Skip to main content
Log in

Asymmetric Multidimensional Scaling of N-Mode M-Way Categorical Data using a Log-Linear Model

  • Published:
Behaviormetrika Aims and scope Submit manuscript

Abstract

Asymmetric multidimensional scaling (AMDS) is a visualization method that can be applied to asymmetric (dis)similarity data, which are adopted in several areas such as marketing research, psychometrics, and information science. Several researchers within these fields have examined and applied AMDS. However, the combination of the number of modes and ways remains fixed across these AMDS. Thus, the choice of model is largely contingent on the number of modes and ways. To overcome this problem, we propose an AMDS using a log-linear model with an m-way frequency table. Using the log-linear model, we apply AMDS to (dis)similarity data without a fixed number of modes and ways. In addition, we were able to simultaneously visualize two types of circles.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Borg, I., & Groenen, P. J. (2005). Modern Multidimensional Scaling: Theory and Applications (2nd ed.). New York: Springer.

    MATH  Google Scholar 

  • Carroll, J. D., & Chang, J. J. (1970). Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition. Psychometrika, 35, 283–319.

    Article  Google Scholar 

  • Carroll, J. D., Pruzansky, S., & Kruskal, J. B. (1980). CANDELINK: A general approach to multidimensional analysis of many-way arrays with linear constraints on parameters. Psy-chometrika, 45, 3–24.

    MATH  Google Scholar 

  • Chino, N., & Okada, A. (1996). Hitaisho tajigen syakudo koseiho to sono syuhen [Asymmetric multidimensional scaling and related topics]. KōdōKeiryōgaku, 23, 130–152 (in Japanese).

    Google Scholar 

  • Cox, T. F., & Cox, M. A. (2000). Multidimensional Scaling (2nd ed.). London: Chapman & Hall/CRC.

    Book  Google Scholar 

  • Cox, T. F., Cox, M. A., & Branco, J. A. (1991). Multidimensional scaling for n-tuples. British Journal of Mathematical and Statistical Psychology, 44, 195–206.

    Article  Google Scholar 

  • De Leeuw, J., & Heiser, W. J. (1982). Theory of multidimensional scaling, In P. R. Krishnaiah & L. N. Kanal (Eds.), Multivariate analysis (Vol. V, pp. 501–522). Amsterdam, The Netherlands: North-Holland.

    Google Scholar 

  • De Rooij, M. (2001). Distance association models for the analysis of repeated transition frequency tables. Statistica Neerlandica, 55, 157–181.

    Article  MathSciNet  Google Scholar 

  • De Rooij, M., & Gower, J. C. (2003). The geometry of triadic distances. Journal of Classification, 20, 181–220.

    Article  MathSciNet  Google Scholar 

  • De Rooij, M., & Heiser, W. J. (2003). A distance representation of the quasi-symmetry model and related distance models. In Yanai, H., Okada, A., Shigemasu, K., Kano, Y., Meulman, J. (Eds.), New Developments in Psychometrics: Proceedings of the International Meeting of the Psychometric Society IMPS2001. Osaka, Japan, July 15–19, 2001 (pp. 487–494). Tokyo: Springer.

    Chapter  Google Scholar 

  • De Rooij, M., & Heiser, W. J. (2005). Graphical representations and odds ratios in a distance-association model for the analysis of cross-classified data. Psychometrika, 70, 99–122.

    Article  MathSciNet  Google Scholar 

  • Heiser, W. J., & Bennani, M. (1997). Triadic distance models: axiomatization and least squares representation. Journal of Mathematical Psychology, 41, 189–206.

    Article  MathSciNet  Google Scholar 

  • Lebart, L. (1998). Correspondence analysis, discrimination, and neural networks. In Hayashi, C., Ohsumi, N. Yajima, K., Tanaka, Y., Bock H.-H., & Baba, Y. (Eds.), Data Science, Classification, and Related Methods (pp. 423–430). Tokyo: Springer.

    Google Scholar 

  • Le Roux, B., & Rouanet, H. (2010). Multiple Correspondence Analysis (Vol. 163). Thousand Oaks: Sage Inc.

    Book  Google Scholar 

  • Nakayama, A. (2005). A multidimensional scaling model for three-way data analysis, Behav-iormetrika, 32, 95–110.

    MathSciNet  MATH  Google Scholar 

  • Nakayama, A., & Okada, A. (2012). Reconstructing one-mode three-way asymmetric data for multidimensional scaling. In Gaul, W., Geyer-Schulz, A., Schmidt-Thieme, L., & Kunze, J. (Eds.), Challenges at the Interface of Data Analysis, Computer Science, and Optimization (pp. 133–141). Heidelberg: Springer-Verlag.

    Chapter  Google Scholar 

  • Okada, A., & Imaizumi, T. (1987). Nonmetric multidimensional scaling of asymmetric proximities. Behaviormetrika, 14, 81–96.

    Article  Google Scholar 

  • Okada, A., & Imaizumi, T. (1997). Asymmetric multidimensional scaling of two-mode three-way proximities. Journal of Classification, 14, 195–224.

    Article  Google Scholar 

  • Okada, A., & Tsurumi, H. (2012). Asymmetric multidimensional scaling of brand switching among margarine brands. Behaviormetrika, 39, 111–126.

    Article  Google Scholar 

  • Okada, K. (2012). A bayesian approach to asymmetric multidimensional scaling. Behaviormetrika, 39, 49–62.

    Article  Google Scholar 

  • Okada, K., & Mayekawa, S. (2011). Bayesian nonmetric successive categories multidimensional scaling. Behaviormetrika, 38, 17–31.

    Article  Google Scholar 

  • Saito, T., & Takeda, S.-i. (1990). Multidimensional scaling of asymmetric proximity: model and method. Behaviormetrika, 17, 49–80.

    Article  Google Scholar 

  • Tversky, A. (1977). Features of similarity. Psychological Review, 84, 327–352.

    Article  Google Scholar 

  • Tversky, A., & Gati, I. (1978). Studies of similarity. In E. Rosch & B. Lloyd (Eds.) Cognition and Categorization (pp. 79-98), Hillsdale: Erlbaum.

    Google Scholar 

  • Warrens, M. J. (2010). n-way metrics. Journal of Classification, 27, 173–190.

    Article  MathSciNet  Google Scholar 

  • Young, F. W. (1975). An asymmetric euclidean model for multi-process asymmetric data. In: Proceedings of the US-Japan Seminar of the Theory, Methods and Applications of Multidimensional Scaling and Related Techniques at the University of California San Diego, San Diego, U.S.A. 79–88.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jun Tsuchida.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tsuchida, J., Yadohisa, H. Asymmetric Multidimensional Scaling of N-Mode M-Way Categorical Data using a Log-Linear Model. Behaviormetrika 43, 103–138 (2016). https://doi.org/10.2333/bhmk.43.103

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2333/bhmk.43.103

Key Words and Phrases

Navigation