Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2025-01-06T02:37:11.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic Ordering Of the Latent Trait by the Sum Score Under Various Polytomous IRT Models

Published online by Cambridge University Press:  01 January 2025

L. Andries van der Ark
L. Andries van der Ark*
Affiliation:
Tilburg University
*
Requests for reprints should be sent to L. Andries van der Ark, Department of Methodology and Statistics, Tilburg University, P.O. Box 90153, 5000 LE, Tilburg, The Netherlands. E-mail: a.vdark@uvt.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

The sum score is often used to order respondents on the latent trait measured by the test. Therefore, it is desirable that under the chosen model the sum score stochastically orders the latent trait. It is known that unlike dichotomous item response theory (IRT) models, most polytomous IRT models do not imply stochastic ordering. It is unknown, however, (1) whether stochastic ordering is often or rarely violated and (2) whether violations yield a serious problem for practical data analysis. These are the central issues of this paper. First, some unanswered questions that pertain to polytomous IRT models implying stochastic ordering were investigated. Second, simulation studies were conducted to evaluate stochastic ordering in practical situations. It was found that for most polytomous IRT models that do not imply stochastic ordering, the sum score can be used safely to order respondents on the latent trait.

Keywords

monotone likelihood rationonparametric item response theoryparametric item response theorypolytomous itemsstochastic orderingsum score
Type
Original Paper
Information
Psychometrika , Volume 70 , Issue 2 , June 2005 , pp. 283 - 304
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

The author would like to thank Klaas Sijtsma for commenting on earlier drafts of this paper.

References

Agresti, A. (1990). Categorical Data Analysis. New York: Wiley.Google Scholar
Andrich, D. (1978). A rating scale formulation for ordered response categories. Psychometrika, 43, 561–573.CrossRefGoogle Scholar
Birnbaum, A. (1968). Some latent trait models and their use 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
Douglas, R., Fienberg, S.E., Lee, M.T., Sampson, A.R., & Whitaker, L.R. (1991). Positive dependence concepts for ordinal contingency tables. In Block, H.W., Sampson, A.R., & Savits, T.H. (Eds.), Topics in Statistical Dependence (pp. 189–202). Hayward, CA: IMS.Google Scholar
Grayson, D.A. (1988). Two group classification in latent trait theory: scores with monotone likelihood ratio. Psychometrika, 53, 383–392.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 and the sum score in polytomous IRT models. Psychometrika, 62, 331–347.CrossRefGoogle Scholar
Hemker, B.T., Van der Ark, L.A., & Sijtsma, K. (2001). On measurement properties of continuation ratio models. Psychometrika, 66, 487–506.CrossRefGoogle Scholar
Huynh, H. (1994). A new proof for monotone likelihood ratio for the sum of independent variables. Psychometrika, 59, 77–79.CrossRefGoogle Scholar
Huynh, H. (1994). On equivalence between a partial credit item and a set of independent Rasch binary items. Psychometrika, 59, 111–119.CrossRefGoogle Scholar
Huynh, H. (1996). Decomposition of a Rasch partial credit item into independent and indecomposable trinary items. Psychometrika, 61, 31–39.CrossRefGoogle Scholar
Junker, B.W. (1991). Essential independence and likelihood-based ability estimation for polytomous items. Psychometrika, 56, 255–278.CrossRefGoogle Scholar
Junker, B.W. (1998). Some remarks on Scheiblechner’s treatment of ISOP models. Psychometrika, 63, 73–82.CrossRefGoogle Scholar
Lehmann, E.L., Rojo, R. (1992). Invariant directional orderings. The Annals of Statistics, 20, 2100–2110.CrossRefGoogle Scholar
Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.CrossRefGoogle Scholar
Mellenbergh, G.J. (1995). Conceptual notes on models for discrete polytomous item responses. Applied Psychological Measurement, 19, 91–100.CrossRefGoogle Scholar
Mokken, R.J. (1971). A Theory and Procedure of Scale Analysis. New York/Berlin: De Gruyter.CrossRefGoogle Scholar
Molenaar, I.W. (1983) Item Steps (Heymans Bulletin HB-83-630-EX). Groningen, The Netherlands: University of Groningen.Google Scholar
Molenaar, I.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: Springer.CrossRefGoogle Scholar
Molenaar, I.W., & Sijtsma, K. (2000). MSP for Windows [Software manual]. Groningen, The Netherlands: iec ProGAMMA.Google Scholar
Muraki, E. (1990). Fitting a polytomous item response model to Likert-type data. Applied Psychological Measurement, 14, 59–71.CrossRefGoogle Scholar
Muraki, E. (1992). A generalized partial credit model: Application of an EM-algorithm. Applied Psychological Measurement, 16, 159–176.CrossRefGoogle Scholar
Rasch, G. (1960). Probabilistic Models for Some Intelligence and Attainment Tests. Copenhagen: Danish Institute for Educational Research.Google Scholar
Ross, S.M. (1996). Stochastic processes 2nd edn, New York: Wiley.Google Scholar
Samejima, F. (1969) Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph 17.CrossRefGoogle Scholar
Samejima, F. (1995). Acceleration model in the heterogeneous case of the general graded response model. Psychometrika, 60, 549–572.CrossRefGoogle Scholar
Samejima, F. (2001) Substantiveness in psychometrics. Paper presented at the First International Meeting of the Psychometric Society, Osaka, JapanGoogle Scholar
Scheiblechner, H. (1995). Isotonic ordinal probabilistic models (ISOP). Psychometrika, 60, 281–304.CrossRefGoogle Scholar
Sijtsma, K., & Hemker, B.T. (1998). Nonparametric polytomous IRT models for invariant item ordering, with results for parametric models. Psychometrika, 63, 183–200.CrossRefGoogle Scholar
Sijtsma, K., & Hemker, B.T. (2000). A taxonomy of IRT models for ordering persons and items using simple sum scores. Journal of Educational and Behavioral Statistics, 25, 391–415.CrossRefGoogle Scholar
Sijtsma, K., & Van der Ark, L.A. (2001). Progress in NIRT analysis of polytomous item scores: Dilemmas and practical solutions. In Boomsma, A., van Duijn, M.A.J., & Snijders, T.A.B. (Eds.), Essays on Item Response Theory (pp. 297–318). New York: Springer.CrossRefGoogle Scholar
Tutz, G. (1990). Sequential item response models with an ordered response. British Journal of Mathematical and Statistical Psychology, 43, 39–55.CrossRefGoogle Scholar
Van der Ark, L.A. (2001). Relationships and properties of polytomous item response theory models. Applied Psychological Measurement, 25, 273–282.CrossRefGoogle Scholar
Van der Ark, L.A., Hemker, B.T., & Sijtsma, K. (2002). Hierarchically related nonparametric IRT models, and practical data analysis. In Marcoulides, G.A., & Moustaki, I. (Eds.), Latent Variable and Latent Structure Models (pp. 41–62). Mahwah: Erlbaum.Google Scholar
Van Engelenburg, G (1997) On Psychometric Models for Polytomous Items with Ordered Categories within the Framework of Item Response Theory, Unpublished doctoral dissertation, University of Amsterdam, Amsterdam.Google Scholar
Verhelst, N.D. (1992) Het Eenparameter Logistisch Model [The one parameter logistic model (OPLM)] (OPD Memorandum 92-3). Arnhem, The Netherlands: CITO National Institute for Educational Measurement.Google Scholar