Skip to main content
SpringerLink
Log in

Quantification Theory: Categories, Variables and Modal Analysis

  • Chapter
  • First Online:
Advanced Studies in Behaviormetrics and Data Science

Part of the book series: Behaviormetrics: Quantitative Approaches to Human Behavior ((BQAHB,volume 5))

Abstract

Quantification theory (QT) is known by many names such as dual scaling, Hayashi’s quantification theory, optimal scaling, homogeneity analysis and correspondence analysis. It is in essence singular value decomposition of categorical data. As Torgerson [29] called QT as principal component analysis (PCA) of categorical data, one may get some ideas about what QT is. The fact that there are so many aliases is interesting and suggests its versatility. Some names reflect certain mathematical aspects and the others quite different characteristics. No matter what aliases one may adopt, it is certain that QT has many hidden characteristics. As we glance at its general developments, we cannot help but wonder why there are still so many problems associated with QT unsolved. The current paper is an essay with its focus on QT’s very basic foundation problems, which have somehow escaped enough attention of researchers. Following Torgerson’s naming of QT as PCA of categorical data, we first look at the most fundamental difference between PCA and QT, and then move on to look at some aspects peculiar to QT. We will conclude the paper with some warnings on the characteristics of input data for QT, to avoid the situation of garbage in garbage out.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Beh, E. J., & Lombardo, R. (2014). Correspondence analysis: Theory, practive and new strategies. Wiley.

    Google Scholar 

  2. Beltrami, E. (1873). Sulle funzioni bilineari (On the bilinear functions). In G. Battagline, & E. Fergola (Eds.), Giornale di Mathematiche (Vol. 11, pp.98–106).

    Google Scholar 

  3. Benzécri, J. P., et al. (1973). L’analyse des données: II. L’analyse des correspondances. Paris: Dunod.

    Google Scholar 

  4. Carroll, J. D., Green, P. E., & Schaffer, C. M. (1986). Interpoint distance comparisons in correspondence analysis. Journal of Marketing Research, 23, 271–280.

    Article  Google Scholar 

  5. Carroll, J. D., Green, P. E., & Schaffer, C. M. (1987). Comparing interpoint distances in correspondence analysis: A clarification. Journal of Marketing Research, 24, 445–450.

    Article  Google Scholar 

  6. Carroll, J. D., Green, P. E., & Schaffer, C. M. (1989). Reply to Greenacre’s commentary on the carroll-green-schaffer scaling of two-way correspondence analysis solutions. Journal of Marketing Research, 26, 366–368.

    Article  Google Scholar 

  7. Eckart, C., & Young, G. (1936). The approximation of one matrix by another of lower rank. Psychometrika, 1, 211–218.

    Article  MATH  Google Scholar 

  8. Garmize, L. M., & Rychlak, J. F. (1964). Role-play validation of a socio-cultural theory of symbolism. Journal of Consulting Psychology, 28, 107–116.

    Google Scholar 

  9. Gifi, A. (1990). Nonlinear multivariate analysis. Wley.

    Google Scholar 

  10. Greenacre, M. J. (1989). The Carroll-Green-Schaffer scaling in correspondence analysis: A theoretical and empirical appraisal. Journal of Marketing Research, 26, 358–365.

    Article  Google Scholar 

  11. Hotelling, H. (1933). Analysis of complex of statistical variables into principal components. Journal of Educational Psychology, 24, 417–441, and 498–520.

    Google Scholar 

  12. Jordan, C. (1874). Mémoire sur les formes bilinieres (Note on bilinear forms). Journal de Mathématiques Pures et Appliquées, deuxiéme Série, 19, 35–54.

    Google Scholar 

  13. Nishisato, S. (1980). Analysis of categorical data: Dual scaling and its applications. Toronto: University of Toronto Press.

    Book  MATH  Google Scholar 

  14. Nishisato, S. (1984). Forced classification: A simple application of a quantification technique. Psychometrika, 49, 25–36.

    Article  Google Scholar 

  15. Nishisato, S. (1986). Generalized forced classification for quantifying categorical data. In E. Deday, et al. (Eds.), Data analysis and informatics (pp. 351–362). Amsterdam: North-Holland.

    Google Scholar 

  16. Nishisato, S. (1988a). Forced classificaiton procedure of dual scaling: its mathematical properties. In H. H. Bock (Ed.), Classificaiton and related methods (pp. 523–532). Amsterdam: North-Holland.

    Google Scholar 

  17. Nishisato, S. (1988b). Market segmentation by dual scaling through generalized forced classification. In W. Gaul & M. Schader (Eds.), Data, expert knowledge and decisions (pp. 268–278). Berlin: Springer-Verlag.

    Google Scholar 

  18. Nishisato, S. (1991). Standardizing multidimensional space for dual scaling. In Proceedings of the 20th Annual Meeting of the German Operations Research Soceity (pp. 584–591). Hohenheim University.

    Google Scholar 

  19. Nishisato, S. (2007). Multidimensional nonlinear descriptive analysis. London: Chapman and Hall/CRC.

    MATH  Google Scholar 

  20. Nishisato, S. (2019). Reflection: Quantification theory and graphs. Theory and Applications of Data Analysis, 8(1), 47–57. (in Japanese).

    Google Scholar 

  21. Nishisato, S., & Baba, Y. (1999). On contingency, projection and forced classification of dual scaling. Behaviormetrika, 26, 207–219.

    Article  Google Scholar 

  22. Nishisato, S., Beh, E., Lombardo, R., & Clavel, J.G. (2020, in press). Modern Quantifiation Theory: Joint Grapphical Display, Biplots and Alternatives. Springer: Singapore.

    Google Scholar 

  23. Nishisato, S., & Clavel, J. G. (2002). A note on between-set distances in dual scaling and correspondence analaysis. Bahaviormetrika, 30, 207–219.

    Google Scholar 

  24. Nishisato, S., & Gaul, W. (1990). An approach to marketing data analysis. Journal of Marketing Research, 27, 354–360.

    Article  Google Scholar 

  25. Nishisato, S. & Hilsdale, N. J. (1994). Elements of dual scaling: An introduction to practical data analysis. Lawrence Erlbaum Associates.

    Google Scholar 

  26. Nishisato, S., & Lawrence, D. R. (1989). Dual scaling of multiway data matrices: Several variants. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 317–326). Amsterdam: Elsevier Science Publishers.

    Google Scholar 

  27. Nishisato, S., & Yamauchi, H. (1974). Principal components of deviation scores and standard scores. Japanese Psychological Research, 16, 162–170.

    Google Scholar 

  28. Pearson, K. (1901). On lines and planes of closest fit to systems of points in space. Philosophical Magazines and Journal of Science, Series, 6(2), 559–572.

    Article  MATH  Google Scholar 

  29. Torgerson, W. S. (1958). Theory and methods of scaling. New York: Wiley.

    Google Scholar 

  30. Young, G., & Householder, A. S. (1938). Discussion of a set of points in terms of their mutual distances. Psychometrika, 3, 19–22.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Toronto, Toronto, Canada

    Shizuhiko Nishisato

Authors
  1. Shizuhiko Nishisato
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shizuhiko Nishisato .

Editor information

Editors and Affiliations

  1. School of Management and Information Sciences, Tama University, Tokyo, Japan

    Tadashi Imaizumi

  2. Graduate School of Management, Tokyo Metropolitan University, Tokyo, Japan

    Atsuho Nakayama

  3. School of Business, Aoyama Gakuin University, Tokyo, Japan

    Satoru Yokoyama

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Nishisato, S. (2020). Quantification Theory: Categories, Variables and Modal Analysis. In: Imaizumi, T., Nakayama, A., Yokoyama, S. (eds) Advanced Studies in Behaviormetrics and Data Science. Behaviormetrics: Quantitative Approaches to Human Behavior, vol 5. Springer, Singapore. https://doi.org/10.1007/978-981-15-2700-5_15

Download citation

Publish with us

Policies and ethics

Search

Navigation