag, The Netherlands: Mouton.CrossRefGoogle Scholar
Mokken, R. J. (1997). Nonparametric models for dichotomous responses. In van der Linden, W. J., & Hambleton, R. K. (Eds.), Handbook of modern item response theory (pp. 351–368). New York, NY: Springer-Verlag.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
Molenaar, W. (1997). Nonparametric models for polytomous responses. In van der Linden, W. J., & Hambleton, R. K. (Eds.), Handbook of modern item response theory (pp. 369–380). New York, NY: Springer-Verlag.CrossRefGoogle Scholar
Ramsay, J. O. (1991). Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika, 56, 611–630.CrossRefGoogle Scholar
Ramsay, J. O. (1997). A functional approach to modeling test data. In van der Linden, W. J., & Hambleton, R. K. (Eds.), Handbook of modern item response theory (pp. 381–394). New York, NY: Springer-Verlag.CrossRefGoogle Scholar
Reckase, M. D. (1997). A linear logistic multidimensional model for dichotomous item response data. In van der Linden, W. J., & Hambleton, R. K. (Eds.), Handbook of modern item response theory (pp. 271–286). New York, NY: Springer-Verlag.CrossRefGoogle 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. (1985). Comparing distributions of item responses for two groups. British Journal of Mathematical and Statistical Psychology, 38, 206–215.CrossRefGoogle Scholar
Stout, W. F. (1987). A nonparametric approach for assessing latent trait unidimensionality. Psychometrika, 52, 589–617.CrossRefGoogle Scholar
Stout, W. F. (1990). A new item response theory modeling approach with applications to unidimensionality assessment and ability estimation. Psychometrika, 55, 293–325.CrossRefGoogle Scholar
Sijtsma, K. (1988). Contributions to Mokken's nonparametric item response theory. Unpublished doctoral dissertation, University of Groningen, The Netherlands.Google Scholar
Sijtsma, K., & Junker, B. W. (1994). A survey of theory and methods of invariant item ordering, Utrecht, The Netherlands: Department of Methodology and Statistics, Faculty of Social Sciences, Utrecht University.Google Scholar
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, 149–158.CrossRefGoogle Scholar
Stout, W. F. (1987). A nonparametric approach for assessing latent trait unidimensionality. Psychometrika, 52, 589–617.CrossRefGoogle Scholar
Stout, W. F. (1990). A new item response theory modeling approach with applications to unidimensionality assessment and ability estimation. Psychometrika, 55, 293–325.CrossRefGoogle Scholar
Tsutakawa, R. K., & Johnson, J. C. (1990). The effect of uncertainty of item parameter estimation on ability estimates. Psychometrika, 55, 371–390.CrossRefGoogle Scholar
van der Linden, W. J. (1998). A decision theory model for course placement. Journal of Educational and Behavioral Statistics, 23, 18–34.CrossRefGoogle Scholar
van der Linden, W. J., & Vos, H. J. (1996). A compensatory approach to optimal selection with mastery scores. Psychometrika, 61, 155–172.CrossRefGoogle Scholar