Abstract
A Monte Carlo study was conducted to investigate the robustness of the assumed error distribution in maximum likelihood estimation models for multidimensional scaling. Data sets generated according to the lognormal, the normal, and the rectangular distribution were analysed with the log-normal error model in Ramsay's MULTISCALE program package. The results show that violations of the assumed error distribution have virtually no effect on the estimated distance parameters. In a comparison among several dimensionality tests, the corrected version of thex 2 test, as proposed by Ramsay, yielded the best results, and turned out to be quite robust against violations of the error model.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akaike, H. (1974). A new look at the statistical model identification.IEEE: Transactions and Automatic Control, 19, 716–723.
Bozdogan, H. (1987). Model selection and Akaike's information criterion (AIC): The general theory and its analytical extensions.Psychometrika, 52, 345–370.
De Soete, G., & Carroll, J. D. (1983). A maximum likelihood method for fitting the wandering vector model.Psychometrika, 48, 553–566.
De Soete, G., Carroll, J. D., & DeSarbo, W. S. (1986). The wandering ideal point model: A probabilistic multidimensional unfolding model for paired comparisons data.Journal of Mathematical Psychology, 30, 28–41.
Johnson, N. L., & Kotz, S. (1970).Distributions in statistics: Continuous univariate distributions—2. Boston: Houghton Miffin.
Kirk, R. E. (1982).Experimental design. Procedures for the behavioral sciences (2nd ed.). Belmont, CA: Brooks/Cole.
Kruskall, J. B. (1964a). Multidimensional scaling by optimizing goodness-of-fit to a nonmetric hypothesis.Psychometrika, 29, 1–27.
Kruskall, J. B. (1964b). Multidimensional scaling: A numerical method.Psychometrika, 29, 115–129.
Mood, A. M., Graybill, F. A., & Boes, D. C. (1974).Introduction to the theory of statistics (3rd ed.). Singapore: McGraw-Hill.
Ramsay, J. O. (1977). Maximum likelihood estimation in multidimensional scaling.Psychometrika, 42, 241–266.
Ramsay, J. O. (1978). Confidence regions for multidimensional scaling analysis.Psychometrika, 43, 145–160.
Ramsay, J. O. (1980a). Some small sample results for maximum likelihood estimation in multidimensional scaling.Psychometrika, 45, 139–144.
Ramsay, J. O. (1980b). The joint analysis of direct ratings, pairwise preferences and dissimilarities.Psychometrika, 45, 149–165.
Schwarz, G. (1978). Estimating the dimensions of a model.Annals of Statistics, 6, 461–464.
Shepard, R. N. (1962). The analysis of proximities: Multidimensional scaling with an unknown distance function.Psychometrika, 27, 125–140.
Shiffman, S. S., Reynolds, M. L., & Young, F. W. (1981).Introduction to multidimensional scaling: Theory, methods and applications. New-York: Academic Press.
Storms, G. (1992).Robustness of the normal distribution for choice processes in maximum likelihood scaling. Paper presented at the 23rd Meeting of the European Mathematical Psychology Group, Vrije Universiteit, Brussels.
Storms, G., & Delbeke, L. (1992). The irrelevance of distributional assumptions on reaction times in multidimensional scaling of same/different judgment tasks.Psychometrika, 57, 599–614.
Takane, Y. (1978). A maximum likelihood method for non-metric multidimensional scaling: I. The case in which all empirical pairwise orderings are independent—theory and evaluations.Japanese Psychological Research, 20, 7–17 and 105–114.
Takane, Y. (1980). Maximum likelihood estimation in the generalized case of Thurstone's model of comarative judgment.Japanese Psychological Research, 22, 188–196.
Takane, Y. (1981). Multidimensional successive categories scaling: A maximum likelihood method.Psychometrika, 46, 9–28.
Takane, Y., & Carroll, J. D. (1981). Nonmetric maximum likelihood multidimensional scaling from directional rankings of similarities.Psychometrika, 46, 389–405.
Takane, Y., & Sergent, J. (1983). Multidimensional scaling models for reaction times and same-different judgments.Psychometrika, 48, 393–423.
Takane, Y., Young, F. W., & de Leeuw, J. (1977). Nonmetric individual differences multidimensional scaling: An alternative least squares method with optimal scaling features.Psychometrika, 42, 7–67.
Winsberg, S., & Ramsay, J. O. (1981). Analysis of pairwise preferences data using integrated B-splines.Psychometrika, 46, 171–186.
Young, F. W. (1984). Scaling.Annual Review of Psychology, 35, 55–81.
Zinnes, J. L., & MacKay, D. B. (1983). Probabilistic multidimensional scaling: Complete and incomplete data.Psychometrika, 48, 27–48.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author thanks Paul De Boeck, Luc Delbeke and Stef De Coene for their useful comments on an earlier version of this manuscript.
Rights and permissions
About this article
Cite this article
Storms, G. On the robustness of maximum likelihood scaling for violations of the error model. Psychometrika 60, 247–258 (1995). https://doi.org/10.1007/BF02301415
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02301415