• Ingwer Borg
  • Patrick J. F. Groenen
  • Patrick Mair
Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)


Unfolding is discussed again in a realistic and more complex application that requires a 3d solution with a special rotation. For mixed samples, multidimensional unfolding can sometimes be replaced by multiple low-dimensional unfolding. One must also clarify whether the data are unconditionally comparable. The stability of unfolding solutions is discussed, and some special unfolding models such as the vector model and circular unfolding are introduced.


Unfolding External unfolding Internal unfolding Conditionalities Vector model unfolding Circular unfolding 


  1. Borg, I., & Groenen, P. J. F. (2005). Modern multidimensional scaling (2nd ed.). New York: Springer.zbMATHGoogle Scholar
  2. Borg, I., & Staufenbiel, T. (2007). Theorien und methoden der skalierung (4th ed.). Berne: Huber.Google Scholar
  3. Busing, F. M. T. A., Groenen, P. J. F., & Heiser, W. J. (2005). Avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation. Psychometrika, 70, 71–98.MathSciNetCrossRefGoogle Scholar
  4. Carroll, J. D. (1980). Models and methods for multidimensional analysis of preferential choice or other dominance data. In E. D. Lantermann & H. Feger (Eds.), Similarity and choice. Bern: Huber.Google Scholar
  5. Coombs, C. H. (1975). A note on the relation between the vector model and the unfolding model for preferences. Psychometrika, 40, 115–116.CrossRefGoogle Scholar
  6. Norpoth, H. (1979). The parties come to order! Dimensions of preferential choice in the West German electorate, 1961–1976. The American Political Science Review, 73, 724–736.CrossRefGoogle Scholar
  7. Schwartz, S. H. (1992). Universals in the content and structure of values: Theoretical advances and empirical tests in 20 countries. Advances in Experimental Social Psychology, 25, 1–65.Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Ingwer Borg
    • 1
  • Patrick J. F. Groenen
    • 2
  • Patrick Mair
    • 3
  1. 1.Westfälische Wilhelms-UniversitätMuensterGermany
  2. 2.Econometric InstituteErasmus University RotterdamRotterdamThe Netherlands
  3. 3.Department of PsychologyHarvard UniversityCambridgeUSA

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