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Testing Manifest Monotonicity Using Order-Constrained Statistical Inference

Published online by Cambridge University Press:  01 January 2025

Jesper Tijmstra ,
David J. Hessen ,
Peter G. M. van der Heijden  and
Klaas Sijtsma
Jesper Tijmstra*
Affiliation:
Utrecht University
David J. Hessen
Affiliation:
Utrecht University
Peter G. M. van der Heijden
Affiliation:
Utrecht University
Klaas Sijtsma
Affiliation:
Tilburg University
*
Requests for reprints should be sent to Jesper Tijmstra, Department ofMethodology and Statistics, Faculty of Social Sciences, Utrecht Univeristy, PO Box 80140, 3508 TC Utrecht, The Netherlands. E-mail: j.tijmstra@uu.nl
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Abstract

Most dichotomous item response models share the assumption of latent monotonicity, which states that the probability of a positive response to an item is a nondecreasing function of a latent variable intended to be measured. Latent monotonicity cannot be evaluated directly, but it implies manifest monotonicity across a variety of observed scores, such as the restscore, a single item score, and in some cases the total score. In this study, we show that manifest monotonicity can be tested by means of the order-constrained statistical inference framework. We propose a procedure that uses this framework to determine whether manifest monotonicity should be rejected for specific items. This approach provides a likelihood ratio test for which the p-value can be approximated through simulation. A simulation study is presented that evaluates the Type I error rate and power of the test, and the procedure is applied to empirical data.

Keywords

item response theorylatent monotonicitymanifest monotonicitymonotone homogeneity modelorder-constrained statistical inference
Type
Original Paper
Information
Psychometrika , Volume 78 , Issue 1 , January 2013 , pp. 83 - 97
Copyright
Copyright © The Psychometric Society

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References

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, Reading: Addison-Wesley 397479Google Scholar
Dekovic, M. (2003). Aggressive and nonaggressive antisocial behavior in adolescence. Psychological Reports, 93, 610–616CrossRefGoogle ScholarPubMed
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, New York: Springer 6996CrossRefGoogle Scholar
Grayson, D.A. (1998). Two-group classification in latent trait theory: scores with monotone likelihood ratio. Psychometrika, 53, 383–392CrossRefGoogle 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
Hessen, D.J. (2005). Constant latent odds-ratios models and the Mantel-Haenszel null hypothesis. Psychometrika, 70, 497–516CrossRefGoogle 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., & Sijtsma, K. (2000). Latent and manifest monotonicity in item response models. Applied Psychological Measurement, 24, 65–81CrossRefGoogle Scholar
Karabatsos, G., & Sheu, C.-F. (2004). Order-constrained Bayes inference for dichotomous models of unidimensional nonparametric IRT. Applied Psychological Measurement, 28, 110–125CrossRefGoogle Scholar
Mokken, R.J. (1971). A theory and procedure of scale analysis, Berlin: de GruyterCrossRefGoogle Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests, Copenhagen: Nielsen and LydicheGoogle Scholar
Rosenbaum, P.R. (1984). Testing the conditional independence and monotonicity assumptions of item response theory. Psychometrika, 49, 425–435CrossRefGoogle Scholar
Rosenbaum, P.R. (1987). Comparing item characteristic curves. Psychometrika, 52, 217–233CrossRefGoogle Scholar
Sijtsma, K., & Molenaar, I.W. (2002). Introduction to nonparametric item response theory, Thousand Oaks: SAGECrossRefGoogle Scholar
Silvapulle, M.J., & Sen, P.K. (2005). Constrained statistical inference: inequality, order, and shape restrictions, Hoboken: WileyGoogle Scholar
Tijmstra, J., Hessen, D.J., Heijden, P.G.M., & Sijtsma, K. (2011). Invariant ordering of item-total regressions. Psychometrika, 76, 217–227CrossRefGoogle 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., Croon, M.A., & Sijtsma, K. (2008). Mokken scale analysis for dichotomous items using marginal models. Psychometrika, 73, 183–208CrossRefGoogle ScholarPubMed