Skip to main content
SpringerLink
Log in

Homogeneity Analysis of Transition Matrices for Spatially Representing a Transition Trend

  • Published:
Behaviormetrika Aims and scope Submit manuscript

Abstract

An optimal scaling method is proposed for spatially representing the trend of individuals’ transition described by a square contingency table. The method gives the low dimensional configuration of category points and the trend vector representing an overall tendency of the individuals’ transition under the framework of homogeneity analysis. The configuration is obtained analytically with simple eigen-decomposition. Some examples are given to illustrate the method and its relationships to ordinary homogeneity analysis, correspondence analysis and the slide vector model of asymmetric multidimensional scaling are discussed.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Adachi, K. (1995) An unfolding model for confusion matrices. Japanese Journal of Psychology, 66, 58–62 (in Japanese with English summary).

    Article  Google Scholar 

  • Andersen, E.B. (1990) The statistical analysis of categorical data (Second edition). Berlin: Springer.

    Book  Google Scholar 

  • Bekker, P., & de Leeuw, J. (1988) Relations between variants of non-linear principal component analysis. In J.L.A. van Rijckevorsel & J. de Leeuw (Eds.), Component and correspondence analysis (pp. 1–31), Chichester: Wiley.

    Google Scholar 

  • Bishop, Y.M.M., Fienberg, S.E., & Holland, P.W. (1975) Discrete multivariate analysis: Theory and practice. Cambridge: MIT Press.

    MATH  Google Scholar 

  • Chino, N., & Okada, A. (1996) Asymmetric multidimensional scaling and related topics. Japanese Journal of Behaviormetrics, 23, 130–152 (in Japanese with English Summary).

    Article  Google Scholar 

  • Garner, W.R., & Haun, F. (1978) Letter identification as a function of type of perceptual limitation and type of attribute. Journal of Experimental Psychology: Human Perception and Performance, 4, 199–209.

    Google Scholar 

  • Gin, A. (1990) Nonlinear multivariate analysis. Chichester: Wiley.

    Google Scholar 

  • Goodman, L.A. (1985) The analysis of cross-classified data having ordered and/or unordered categories: Association models, correlation models, and asymmetry models for contingency tables with or without missing entries. The Annals of Statistics, 13, 10–69.

    Article  MathSciNet  Google Scholar 

  • Greenacre, M.J. (1984) Theory and applications of correspondence analysis. London: Academic Press.

    MATH  Google Scholar 

  • Harshman, R.A., Green, P.E., Wind, Y., & Lundy, M.E. (1982) A model for the analysis of asymmetric data in marketing research. Marketing Science, 1, 205–242.

    Article  Google Scholar 

  • Hayashi, C. (1952) On the prediction of phenomena from qualitative data and the quantification of qualitative data from the mathematico-statistical point of view. Annals of the Institute of Statistical Mathematics, 3, 69–98.

    Article  Google Scholar 

  • Heiser, W.J., & Meulman, JJ. (1995) Nonlinear methods for the analysis of homogeneity and heterogeneity. In W.J. Krzanowski (Ed.), Recent advances in descriptive multivariate analysis (pp. 51–89), Oxford: Oxford University Press.

    Google Scholar 

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

    MATH  Google Scholar 

  • Seiyama, K., Naoi, A., Sato, Y., Tsuzuki, K., & Kojima, H. (1990) Gendai nippon no kaisou kouzou to sono susei (Hierarchical structure and trends in modern Japan). In A. Naoi & K. Seiyama (Eds.), Gendai nippon no kaisou kouzou 1: Shakai kaisou no kouzou to katei (Hierarchical structure in modern Japan 1: Structure and process of social hierarchy) (pp. 15–50), Tokyo: University of Tokyo Press (in Japanese).

    Google Scholar 

  • ten Berge, J.M.F. (1993) Least squares optimization in multivariate analysis. Leiden: DSWO press.

    Google Scholar 

  • Weeks, D.G., & Bentler, P.M. (1982) Restricted multidimensional scaling models for asymmetric proximities. Psychometrika, 47, 201–208.

    Article  Google Scholar 

  • Yanai, H. (1994) Tahenryo deta kaiseki (Multivariate data analysis). Tokyo: Asakura Shoten (in Japanese).

    Google Scholar 

  • Zielman, B., & Heiser, W.J. (1993) Analysis of asymmetry by a slide-vector. Psychometrika, 58, 101–114.

    Article  Google Scholar 

  • Zielman, B., & Heiser, W.J. (1996) Models for asymmetric similarities. British Journal of Mathematical and Statistical Psychology, 49, 127–146.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Humanities, Koshien University, 10-1, Momijigaoka, Takarazuka-shi, Hyogo, 665, Japan

    Kohei Adachi

Authors
  1. Kohei Adachi
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Adachi, K. Homogeneity Analysis of Transition Matrices for Spatially Representing a Transition Trend. Behaviormetrika 24, 159–178 (1997). https://doi.org/10.2333/bhmk.24.159

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2333/bhmk.24.159

Key Words and Phrases

Search

Navigation