Skip to main content
Log in

A Model-Based Approach for Visualizing the Dimensional Structure of Ordered Successive Categories Preference Data

  • Theory and Methods
  • Published:
Psychometrika Aims and scope Submit manuscript

Abstract

A cyclical conditional maximum likelihood estimation procedure is developed for the multidimensional unfolding of two- or three-way dominance data (e.g., preference, choice, consideration) measured on ordered successive category rating scales. The technical description of the proposed model and estimation procedure are discussed, as well as the rather unique joint spaces derived. We then conduct a modest Monte Carlo simulation to demonstrate the parameter recovery of the proposed methodology, as well as investigate the performance of various information heuristics for dimension selection. A consumer psychology application is provided where the spatial results of the proposed model are compared to solutions derived from various traditional multidimensional unfolding procedures. This application deals with consumers intending to buy new luxury sport-utility vehicles (SUVs). Finally, directions for future research are discussed.

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

  • Aaker, D. (1991). Managing brand equity. New York: The Free Press.

    Google Scholar 

  • Aaker, D. (1996). Building strong brands. New York: The Free Press.

    Google Scholar 

  • Adams, E., & Messick, S. (1958). An axiomatic formulation and generalization of successive intervals scaling. Psychometrika, 23, 355–368.

    Article  Google Scholar 

  • Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19, 716–723.

    Article  Google Scholar 

  • Benzécri, J.P. (1973). L’analyse des données: Tome II. Analyse de correspondances. Paris: Dunod.

    Google Scholar 

  • Benzécri, J.P. (1992). Correspondence analysis handbook. New York: Dekker.

    Google Scholar 

  • Borg, I., & Groenen, P. (2005). Modern multidimensional scaling: Theory and application (2nd edn.). New York: Springer.

    Google Scholar 

  • Bozdogan, H. (1987). Model selection and Akaike’s Information Criterion (AIC): The general theory and its analytical extensions. Psychometrika, 52, 345–370.

    Article  Google Scholar 

  • Busing, F.M.T.A., Groenen, P.J.F., & Heiser, W. (2005). Avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation. Psychometrika, 70, 71–98.

    Article  Google Scholar 

  • Carroll, J.D. (1972). Individual differences and multidimensional scaling. In R.N. Shepard, A.K. Romney, & S. Nerlove (Eds.), Multidimensional scaling: Theory and applications in the behavior sciences: Vol. I. Theory. New York: Seminar Press.

    Google Scholar 

  • 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. Vienna: Hans Huber.

    Google Scholar 

  • Cliff, N. (1973). Scaling. Annual Review of Psychology, 24, 473–506.

    Article  Google Scholar 

  • Consumer quide 2002 automobile book (2002). Lincolnwood, IL: Publications International Ltd.

  • Coombs, C.H., Dawes, R.M., & Tversky, A. (1970). Mathematical psychology. Englewood Cliffs: Prentice-Hall.

    Google Scholar 

  • Cox, T.F., & Cox, M.A. (2001). Multidimensional scaling (2nd edn.). London: Chapman & Hall.

    Google Scholar 

  • DeSarbo, W.S., & Carroll, J.D. (1985). Three-way metric unfolding via weighted least-squares. Psychometrika, 50, 275–300.

    Article  Google Scholar 

  • DeSarbo, W.S., & Hoffman, D. (1986). Simple and weighted unfolding MDS threshold models for the spatial analysis of binary data. Applied Psychological Measurement, 10, 247–264.

    Article  Google Scholar 

  • DeSarbo, W.S., & Rao, V.R. (1984). GENFOLD2: A set of models and algorithms for the general unfolding analysis of preference/dominance data. Journal of Classification, 1, 147–186.

    Article  Google Scholar 

  • DeSarbo, W.S., & Rao, V.R. (1986). A constrained unfolding methodology for product positioning. Marketing Science, 5, 1–19.

    Article  Google Scholar 

  • DeSarbo, W.S., Young, M.R., & Rangaswamy, A. (1997). A parametric multidimensional unfolding procedure for incomplete nonmetric preference/choice set data in marketing research. Journal of Marketing Research, 34, 499–516.

    Article  Google Scholar 

  • Farquhar, P.H. (1989). Managing brand equity. Marketing Research, 1, 24–33.

    Google Scholar 

  • Fletcher, R. (1987). Practical methods of optimization. New York: Wiley.

    Google Scholar 

  • Galanter, E., & Messick, S. (1961). The relation between category and magnitude scales of loudness. Psychological Review, 68, 363–372.

    Article  PubMed  Google Scholar 

  • Gifi, A. (1990). Nonlinear multivariate analysis. New York: Wiley.

    Google Scholar 

  • Gilbert, J., & Nocedal, J. (1992). Global convergence properties of conjugate gradient methods for optimization. SIAM Journal on Optimization, 2, 21–42.

    Article  Google Scholar 

  • Good, P. (2005). Permutation, parametric, and bootstrap tests of hypotheses (3rd edn.). New York: Springer.

    Google Scholar 

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

    Google Scholar 

  • Greenacre, M.J., & Browne, M.W. (1986). An efficient alternating least-squares algorithm to perform multidimensional unfolding. Psychometrika, 51, 241–250.

    Article  Google Scholar 

  • Greene, W.H. (2003). Econometric analysis (5th edn.). Upper Saddle River, Prentice-Hall.

    Google Scholar 

  • Heiser, W.J. (1981). Unfolding analysis of proximity data. Dissertation, University of Leiden.

  • J.D. Power and Associates (2002). Automobile sales report. www.jdpower.com/corporate/library/publications.

  • Jedidi, K., & DeSarbo, W.S. (1991). A stochastic multidimensional scaling procedure for the spatial representation of three-mode, three-way pick any/J data. Psychometrika, 56, 471–494.

    Article  Google Scholar 

  • Johnson, M.S., & Junker, B.W. (2003). Using data augmentation and Markov chain Monte Carlo for the estimation of unfolding response models. Journal of Educational and Behavioral Statistics, 28, 195–230.

    Article  Google Scholar 

  • Kamakura, W.A., & Russell, G.J. (1993). Measuring brand value with scanner data. International Journal of Research in Marketing, 10, 9–22.

    Article  Google Scholar 

  • Keller, K.L. (1993). Conceptualizing, measuring, and managing customer-based brand equity. Journal of Marketing, 57, 1–22.

    Article  Google Scholar 

  • Keller, K.L. (2003). Strategic brand management (2nd edn.). Englewood Cliffs: Prentice-Hall.

    Google Scholar 

  • Kim, C., Rangaswamy, A., & DeSarbo, W.S. (1999). A quasi-metric approach to multidimensional unfolding for reducing the occurrence of degenerate solutions. Multivariate Behavioral Research, 34(2), 143–180.

    Article  Google Scholar 

  • Kim, C., Rangaswamy, A., & DeSarbo, W.S. (2000). A fixed point non-metric unfolding procedure. Multivariate Behavioral Research, 34, 143–180.

    Article  Google Scholar 

  • Kruskal, J.B. (1964a). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1–27.

    Article  Google Scholar 

  • Kruskal, J.B. (1964b). Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115–129.

    Article  Google Scholar 

  • Kruskal, J.B., & Carroll, J.D. (1969). Geometric models and badness of fit functions. In P.R. Krishnaiah (Ed.), Multivariate analysis II. New York: Academic Press.

    Google Scholar 

  • Lingoes, J.C. (1972). A general survey of the Guttman–Lingoes nonmetric program series. In R.N. Shepard, A.K. Romney, & S. Nerlove (Eds.), Theory and applications in the behavior sciences: Vol. I. Theory. New York: Seminar Press.

    Google Scholar 

  • Lingoes, J.C. (1973). The Guttman–Lingoes nonmetric program series. Ann Arbor: Mathesis Press.

    Google Scholar 

  • Luce, R.D., & Edwards, W. (1958). The derivation of subjective scales from just noticeable differences. Psychological Review, 65, 227–237.

    Article  Google Scholar 

  • Maurin, M. (1983). Another least-squares solution for the successive intervals following Adams and Messick. In Third European Meeting of the Psychometric Society. Jouy-en-Josas.

  • Messick, S.J. (1956). An empirical evaluation of multidimensional successive categories. Psychometrika, 21, 367–375.

    Article  Google Scholar 

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

    Google Scholar 

  • Nocedal, J., & Wright, S.J. (1999). Numerical optimization. New York: Springer.

    Book  Google Scholar 

  • Park, C.S., & Srinivasan, V. (1994). A survey-based method for measuring and understanding brand equity and its extendibility. Journal of Marketing Research, 31, 271–288.

    Article  Google Scholar 

  • Polak, E., & Ribiére, G. (1969). Note sur la convergence de methods de directions conjuguées. Revue Franşaise d’Informatique et de Recherche Opérationnelle, 16, 35–43.

    Google Scholar 

  • Powell, M.J.D. (1977). Restart procedures for the conjugate gradient method. Mathematical Programming, 12, 241–254.

    Article  Google Scholar 

  • Roskam, E.E. (1973). Fitting ordinal relational data to a hypothesized structure. Technical Report #73MA06, Catholic University, Nijmegen.

  • Schönemann, P.H. (1970). On metric multidimensional unfolding. Psychometrika, 35, 349–366.

    Article  Google Scholar 

  • Schönemann, P.H., & Tucker, L.R. (1967). A maximum likelihood solution for the method of successive intervals allowing for unequal stimulus dispersions. Psychometrika, 32, 403–418.

    Article  Google Scholar 

  • Slater, P. (1960). Inconsistencies in a schedule of paired comparisons. Biometrika, 48, 303–312.

    Google Scholar 

  • Srinivasan, V. (1979). Network models for estimating brand-specific effects in multi-attribute marketing models. Management Science, 25, 11–21.

    Article  Google Scholar 

  • Srinivasan, V., Park, C.S., & Chang, D.R. (2005). An approach to the measurement, analysis, and prediction of brand equity and its sources. Management Science, 51, 1433–1448.

    Article  Google Scholar 

  • Stevens, S.S. (1966). A metric for the social consensus. Science, 151, 530–541.

    Article  PubMed  Google Scholar 

  • Stevens, S.S. (1971). Issues in psychophysics. Psychological Review, 78, 426–450.

    Article  Google Scholar 

  • Swait, J., Erdem, T., Louvier, T., & Dubelaar, C. (1993). The equalization price: A measure of consumer perceived brand equity. International Journal of Research in Marketing, 10, 23–45.

    Article  Google Scholar 

  • Takane, Y. (1981). Multidimensional successive categories scaling: A maximum likelihood method. Psychometrika, 46, 9–28.

    Article  Google Scholar 

  • Takane, Y. (1997). Choice model analysis of the pick any/n type of binary data. Japanese Psychological Research, 40, 31–39.

    Article  Google Scholar 

  • Takane, Y., & Carroll, J.D. (1981). Maximum likelihood multidimensional scaling from directional rankings of similarities. Psychometrika, 46, 389–406.

    Article  Google Scholar 

  • Takane, Y., Young, F.W., & DeLeeuw, J. (1977). Nonmetric individual differences multidimensional scaling: An alternating least-squares method with optimal scaling features. Psychometrika, 42, 7–67.

    Article  Google Scholar 

  • Torgerson, W.S. (1952). Multidimensional scaling: I. Theory and method. Psychometrika, 17, 401–419.

    Article  Google Scholar 

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

    Google Scholar 

  • Tucker, L.R. (1960). Intra-individual and inter-individual multidimensionality. In H. Gulliksen, & S. Messick (Eds.), Psychological scaling: Theory and applications (pp. 110–123). New York: Wiley.

    Google Scholar 

  • Young, F.W., & Torgerson, W.S. (1967). TORSCA, a FORTRAN IV program for Shepard–Kruskal multidimensional scaling analysis. Behavioral Science, 12, 498.

    Article  Google Scholar 

  • Zinnes, J.L., & Wolff, R.P. (1977). Single and multidimensional same–different judgments. Journal of Mathematical Psychology, 16, 30–50.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wayne S. DeSarbo.

Additional information

The author names are presented alphabetically as all coauthors contributed equally to this manuscript.

The authors wish to thank the editor, the associate editor, and three anonymous referees for their excellent constructive comments which resulted in the considerable improvement of this manuscript.

Rights and permissions

Reprints and permissions

About this article

Cite this article

DeSarbo, W.S., Park, J. & Scott, C.J. A Model-Based Approach for Visualizing the Dimensional Structure of Ordered Successive Categories Preference Data. Psychometrika 73, 1–20 (2008). https://doi.org/10.1007/s11336-007-9015-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11336-007-9015-2

Keywords

Navigation