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An Isotonic Partial Credit Model for Ordering Subjects on the Basis of Their Sum Scores

Published online by Cambridge University Press:  01 January 2025

Rudy Ligtvoet
Rudy Ligtvoet*
Affiliation:
University of Amsterdam
*
Requests for reprints should be sent to Rudy Ligtvoet, Department of Pedagogical and Educational Sciences, University of Amsterdam, Nieuwe Prinsengracht 130, 1018 VZ, Amsterdam, The Netherlands. E-mail: r.ligtvoet@uva.nl
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Abstract

In practice, the sum of the item scores is often used as a basis for comparing subjects. For items that have more than two ordered score categories, only the partial credit model (PCM) and special cases of this model imply that the subjects are stochastically ordered on the common latent variable. However, the PCM is very restrictive with respect to the constraints that it imposes on the data. In this paper, sufficient conditions for the stochastic ordering of subjects by their sum score are obtained. These conditions define the isotonic (nonparametric) PCM model. The isotonic PCM is more flexible than the PCM, which makes it useful for a wider variety of tests. Also, observable properties of the isotonic PCM are derived in the form of inequality constraints. It is shown how to obtain estimates of the score distribution under these constraints by using the Gibbs sampling algorithm. A small simulation study shows that the Bayesian p-values based on the log-likelihood ratio statistic can be used to assess the fit of the isotonic PCM to the data, where model-data fit can be taken as a justification of the use of the sum score to order subjects.

Keywords

item response theorymonotone likelihood ratioordinal inferencespartial credit modelpolytomously scored itemsstochastic orderingsum score
Type
Original Paper
Information
Psychometrika , Volume 77 , Issue 3 , July 2012 , pp. 479 - 494
Copyright
Copyright © 2012 The Psychometric Society

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References

Agresti, A. (1990). Categorical data analysis, New York: WileyGoogle Scholar
Andersen, E.B. (1973). A goodness of fit test for the Rasch model. Psychometrika, 38, 123–140CrossRefGoogle Scholar
Douglas, R., Fienberg, S.E., Lee, M.-L.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. Topics in statistical dependence, Hayward: Institute of Mathematical Statistics 189202Google Scholar
Gelman, A., Meng, X.L., Stern, H. (1996). Posterior predictive assessment of model fitness via realized discrepancies. Statistica Sinica, 6, 733–807Google Scholar
Grayson, D.A. (1988). Two-group classification in latent trait theory: Scores with monotone likelihood ratio. Psychometrika, 53, 383–392CrossRefGoogle Scholar
Hemker, B.T. (1996). Unidimensional IRT models for polytomous items, with results for Mokken scale analysis. Unpublished doctoral dissertation, Utrecht University, The Netherlands. Google 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, 679–693CrossRefGoogle 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, 331–347CrossRefGoogle Scholar
Hoijtink, H. (1998). Constrained latent class analysis using the Gibbs sampler and posterior predictive p-values: Applications to educational testing. Statistica Sinica, 8, 691–711Google Scholar
Hoijtink, H., Molenaar, I.W. (1997). A multidimensional item response model: Constrained latent class analysis using Gibbs sampler and posterior predictive checks. Psychometrika, 62, 171–189CrossRefGoogle Scholar
Holland, P.W., Rosenbaum, P.R. (1986). Conditional association and unidimensionality in monotone latent variable models. Annals of Statistics, 14, 1523–1543CrossRefGoogle Scholar
Huynh, H. (1994). A new proof for monotone likelihood ratio for the sum of independent Bernoulli random variables. Psychometrika, 59, 77–79CrossRefGoogle Scholar
Junker, B.W. (1993). Conditional association, essential independence and monotone unidimensional item response models. Annals of Statistics, 21, 1359–1378CrossRefGoogle Scholar
Karabatsos, G. (2001). The Rasch model, additive conjoint measurement, and new models of probabilistic measurement theory. Journal of Applied Measurement, 2, 389–423Google ScholarPubMed
Karabatsos, G., Sheu, C.-F. (2004). Order-constrained Bayes inference for dichotomous models of unidimensional nonparametric IRT. Applied Psychological Measurement, 28, 110–125CrossRefGoogle Scholar
Karlin, S. (1968). Total positivity, Palo Alto: Stanford University PressGoogle Scholar
Karlin, S., Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. Journal of Multivariate Analysis, 10, 467–498CrossRefGoogle Scholar
Laudy, O., Hoijtink, H. (2007). Bayesian methods for the analysis of inequality constrained contingency tables. Statistical Methods in Medical Research, 16, 123–138CrossRefGoogle ScholarPubMed
Lazarsfeld, P.F., Henry, N.W. (1968). Latent structure analysis, Boston: Houghton MifflinGoogle Scholar
Lehmann, E.L. (1959). Testing statistical hypotheses, New York: WileyGoogle Scholar
Ligtvoet, R., Van der Ark, L.A., Bergsma, W.P., Sijtsma, K. (2011). Polytomous latent scales for the investigation of the ordering of items. Psychometrika, 73, 200–216CrossRefGoogle Scholar
Ligtvoet, R., Vermunt, J.K. (2012). Latent class models for testing monotonicity and invariant item ordering for polytomous items. British Journal of Mathematical & Statistical Psychology, 65, 237–250CrossRefGoogle ScholarPubMed
Masters, G. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174CrossRefGoogle Scholar
Meng, X.L. (1994). Posterior predictive p-values. Annals of Statistics, 22, 1142–1160CrossRefGoogle Scholar
Mokken, R.J. (1971). A theory and procedure of scale analysis, Berlin: De GruyterCrossRefGoogle Scholar
Molenaar, I.W. (1997). Nonparametric models for polytomous responses. In van der Linden, W.J., Hambleton, R.K. Handbook of modern item response theory, New York: Springer 369380CrossRefGoogle Scholar
Molenaar, I.W., & Sijtsma, K. (2000). User’s manual MSP5 for Windows [Software manual]. Groningen, The Netherlands: IEC ProGAMMA. Google Scholar
Muraki, E. (1992). A generalized partial credit model: Applications for an EM algorithm. Applied Psychological Measurement, 16, 159–177CrossRefGoogle Scholar
Nadarajah, S., Kotz, S. (2006). R Programs for computing truncated distributions. Journal of Statistical Software, 16(2), 18CrossRefGoogle Scholar
R Development Core Team (2009). R: A language and environment for statistical computing. Vienna: R Foundation for Statistical Computing. http://www.R-project.org. Google Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests, Copenhagen: Nielsen and LydicheGoogle Scholar
Samejima, F. (1969). Estimation of latent trait ability using a response pattern of graded scores. Psychometrika Monograph, No. 17. CrossRefGoogle Scholar
Samejima, F. (1972). A general model for free-response data. Psychometrika Monograph, No. 18. Google Scholar
Scheiblechner, H. (1995). Isotonic ordinal probabilistic models (ISOP). Psychometrika, 60, 281–304CrossRefGoogle Scholar
Scheiblechner, H. (1999). Additive conjoint isotonic probabilistic models (ADISOP). Psychometrika, 64, 295–316CrossRefGoogle Scholar
Scheiblechner, H. (2007). A unified nonparametric IRT model for d-dimensional psychological test data (d-ISOP). Psychometrika, 72, 43–67CrossRefGoogle Scholar
Ünlü, A. (2008). A note on monotone likelihood ratio of the total score variable in unidimensional item response theory. British Journal of Mathematical & Statistical Psychology, 61, 179–187CrossRefGoogle ScholarPubMed
Van der Ark, L.A. (2005). Stochastic ordering of the latent trait by the sum score under various polytomous IRT models. Psychometrika, 70, 283–304CrossRefGoogle Scholar
Van der Ark, L.A., Bergsma, W.P. (2010). A note on stochastic ordering of the latent trait using the sum of polytomous item scores. Psychometrika, 75, 272–279CrossRefGoogle Scholar
Van der Ark, L.A., Hemker, B.T., Sijtsma, K. (2002). Hierarchically related nonparametric IRT models, and practical data analysis methods. In Marcoulides, G.A., Moustaki, I. Latent variable and latent structure models, Mahwah: Erlbaum 4162Google Scholar
Van Onna, H.J.M. (2002). Bayesian estimation and model selection in ordered latent class models for polytomous items. Psychometrika, 67, 519–538CrossRefGoogle Scholar