Abstract
This article addresses the issue of assigning items to different test dimensions (e.g., determining which dimension an item belongs to) with cluster analysis. Previously, hierarchical methods have been used (Roussos et al. 1997); however, the findings here suggest that an iterative reallocation partitioning (IRP) algorithm provides interpretively similar solutions and statistically better solutions to the problem. More importantly, it is shown that the inherent nature of locally optimal solutions in the IRP algorithm leads to a method that aids in determining the appropriateness of performing a cluster analysis—a feature that is lacking in the standard hierarchical methods currently in the literature.
Similar content being viewed by others
References
Arabie, P., & Hubert, L. (1996). An overview of combinatorial data analysis. In Arabie, P., Hubert, L.J., De Soete, G., et al. (Eds.) Clustering and classification (pp. 5–63). River Edge: World Scientific.
Baker, F.B., & Hubert, L.J. (1975). Measuring the power of hierarchical cluster analyis. Journal of the American Statistical Association, 70, 31–38.
Batagelj, V., Ferligoj, A., Doreian, P. (1992). Direct and indirect methods for structural equivalence. Social Networks, 14, 63–90.
Brusco, M.J. (2004). Clustering binary data in the presence of masking variables. Psychological Methods, 9, 510–523.
Cormack, R.M. (1971). A review of classification. Journal of the Royal Statistical Society, Series A, 134, 321–367.
De La Torre, J., & Douglas, J.A. (2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69, 333–353.
Ferligoj, A., Batagelj, V., Doreian, P. (1994). On connecting network analysis and cluster analysis. In Fischer, G.H., Laming, D., et al. (Eds.) Contributions to mathematical psychology, psychometrics, and methodology (pp. 329–344). New York: Springer.
Gordon, A. D. (1987). A review of hierarchical classification. Journal of the Royal Statistical Society, Series A, 150, 119–137.
Gordon, A.D. (1996). Hierarchical classification. In Arabie, P., Hubert, L.J., De Soete, G., et al. (Eds.) Clustering and classification (pp. 65–121). River Edge: World Science.
Gower, J.C., & Legendre, P. (1986). Metric and Euclidean properties of dissimilarity coefficients. Journal of Classification, 3, 5–48.
Hartigan, J.A. (1975). Clustering algorithms. New York: Wiley.
Hartigan, J., & Wong, M.A. (1979). Algorithm AS136: a k-means clustering algorithm. Applied Statistics, 28, 100–108.
Hubert, L., & Arabie, P. (1985). Comparing partitions. Journal of Classification, 2, 193–218.
Hubert, L., & Arabie, P. (1986). Comparing partitions. In Gaul, W., Schader, M., et al. (Eds.) Classification as a tool of research (pp. 209–215). North-Holland: Elsevier Science.
Hubert, L.J., & Levin, J.R. (1976). A general statistical framework for assessing categorical clustering in free recall. Psychological Bulletin, 83, 1072–1080.
Lance, G.N., & Williams, W.T. (1966). A generalised sorting strategy for computer classifications. Nature, 212, 218.
Lance, G.N., & Williams, W.T. (1967). A general theory of classificatory strategies I. Hierarchical systems. The Computer Journal, 9, 373–380.
MacQueen, J. (1967). Some methods for classification and analysis of multivariate observations. In Le, L.M., Neyman, C.J., et al. (Eds.) Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (pp. 281–297). Berkeley: University of California Press.
McDonald, R.P. (1967). Nonlinear factor analysis. Psychometric monographs (No. 15).
McQuitty, L.L. (1960). Hierarchical linkage analysis for the isolation of types. Educational and Psychological Measurement, 20, 55–67.
Milligan, G.W. (1996). Clustering validation, results and implications for applied analysis. In Arabie, P., Hubert, L.J., De Soete, G., et al. (Eds.) Clustering and classification (pp. 341–375). River Edge: World Scientific.
Milligan, G.W., & Cooper, M.C. (1985). An examination of procedures for determining the number of clusters in a data set. Psychometrika, 50, 159–179.
Milligan, G.W., & Cooper, M.C. (1986). A study of the comparability of external criteria for hierarchical cluster analysis. Multivariate Behavioral Research, 21, 441–458.
Nandakumar, R., & Stout, W.F. (1993). Refinements of Stout’s procedure for assessing latent trait unidimensionality. Journal of Educational Statistics, 18, 41–68.
Reckase, M.D., & McKinley, R.L. (1991). The discriminating power of items that measure more than one dimension. Applied Psychological Measurement, 15, 361–373.
Roussos, L.A., Stout, W.F., Marden, J.I. (1997). Using new proximity measures with hierarchical cluster analysis to detect multidimensionality. Journal of Educational Measurement, 35, 1–30.
Rutherford, A. (2001). Introducing ANOVA and ANCOVA: A GLM approach. Thousand Oaks: Sage.
Sneath, P.H.A. (1957). The application of computers in taxonomy. Journal of General Microbiology, 17, 201–226.
Sokal, R.R., & Michener, C.D. (1958). A statistical method for evaluating systematic relationships. University of Kansas Science Bulletin, 38, 1409–1438.
Steinley, D. (2003). K-means clustering: What you don’t know may hurt you. Psychological Methods, 8, 294–304.
Steinley, D. (2004). Properties of the Hubert-Arabie adjusted Rand index. Psychological Methods, 9, 386–396.
Steinley, D. (2006b). Profiling local optima in K-means clustering: Developing a diagnostic technique. Psychological Methods, 11, 178–192.
Steinley, D., & Henson, R. (2005). An analytic method for clusters with known overlap. Manuscript submitted for publication.
Stout, W.F. (1987). A nonparametric approach for assessing latent trait dimensionality. Psychometrika, 52, 589–617.
Stout, W.F., Habing, B., Douglas, J., Kim, H.R., Roussos, L., Zhang, J. (1996). Conditional covariance based nonparametric multidimensionality assessment. Applied Psychological Measurement, 20, 331–354.
Tatsouka, C. (2002). Data analytic methods for latent partially ordered classification models. Journal of the Royal Statistical Society Series C, 51, 337–350.
Tatsouka, K. (1990). Toward an integration of item-response theory and cognitive error diagnosis. In Frederiksen, N., Glaser, R., Lesgold, A., Safto, M., et al. (Eds.) Monitoring skills and knowledge acquisition (pp. 453–488). Hillsdale: Erlbaum.
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.
Ward, J.H. Jr. (1963). Hierarchical grouping to optimize an objective function. Journal of the American Statistical Association, 58, 236–244.
Zhang, J., & Stout, W. (1999). The theoretical detect index of dimensionality and its application to approximate simple structure. Psychometrika, 63, 213–249.
Acknowledgments
I would like to thank Lawrence Hubert and Louis Roussos for comments and suggestions on earlier versions of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Steinley, D.L., Brusco, M.J. Using an Iterative Reallocation Partitioning Algorithm to Verify Test Multidimensionality. J Classif 36, 397–413 (2019). https://doi.org/10.1007/s00357-019-09347-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00357-019-09347-z