Abstract

Fundamental concepts of MDS models are discussed. Since MDS includes a family of different models and various terms are used to describe these models as well as their corresponding elements, I explain these models and their associated terms using more understandable language.

Keyword

MDS models Vector representation Metric model Non-metric model Preference model Unfolding model Individual differences model 

References

  1. Borg, I. (1977). Some basic concepts of facet theory. In J. C. Lingoes (Ed.), Progressively complex linear transformations for finding geometric similarities among data structures (pp. 65–102). Mimeo.Google Scholar
  2. Borg, I., & Groenen, P. J. F. (2005). Modern multidimensional scaling: Theory and applications (2nd ed.). New York, NY: Springer.MATHGoogle Scholar
  3. Bozdogan, H. (1987). Model selection and Akaike’s information criterion (AIC): The general theory and its analytical extensions. Psychometrika, 52(3), 345–370.MathSciNetCrossRefGoogle Scholar
  4. 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 behavioral sciences (Vol. I). New York, NY: Academic Press.Google Scholar
  5. 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, 238–319.CrossRefGoogle Scholar
  6. Carroll, J. D., & Chang, J. J. (1972, March). IDIOSCAL (Individual Differences in Orientation Scaling): A generalization of INDSCAL allowing idosyncratic reference systems as well as analytic approximation to INDSCAL. Paper presented at the The Psychometric Society, Princeton, NJ.Google Scholar
  7. Carroll, J. D., & Wish, M. (1973). Models and methods for three-way multidimensional scaling. In R. C. Atkinson, D. H. Krantz, R. D. Luce, & P. Suppes (Eds.), Contemporary developments in mathematical psychology. San Francisco, CA: W. H. Freeman.Google Scholar
  8. Coombs, C. H. (1950). Psychological scaling without a unit of measurement. Psychological Review, 57(3), 145–158.CrossRefGoogle Scholar
  9. Coombs, C. H. (1964). A theory of data. New York, NY: Wiley.Google Scholar
  10. Cox, T. F., & Cox, M. A. A. (2001). Multidimensional scaling (2nd ed.). Boca Raton, FL: Chapman & Hall/CRC.MATHGoogle Scholar
  11. Coxon, A. P. M. (1982). The user’s guide to multidimensional scaling. London: Heinemann Educational Books.Google Scholar
  12. Coxon, A. P. M., Brier, A. P., & Hawkins, P. K. (2005). The New MDSX program series, version 5. Edinburgh: London: New MDSX Project.Google Scholar
  13. Davison, M. L. (1983). Multidimensional scaling. New York: Wiley.MATHGoogle Scholar
  14. Guttman, L. (1968). A general non-metric technique for finding the smallest co-ordinate space for a configuration of points. Psychometrika, 33, 469–506.CrossRefGoogle Scholar
  15. Kruskal, J. B. (1964). Nonmetric scaling: A numerical method. Psychometrika, 29, 28–42.MathSciNetMATHGoogle Scholar
  16. Kuhfeld, W., Young, F. W., & Kent, D. P. (1987). New developments in psychometric and market research procedures. SUGI, 12, 1101–1106.Google Scholar
  17. Lingoes, J. C. (1977). Progressively complex linear transformations for finding geometric similarities among data structures. Mimeo.Google Scholar
  18. Lingoes, J. C., & Borg, I. A. (1978). A direct approach to individual differences scaling using increasingly complex transformations. Psychometrika, 43, 491–519.MathSciNetCrossRefGoogle Scholar
  19. MacCallum, R. C. (1977). Effects of conditionality on INOSCALand ALSCALweights. Psychometrika, 42, 297–305.CrossRefGoogle Scholar
  20. MacKay, D. B. (1989). Probabilistic multidimensional scaling: An anisotropic model for distance judgments. Journal of Mathematical Psychology, 33, 187–205.MathSciNetCrossRefGoogle Scholar
  21. MacKay, D. B. (2007). Internal multidimensional unfolding about a single-idea--A probabilistic solution. Journal of Mathematical Psychology, 51(5), 305–318.MathSciNetCrossRefGoogle Scholar
  22. MacKay, D. B., Easley, R. F., & Zinnes, J. L. (1995). A single ideal point model for market structure analysis. Journal of Marketing Research, XXXII, 433–443.CrossRefGoogle Scholar
  23. MacKay, D. B., & Zinnes, J. (2014). PROSCAL professional: A program for probabilistic scaling: www.proscal.com.
  24. MacKay, D. B., & Zinnes, J. L. (1986). A probabilistic model for the multidimensional scaling of proximity and preference data. Marketing Science, 5, 325–344.CrossRefGoogle Scholar
  25. Ramsay, J. O. (1977). Maximum likelihood estimation in multidimensional scaling. Psychometrika, 42, 241–266.CrossRefGoogle Scholar
  26. Sammon, J. W. (1969). A nonlinear mapping for data structure analysis. IEEE Transportation & Computing, 18, 401–409.CrossRefGoogle Scholar
  27. SAS Institute. (2010). SAS/STAT(R) 9.22 user’s guide. Cary, NC: SAS Institute Inc..Google Scholar
  28. Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6(2), 461–464.MathSciNetCrossRefGoogle Scholar
  29. Shepard, L. (1962). The analysis of proximities: Multidimensional scaling with an unknown distance. I and II. Psychometrika, 27, 323–355.MathSciNetCrossRefGoogle Scholar
  30. Inc, S. P. S. S. (2007). SPSS Statistics 17.0: Command syntax reference. Chicago, IL: SPSS Inc..Google Scholar
  31. Takane, Y. (1978). A maximum likelihood method for nonmetric multidimensional scaling: I The case in which all empirical pairwise orderings are independent-theory. Japanese Psychological Research, 20, 7–17.CrossRefGoogle Scholar
  32. Takane, Y., & Carroll, J. D. (1981). Nonmetric maximum likelihood multidimensional scaling from directional rankings of similarities. Psychometrika, 46, 389–405.MathSciNetCrossRefGoogle Scholar
  33. Thurstone, L. L. (1928). Attitudes can be measured. American Journal of Sociology, 33, 529–554.CrossRefGoogle Scholar
  34. Torgerson, W. S. (1952). Multidimensinoal scaling: I. Theory and method. Psychometrika, 17(4), 401–419.MathSciNetCrossRefGoogle Scholar
  35. Tucker, L. R. (1960). Intra-individual and inter-individual multidimensionality. In H. Gulliksen & S. Messick (Eds.), Psychological scaling: Theory and applications (pp. 155–167). New York: Wiley.Google Scholar
  36. Tucker, L. R. (1972). Relations between multidimensional scaling and three-mode factor analysis. Psychometrika, 37, 3–27.MathSciNetCrossRefGoogle Scholar
  37. Tversky, A., & Krantz, D. H. (1970). Dimensional representation and the metric structure of similarity data. Journal of Mathematical Psychology, 7, 572–596.MathSciNetCrossRefGoogle Scholar
  38. Young, G., & Householder, A. S. (1941). Note on multidimensional psychophysical analysis. Psychometrika, 6, 331–333.CrossRefGoogle Scholar
  39. Zinnes, J. L., & MacKay, D. B. (1983). Probabilistic multidimensional scaling: Complete and incomplete data. Psychometrika, 48, 24–48.CrossRefGoogle Scholar
  40. Zinnes, J. L., & MacKay, D. B. (1992). A probabilistic multidimensional scaling approach: Properties and procedures. In F. G. Ashby (Ed.), Multidimensional models of perception and cognition (pp. 35–60). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Cody S. Ding
    • 1
    • 2
  1. 1.Department of Education Science and Professional ProgramUniversity of Missouri-St. LouisSt. LouisUSA
  2. 2.Center for NeurodynamicsUniversity of Missouri-St. LouisSt. LouisUSA

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