Mokken scale analysis can be used for scaling under nonparametric item response theory models. The results may, however, not reflect the underlying dimensionality of data. Various features of Mokken scale analysis—the H coefficient, Mokken scale conditions, and algorithms—may explain this result. In this article, three new H-based objective functions with slight reformulations of Mokken scale analysis in the unidimensional and multidimensional cases are introduced. Deterministic and stochastic nonhierarchical clustering algorithms reduced the probability of obtaining suboptimal solutions. A simulation study investigated whether these methods can determine the dimensionality structure of data sets that vary with respect to item discrimination, item difficulty, number of items per trait, and numbers of observations per test. Furthermore, it was investigated whether deterministic and stochastic algorithms can generate approximately global optimal solutions. The method based on the average within-scale Hi combined with a stochastic nonhierarchical clustering algorithm was the most successful in dimensionality assessment.
Berthold, M. , & Hand, D.J. (1999). Intelligent data analysis: An introduction. New York: Springer .
2.
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability . In F. M. Lord & M. R. Novick (Eds.), Statistical theories of mental test scores (pp. 397-479). Reading, MA: Addison-Wesley .
3.
Danzig, G.B. , & Thapa, M.N. (1997). Linear programming. New York / Berlin : Springer-Verlag .
4.
Douglas, J. , & Cohen, A. (2001). Nonparametric item response function estimation for assessing parametric model fit . Applied Psychological Measurement, 25, 234-243 .
5.
Emons, W.H.M. (2003). Detection and diagnosis of misfitting item-score patterns. Amsterdam, Netherlands: DutchUniversity Press .
6.
Grayson, D.A. (1988). Two-group classification in latent trait theory: Scores with monotone likelihood ratio . Psychometrika, 53, 383-392 .
7.
Guttman, L. (1950). The basis for scalogram analysis. In S. A. Stouffer , L. Guttman , E. A. Suchman , P. F. Lazersfeld , S. A. Star , & J. A. Clausen (Eds.), Measurement and prediction (pp. 60-90), Princeton, NJ: Princeton University Press .
8.
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 .
9.
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 .
10.
Holland, P.W. , & Rosenbaum, P.R. (1986). Conditional association and unidimensionality in monotone latent variable models . Annals of Statistics, 14, 1523-1543 .
11.
Hoogland, J.J. (1999). The robustness of estimation methods for covariance structure analysis. Unpublished doctoral dissertation, University of Groningen , Netherlands.
12.
Junker, B.W. (1993). Conditional association, essential independence and monotone unidimensional item response models . Annals of Statistics, 21, 1359-1378 .
13.
Karabatsos, G. (2003). Comparing the aberrant response detection performance of thirty-six person-fit statistics . Applied Measurement in Education, 16, 277-298 .
14.
Kim, H.R. (1994). New techniques for the dimensionality assessment of standardized test data. Unpublished doctoral dissertation, University of Illinois, Urbana-Champaign .
15.
Loevinger, J. (1948). The technique of homogenous tests compared with some aspects of scale analysis and factor analysis . Psychological Bulletin, 45, 507-530 .
16.
McDonald, R.P. (1985). Factor analysis and related models. Hillsdale, NJ: Lawrence Erlbaum .
17.
Meijer, R.R. (1994). Nonparametric person fit analysis. Unpublished doctoral dissertation, Vrije Universiteit, Amsterdam , Netherlands.
18.
Michalewicz, Z. (1996). Genetic algorithms + data structures = evolution programs. New York: Springer .
19.
Mokken, R.J. (1971). A theory and procedure of scale analysis. The Hague, Netherlands: Mouton .
20.
Molenaar, I.W. (1991). A weighted Loevinger H-coefficient extending Mokken scaling to multi-category items . Kwantitatieve Methoden, 12(37), 97-117 .
21.
Molenaar, I.W. , & Sijtsma, K. (2000). Users manual MSP5 for Windows. A program for Mokken scale analysis for polytomous items [Computer software manual]. Groningen , Netherlands: iec ProGAMMA .
22.
Muthén, L. , & Muthén, B. (2003). Mplus version 2.13 [Computer software]Los Angeles: Authors .
23.
Nandakumar, R. , & Stout, W.F. (1993). Refinements of Stout's procedure for assessing latent trait unidimensionality . Journal of Educational Statistics , 18, 41-68 .
24.
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 .
25.
Roskam, E.E. , Van den Wollenberg, A.L. , & Jansen, P.G.W. (1986). The Mokken scale: a critical discussion . Applied Psychological Measurement, 10, 265-277 .
26.
Roussos, L. , & Ozbek, O. (2003, April). Formulation of the DETECT population parameter and evaluation of DETECT estimator bias . Paper presented at the annual meeting of the National Council on Measurement in Education, Chicago, IL.
27.
Roussos, L.A. , Stout, W.F. , & Marden, J.I. (1998). Using new proximity measures with hierarchical cluster analysis to detect multidimensionality . Journal of Educational Measurement, 35, 1-30 .
28.
Samejima, F. (2000). Logistic positive exponent family of models: Virtue of asymmetrical item characteristic curves . Psychometrika , 65, 319-335 .
29.
Sijtsma, K. , & Molenaar, I.W. (2002). Introduction to nonparametric item response theory. Thousand Oaks: Sage .
30.
Stout, W.F. (1987). A nonparametric approach for assessing latent trait unidimensionality . Psychometrika, 52, 589-617 .
31.
Stout, W.F. (2002). Psychometrics: From practice to theory and back: 15 years of nonparametric multidimensional IRT, DIF/test equity, and skills diagnostic assessment . Psychometrika, 67, 485-518 .
32.
Stout, W. , Goodwin Froelich, A. , & Gao, F. (2001). Using resampling methods to produce an improved DIMTEST procedure. In A. Boomsma , M. A. J. van Duijn , & T. A. B. Snijders (Eds.), Essays on item response theory (pp. 357-375). New York: Springer .
33.
van Abswoude, A.A.H. (2004). Dimensionality assessment under nonparametric IRT models. Unpublished doctoral dissertation, Tilburg University , Netherlands.
34.
van Abswoude, A.A.H. , Van der Ark, L.A. , & Sijtsma, K. (2004). A comparative study of test data dimensionality assessment procedures under nonparametric IRT models . Applied Psychological Measurement, 28, 3-24 .
35.
Vermunt, J.K. (1997). LEM: A general program for the analysis of categorical data [Computer software]. Tilburg, Netherlands: Tilburg University .
36.
Zhang, J. , & Stout, W.F. (1999). The theoretical DETECT index of dimensionality and its application to approximate simple structure . Psychometrika , 64, 213-249 .
37.
Zhang, Y.O. , Yu, F. , & Nandakumar, R. (2003, April). The impact of conditional scores on the performance of DETECT . Paper presented at the annual meeting of the National Council on Measurement in Education , Chicago, IL.
