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Mokken Scale Analysis for Dichotomous Items Using Marginal Models

Published online by Cambridge University Press:  01 January 2025

L. Andries van der Ark,
Marcel A. Croon and
Klaas Sijtsma
L. Andries van der Ark*
Affiliation:
Tilburg University
Marcel A. Croon
Affiliation:
Tilburg University
Klaas Sijtsma
Affiliation:
Tilburg University
*
Requests for reprints should be sent to L. Andries van der Ark, Department of Methodology and Statistics, FSW, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands. E-mail: a.vdark@uvt.nl
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Abstract

Scalability coefficients play an important role in Mokken scale analysis. For a set of items, scalability coefficients have been defined for each pair of items, for each individual item, and for the entire scale. Hypothesis testing with respect to these scalability coefficients has not been fully developed. This study introduces marginal modelling as a framework to derive the standard errors for the scaling coefficients and test hypotheses about these coefficients. Several examples demonstrate the possibilities of marginal modelling in Mokken scale analysis. These possibilities include testing whether Mokken’s criteria for a scale are satisfied, testing whether scalability coefficients of different items are equal, and testing whether scalability coefficients are equal across different groups.

Keywords

marginal modelsMokken scale analysisscalability coefficientstest construction

Information

Type
Original Paper
Information
Psychometrika , Volume 73 , Issue 2 , June 2008, pp. 183 - 208
Copyright
Copyright © 2007 The Psychometric Society

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References

Bartolucci, F., & Forcina, A. (2002). Extended RC association models allowing for order restrictions and marginal modeling. Journal of the American Statistical Association, 97, 1192–1199.CrossRefGoogle Scholar
Bartolucci, F., Forcina, A., & Dardanoni, V. (2001). Positive quadrant dependence and marginal modeling in two-way dependence with ordered margins. Journal of the American Statistical Association, 96, 1497–1505.CrossRefGoogle Scholar
Bergsma, W.P. (1997). Marginal models for categorical data, Tilburg: Tilburg University Press.Google Scholar
Bergsma, W.P. (1997b). marg_mod.nb [Mathematica computer code]. Retrieved from http://www.uvt.nl/mto/software2.html.Google Scholar
Bergsma, W.P., & Croon, M.A. (2005). Analyzing categorical data by marginal models. In van der Ark, L.A., Croon, M.A., & Sijtsma, K. (Eds.), New developments in categorical data analysis for the social and behavioral sciences (pp. 83–101). Mahwah, NJ: Erlbaum.Google Scholar
Bergsma, W.P., & Rudas, T. (2002). Marginal models for categorical data. The Annals of Statistics, 30, 140–159.CrossRefGoogle Scholar
Ellis, J.L., & Van den Wollenberg, A.L. (1993). Local homogeneity in latent trait models: A characterization of the homogeneous monotone latent trait model. Psychometrika, 58, 417–429.CrossRefGoogle Scholar
Goodman, L.A., & Kruskal, W.H. (1954). Measures of association for cross classification. Journal of the American Statistical Association, 49, 732–764.Google Scholar
Grayson, D.A. (1988). Two-group classification in latent trait theory: Scores with monotone likelihood ratio. Psychometrika, 53, 383–392.CrossRefGoogle Scholar
Guttman, L. (1950). The basis for scalogram analysis. In Stouffer, S.A., Guttman, L., Suchman, E.A., Lazarsfeld, P.F., Star, S.A., & Clausen, J.A. (Eds.), Measurement and prediction (pp. 60–90). Princeton, NJ: Princeton University Press.Google Scholar
Hemker, B.T., & Sijtsma, K., Molenaar, I.W. (1995). Selection of unidimensional scales from a multidimensional item bank in the polytomous Mokken IRT model. Applied Psychological Measurement, 19, 337–352.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., & Rosenbaum, P.R. (1986). Conditional association and unidimensionality in monotone latent variable models. The Annals of Statistics, 14, 1523–1543.CrossRefGoogle Scholar
Junker, B.W., & Sijtsma, K. (2000). Latent and manifest monotonicity in item response models. Applied Psychological Measurement, 24, 65–81.CrossRefGoogle Scholar
Kritzer, H.M. (1977). Analyzing measures of association derived from contingency tables. Sociological Methods and Research, 5, 35–50.CrossRefGoogle Scholar
Kullback, S. (1971). Marginal homogeneity of multidimensional contingency tables. Annals of Mathematical Statistics, 42, 594–606.CrossRefGoogle Scholar
Lang, J.B., & Agresti, A. (1994). Simultaneously modeling the joint and marginal distributions of multivariate categorical responses. Journal of the American Statistical Association, 89, 625–632.CrossRefGoogle Scholar
Loevinger, J. (1948). The technique of homogeneous tests compared with some aspects of ‘scale analysis’ and factor analysis. Psychological Bulletin, 45, 507–529.CrossRefGoogle Scholar
Mokken, R.J. (1971). A theory and procedure of scale analysis, The Hague/Berlin: Mouton/De Gruyter.CrossRefGoogle 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). User’s manual MSP5 for Windows, Groningen, The Netherlands: iec ProGAMMA [software manual]Google Scholar
Ramsay, J.O. (1991). Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika, 56, 611–630.CrossRefGoogle Scholar
Read, T.R.C., & Cressie, N.C. (1988). Goodness of fit statistics for discrete multivariate analysis, New York: Springer.CrossRefGoogle Scholar
Rudas, T., & Bergsma, W.P. (2004). On applications of marginal models for categorical data. Metron, 62, 1–23.Google Scholar
Scheiblechner, H. (2007). A unified nonparametric IRT model for d-dimensional psychological test data (d-ISOP). Psychometrika, 72, 43–67.CrossRefGoogle Scholar
Sijtsma, K., & Meijer, R.R. (2007). Nonparametric item response theory and related topics. In Rao, C.R., & Sinharay, S. (Eds.), Psychometrics (pp. 719–746). Amsterdam: Elsevier.Google Scholar
Sijtsma, K., & Molenaar, I.W. (2002). Introduction to nonparametric item response theory, Thousand Oaks, CA: Sage.CrossRefGoogle Scholar
Stout, W.F. (1990). A new item response modelling approach with applications to unidimensionality assessment and ability estimation. Psychometrika, 55, 293–325.CrossRefGoogle Scholar
Van Abswoude, A.A.H., Van der Ark, L.A., & Sijtsma, K. (2004). A comparative study of test dimensionality assessment procedures under nonparametric IRT models. Applied Psychological Measurement, 28, 3–24.CrossRefGoogle Scholar
Van der Ark, L.A. (2007). Mokken scale analysis in R. Journal of Statistical Software, 20(11), 1–19.Google Scholar
Van Maanen, L., Been, P.H., & Sijtsma, K. (1989). The linear logistic test model and heterogeneity of cognitive strategies. In Roskam, E.E. (Eds.), Mathematical psychology in progress (pp. 267–288)). Berlin: Springer.CrossRefGoogle Scholar
Wolfram, S. (1999). The Mathematica book, (4th ed.). Cambridge: Wolfram Media/Cambridge University Press.Google Scholar