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On the robustness of maximum likelihood scaling for violations of the error model

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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.

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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.

    Article  Google Scholar 

  • De Soete, G., & Carroll, J. D. (1983). A maximum likelihood method for fitting the wandering vector model.Psychometrika, 48, 553–566.

    Google 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.

    Google 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. (1964a). Multidimensional scaling by optimizing goodness-of-fit to a nonmetric hypothesis.Psychometrika, 29, 1–27.

    Google Scholar 

  • Kruskall, J. B. (1964b). Multidimensional scaling: A numerical method.Psychometrika, 29, 115–129.

    Google 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.

    Google Scholar 

  • Ramsay, J. O. (1978). Confidence regions for multidimensional scaling analysis.Psychometrika, 43, 145–160.

    Google Scholar 

  • Ramsay, J. O. (1980a). Some small sample results for maximum likelihood estimation in multidimensional scaling.Psychometrika, 45, 139–144.

    Google Scholar 

  • Ramsay, J. O. (1980b). The joint analysis of direct ratings, pairwise preferences and dissimilarities.Psychometrika, 45, 149–165.

    Google Scholar 

  • Schwarz, G. (1978). Estimating the dimensions of a model.Annals of Statistics, 6, 461–464.

    Google Scholar 

  • Shepard, R. N. (1962). The analysis of proximities: Multidimensional scaling with an unknown distance function.Psychometrika, 27, 125–140.

    Google 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. Paper presented at the 23rd Meeting of the European Mathematical Psychology Group, Vrije Universiteit, Brussels.

    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.

    Article  Google 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 and 105–114.

    Google Scholar 

  • Takane, Y. (1980). Maximum likelihood estimation in the generalized case of Thurstone's model of comarative judgment.Japanese Psychological Research, 22, 188–196.

    Google Scholar 

  • Takane, Y. (1981). Multidimensional successive categories scaling: A maximum likelihood method.Psychometrika, 46, 9–28.

    Google Scholar 

  • Takane, Y., & Carroll, J. D. (1981). Nonmetric maximum likelihood multidimensional scaling from directional rankings of similarities.Psychometrika, 46, 389–405.

    Google Scholar 

  • Takane, Y., & Sergent, J. (1983). Multidimensional scaling models for reaction times and same-different judgments.Psychometrika, 48, 393–423.

    Article  Google 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.

    Article  Google Scholar 

  • Winsberg, S., & Ramsay, J. O. (1981). Analysis of pairwise preferences data using integrated B-splines.Psychometrika, 46, 171–186.

    Article  Google Scholar 

  • Young, F. W. (1984). Scaling.Annual Review of Psychology, 35, 55–81.

    Article  Google Scholar 

  • Zinnes, J. L., & MacKay, D. B. (1983). Probabilistic multidimensional scaling: Complete and incomplete data.Psychometrika, 48, 27–48.

    Google Scholar 

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Correspondence to Gert Storms.

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

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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

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  • DOI: https://doi.org/10.1007/BF02301415

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