Skip to main content
Top
Published in: Journal of Classification 3/2019

21-11-2019

Using an Iterative Reallocation Partitioning Algorithm to Verify Test Multidimensionality

Authors: Douglas L. Steinley, M. J. Brusco

Published in: Journal of Classification | Issue 3/2019

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
go back to reference 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. 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.
go back to reference Baker, F.B., & Hubert, L.J. (1975). Measuring the power of hierarchical cluster analyis. Journal of the American Statistical Association, 70, 31–38.CrossRef Baker, F.B., & Hubert, L.J. (1975). Measuring the power of hierarchical cluster analyis. Journal of the American Statistical Association, 70, 31–38.CrossRef
go back to reference Batagelj, V., Ferligoj, A., Doreian, P. (1992). Direct and indirect methods for structural equivalence. Social Networks, 14, 63–90.CrossRef Batagelj, V., Ferligoj, A., Doreian, P. (1992). Direct and indirect methods for structural equivalence. Social Networks, 14, 63–90.CrossRef
go back to reference Brusco, M.J. (2004). Clustering binary data in the presence of masking variables. Psychological Methods, 9, 510–523.CrossRef Brusco, M.J. (2004). Clustering binary data in the presence of masking variables. Psychological Methods, 9, 510–523.CrossRef
go back to reference Cormack, R.M. (1971). A review of classification. Journal of the Royal Statistical Society, Series A, 134, 321–367.MathSciNetCrossRef Cormack, R.M. (1971). A review of classification. Journal of the Royal Statistical Society, Series A, 134, 321–367.MathSciNetCrossRef
go back to reference De La Torre, J., & Douglas, J.A. (2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69, 333–353.MathSciNetCrossRef De La Torre, J., & Douglas, J.A. (2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69, 333–353.MathSciNetCrossRef
go back to reference 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. 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.
go back to reference Gordon, A. D. (1987). A review of hierarchical classification. Journal of the Royal Statistical Society, Series A, 150, 119–137.MathSciNetCrossRef Gordon, A. D. (1987). A review of hierarchical classification. Journal of the Royal Statistical Society, Series A, 150, 119–137.MathSciNetCrossRef
go back to reference 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. 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.
go back to reference Gower, J.C., & Legendre, P. (1986). Metric and Euclidean properties of dissimilarity coefficients. Journal of Classification, 3, 5–48.MathSciNetCrossRef Gower, J.C., & Legendre, P. (1986). Metric and Euclidean properties of dissimilarity coefficients. Journal of Classification, 3, 5–48.MathSciNetCrossRef
go back to reference Hartigan, J.A. (1975). Clustering algorithms. New York: Wiley.MATH Hartigan, J.A. (1975). Clustering algorithms. New York: Wiley.MATH
go back to reference Hartigan, J., & Wong, M.A. (1979). Algorithm AS136: a k-means clustering algorithm. Applied Statistics, 28, 100–108.CrossRef Hartigan, J., & Wong, M.A. (1979). Algorithm AS136: a k-means clustering algorithm. Applied Statistics, 28, 100–108.CrossRef
go back to reference Hubert, L., & Arabie, P. (1985). Comparing partitions. Journal of Classification, 2, 193–218.CrossRef Hubert, L., & Arabie, P. (1985). Comparing partitions. Journal of Classification, 2, 193–218.CrossRef
go back to reference 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., & 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.
go back to reference Hubert, L.J., & Levin, J.R. (1976). A general statistical framework for assessing categorical clustering in free recall. Psychological Bulletin, 83, 1072–1080.CrossRef Hubert, L.J., & Levin, J.R. (1976). A general statistical framework for assessing categorical clustering in free recall. Psychological Bulletin, 83, 1072–1080.CrossRef
go back to reference Lance, G.N., & Williams, W.T. (1966). A generalised sorting strategy for computer classifications. Nature, 212, 218.CrossRef Lance, G.N., & Williams, W.T. (1966). A generalised sorting strategy for computer classifications. Nature, 212, 218.CrossRef
go back to reference Lance, G.N., & Williams, W.T. (1967). A general theory of classificatory strategies I. Hierarchical systems. The Computer Journal, 9, 373–380.CrossRef Lance, G.N., & Williams, W.T. (1967). A general theory of classificatory strategies I. Hierarchical systems. The Computer Journal, 9, 373–380.CrossRef
go back to reference 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. 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.
go back to reference McDonald, R.P. (1967). Nonlinear factor analysis. Psychometric monographs (No. 15). McDonald, R.P. (1967). Nonlinear factor analysis. Psychometric monographs (No. 15).
go back to reference McQuitty, L.L. (1960). Hierarchical linkage analysis for the isolation of types. Educational and Psychological Measurement, 20, 55–67.CrossRef McQuitty, L.L. (1960). Hierarchical linkage analysis for the isolation of types. Educational and Psychological Measurement, 20, 55–67.CrossRef
go back to reference 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.CrossRef 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.CrossRef
go back to reference 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.CrossRef 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.CrossRef
go back to reference 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.CrossRef 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.CrossRef
go back to reference Nandakumar, R., & Stout, W.F. (1993). Refinements of Stout’s procedure for assessing latent trait unidimensionality. Journal of Educational Statistics, 18, 41–68. Nandakumar, R., & Stout, W.F. (1993). Refinements of Stout’s procedure for assessing latent trait unidimensionality. Journal of Educational Statistics, 18, 41–68.
go back to reference Reckase, M.D., & McKinley, R.L. (1991). The discriminating power of items that measure more than one dimension. Applied Psychological Measurement, 15, 361–373.CrossRef Reckase, M.D., & McKinley, R.L. (1991). The discriminating power of items that measure more than one dimension. Applied Psychological Measurement, 15, 361–373.CrossRef
go back to reference 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.CrossRef 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.CrossRef
go back to reference Rutherford, A. (2001). Introducing ANOVA and ANCOVA: A GLM approach. Thousand Oaks: Sage.MATH Rutherford, A. (2001). Introducing ANOVA and ANCOVA: A GLM approach. Thousand Oaks: Sage.MATH
go back to reference Sneath, P.H.A. (1957). The application of computers in taxonomy. Journal of General Microbiology, 17, 201–226.CrossRef Sneath, P.H.A. (1957). The application of computers in taxonomy. Journal of General Microbiology, 17, 201–226.CrossRef
go back to reference Sokal, R.R., & Michener, C.D. (1958). A statistical method for evaluating systematic relationships. University of Kansas Science Bulletin, 38, 1409–1438. Sokal, R.R., & Michener, C.D. (1958). A statistical method for evaluating systematic relationships. University of Kansas Science Bulletin, 38, 1409–1438.
go back to reference Steinley, D. (2003). K-means clustering: What you don’t know may hurt you. Psychological Methods, 8, 294–304.CrossRef Steinley, D. (2003). K-means clustering: What you don’t know may hurt you. Psychological Methods, 8, 294–304.CrossRef
go back to reference Steinley, D. (2004). Properties of the Hubert-Arabie adjusted Rand index. Psychological Methods, 9, 386–396.CrossRef Steinley, D. (2004). Properties of the Hubert-Arabie adjusted Rand index. Psychological Methods, 9, 386–396.CrossRef
go back to reference Steinley, D. (2006b). Profiling local optima in K-means clustering: Developing a diagnostic technique. Psychological Methods, 11, 178–192.CrossRef Steinley, D. (2006b). Profiling local optima in K-means clustering: Developing a diagnostic technique. Psychological Methods, 11, 178–192.CrossRef
go back to reference Steinley, D., & Henson, R. (2005). An analytic method for clusters with known overlap. Manuscript submitted for publication. Steinley, D., & Henson, R. (2005). An analytic method for clusters with known overlap. Manuscript submitted for publication.
go back to reference Stout, W.F. (1987). A nonparametric approach for assessing latent trait dimensionality. Psychometrika, 52, 589–617.MathSciNetCrossRef Stout, W.F. (1987). A nonparametric approach for assessing latent trait dimensionality. Psychometrika, 52, 589–617.MathSciNetCrossRef
go back to reference 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.CrossRef 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.CrossRef
go back to reference Tatsouka, C. (2002). Data analytic methods for latent partially ordered classification models. Journal of the Royal Statistical Society Series C, 51, 337–350.MathSciNetCrossRef Tatsouka, C. (2002). Data analytic methods for latent partially ordered classification models. Journal of the Royal Statistical Society Series C, 51, 337–350.MathSciNetCrossRef
go back to reference 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. 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.
go back to reference 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.MathSciNetCrossRef 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.MathSciNetCrossRef
go back to reference Ward, J.H. Jr. (1963). Hierarchical grouping to optimize an objective function. Journal of the American Statistical Association, 58, 236–244.MathSciNetCrossRef Ward, J.H. Jr. (1963). Hierarchical grouping to optimize an objective function. Journal of the American Statistical Association, 58, 236–244.MathSciNetCrossRef
go back to reference Zhang, J., & Stout, W. (1999). The theoretical detect index of dimensionality and its application to approximate simple structure. Psychometrika, 63, 213–249.MathSciNetCrossRef Zhang, J., & Stout, W. (1999). The theoretical detect index of dimensionality and its application to approximate simple structure. Psychometrika, 63, 213–249.MathSciNetCrossRef
Metadata
Title
Using an Iterative Reallocation Partitioning Algorithm to Verify Test Multidimensionality
Authors
Douglas L. Steinley
M. J. Brusco
Publication date
21-11-2019
Publisher
Springer US
Published in
Journal of Classification / Issue 3/2019
Print ISSN: 0176-4268
Electronic ISSN: 1432-1343
DOI
https://doi.org/10.1007/s00357-019-09347-z

Other articles of this Issue 3/2019

Journal of Classification 3/2019 Go to the issue

Premium Partner