Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-08T12:09:39.835Z Has data issue: false hasContentIssue false
  • English
  • Français

Fuzzy Clusterwise Generalized Structured Component Analysis

Published online by Cambridge University Press:  01 January 2025

Heungsun Hwang ,
Wayne S. DeSarbo  and
Yoshio Takane
Heungsun Hwang*
Affiliation:
McGill University
Wayne S. DeSarbo
Affiliation:
Pennsylvania State University
Yoshio Takane
Affiliation:
McGill University
*
Requests for reprints should be sent to Heungsun Hwang, Department of Psychology, McGill University, 1205 Dr. Penfield Avenue, Montreal, QC, H3A 1B1, Canada. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Generalized Structured Component Analysis (GSCA) was recently introduced by Hwang and Takane (2004) as a component-based approach to path analysis with latent variables. The parameters of GSCA are estimated by pooling data across respondents under the implicit assumption that they all come from a single, homogenous group. However, as has been empirically demonstrated by various researchers across a number of areas of inquiry, such aggregate analyses can often mask the true structure in data when respondent heterogeneity is present. In this paper, GSCA is generalized to a fuzzy clustering framework so as to account for potential group-level respondent heterogeneity. An alternating least-squares procedure is developed and technically described for parameter estimation. A small-scale Monte Carlo study involving synthetic data is carried out to compare the performance between the proposed method and an extant approach. In addition, an empirical application concerning alcohol use among adolescents from US northwestern urban areas is presented to illustrate the usefulness of the proposed method. Finally, a number of directions for future research are provided.

Keywords

generalized structured component analysisfuzzy clusteringrespondent heterogeneityalternating least squareslatent curve modelsalcohol use
Type
Original Paper
Information
Psychometrika , Volume 72 , Issue 2 , June 2007 , pp. 181 - 198
Copyright
Copyright © 2007 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The work reported in this paper was supported by Grant 290439 and Grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the first and third authors, respectively. We wish to thank Terry Duncan for generously providing us with his alcohol use data. We also wish to thank the Editor, Associate Editor, and two anonymous reviewers for their constructive comments which helped improve the overall quality and readability of this manuscript.

References

Arabie, P., Carroll, J.D., DeSarbo, W.S., Wind, J. (1981). Overlapping clustering: A new method for product positioning. Journal of Marketing Research, 18, 310–317.CrossRefGoogle Scholar
Arabie, P., Hubert, L. (1994). Cluster analysis in marketing research. In Bagozzi, R.P. (Eds.), Advanced methods of marketing research (pp. 160–189). Oxford: Blackwell.Google Scholar
Bagozzi, R.P. (1982). A field investigation of causal relations among cognition, affect, intensions, and behavior. Journal of Marketing Research, 19, 562–584.CrossRefGoogle Scholar
Bauer, D.J., Curran, P.J. (2003). Distributional assumptions of growth mixture models: Implications for overextraction of latent trajectory classes. Psychological Methods, 8, 338–363.CrossRefGoogle ScholarPubMed
Bentler, P.M., Weeks, D.G. (1980). Linear structural equations with latent variables. Psychometrika, 45(3), 289–308.CrossRefGoogle Scholar
Bezdek, J.C. (1974a). Numerical taxonomy with fuzzy sets. Journal of Mathematical Biology, 1, 57–71.CrossRefGoogle Scholar
Bezdek, J.C. (1974b). Cluster validity with fuzzy set. Journal of Cybernetics, 3, 58–72.CrossRefGoogle Scholar
Bezdek, J.C. (1981). Pattern recognition with fuzzy objective function algorithms, New York: Plenum Press.CrossRefGoogle Scholar
Bezdek, J.C., Coray, C., Gunderson, R., Watson, J. (1981). Detection and characteristics of cluster substructure. II. Fuzzy c-varieties and convex combinations thereof. SIAM Journal on Applied Mathematics, 40, 358–372.CrossRefGoogle Scholar
Bock, R.D. (1989). Multilevel analysis of educational data, San Diego: Academic Press.Google Scholar
Bock, R.D., Bargmann, R.E. (1966). Analysis of covariance structures. Psychometrika, 31, 507–534.CrossRefGoogle ScholarPubMed
Chin, W.W. (1998). Issues and opinion on structural equation modeling. Management Information Systems Quarterly, 22, 7–16.Google Scholar
de Leeuw, J., Young, F.W., Takane, Y. (1976). Additive structure in qualitative data: An alternating least squares method with optimal scaling features. Psychometrika, 41, 471–503.CrossRefGoogle Scholar
Dempster, A.P., Laird, N.M., Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM-algorithm. Journal of the Royal Statistical Society, Series B, 39, 1–18.CrossRefGoogle Scholar
DeSarbo, W.S., Cron, W.L. (1988). A conditional mixture maximum likelihood methodology for clusterwise linear regression. Journal of Classification, 5, 249–289.CrossRefGoogle Scholar
DeSarbo, W.S., Grewal, R., & Hwang, H. (2006). A clusterwise bilinear multidimensional scaling methodology for marketing research: An application to the estimation of strategic groups. Unpublished manuscript.Google Scholar
Duncan, T.E., Duncan, S.C., Alpert, A., Hops, H., Stoolmiller, M., Muthén, B. (1997). Latent variable modeling of longitudinal and multilevel substance use data. Multivariate Behavioral Research, 32, 275–318.CrossRefGoogle ScholarPubMed
Dunn, J.C. (1974). A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters. Journal of Cybernetics, 3, 32–57.CrossRefGoogle Scholar
Efron, B. (1982). The jackknife, the bootstrap and other resampling plans, Philadelphia: SIAM.CrossRefGoogle Scholar
Everitt, B.S., Landau, S., & Leese, M. (2001). Cluster analysis (4th ed.). London: Arnold Press.Google Scholar
Fornell, C., Bookstein, F.L. (1982). Two structural equation models: LISREL and PLS applied to consumer exit-voice theory. Journal of Marketing Research, 19, 440–452.CrossRefGoogle Scholar
Gifi, A. (1990). Nonlinear multivariate analysis, Chichester: Wiley.Google Scholar
Goldstein, H.I. (1987). Multilevel models in educational and social research, London: Oxford University Press.Google Scholar
Gordon, A.D. (1999). Classification, London: Chapman & Hall/CRC.CrossRefGoogle Scholar
Hathaway, R.J., Bezdek, J.C. (1993). Switching regression models and fuzzy clutering. IEEE Transactions on Fuzzy Systems, 1, 195–204.CrossRefGoogle Scholar
Heiser, W.J., Groenen, P.J.F. (1997). Cluster differences scaling with a within-clusters loss component and a fuzzy successive approximation strategy to avoid local minima. Psychometrika, 62, 63–83.CrossRefGoogle Scholar
Hox, J.J. (1995). Applied multilevel analysis, Amsterdam: T-T Publikaties.Google Scholar
Hruschka, H. (1986). Market definition and segmentation using fuzzy clustering methods. International Journal of Research in Marketing, 3, 117–134.CrossRefGoogle Scholar
Hwang, H., & Takane, Y. (2002). Structural equation modeling by extended redundancy analysis. In Nishisato, S., Baba, Y., Bozdogan, H. & Kanefuji, K. (Eds.), Measurement and multivariate analysis (pp. 115–124). Tokyo: Springer-Verlag.CrossRefGoogle Scholar
Hwang, H., Takane, Y. (2004). Generalized structured component analysis. Psychometrika, 69, 81–99.CrossRefGoogle Scholar
Jedidi, K., Jagpal, H.S., DeSarbo, W. S. (1997). Finite-mixture structural equation models for response-based segmentation and unobserved heterogeneity. Marketing Science, 16, 39–59.CrossRefGoogle Scholar
Jöreskog, K.G. (1970). A general method for analysis of covariance structures. Biometrika, 57, 409–426.CrossRefGoogle Scholar
Jöreskog, K. G. (1973). A generating method for estimating a linear structural equation system. In Goldberger, A.S. & Duncan, O.D. (Eds.), Structural equation models in the social sciences (pp. 85–112). New York: Academic Press.Google Scholar
Kamakura, W.A., Kim, B., Lee, J. (1996). Modeling preference and structural heterogeneity in consumer choice. Marketing Science, 15, 152–172.CrossRefGoogle Scholar
MacCallum, R.C., Kim, C., Malarkey, W.B., Kiecolt-Glaser, J.K. (1997). Studying multivariate change using multilevel models and latent curve models. Multivariate Behavioral Research, 32, 215–253.CrossRefGoogle ScholarPubMed
Manton, K.G., Woodbury, M.A., Tolley, H.D. (1994). Statistical applications using fuzzy sets, New York: Wiley.Google Scholar
McArdle, J.J., McDonald, R.P. (1984). Some algebraic properties of the reticular action model for moment structures. British Journal of Mathematical and Statistical Psychology, 37, 234–251.CrossRefGoogle ScholarPubMed
McBratney, A.B., Moore, A.W. (1985). Application of fuzzy sets to climatic classification. Agricultural and Forest Meteorology, 35, 165–185.CrossRefGoogle Scholar
McLachlan, G., Peel, D. (2000). Finite mixture models, New York: Wiley.CrossRefGoogle Scholar
Meredith, W., Tisak, J. (1990). Latent curve analysis. Psychometrika, 55, 107–122.CrossRefGoogle Scholar
Moffitt, T.E. (1993). Adolescent-limited and life-course-persistent antisocial behavior: A developmental taxonomy. Psychological Review, 100, 674–701.CrossRefGoogle Scholar
Mulaik, S.A. (1972). The foundations of factor analysis, New York: McGraw-Hill.Google Scholar
Muthén, B.O. (2001). Latent variable mixture modeling. In Marcoulides, G.A. & Schumacker, R.E. (Eds.), New developments and techniques in structural equation modeling (pp. 1–13). Mahwah, NJ: Erlbaum.Google Scholar
Muthén, L.K., & Muthén, B.O. (2001). Mplus user’s guide (2nd ed.). Los Angeles: Muthén & Muthén.Google Scholar
Nagin, D. (1999). Analyzing developmental trajectories: A semi-parametric, group-based approach. Psychological Methods, 4, 139–157.CrossRefGoogle Scholar
Okeke, F., Karnieli, A. (2006). Linear mixture model approach for selecting fuzzy exponent values in fuzzy c-means algorithm. Ecological Informatics, 1, 117–124.CrossRefGoogle Scholar
Ramaswamy, V., DeSarbo, W.S., Reibstein, D., Robinson, W. (1993). An empirical pooling approach for estimating marketing mix elasticities with PIMS data. Marketing Science, 12, 103–124.CrossRefGoogle Scholar
Raudenbush, S.W., & Bryk, A.S. (2002). Hierarchical linear models: Applications and data analysis methods (2nd ed.). Thousand Oaks, CA: Sage.Google Scholar
Roubens, M. (1982). Fuzzy clustering algorithms and their cluster validity. European Journal of Operational Research, 10, 294–301.CrossRefGoogle Scholar
Tucker, L.R. (1951). A method for synthesis of factor analysis studies (Personnel Research Section Report No. 984). Washington, DC: US Department of the Army.CrossRefGoogle Scholar
Velicer, W.F., Jackson, D.N. (1990). Component analysis versus common factor analysis: Some issues in selecting appropriate procedure. Multivariate Behavioral Research, 25, 1–28.CrossRefGoogle ScholarPubMed
Wedel, M., Kamakura, W.A. (1998). Market segmentation: Conceptual and methodological foundations, Boston: Kluwer Academic.Google Scholar
Wedel, M., Steenkamp, J.-B.E.M. (1989). Fuzzy clusterwise regression approach to benefit segmentation. International Journal of Research in Marketing, 6, 241–258.CrossRefGoogle Scholar
Wedel, M., Steenkamp, J.-B.E.M. (1991). A clusterwise regression method for simultaneous fuzzy market structuring and benefit segmentation. Journal of Marketing Research, 28, 385–396.CrossRefGoogle Scholar
Wind, Y. (1978). Issues and advances in segmentation research. Journal of Marketing Research, 15, 317–337.CrossRefGoogle Scholar
Yang, C.C. (1998). Finite mixture model selection with psychometric applications. Unpublished doctoral dissertation. University of California, Los Angeles.Google Scholar
Young, F.W. (1981). Quantitative analysis of qualitative data. Psychometrika, 46, 357–388.CrossRefGoogle Scholar
Zadeh, L.A. (1965). Fuzzy sets. Information and Control, 8, 338–353.CrossRefGoogle Scholar