Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-07T18:39:48.778Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogeneity Analysis with k Sets of Variables: An Alternating Least Squares Method with Optimal Scaling Features

Published online by Cambridge University Press:  01 January 2025

Eeke van der Burg ,
Jan de Leeuw  and
Renée Verdegaal
Eeke van der Burg*
Affiliation:
Department of Education, University of Twente
Jan de Leeuw
Affiliation:
Department of Data Theory, Leiden University
Renée Verdegaal
Affiliation:
Department of Data Theory, Leiden University
*
Requests for reprints should be sent to Eeke van der Burg, University of Twente, Department of Education, PO Box 217, 7500 AE Enschede, THE NETHERLANDS.
Get access Rights & Permissions [Opens in a new window]

Abstract

Homogeneity analysis, or multiple correspondence analysis, is usually applied to k separate variables. In this paper we apply it to sets of variables by using sums within sets. The resulting technique is called OVERALS. It uses the notion of optimal scaling, with transformations that can be multiple or single. The single transformations consist of three types: nominal, ordinal, and numerical. The corresponding OVERALS computer program minimizes a least squares loss function by using an alternating least squares algorithm. Many existing linear and nonlinear multivariate analysis techniques are shown to be special cases of OVERALS. An application to data from an epidemiological survey is presented.

Keywords

homogeneity analysiscorrespondence analysisoptimal scalingtransformationalternating least squarescanonical correlation analysisprincipal component analysis
Type
Original Paper
Information
Psychometrika , Volume 53 , Issue 2 , June 1988 , pp. 177 - 197
Copyright
Copyright © 1988 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

This research was partly supported by SWOV (Institute for Road Safety Research) in Leidschendam, The Netherlands.

The authors wish to thank Wilfrid van Pelt, Department of Physiology and Physiological Physics, University Medical Centre, Leiden, for his help and cooperation with the chronic lung disease application. The data were obtained from the Department of Epidemiology, Institute for Social Medical Research, University of Groningen. The authors kindly thank R. van der Lende who made the data available to them.

We also thank the editor and referees of Psychornetrika for their very helpful comments on previous versions of this paper.

References

Benzécri, J. P.et al. (1973). L'Analyse des données, Paris: Dunod.Google Scholar
Benzécri, J. P.et al. (1980). Pratique de l'Analyse des données [Practice of data analysis], Paris: Dunod.Google Scholar
Carroll, J. D. (1968). Generalization of canonical correlation analysis to three or more sets of variables. Proceedings of the 76th Annual Convention of the American Psychological Association, 5, 227–228.Google Scholar
Cliff, N. (1966). Orthogonal rotation to congruence. Psychometrika, 31, 33–42.CrossRefGoogle Scholar
de Leeuw, J. (1983, July). Nonlinear joint bivariate analysis. Paper presented at the meeting of the Psychometric Society, Jouy-en-Josas, France.Google Scholar
de Leeuw, J. (1984). The Gifi-system of nonlinear multivariate analysis. In Diday, E., Jambu, M., Lebart, L., Pagès, J., Thomassone, R. (Eds.), Data Analysis and Informatics III (pp. 415–424). Amsterdam: North Holland.Google Scholar
de Leeuw, J. (1984). Canonical analysis of categorical data, Leiden: DSWO-Press.Google Scholar
de Leeuw, J., van der Burg, E. (1986). The Permutational limit distribution of generalized canonical correlations. In Diday, E., Escoufier, Y., Lebart, L., Pagès, J. P., Schektman, Y., Thomassone, R. (Eds.), Data analysis and informatics IV (pp. 509–521). Amsterdam: North Holland.Google Scholar
de Leeuw, J., van Rijckevorsel, J., van der Wouden, H. (1981). Nonlinear principal component analysis using B-splines. Methods of operations research, 23, 211–234.Google Scholar
Gifi, A. (1981). Nonlinear multivariate analysis, Leiden: DSWO-Press.Google Scholar
Gifi, A. (1985). PRINCALS. (User's Guide UG-85-03). Department of Data Theory, University of Leiden.Google Scholar
Greenacre, M. J. (1984). Theory and applications of correspondence analysis, New York: Academic Press.Google Scholar
Guttman, L. (1941). The quantification of a class of attributes: A theory and method of scale construction. In Horst, P. (Eds.), The prediction of personal adjustment, New York: Social Science Research Council.Google Scholar
Horst, P. (1961). Relations amongm sets of measures. Psychometrika, 26, 129–149.CrossRefGoogle Scholar
Kettenring, J. R. (1971). Canonical analysis of several sets of variables. Biometrika, 56, 433–451.CrossRefGoogle Scholar
Kruskal, J. B. (1964). Multidimensional scaling by optimizing goodness-of-fit to a nonmetric hypothesis. Psychometrika, 29, 1–28.CrossRefGoogle Scholar
Kuhfeld, W. F. (1985). Principal components of ordered categorical data. Unpublished doctoral dissertation, University of North Carolina.Google Scholar
Kuhfeld, W. F., Sarle, W. S., Young, F. W. (1985). Methods in generating model estimates in the PRINQUAL macro (pp. 962–971). Cary, NC: SAS-Institute.Google Scholar
Lebart, L., Morineau, A., Warwick, K. M. (1984). Multivariate descriptive analysis, New York: Wiley.Google Scholar
Meulman, J. (1982). Homogeneity analysis of incomplete data, Leiden: DSWO-Press.Google Scholar
Nishisato, S. (1980). Analysis of categorical data: Dual scaling and its applications, Toronoto: University of Toronto Press.CrossRefGoogle Scholar
Rutishauser, H. (1969). Computational aspects of F. L. Bauer's simultaneous iteration method. Numerische Mathematik, 13, 4–13.CrossRefGoogle Scholar
Segijn, R. (1984). Lokale minima in PRINCALS [Local minima in PRINCALS]. Unpublished master's Thesis, Department of Data Theory, University of Leiden.Google Scholar
Takane, Y., Young, F. W., de Leeuw, J. (1980). An individual differences additive model: An alternating least squares method with optimal scaling features. Psychometrika, 45, 183–209.CrossRefGoogle Scholar
Tenenhaus, M., Young, F. W. (1985). An analysis and synthesis of multiple correspondence analysis, optimal scaling, dual scaling, homogeneity analysis, and other methods for quantifying categorical multivariate data. Psychometrika, 50, 91–120.CrossRefGoogle Scholar
Thorndike, R. M. (1977). Canonical analysis and predictor selection. Multivariate Behavioral Research, 12, 75–87.CrossRefGoogle ScholarPubMed
van de Geer, J. P. (1985). HOMALS. (User's guide UG-85-02). Department of data theory, University of Leiden.Google Scholar
van de Geer, J. P. (1986). Introduction to multivariate data analysis (2 vols.), Leiden: DSWO-Press.Google Scholar
van der Burg, E. (1983). CANALS (User's guide UG-85-05). Department of Data Theory, University of Leiden.Google Scholar
van der Burg, E., de Leeuw, J. (1983). Non-linear canonical correlation. British Journal of Mathematical and Statistical Psychology, 36, 54–80.CrossRefGoogle Scholar
van der Burg, E., & de Leeuw, J. (1985). Use of the multinomial jackknife in generalized canonical correlation analysis. Paper presented at the Multidimensional Data Analysis Workshop, Cambridge, G.B.Google Scholar
van der Burg, E., de Leeuw, J., Verdegaal, R. (1984). Nonlinear canonical correlation with m sets of variables, Leiden: University of Leiden, Department of Data Theory.Google Scholar
van der Lende, R., Kok, T. J., Peset Reig, R., Quanjer, Ph. H., Schouten, J. P., Orie, N. G. M. (1981). Decreases in VC and FEV with time: Indicators for effects of smoking and air pollution. Bulletin Européen de Psychopathologie Respiratoire, 17, 775–792.Google Scholar
van Pelt, W. J., Quanjer, Ph. W., Wise, M. E., van der Burg, E., van der Lende, R. (1985). Analysis of maximum expiratory flow-volume curves using canonical correlation analysis. Methods of Information in Medicine, 24, 91–100.Google ScholarPubMed
Verdegaal, R. (1985). Meer-sets analyse voor kwalitatieve gegevens [Multi-set analysis of qualitative data] (Research Report RR-85-14). Department of Data Theory, University of Leiden.Google Scholar
Verdegaal, R. (1986). OVERALS (User's guide UG-86-01). Department of Data Theory, University of Leiden.Google Scholar
Young, F. W. (1981). Quantitative analysis of qualitative data. Psychometrika, 46, 347–388.CrossRefGoogle Scholar
Young, F. W., de Leeuw, J., Takane, Y. (1976). Regression with qualitative and quantitative variables: An alternating least squares method with optimal scaling features. Psychometrika, 41, 505–529.CrossRefGoogle Scholar
Young, F. W., Takane, Y., de Leeuw, J. (1978). The principal components of mixed measurement multivariate data: An alternating least squares method with optimal scaling features. Psychometrika, 43, 279–281.CrossRefGoogle Scholar