Hostname: page-component-7dd5485656-glrdx Total loading time: 0 Render date: 2025-10-30T14:47:22.798Z Has data issue: false hasContentIssue false
  • English
  • Français

Multidimensional Successive Categories Scaling: A Maximum Likelihood Method

Published online by Cambridge University Press:  01 January 2025

Yoshio Takane
Yoshio Takane*
Affiliation:
McGill University
*
Requests for reprints should be sent to Yoshio Takane, Department of Psychology, McGill University, 1205 Docteur Penfield Avenue, Montreal, Quebec H3A 1B1, Canada.
Get access Rights & Permissions [Opens in a new window]

Abstract

A single-step maximum likelihood estimation procedure is developed for multidimensional scaling of dissimilarity data measured on rating scales. The procedure can fit the euclidian distance model to the data under various assumptions about category widths and under two distributional assumptions. The scoring algorithm for parameter estimation has been developed and implemented in the form of a computer program. Practical uses of the method are demonstrated with an emphasis on various advantages of the method as a statistical procedure.

Keywords

similarity ratingsmaximum likelihood multidimensional scaling (MDS)method of successive categories

Information

Type
Original Paper
Information
Psychometrika , Volume 46 , Issue 1 , March 1981 , pp. 9 - 28
Copyright
Copyright © 1981 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.)

Article purchase

Temporarily unavailable

Footnotes

The research reported here was partly supported by Grant A6394 to the author by Natural Sciences and Engineering Research Council of Canada. Portions of this research were presented at the Psychometric Society meeting in Uppsala, Sweden, in June, 1978. MAXSCAL-2.1, a program to perform the computations discussed in this paper may be obtained from the author. Thanks are due to Jim Ramsay for his helpful comments.

References

Reference Notes

Takane, Y., & Carroll, J. D. Maximum likelihood multidimensional scaling from directional rankings of similarities. Paper submitted for publication, 1980.CrossRefGoogle Scholar
Jennrich, R. I., &Moore, R. H. Maximum likelihood estimation by means of nonlinear least squares, 1975, Princeton, N.J.: Educational Testing Service.CrossRefGoogle Scholar
Takane, Y., & Carroll, J. D. On the robustness of AIC in the context of maximum likelihood multidimensional scaling. Manuscript in preparation.Google Scholar
Steiger, J. H., & Lind, J. M. Statistically based tests for the number of common factors. Handout for a paper presented at the Psychometric Society meeting, Iowa, 1980.Google Scholar
Krantz, D. H., & Tversky, A. Similarities of rectangles: An analysis of subjective dimensions. Ann Arbor, Mich.: Michigan Mathematical Psychology Program. 1973, 73–78.Google Scholar
Okada, A. Nonmetric multidimensional scaling and rating scales. San Diego, California: Proceedings of the U.S.-Japan Seminar on Theory, Methods and Applications of Multidimensional Scaling and Related Techniques, 1975, 141–150.Google Scholar

References

Akaike, H. Information theory and an extension of the maximum likelihood principle. In Petrov, B. N. & Csaki, F. (Eds.), The second international symposium on information theory, 1973, Budapest: Akadémiai Kiado.Google Scholar
Akaike, H. A new look at the statistical model identification. IEEE Transactions on Automatic Control, 1974, 19, 716–723.CrossRefGoogle Scholar
Akaike, H. On entropy maximization principle. In Krishnaiah, P. R. (Eds.), Applications of Statistics, 1977, Holland: North-Holland Publishing Co..Google Scholar
Beck, J. V., & Arnold, K. J. Parameter estimation in engineering and science, 1977, New York: Wiley.Google Scholar
Bentler, P. M., & Weeks, D. G. Restricted multidimensional scaling models. Journal of Mathematical Psychology, 1978, 17, 138–151.CrossRefGoogle Scholar
Bloxom, B. Constrained multidimensional scaling in N spaces. Psychometrika, 1978, 43, 397–408.CrossRefGoogle Scholar
Bock, R. D., & Jones, L. V. The measurement and prediction of judgment and choice, 1968, San Francisco: Holden Day.Google Scholar
Carroll, J. D. Spatial, non-spatial and hybrid models of scaling. Psychometrika, 1976, 41, 439–463.CrossRefGoogle Scholar
Green, P. E., & Rao, V. R. Rating scales and information recovery—how many scales and response categories to use. Journal of Marketing, 1970, 34, 33–39.Google Scholar
Hastings, C. Approximation for digital computers, 1955, Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
Johnson, S. C. Hierarchical clustering schemes. Psychometrika, 1967, 32, 241–254.CrossRefGoogle ScholarPubMed
Kitagawa, G. On the use of AIC for the detection of outliers. Technometrics, 1979, 21, 193–199.CrossRefGoogle Scholar
Krantz, D. H., & Tversky, A. Similarity of rectangles; An analysis of subjective dimensions. Journal of Mathematical Psychology, 1975, 12, 4–34.CrossRefGoogle Scholar
Kruskal, J. B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 1964, 29, 1–27.CrossRefGoogle Scholar
Kruskal, J. B. Nonmetric multidimensional scaling: A numerical method. Psychometrika, 1964, 29, 115–129.CrossRefGoogle Scholar
Kruskal, J. B., & Wish, M. Multidimensional scaling, 1978, Beverly Hills, Calif.: Sage Publications.CrossRefGoogle Scholar
Lord, F. M., & Novick, M. R. Statistical theories of mental test scores, 1968, Reading, Massachusetts: Addison-Wesley.Google Scholar
Messick, S. J. An empirical evaluation of multidimensional successive categories. Psychometrika, 1956, 21, 367–375.CrossRefGoogle Scholar
Nakatani, L. H. Confusion-choice model for multidimensional psychophysics. Journal of Mathematical Psychology, 1972, 9, 104–127.CrossRefGoogle Scholar
Ramsay, J. O. The effect of number of categories in rating scales on precision of estimation of scale values. Psychometrika, 1973, 38, 513–532.CrossRefGoogle Scholar
Ramsay, J. O. Maximum likelihood estimation in multidimensional scaling. Psychometrika, 1977, 42, 241–266.CrossRefGoogle Scholar
Ramsay, J. O. Confidence regions for multidimensional scaling analysis. Psychometrika, 1978, 43, 145–160.CrossRefGoogle Scholar
Ramsay, J. O. Joint analysis of direct ratings, pairwise preferences and dissimilarities. Psychometrika, 1980, 45, 149–165.CrossRefGoogle Scholar
Ramsay, J. O. Some small sample results for maximum likelihood estimation in multidimensional scaling. Psychometrika, 1980, 45, 139–144.CrossRefGoogle Scholar
Rao, C. R. Advanced statistical methods in biometric research, 1952, New York: Wiley.Google Scholar
Rothkopf, E. Z. A measure of stimulus similarity and errors in some paired associate learning. Journal of Experimental Psychology, 1957, 53, 94–101.CrossRefGoogle ScholarPubMed
Sakamoto, Y. & Akaike, H. Analysis of cross classified data by AIC. Annals of the Institute of Statistical Mathematics, 1978, B30, 185–197.CrossRefGoogle Scholar
Schönemann, P. H. Similarity of rectangles. Journal of Mathematical Psychology, 1977, 16, 161–165.CrossRefGoogle Scholar
Schönemann, P. H., & Tucker, L. R. A maximum likelihood solution for the method of successive intervals allowing for unequal stimulus dispersions. Psychometrika, 1967, 32, 403–418.CrossRefGoogle Scholar
Schwartz, G. Estimating the dimensions of a model. The Annals of Statistics, 1978, 6, 461–464.Google Scholar
Shepard, R. N. The analysis of proximities: Multidimensional scaling with an unknown distance function, I & II. Psychometrika, 1962, 27, 125–140.CrossRefGoogle Scholar
Shimizu, R. Entropy maximization principle and selection of the order of an autoregressive Gaussian process. Annals of the Institute of Statistical Mathematics, 1978, A30, 263–270.CrossRefGoogle Scholar
Takane, Y. A maximum likelihood method for nonmetric multidimensional scaling: I. The case in which all empirical pairwise orderings are independent—theory. Japanese Psychological Research, 1978, 20, 7–17.CrossRefGoogle Scholar
Takane, Y. A maximum likelihood method for nonmetric multidimensional scaling: II. The case in which all empirical pairwise orderings are independent—evaluations. Japanese Psychological Research, 1978, 20, 105–114.CrossRefGoogle Scholar
Thurstone, L. L. The measurement of values, 1959, Chicago, Ill.: University of Chicago Press.Google Scholar
Torgerson, W. S. Multidimensional scaling: I. Theory and method. Psychometrika, 1952, 17, 401–419.CrossRefGoogle Scholar
Torgerson, W. S. Theory and methods of scaling, 1958, New York: Wiley.Google Scholar
Tversky, A. Features of similarity. Psychological Review, 1977, 84, 327–352.CrossRefGoogle Scholar
Wilks, S. S. Mathematical statistics, 1962, New York: Wiley.Google Scholar
Young, G., & Householder, A. S. Discussion of a set of points in terms of their mutual distances. Psychometrika, 1938, 3, 19–22.CrossRefGoogle Scholar
Zinnes, J. L., & Wolff, R. P. Single and multidimensional same-different judgments. Journal of Mathematical Psychology, 1977, 16, 30–50.CrossRefGoogle Scholar