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Points of View Analysis Revisited: Fitting Multidimensional Structures to Optimal Distance Components with Cluster Restrictions on the Variables

Published online by Cambridge University Press:  01 January 2025

Jacqueline J. Meulman  and
Peter Verboon
Jacqueline J. Meulman*
Affiliation:
Department of Data Theory, University of Leiden
Peter Verboon
Affiliation:
Department of Data Theory, University of Leiden
*
Requests for reprints should be sent to Jacqueline Meulman, Department of Data Theory, Faculty of Social Sciences, University of Leiden, PO Box 9555, 2300 RB Leiden, THE NETHERLANDS.
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Abstract

Points of view analysis (PVA), proposed by Tucker and Messick in 1963, was one of the first methods to deal explicitly with individual differences in multidimensional scaling, but at some point was apparently superceded by the weighted Euclidean model, well-known as the Carroll and Chang INDSCAL model. This paper argues that the idea behind points of view analysis deserves new attention, especially as a technique to analyze group differences. A procedure is proposed that can be viewed as a streamlined, integrated version of the Tucker and Messick Process, which consisted of a number of separate steps. At the same time, our procedure can be regarded as a particularly constrained weighted Euclidean model. While fitting the model, two types of nonlinear data transformations are feasible, either for given dissimilarities, or for variables from which the dissimilarities are derived. Various applications are discussed, where the two types of transformation can be mixed in the same analysis; a quadratic assignment framework is used to evaluate the results.

Keywords

points of viewindividual differencesgroup differencesnonlinear multivariate analysisnonmetric multidimensional scalingdistance componentscomposite dissimilaritiesvariable clustering
Type
Original Paper
Information
Psychometrika , Volume 58 , Issue 1 , March 1993 , pp. 7 - 35
Copyright
Copyright © 1993 The Psychometric Society

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Footnotes

The research of the first author was supported by the Royal Netherlands Academy of Arts and Sciences (KNAW); the research of the second author by the Netherlands Organization for Scientific Research (NWO Grant 560-267-029). An earlier version of this paper was presented at the European Meeting of the Psychometric Society, Leuven, 1989. We wish to thank Willem J. Heiser for his stimulating comments to earlier versions of this paper, and we are grateful to the Editor and anonymous referees for their helpful suggestions.

References

Bloxom, B. (1978). Constrained multidimensional scaling in N spaces. Psychometrika, 43, 397–408.CrossRefGoogle Scholar
Carroll, J. D., Chang, J. J. (1970). Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition. Psychometrika, 35, 283–319.CrossRefGoogle Scholar
Cliff, N. (1968). The “idealized individual” interpretation of individual differences in multidimensional scaling. Psychometrika, 33, 225–232.CrossRefGoogle Scholar
de Leeuw, J. (1988). Convergence of the majorization algorithm for multidimensional scaling. Journal of Classification, 5, 163–180.CrossRefGoogle Scholar
de Leeuw, J., Heiser, W. J. (1980). Multidimensional scaling with restrictions on the configuration. In Krisnaiah, P. R. (Eds.), Multivariate analysis, Vol. V (pp. 501–522). Amsterdam: North-Holland.Google Scholar
De Soete, G., DeSarbo, W. S., Carroll, J. D. (1985). Optimal variable weighting for hierarchical clustering: An alternative least-squares algorithm. Journal of Classification, 2, 173–192.CrossRefGoogle Scholar
Escoufier, Y. (1973). Le traitement des variables vectorielles [The treatment of vector-valued variables]. Biometrics, 29, 751–760.CrossRefGoogle Scholar
Escoufier, Y. (1980). Exploratory data analysis when data are matrices. In Matusita, K. (Eds.), Recent developments in statistical inference and data analysis (pp. 45–53). Amsterdam: North-Holland.Google Scholar
Escoufier, Y. (1988). Beyond correspondence analysis. In Bock, H. H. (Eds.), Classification and related methods of data analysis (pp. 505–514). Amsterdam: North-Holland.Google Scholar
Gifi, A. (1990). Nonlinear multivariate analysis First edition,, Leiden: The University of Leiden, Department of Data Theory.Google Scholar
Gower, J. C. (1971). A general coefficient of similarity and some of its properties. Biometrics, 27, 857–871.CrossRefGoogle Scholar
Guttman, L. (1968). A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika, 33, 469–506.CrossRefGoogle Scholar
Heiser, W. J., Stoop, I. (1986). Explicit SMACOF algorithms for individual differences scaling, Leiden: Department of Data Theory.Google Scholar
Horan, C. B. (1969). Multidimensional scaling: Combining observations when individuals have different perceptual structures. Psychometrika, 34, 139–165.CrossRefGoogle Scholar
Hubert, L. J. (1987). Assignment methods in combinatorial data analysis, New York: Marcel Dekker.Google Scholar
Jones, C. L. (1983). A note on the use of directional statistics in weighted Euclidean distances multidimensional scaling models. Psychometrika, 48, 473–476.CrossRefGoogle Scholar
Kiers, H. A. L. (1989). Three-way methods for the analysis of qualitative and quantitative two-way data, Leiden: DSWO Press.Google Scholar
Kruskal, J. B. (1964). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1–28.CrossRefGoogle Scholar
Kruskal, J. B. (1964). Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115–129.CrossRefGoogle Scholar
Kruskal, J. B., Young, F. W., Seery, J. B. (1973). How to use KYST, a very flexible program to do multidimensional scaling and unfolding, Murray Hill: Bell Laboratories.Google Scholar
Meulman, J. J. (1986). A distance approach to nonlinear multivariate analysis, Leiden: DSWO Press.Google Scholar
Meulman, J. J. (1992). The integration of multidimensional scaling and multivariate analysis with optimal transformations. Psychometrika, 57, 539–565.CrossRefGoogle Scholar
Ramsay, J. O. (1982). Some statistical approaches to multidimensional scaling data. Journal of the Royal Statistical Society, Series A, 145, 285–312.CrossRefGoogle Scholar
Ramsay, J. O. (1982). MULTISCALE II manual, Montréal: McGill University, Department of Psychology.Google Scholar
Ramsay, J. O. (1989). Monotone regression splines in action. Statistical Science, 4, 425–441.Google Scholar
Ross, J. (1966). A remark on Tucker and Messick's “Points of view” analysis. Psychometrika, 31, 27–32.CrossRefGoogle Scholar
Saporta, G. (1975). Pondération optimale de variables qualitatives en analyse des données [Optimal weighting of qualitative variables in ‘analyse des données’]. Statistique et Analyse des Données, 3, 19–31.Google Scholar
Shepard, R. N. (1962). The analysis of proximities: Multidimensional scaling with an unknown distance function. I. Psychometrika, 27, 125–140.CrossRefGoogle Scholar
Shepard, R. N. (1962). The analysis of proximities: Multidimensional scaling with an unknown distance function. II. Psychometrika, 27, 219–246.CrossRefGoogle Scholar
Schiffman, S. S., Reynolds, M. L., Young, F. W. (1981). Introduction to multidimensional scaling, New York: Academic Press.Google Scholar
Tucker, L. R. (1951). A method of synthesis of factor analysis studies, Washington, D. C.: Department of the Army.CrossRefGoogle Scholar
Tucker, L. R., Messick, S. (1963). An individual differences model for multidimensional scaling. Psychometrika, 28, 333–367.CrossRefGoogle Scholar
Verboon, P., van der Kloot, W. A. (1989). On the relation between INDSCAL and nonlinear points of view analysis, Leiden: University of Leiden, Department of Psychometrics.Google Scholar
Winsberg, S., & De Soete, G. (in press). A latent class approach to fitting the INDSCAL model. Psychometrika.Google Scholar
Winsberg, S., Ramsay, J. O. (1983). Monotone spline transformations for dimension reduction. Psychometrika, 48, 575–595.CrossRefGoogle Scholar
Zegers, F. E., ten Berge, J. M. F. (1985). A family of association coefficients for metric scales. Psychometrika, 50, 17–24.CrossRefGoogle Scholar