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Multivariate Analysis with Linearizable Regressions

Published online by Cambridge University Press:  01 January 2025

Jan de Leeuw
Jan de Leeuw*
Affiliation:
Departments of Psychology and Mathematics, University of California, Los Angeles
*
Requests for reprints should be sent to Jan de Leeuw, Department of Psychology and Mathematics, University of California, Los Angeles, Los Angeles, CA 90024-1563.
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Abstract

We study the class of multivariate distributions in which all bivariate regressions can be linearized by separate transformation of each of the variables. This class seems more realistic than the multivariate normal or the elliptical distributions, and at the same time its study allows us to combine the results from multivariate analysis with optimal scaling and classical multivariate analysis. In particular a two-stage procedure which first scales the variables optimally, and then fits a simultaneous equations model, is studied in detail and is shown to have some desirable properties.

Keywords

multivariate analysisoptimal scalingcorrespondence analysisstructural modelssimultaneous equationsfactor analysisLISRELtransformation

Information

Type
Original Paper
Information
Psychometrika , Volume 53 , Issue 4 , December 1988 , pp. 437 - 454
Copyright
Copyright © 1988 The Psychometric Society

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References

Anderson, T. W. (1958). An introduction to multivariate statistical analysis, New York: Wiley.Google Scholar
Bakker, B. F. M., Dronkers, J., & Ganzeboom, H. B. G. (1984). Social stratification and mobility in The Netherlands, Amsterdam: SISWO.Google Scholar
Bekker, P. & de Leeuw, J. (1988). Relations between various forms of nonlinear principal component analysis. In van Rijckevorsel, J., de Leeuw, J. (Eds.), Progress in component and correspondence analysis, New York: Wiley.Google Scholar
Bentler, P. M. (1983). Some contributions to efficient statistics in structural models: specification and estimation of moment structures. Psychometrika, 48, 493–518.CrossRefGoogle Scholar
Besse, P., Ramsay, J. O. (1986). Principal components analysis of sampled functions. Psychometrika, 51, 285–311.CrossRefGoogle Scholar
Breiman, L., & Friedman, J. H. (1985). Estimating optimal transformations for multiple regression and correlation. Journal of the American Statistical Association, 80, 580–598.CrossRefGoogle Scholar
de Leeuw, J. (1968). Canonical discriminant analysis of relational data (Report RN 007-68), Leiden: University of Leiden, Department of Data Theory.Google Scholar
de Leeuw, J. (1982). Nonlinear principal component analysis. In Caussinus, H. (Eds.), COMPSTAT 1982 (pp. 77–86). Wien: Physika Verlag.Google Scholar
de Leeuw, J. (1983). On the prehistory of correspondence analysis. Statistica Neerlandica, 37, 161–164.CrossRefGoogle Scholar
de Leeuw, J. (1983). Models and methods for the analysis of correlation coefficients. Journal of Econometrics, 22, 113–137.CrossRefGoogle Scholar
de Leeuw, J. (1984). Models of data. Kwantitatieve Methoden, 5, 17–30.Google Scholar
de Leeuw, J. (1984). The Gift-system of nonlinear multivariate analysis. In Diday, E. (Eds.), Data analysis and informatics II (pp. 415–424). Amsterdam: North Holland.Google Scholar
de Leeuw, J. (1984). Discrete normal linear regression models. In Dijkstra, T. K. (Eds.), Misspecification analysis (pp. 56–71). Berlin: Spring Verlag.CrossRefGoogle Scholar
de Leeuw, J. (1986). Regression with optimal scaling of the dependent variable. In Bunke, O. (Eds.), Proceedings of the 7th International Summer School on Problems of Model Choice and Parameter Estimation in Regression Analysis (pp. 99–111). Berlin, GDR: Humboldt University, Department of Mathematics.Google Scholar
de Leeuw, J. (1988). Model selection in multinomial experiments. In Dijkstra, T. K. (Eds.), On model uncertainty and its statistical implications (pp. 118–138). Berlin: Springer Verlag.CrossRefGoogle Scholar
de Leeuw, J. (1988). Models and techniques. Statistica Neerlandica, 42, 91–98.CrossRefGoogle Scholar
de Leeuw, J. (in press). Multivariate analysis with optimal scaling. In Das Gupta, S. (Ed.), Progress in multivariate analysis. Calcutta: Indian Statistical Institute.Google Scholar
de Leeuw, J. & van der Heijden, P. G. M. (1988). Correspondence analysis of incomplete contingency tables. Psychometrika, 53, 223–233.CrossRefGoogle Scholar
de Leeuw, J. & van der Heijden, P. G. M. (in press). The analysis of time budgets with a latent time budget model. In Diday, E. (Eds.), Data analysis and informatics V. Amsterdam: North Holland.Google Scholar
de Leeuw, J. & van Rijckevorsel, J. L. A. (1988). Beyond homogeneity analysis. In van Rijckevorsel, J., de Leeuw, J. (Eds.), Progress in component and correspondence analysis (pp. 55–81). New York: Wiley.Google Scholar
Dijkstra, T. (1983). Some comments on maximum likelihood and partial least squares methods. Journal of Econometrics, 22, 67–90.CrossRefGoogle Scholar
Freedman, D. A. (1987). As others see us: A case study in path analysis. Journal of Educational Statistics, 12, 101–129.CrossRefGoogle Scholar
Gifi, A. (1980). Data analyse en statistiek [Data analysis and statistics]. Bulletin VVS, 13(5), 10–16.Google Scholar
Gifi, A. (1988). Nonlinear multivariate analysis, Leiden: DSWO-Press.Google Scholar
Gilula, Z., & Haberman, S. J. (1986). Canonical analysis of contingency tables by maximum likelihood. Journal of the American Statistical Association, 81, 780–788.CrossRefGoogle Scholar
Goodman, L. A. (1986). Some useful extensions of the usual correspondence analysis approach and the usual loglinear models approach in the analysis of contingency tables. International Statistical Review, 54, 243–309.CrossRefGoogle Scholar
Greenacre, M. J. (1984). Theory and applications of correspondence analysis, New York: Academic Press.Google Scholar
Guttman, L. (1955). The determinacy of factor score matrices with implications for five other basic problems of common-factor theory. British Journal of Statistical Psychology, 8, 65–82.CrossRefGoogle Scholar
Guttman, L. (1959). Metricizing rank-ordered or unordered data for a linear factor analysis. Sankhya, 21, 257–268.Google Scholar
Guttman, L. (1971). Measurement as structural theory. Psychometrika, 36, 329–347.CrossRefGoogle Scholar
Hirschfeld, H. O. (1935). A connection between correlation and contingency. Proceedings of the Cambridge Philosophical Society, 31, 520–524.CrossRefGoogle Scholar
Isserlis, L. (1916). On certain probable errors and correlation coefficients of multiple frequency distributions with skew regression. Biometrika, 11, 185–190.CrossRefGoogle Scholar
Kiiveri, H. T. (1987). An incomplete data approach to the analysis of covariance structures. Psychometrika, 52, 539–554.CrossRefGoogle Scholar
Koyak, R. A. (1987). On measuring internal dependence in a set of random variables. Annals of Statistics, 15, 1215–1229.CrossRefGoogle Scholar
Lebart, L., Morineau, A., & Warwick, K. M. (1984). Multivariate descriptive statistical analysis, New York: Wiley.Google Scholar
Little, R. J. A., & Rubin, D. B. (1983). On jointly estimating parameters and missing data by maximizing the complete-data likelihood. American Statistician, 37, 218–220.CrossRefGoogle Scholar
Little, R. J. A., & Rubin, D. B. (1987). Statistical analysis with missing data, New York: Wiley.Google Scholar
Molenaar, I. (1988). Formal statistics and informal data analysis, or why laziness should be discouraged. Statistica Neerlandica, 83–90.CrossRefGoogle Scholar
Mooijaart, A., Meijerink, F., & de Leeuw, J. (1988). Nonlinear path models. Unpublished manuscript.Google Scholar
Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49, 115–132.CrossRefGoogle Scholar
Pearson, K. (1906). On certain points connected with scale order in the case of a correlation of two characters which for some arrangement give a linear regression line. Biometrika, 5, 176–178.Google Scholar
Peschar, J. L. (1973). School, milieu, beroep, Groningen, The Netherlands: Wolters.Google Scholar
Steiger, J. H., & Browne, M. W. (1984). The comparison of independent correlations between optimal linear composites. Psychometrika, 49, 11–24.CrossRefGoogle Scholar
Takane, Y., & Young, F. W., de Leeuw, J. (1979). Nonmetric common factor analysis: An alternating least squares method with optimal scaling features. Behaviormetrika, 6, 45–56.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, 91–119.CrossRefGoogle Scholar
van der Burg, E., de Leeuw, J., & Verdegaal, R. (1988). Homogeneity analysis withk sets of variables. Psychometrika, 53, 177–197.CrossRefGoogle Scholar
van der Heijden, P. G. M., & de Leeuw, J. (1985). Correspondence analysis used complementary to loglinear analysis. Psychometrika, 50, 429–447.CrossRefGoogle Scholar
van Praag, B. M. S., de Leeuw, J., & Kloek, T. (1986). The population sample decomposition approach to multivariate estimation methods. Applied Stochastic Models and Data Analysis, 2, 99–120.CrossRefGoogle Scholar
van Rijckevorsel, J. L. A. (1987). The application of horseshoes and fuzzy coding in multiple correspondence analysis, Leiden: DSWO-Press.Google Scholar
van Rijckevorsel, J. L. A., de Leeuw, J. (1988). Progress in component and correspondence analysis, New York: Wiley.Google Scholar
Winsberg, S., & Ramsay, J. O. (1980). Monotonic transformations to additivity using splines. Biometrika, 67, 669–674.CrossRefGoogle Scholar
Young, F. W. (1981). Quantitative analysis of qualitative data. Psychometrika, 46, 357–388.CrossRefGoogle Scholar