Multidimensional Successive Categories Scaling: A Maximum Likelihood Method | Psychometrika | Cambridge Core (original) (raw)

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.

References

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

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

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

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