Psychometrika

, Volume 57, Issue 1, pp 43–69 | Cite as

Tscale: A new multidimensional scaling procedure based on tversky's contrast model

  • Wayne S. DeSarbo
  • Michael D. Johnson
  • Ajay K. Manrai
  • Lalita A. Manrai
  • Elizabeth A. Edwards
Article

Abstract

Tversky's contrast model of proximity was initially formulated to account for the observed violations of the metric axioms often found in empirical proximity data. This set-theoretic approach models the similarity/dissimilarity between any two stimuli as a linear (or ratio) combination of measures of the common and distinctive features of the two stimuli. This paper proposes a new spatial multidimensional scaling (MDS) procedure called TSCALE based on Tversky's linear contrast model for the analysis of generally asymmetric three-way, two-mode proximity data. We first review the basic structure of Tversky's conceptual contrast model. A brief discussion of alternative MDS procedures to accommodate asymmetric proximity data is also provided. The technical details of the TSCALE procedure are given, as well as the program options that allow for the estimation of a number of different model specifications. The nonlinear estimation framework is discussed, as are the results of a modest Monte Carlo analysis. Two consumer psychology applications are provided: one involving perceptions of fast-food restaurants and the other regarding perceptions of various competitive brands of cola soft-drinks. Finally, other applications and directions for future research are mentioned.

Key words

multidimensional scaling asymmetric proximity data Tversky's contrast model consumer psychology 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abelson, R. P., & Levi, A. (1985). Decision making and decision theory. In G. Lindsey & E. Aronson (Eds.),The handbook of social psychology, Vol. 3 (pp. 231–309). New York: Academic Press.Google Scholar
  2. Addelman, S. (1962). Orthogonal main-effect plans for asymmetrical factorial experiments.Technometrics, 4, 21–46.Google Scholar
  3. Bennett, J. F., & Hays, W. L. (1960). Multidimensional unfolding: Determining the dimensionality of ranked preference data.Psychometrika, 25, 27–43.Google Scholar
  4. Bentler, P. M., & Weeks, D. G. (1978). Restricted multidimensional scaling models.Journal of Mathematical Psychology, 17, 138–151.Google Scholar
  5. Bettman, J. R. (1986). Consumer psychology.Annual Review of Psychology, 37, 257–289.Google Scholar
  6. Bettman, J. R., & Park, C. W. (1980). Effects of prior knowledge and experience and phase of the choice process on consumer decision processes: A protocol analysis.Journal of Consumer Research, 7, 234–248.Google Scholar
  7. Bishop, Y. M. M., Fienberg, S. E., & Holland, P. W. (1975).Discrete multivariate analysis: Theory and practice. Cambridge: MIT Press.Google Scholar
  8. Bloxom, B. (1978). Constrained multidimensional scaling inN spaces.Psychometrika, 43, 397–408.Google Scholar
  9. Bush, R. R., & Mosteller, F. (1951). A model for stimulus generalization and discrimination.Psychological Review, 58, 413–423.Google Scholar
  10. 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 (pp. 234–289). Bern: Huber.Google Scholar
  11. Carroll, J. D., Pruzansky, S., & Kruskal, J. B. (1980). CANDELINC: A general approach to multidimensional analysis of many-way arrays with linear constraints on parameters.Psychometrika, 45, 3–24.Google Scholar
  12. Chino, N. (1978). A graphical technique for representing the asymmetric relationship betweenN objects.Behaviormetrika, 5, 23–40.Google Scholar
  13. Chino, N. (1979). Extension of Chino's ASYMSCAL into higher dimensions.Paper presented at the annual meeting of the 7th Behaviormetric Society of Japan, Japan.Google Scholar
  14. Chino, N. (1990). A generalized inner product model for the analysis of asymmetry.Behaviormetrika, 27, 25–46.Google Scholar
  15. Constantine, A. G., & Gower, J. C. (1978). Graphical representation of asymmetric matrices.Applied Statistics, 27, 297–304.Google Scholar
  16. Coombs, C. H. (1950). Psychological scaling without a unit of measurement.Psychological Review, 57, 148–158.Google Scholar
  17. Corter, J. E., & Tversky, A. (1986). Extended similarity trees.Psychometrika, 51, 429–451.Google Scholar
  18. Dawes, R. M. (1979). The robust beauty of improper linear models in decision making.American Psychologist, 34, 571–582.Google Scholar
  19. de Leeuw, J., & Heiser, W. (1980). Multidimensional scaling with restrictions on the configuration. In P. R. Krishnaiah (Ed.),Multivariate analysis-V (pp. 501–522). New York: North-Holland.Google Scholar
  20. DeSarbo, W. S. (1982). GENNCLUS: New models for general nonhierarchical cluster analysis.Psychometrika, 47, 446–469.Google Scholar
  21. DeSarbo, W. S., & Carroll, J. D. (1985). Three-way metric unfolding via weighted alternating least squares.Psychometrika, 50, 275–300.Google Scholar
  22. DeSarbo, W. S., Carroll, J. D., Lehmann, D., & O'Shaugnessy, J. (1982). Three-way multivariate conjoint analysis.Marketing Science, 1, 323–350.Google Scholar
  23. DeSarbo, W. S., & Manrai, A. K. (in press). A new multidimensional scaling methodology for the analysis of asymmetric proximity data in marketing research.Marketing Science.Google Scholar
  24. DeSarbo, W. S., Manrai, A. K., & Burke, R. R. (1990). A nonspatial methodology for the analysis of two-way proximity data incorporating the distance-density hypothesis.Psychometrika, 55, 229–253.Google Scholar
  25. 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.Google Scholar
  26. DeSarbo, W. S., & Rao, V. R. (1986). A constrained unfolding methodology for product positioning.Marketing Science, 5, 1–19.Google Scholar
  27. Doyle, P., & Hutchinson, P. (1973). Individual differences in family decision making.Journal of the Marketing Research Society, 15, 193–206.Google Scholar
  28. Einhorn, H. J., Kleinmuntz, D. N., & Kleinmuntz, B. (1979). Linear regression and process-tracing models of judgment.Psychological Review, 86, 465–485.Google Scholar
  29. Eisler, H., & Ekman, G. (1959). A mechanism of subjective similarity.Acta Psychologica, 16, 1–10.Google Scholar
  30. Garner, W. R. (1978). Aspects of a stimulus: Features, dimensions, and configurations. In E. Rosch & B. Lloyd (Eds.),Cognition and categorization (pp. 99–133). New Jersey: Erlbaum.Google Scholar
  31. Gati, I., & Tversky, A. (1982). Representations of qualitative and quantitative dimensions.Journal of Experimental Psychology: Human Perception and Performance, 8, 325–340.Google Scholar
  32. Gati, I., & Tversky, A. (1984). Weighting common and distinctive features in perceptual and conceptual judgments.Cognitive Psychology, 16, 341–370.Google Scholar
  33. Gentry, J. W., Doering, M., & O'Brien, T. V. (1978). Masculinity and femininity factors in product perception and self image. In H. Keith Hunt (Ed.),Advances in consumer research (pp. 326–332). Ann Arbor, MI: Association for Consumer Research.Google Scholar
  34. Gill, P. E., Murray, W., & Wright, M. H. (1981).Practical optimization. Orlando, FL: Academic Press.Google Scholar
  35. Gower J. C. (1978). Unfolding: Some technical problems and novel uses.Presented at European meeting of Psychometric and Mathematical Psychology, Uppsala.Google Scholar
  36. Green, P. E., & Rao, V. R. (1971). Conjoint measurement for quantifying judgmental data.Journal of Marketing Research, 8, 355–363.Google Scholar
  37. Gregson, R. A. M. (1975).Psychometrics of similarity. New York: Academic Press.Google Scholar
  38. Harshman, R. A. (1975). Models for the analysis of asymmetrical relationships amongN objects or stimuli.Presented at US-Japan seminar on multidimensional scaling, University of California, San Diego, La Jolla.Google Scholar
  39. Hartigan, J. A. (1975).Clustering algorithms. New York: John Wiley & Sons.Google Scholar
  40. Himmelblau, D. M. (1972).Applied nonlinear programming. New York, McGraw Hill.Google Scholar
  41. Holman, E. W. (1979). Monotonic models for asymmetric proximities.Journal of Mathematical Psychology, 20, 1–15.Google Scholar
  42. Howard, J. (1977).Consumer behavior: Application of theory. New York: McGraw-Hill.Google Scholar
  43. Johnson, E. J., & Russo, J. E. (1984). Product familiarity and learning new information.Journal of Consumer Research, 11, 542–550.Google Scholar
  44. Johnson, M. D. (1986). Consumer similarity judgments: A test of the contrast model.Psychology and Marketing, 3, 47–60.Google Scholar
  45. Johnson, M. D., & Fornell, C. (1987). The nature and methodological implications of the cognitive representation of products.Journal of Consumer Research, 14, 214–228.Google Scholar
  46. Johnson, S. C. (1967). Hierarchical clustering schemes.Psychometrika, 32, 241–254.Google Scholar
  47. Krumhansl, C. (1978). Concerning the applicability of geometric models to similarity data: The interrelationship between similarity and spatial density.Psychological Review, 85, 445–463.Google Scholar
  48. Kruskal, J. B., Young, F. W., & Seery, J. B. (1973).How to use KYST, a very flexible program to do multidimensional scaling and unfolding. Bell Laboratories, Murray Hill, NJ.Google Scholar
  49. Lawson, C. L., & Hanson, R. J. (1972).Solving least squares problems. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  50. Lopes, L. L., & Johnson, M. D. (1982). Judging similarity among strings described by hierarchical trees.Acta Psychologica, 51, 13–26.Google Scholar
  51. Manrai, A. K. (1986).Similarity, perceptions, and choice. Unpublished doctoral dissertation, Northwestern University, Evanston, IL.Google Scholar
  52. Manrai, A. K. & Manrai, L. A. (1989). Mathematical models for relating proximity to multidimensional scaling. In N. Avlonitis (Ed.),Marketing thought and practice in the 1990's (pp. 853–868). Athens, Greece: European Marketing Academy, The Athens School of Economics and Business Science.Google Scholar
  53. Manrai, A. K., & Sinha, P. K. (1989). Elimination by cutoffs.Marketing Science, 8, 133–152.Google Scholar
  54. Meyers-Levy, J. (1988). The influence of sex roles on judgment.Journal of Consumer Research, 14, 522–530.Google Scholar
  55. Nakatani, L. H. (1972). Confusion-choice model for multidimensional psychophysics.Journal of Mathematical Psychology, 9, 104–127.Google Scholar
  56. Noma, E., & Johnson, J. (1977).Constraining nonmetric multidimensional scaling configurations (Tech. Rep. 60). The University of Michigan, Human Performance Center, Ann Arbor.Google Scholar
  57. Okada, A., & Imaizumi, T. (1987). Nonmetric multidimensional scaling of asymmetric proximities.Behaviormetrika, 21, 81–96.Google Scholar
  58. Prinz, W., & Scheerer-Neumann, G. (1974). Component processes in multiattribute stimulus classification.Psychological Research, 37, 25–50.Google Scholar
  59. Powell, M. J. D. (1977). Restart procedures for the conjugate gradient method.Mathematical Programming, 12, 241–254.Google Scholar
  60. Pruzansky, S., Tversky, A., & Carroll, J. D. (1982). Spatial versus tree representations of proximity data.Psychometrika, 47, 3–24.Google Scholar
  61. Rao, S. S. (1984).Optimization: Theory and applications (2nd ed.). New York: Wiley.Google Scholar
  62. Restle, F. A. (1959). A metric and an ordering on sets.Psychometrika, 24, 207–220.Google Scholar
  63. Restle, F. A. (1961).Psychology of judgment and choice. New York: Wiley.Google Scholar
  64. Rosch, E. (1975). Cognitive representation of semantic categories.Journal of Experimental Psychology: General, 104, 192–233.Google Scholar
  65. Rosch, E., & Mervis, C. B. (1975). Family resemblances: Studies in the internal structure of categories.Cognitive Psychology, 7, 573–603.Google Scholar
  66. Saito, T. (1986). Multidimensional scaling to explore complex aspects in dissimilarity judgment.Behaviormetrika, 20, 35–62.Google Scholar
  67. Sattath, S., & Tversky, A. (1977). Additive similarity trees.Psychometrika, 42, 319–345.Google Scholar
  68. Sattath, S., & Tversky, A. (1987). On the relation between common and distinctive feature models.Psychological Review, 94, 16–22.Google Scholar
  69. Shepard, R. N. (1962). The analysis of proximities: Multidimensional scaling with an unknown distance function. I.Psychometrika, 27, 125–140.Google Scholar
  70. Shepard, R. N., & Arabie, P. (1979). Additive clustering: Representation of similarities as combinations of discrete overlapping properties.Psychological Review, 86, 87–123.Google Scholar
  71. Shor, N. Z. (1979).Minimization methods for non-differentiable functions. New York: Springer-Verlag.Google Scholar
  72. Sjöberg, L. (1972).A cognitive theory of similarity. Goteborg Psychological Reports (No. 10).Google Scholar
  73. Smith, E. E., Shoben, E. J., & Rips, L. J. (1974). Structure and process in semantic memory: A featural model for semantic decisions.Psychological Review, 81, 214–241.Google Scholar
  74. Takane, Y., Young, F. W., & de Leeuw, J. (1977). Nonmetric individual differences multidimensional scaling: An alternating least square method with optimal scaling features.Psychometrika, 42, 7–67.Google Scholar
  75. Turle, J. E., & Falconer, R. (1972). Men and women are different.Journal of the Marketing Research Society, 14, 111–123.Google Scholar
  76. Tversky, A. (1972). Elimination by aspects: A theory of choice.Psychological Review, 79, 281–299.Google Scholar
  77. Tversky, A. (1977). Features of similarity.Psychological Review, 84, 327–352.Google Scholar
  78. Tversky, A., & Gati, I. (1978). Studies of similarity. In E. Rosch & B. Lloyd (Eds.),Cognition and categorization (pp. 79–98). New Jersey: Erlbaum.Google Scholar
  79. Tversky, A., & Gati, I. (1982). Similarity, separability, and the triangle inequality.Psychological Review, 89, 123–154.Google Scholar
  80. Tversky, A., & Hutchinson, J. W. (1986). Nearest neighbor analysis of psychological spaces.Psychological Review, 93, 3–22.Google Scholar
  81. Young, F. W. (1975). An asymmetric Euclidean model for multiprocess asymmetric data.Presented at US-Japan seminar on multidimensional scaling, University of California, San Diego, La Jolla.Google Scholar

Copyright information

© The Psychometric Society 1992

Authors and Affiliations

  • Wayne S. DeSarbo
    • 3
  • Michael D. Johnson
    • 1
  • Ajay K. Manrai
    • 2
  • Lalita A. Manrai
    • 2
  • Elizabeth A. Edwards
    • 3
  1. 1.Marketing Department School of Business AdministrationUniversity of MichiganUSA
  2. 2.Department of Business Administration College of Business and EconomicsUniversity of DelawareUSA
  3. 3.Marketing and Statistics Depts., School of Business AdministrationThe University of MichiganAnn Arbor

Personalised recommendations