Hostname: page-component-5f745c7db-8qdnt Total loading time: 0 Render date: 2025-01-06T05:49:48.486Z Has data issue: true hasContentIssue false
  • English
  • Français

On the Robustness of Maximum Likelihood Scaling for Violations of the Error Model

Published online by Cambridge University Press:  01 January 2025

Gert Storms
Gert Storms*
Affiliation:
University of Leuven
*
Please send requests for reprints to Gert Storms, University of Leuven, Tiensestraat 102, B-3000 Leuyen, BELGIUM. E-mail: FPAAE09@BLEKUL11
Get access Rights & Permissions [Opens in a new window]

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 the x2 test, as proposed by Ramsay, yielded the best results, and turned out to be quite robust against violations of the error model.

Keywords

test for dimensionalitymaximum likelihood estimationmultidimensional scalingx2AICBICCAIC
Type
Original Paper
Information
Psychometrika , Volume 60 , Issue 2 , June 1995 , pp. 247 - 258
Copyright
Copyright © 1995 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.)

Footnotes

The author thanks Paul De Boeck, Luc Delbeke and Stef De Coene for their useful comments on an earlier version of this manuscript.

References

Akaike, H. (1974). A new look at the statistical model identification. IEEE: Transactions and Automatic Control, 19, 716–723.Google Scholar
Bozdogan, H. (1987). Model selection and Akaike's information criterion (AIC): The general theory and its analytical extensions. Psychometrika, 52, 345–370.CrossRefGoogle Scholar
De Soete, G., Carroll, J. D. (1983). A maximum likelihood method for fitting the wandering vector model. Psychometrika, 48, 553–566.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Johnson, N. L., Kotz, S. (1970). Distributions in statistics: Continuous univariate distributions—2, Boston: Houghton Miffin.Google Scholar
Kirk, R. E. (1982). Experimental design. Procedures for the behavioral sciences 2nd ed.,, Belmont, CA: Brooks/Cole.Google Scholar
Kruskall, J. B. (1964). Multidimensional scaling by optimizing goodness-of-fit to a nonmetric hypothesis. Psychometrika, 29, 1–27.CrossRefGoogle Scholar
Kruskall, J. B. (1964). Multidimensional scaling: A numerical method. Psychometrika, 29, 115–129.CrossRefGoogle Scholar
Mood, A. M., Graybill, F. A., Boes, D. C. (1974). Introduction to the theory of statistics 3rd ed.,, Singapore: McGraw-Hill.Google Scholar
Ramsay, J. O. (1977). Maximum likelihood estimation in multidimensional scaling. Psychometrika, 42, 241–266.CrossRefGoogle Scholar
Ramsay, J. O. (1978). Confidence regions for multidimensional scaling analysis. Psychometrika, 43, 145–160.CrossRefGoogle Scholar
Ramsay, J. O. (1980). Some small sample results for maximum likelihood estimation in multidimensional scaling. Psychometrika, 45, 139–144.CrossRefGoogle Scholar
Ramsay, J. O. (1980). The joint analysis of direct ratings, pairwise preferences and dissimilarities. Psychometrika, 45, 149–165.CrossRefGoogle Scholar
Schwarz, G. (1978). Estimating the dimensions of a model. Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
Shepard, R. N. (1962). The analysis of proximities: Multidimensional scaling with an unknown distance function. Psychometrika, 27, 125–140.CrossRefGoogle Scholar
Shiffman, S. S., Reynolds, M. L., Young, F. W. (1981). Introduction to multidimensional scaling: Theory, methods and applications, New-York: Academic Press.Google Scholar
Storms, G. (1992). Robustness of the normal distribution for choice processes in maximum likelihood scaling, Brussels: Vrije Universiteit.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Takane, Y. (1980). Maximum likelihood estimation in the generalized case of Thurstone's model of comarative judgment. Japanese Psychological Research, 22, 188–196.CrossRefGoogle Scholar
Takane, Y. (1981). Multidimensional successive categories scaling: A maximum likelihood method. Psychometrika, 46, 9–28.CrossRefGoogle Scholar
Takane, Y., Carroll, J. D. (1981). Nonmetric maximum likelihood multidimensional scaling from directional rankings of similarities. Psychometrika, 46, 389–405.CrossRefGoogle Scholar
Takane, Y., Sergent, J. (1983). Multidimensional scaling models for reaction times and same-different judgments. Psychometrika, 48, 393–423.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Winsberg, S., Ramsay, J. O. (1981). Analysis of pairwise preferences data using integrated B-splines. Psychometrika, 46, 171–186.CrossRefGoogle Scholar
Young, F. W. (1984). Scaling. Annual Review of Psychology, 35, 55–81.CrossRefGoogle ScholarPubMed
Zinnes, J. L., MacKay, D. B. (1983). Probabilistic multidimensional scaling: Complete and incomplete data. Psychometrika, 48, 27–48.CrossRefGoogle Scholar