This article discusses the use of the nonparametric IRT Mokken models of monotone homogeneity and double monotonicity and the parametric Rasch and Verhelst models for the analysis of binary test data. First, the four IRT models are discussed and compared at the theoretical level, and for each model, methods are discussed for evaluating the fit of the model to test data. Second, each of the four IRT models is used for analyzing the data collected by means of two versions of a test for inductive reasoning. Finally, the results are discussed and recommendations are given about the practical use of each of the IRT models. It is concluded that the simultaneous use of several IRT models for practical data analysis provides more insight into the structure of tests than the rigid use of only one model.
Get full access to this article
View all access options for this article.
References
1.
Bleichrodt, N. , Drenth, P. J. D., Zaal, J. N., & Resing, W. C. M. (1985). Revisie Amsterdamse Kinder-Intelligentie Test (RAKIT) [Revision of the Amsterdam Child Intelligence Test]. Lisse: Swets & Zeitlinger.
2.
Carpenter, P. A. , Just, M. A., & Shell, P. (1990). What one intelligence test measures: A theoretical account of the processing in the Raven Progressive Matrices Test. Psychological Review, 97(3), 404-431.
3.
de Koning,E. ,&Hamers,J.H.M. (1995). Programma Inductief Redeneren 1 [Program Inductive Reasoning I]. Utrecht: Utrecht University Press ISOR.
4.
de Koning, E. , & Hamers, J. H. M. (1999). Teaching inductive reasoning: Theoretical background and educational implications. In J. H. M. Hamers, J. E. H. van Luit, & B. Csapó (Eds.), Teaching and learning thinking skills (pp. 157-188). Lisse: Swets & Zeitlinger.
5.
de Koning, E. , Sijtsma, K., & Hamers, J. H. M. (in press). Construction and validation of a test for inductive reasoning. European Journal of Psychological Assessment.
6.
Embretson, S. E. , & Reise, S. (2000). Item response theory for psychologists. Mahwah, NJ: Lawrence Erlbaum.
7.
Fischer, G. H. , & Molenaar, I. W. (1995). Rasch models: Foundations, recent developments, and applications. New York: Springer.
8.
Glas, C. A. W. (1988). The derivation of some tests for the Rasch model from the multinomial distribution. Psychometrika, 53, 525-546.
9.
Glas, C. A. W. , & Ellis, J. L. (1993). User’s manual RSP: Rasch Scaling Program. Groningen, The Netherlands: iecProGAMMA.
10.
Glas, C. A. W. , & Verhelst, N. D. (1995). Testing the Rasch model. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 69-95). New York: Springer-Verlag.
11.
Hambleton, R. K. , & Swaminathan, H. (1985). Item response theory: Principles and applications. Boston: Kluwer Nijhoff.
12.
Hemker, B. T. , Sijtsma, K., & Molenaar, I. W. (1995). Selection of unidimensional scales from a multidimensional itembank in the polytomous Mokken IRT model. Applied Psychological Measurement, 19, 337-352.
13.
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.
14.
Holland, P. W. , & Rosenbaum, P. R. (1986). Conditional association and unidimensionality in monotone latent variable models. The Annals of Statistics, 14, 1523-1543.
15.
Holland, P. W. , & Wainer, H. (1993). Differential item functioning. Hillsdale, NJ: Lawrence Erlbaum.
16.
Junker, B. W. (1993). Conditional association, essential independence and monotone unidimensional item response models. The Annals of Statistics, 21, 1359-1378.
17.
Junker, B. W. , & Sijtsma, K. (2000). Latent and manifest monotonicity in item response models. Applied Psychological Measurement, 24, 65-81.
18.
Kim, H. R. , Zhang, J., & Stout, W. (1996). A new index of dimensionality—DETECT. Internal Report, Department of Statistics, University of Illinois at Urbana-Champaign.
19.
Klauer, K. J. (1989). Denktraining für Kinder 1. Ein Program zur intellectuellen Förderung [Inductive reasoning. A programme for the stimulation of inductive reasoning]. Göttingen: Hogrefe Verlag.
20.
Meijer, R. R. , & Sijtsma, K. (2001). Methodology review: Evaluating person fit. Applied Psychological Measurement, 25, 107-135.
21.
Meijer, R. R. , Sijtsma, K., & Smid, N. G. (1990). Theoretical and empirical comparison of the Mokken and the Rasch approach to IRT. Applied Psychological Measurement, 14, 283-298.
22.
Mokken, R. J. (1971). A theory and procedure of scale analysis. Berlin: De Gruyter.
23.
Mokken, R. J. , & Lewis, C. (1982). A nonparametric approach to the analysis of dichotomous item responses. Applied Psychological Measurement, 6, 417-430.
24.
Mokken, R. J. , Lewis, C., & Sijtsma, K. (1986). Rejoinder to “The Mokken scale: A critical discussion.”Applied Psychological Measurement, 10, 279-285.
25.
Molenaar, I. W. (1983). Some improved diagnostics for failure of the Rasch model. Psychometrika, 48, 49-72.
26.
Molenaar, I. W. , & Sijtsma, K. (2000). MSP5 for Windows. User’s manual MSP. Groningen, The Netherlands: iecProGAMMA.
27.
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen, Denmark: Nielsen & Lydiche.
28.
Reckase, M. D. (1997). A linear logistic multidimensional model for dichotomous item response data. In W. J. Van der Linden & R. K. Hambleton (Eds.), Handbook of modern item response theory (pp. 271-286). New York: Springer-Verlag.
29.
Robin, F. , Xing, D., & Hambleton, R. K. (1999). Software review: Rasch scaling program (RSP). Applied Psychological Measurement, 23, 90-94.
30.
Rosenbaum, P. R. (1984). Testing the conditional independence and monotonicity assumptions of item response theory. Psychometrika, 49, 425-435.
31.
Sijtsma, K. (1998). Methodology review: Nonparametric IRT approaches to the analysis of dichotomous item scores. Applied Psychological Measurement, 22, 3-31.
32.
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.
33.
Sijtsma, K. , & Junker, B. W. (1997). Invariant item ordering of transitive reasoning tasks. In J. Rost & R. Langeheine (Eds.), Applications of latent trait and latent class models in the social sciences (pp. 100-110). Münster, Germany: Waxmann Verlag.
34.
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, 149-157.
35.
Sijtsma, K. , & Molenaar, I. W. (2002). Introduction to nonparametric item response theory. Thousand Oaks, CA: Sage.
36.
Stout, W. F. (1990). A new item response theory modeling approach with applications to unidimensionality assessment and ability estimation. Psychometrika, 55, 293-325.
37.
Stout, W. F. , Habing, B., Douglas, J., Kim, H., Roussos,L.,&Zhang,J.(1996).Conditional covariance based nonparametric multidimensionality assessment. Applied Psychological Measurement, 20, 331-354.
38.
Suppes, P. , & Zanotti, M. (1981). When are probabilistic explanations possible?Synthese, 48, 191-199.
39.
Van den Wollenberg, A. L. (1982). Two new test statistics for the Rasch model. Psychometrika, 47, 123-140.
40.
Van der Linden, W. J. , & Hambleton, R. K. (1997). Handbook of modern item response theory. New York: Springer.
41.
Verhelst, N. D. (1992). Het eenparameter logistisch model (OPLM) [The one parameter logistic model (OPLM)] (OPD Memorandum 92-3). Arnhem, The Netherlands: Cito.
42.
Verhelst, N. D. , & Glas, C. A. W. (1995). The one parameter logistic model. In G. H. Fischer & I.W. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 215-237). New York: Springer.
