Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2025-01-06T03:01:24.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-Way Metric Unfolding Via Alternating Weighted Least Squares

Published online by Cambridge University Press:  01 January 2025

Wayne S. DeSarbo  and
J. Douglas Carroll
Wayne S. DeSarbo*
Affiliation:
Marketing Department, Wharton School, University of Pennsylvania
J. Douglas Carroll
Affiliation:
AT&T Bell Laboratories
*
Requests for reprints should be sent to Wayne S. DeSarbo, Marketing Department, Wharton School, University of Pennsylvania, Philadelphia, PA 19104.
Get access Rights & Permissions [Opens in a new window]

Abstract

Three-way unfolding was developed by DeSarbo (1978) and reported in DeSarbo and Carroll (1980, 1981) as a new model to accommodate the analysis of two-mode three-way data (e.g., nonsymmetric proximities for stimulus objects collected over time) and three-mode, three-way data (e.g., subjects rendering preference judgments for various stimuli in different usage occasions or situations). This paper presents a revised objective function and new algorithm which attempt to prevent the common type of degenerate solutions encountered in typical unfolding analysis. We begin with an introduction of the problem and a review of three-way unfolding. The three-way unfolding model, weighted objective function, and new algorithm are presented. Monte Carlo work via a fractional factorial experimental design is described investigating the effect of several data and model factors on overall algorithm performance. Finally, three applications of the methodology are reported illustrating the flexibility and robustness of the procedure.

Keywords

multidimensional scalingunfoldingpreference analysis
Type
Original Paper
Information
Psychometrika , Volume 50 , Issue 3 , September 1985 , pp. 275 - 300
Copyright
Copyright © 1985 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

We wish to thank the editor and reviewers for their insightful comments.

References

Addelman, S. (1962). Orthogonal main-effect plans for asymmetrical factorial experiments. Technometrics, 4, 21–46.CrossRefGoogle 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.CrossRefGoogle Scholar
Carroll, J. D. (1972). Individual differences and multidimensional scaling. In Shepard, R. N., Romney, A. K., Nerlove, S. (Eds.), Multidimensional Scaling: Theory and Applications in the Behavioral Sciences (Vol. I), New York: Academic Press.Google Scholar
Carroll, J. D., Pruzansky, S. (1980). Discrete and hybrid scaling models. In Lantermann, E. D., Feger, H. (Eds.), Similarity and choice, Bern: Hans Huber.Google Scholar
Coombs, C. H. (1964). A Theory of Data, New York: Wiley.Google Scholar
Cooper, L., Steinberg, D. (1970). Introduction to Methods of Optimization, Philadelphia, PA: W. B. Saunders Co..Google Scholar
Courant, R. (1965). Differential and Integral Calculus 2nd ed.,, New York: Wiley.Google Scholar
DeSarbo, W. S. (1978). Three-way unfolding and situational dependence in consumer preference analysis, Philadelphia: University of Pennsylvania.Google Scholar
DeSarbo, W. S., Carroll, J. D. (1980). Three-way unfolding and situational dependence in consumer preference analysis. In Bernhardt, K., Dolich, I., Etzel, M., Kinnear, T., Perreault, W., Roering, K. (Eds.), The changing marketing environment: New theories and applications (pp. 321–325). Chicago: American Marketing Association.Google Scholar
DeSarbo, W. S., & Carroll, J. D. (1981). Three-way metric unfolding. Proceedings of the ORSA/TIMS Market Measurement Conference. New York, NY.Google Scholar
DeSarbo, W. S., Rao, V. R. (1983). A constrained unfolding model for product positioning, Murray Hill, NJ: Bell Laboratories.Google Scholar
DeSarbo, W. S., Rao, V. R. (1984). GENFOLD2: A Set of Models and Algorithms for the GENeral Un-FOLDing Analysis of Preference/Dominance Data. Journal of Classification, 1, 147–186.CrossRefGoogle Scholar
Fisher, D. G., Rothkopf, E. Z. (1982). An automated eye movement laboratory for on-line electrooculography. Behavior Research Methods and Instrumentation, 14, 113–120.CrossRefGoogle Scholar
Fletcher, R., Reeves, C. M. (1964). Function minimization by conjugate gradients. Computer Journal, 7, 149–154.CrossRefGoogle Scholar
Graef, J., Spence, I. (1979). Using distance information in the design of large multidimensional scaling experiments. Psychological Bulletin, 86, 60–66.CrossRefGoogle Scholar
Grambsch, P., Clark, L., DeSarbo, W. S., Rothkopf, E. (1983). An analysis of individual differences in reading, Murray Hill, NJ: Bell Laboratories.Google Scholar
Greenstadt, J. (1967). Eigenvalue analysis for gradient procedures. Mathematical Computation, 21, 360–360.CrossRefGoogle Scholar
Heiser, W. J. (1981). Unfolding Analysis of Proximity Data, Leiden, The Netherlands: Reprodienst Psychologie RUL.Google Scholar
Himmelblau, D. M. (1972). Applied Nonlinear Programming, New York: McGraw-Hill.Google Scholar
Isaac, P. D. (1980). Considerations in selection of stimulus pairs for data collection in MDS. In Golledge, R, B., Rayner, J. N. (Eds.), Multidimensional Analysis of Large Data Sets (pp. 326–341). Columbus: Ohio State University Press.Google Scholar
Johnson, S. C. (1967). Hierarchical clustering schemes. Psychometrika, 32, 241–254.CrossRefGoogle ScholarPubMed
Kruskal, J. B., Carroll, J. D. (1969). Geometrical models and badness-of-fit functions. In Krishnaiah, P. R. (Eds.), Multivariate Analysis (Vol. 2 (pp. 639–671). New York: Academic Press.Google Scholar
Miller, G. A., Nicely, P. E. (1955). An analysis of perceptual confusion among some English consonants. Journal of the Acoustical Society of America, 27, 338–352.CrossRefGoogle Scholar
Schönemann, P. H. (1970). On metric multidimensional unfolding. Psychometrika, 35, 349–366.CrossRefGoogle Scholar
Shepard, R. N. (1972). Psychological representation of speech sounds. In David, E. E., Denes, P. B. (Eds.), Human Communication: A Unified View, New York: McGraw Hill.Google Scholar
Snedecor, G. W., Cochran, W. G. (1981). Statistical Methods 7th ed.,, Ames: Iowa State University Press.Google Scholar
Takane, Y., Young, F. W., de Leeuw, J. (1977). Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features. Psychometrika, 42, 7–67.CrossRefGoogle Scholar
Torgerson, W. S. (1958). Theory and methods of scaling, New York: Wiley.Google Scholar
Wold, H. (1966). Estimation of principal components and related models by iterative least squares. In Krishnaiah, P. R. (Eds.), Multivariate analysis, New York: Academic Press.Google Scholar