, Volume 53, Issue 4, pp 437–454 | Cite as

Multivariate analysis with linearizable regressions

  • Jan de Leeuw


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.

Key words

multivariate analysis optimal scaling correspondence analysis structural models simultaneous equations factor analysis LISREL transformation 


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  1. Anderson, T. W. (1958).An introduction to multivariate statistical analysis. New York, Wiley.Google Scholar
  2. Bakker, B. F. M., Dronkers, J., & Ganzeboom, H. B. G. (1984).Social stratification and mobility in The Netherlands. Amsterdam: SISWO.Google Scholar
  3. Bekker, P. & de Leeuw, J. (1988). Relations between various forms of nonlinear principal component analysis. In J. van Rijckevorsel & J. de Leeuw,Progress in component and correspondence analysis. New York: Wiley.Google Scholar
  4. Bentler, P. M. (1983). Some contributions to efficient statistics in structural models: specification and estimation of moment structures.Psychometrika, 48, 493–518.Google Scholar
  5. Besse, P., & Ramsay, J. O. (1986). Principal components analysis of sampled functions.Psychometrika, 51, 285–311.Google Scholar
  6. Breiman, L., & Friedman, J. H. (1985). Estimating optimal transformations for multiple regression and correlation.Journal of the American Statistical Association, 80, 580–598.Google Scholar
  7. de Leeuw, J. (1968).Canonical discriminant analysis of relational data (Report RN 007-68). Leiden: University of Leiden, Department of Data Theory.Google Scholar
  8. de Leeuw, J. (1982). Nonlinear principal component analysis. In H. Caussinus (Ed.),COMPSTAT 1982 (pp. 77–86). Wien: Physika Verlag.Google Scholar
  9. de Leeuw, J. (1983a). On the prehistory of correspondence analysis.Statistica Neerlandica, 37, 161–164.Google Scholar
  10. de Leeuw, J. (1983b). Models and methods for the analysis of correlation coefficients.Journal of Econometrics, 22, 113–137.Google Scholar
  11. de Leeuw, J. (1984a). Models of data.Kwantitatieve Methoden, 5, 17–30.Google Scholar
  12. de Leeuw, J. (1984b). The Gift-system of nonlinear multivariate analysis. In E. Diday (Ed.),Data analysis and informatics II (pp. 415–424). Amsterdam: North Holland.Google Scholar
  13. de Leeuw, J. (1984c). Discrete normal linear regression models. In T. K. Dijkstra (Ed.),Misspecification analysis (pp. 56–71). Berlin: Spring Verlag.Google Scholar
  14. de Leeuw, J. (1986). Regression with optimal scaling of the dependent variable. In O. Bunke (Ed.),Proceedings of the 7th International Summer School on Problems of Model Choice and Parameter Estimation in Regression Analysis (Report No. 84, pp. 99–111). Berlin, GDR: Humboldt University, Department of Mathematics.Google Scholar
  15. de Leeuw, J. (1988a). Model selection in multinomial experiments. In T. K. Dijkstra (Ed.),On model uncertainty and its statistical implications (pp. 118–138). Berlin: Springer Verlag.Google Scholar
  16. de Leeuw, J. (1988b). Models and techniques.Statistica Neerlandica, 42, 91–98.Google Scholar
  17. de Leeuw, J. (in press). Multivariate analysis with optimal scaling. In S. Das Gupta (Ed.),Progress in multivariate analysis. Calcutta: Indian Statistical Institute.Google Scholar
  18. de Leeuw, J. & van der Heijden, P. G. M. (1988). Correspondence analysis of incomplete contingency tables,Psychometrika, 53, 223–233.Google Scholar
  19. de Leeuw, J. & van der Heijden, P. G. M. (in press). The analysis of time budgets with a latent time budget model. In E. Diday (Eds.),Data analysis and informatics V. Amsterdam: North Holland.Google Scholar
  20. de Leeuw, J. &. van Rijckevorsel, J. L. A. (1988). Beyond homogeneity analysis. In J. van Rijckevorsel & J. de Leeuw,Progress in component and correspondence analysis (pp. 55–81). New York: Wiley.Google Scholar
  21. Dijkstra, T. (1983). Some comments on maximum likelihood and partial least squares methods.Journal of Econometrics, 22, 67–90.Google Scholar
  22. Freedman, D. A. (1987). As others see us: A case study in path analysis.Journal of Educational Statistics, 12, 101–129.Google Scholar
  23. Gifi, A. (1980). Data analyse en statistiek [Data analysis and statistics].Bulletin VVS, 13(5), 10–16.Google Scholar
  24. Gifi, A. (1988).Nonlinear multivariate analysis. Leiden, DSWO-Press.Google Scholar
  25. Gilula, Z., & Haberman, S. J. (1986). Canonical analysis of contingency tables by maximum likelihood.Journal of the American Statistical Association, 81, 780–788.Google Scholar
  26. 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.Google Scholar
  27. Greenacre, M. J. (1984).Theory and applications of correspondence analysis. New York: Academic Press.Google Scholar
  28. 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.Google Scholar
  29. Guttman, L. (1959). Metricizing rank-ordered or unordered data for a linear factor analysis.Sankhya, 21, 257–268.Google Scholar
  30. Guttman, L. (1971). Measurement as structural theory.Psychometrika, 36, 329–347.Google Scholar
  31. Hirschfeld, H. O. (1935). A connection between correlation and contingency.Proceedings of the Cambridge Philosophical Society, 31, 520–524.Google Scholar
  32. Isserlis, L. (1916). On certain probable errors and correlation coefficients of multiple frequency distributions with skew regression.Biometrika, 11, 185–190.Google Scholar
  33. Kiiveri, H. T. (1987). An incomplete data approach to the analysis of covariance structures.Psychometrika, 52, 539–554.Google Scholar
  34. Koyak, R. A. (1987). On measuring internal dependence in a set of random variables.Annals of Statistics, 15, 1215–1229.Google Scholar
  35. Lebart, L., Morineau, A., & Warwick, K. M. (1984).Multivariate descriptive statistical analysis. New York: Wiley.Google Scholar
  36. 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.Google Scholar
  37. Little, R. J. A., & Rubin, D. B. (1987).Statistical analysis with missing data. New York: Wiley.Google Scholar
  38. Molenaar, I. (1988). Formal statistics and informal data analysis, or why laziness should be discouraged.Statistica Neerlandica, 83–90.Google Scholar
  39. Mooijaart, A., Meijerink, F., & de Leeuw, J. (1988). Nonlinear path models. Unpublished manuscript.Google Scholar
  40. Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators.Psychometrika, 49, 115–132.Google Scholar
  41. 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
  42. Peschar, J. L. (1973).School, milieu, beroep [School, environment, profession]. Groningen, The Netherlands: Wolters.Google Scholar
  43. Steiger, J. H., & Browne, M. W. (1984). The comparison of independent correlations between optimal linear composites.Psychometrika, 49, 11–24.Google Scholar
  44. 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.Google Scholar
  45. 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.Google Scholar
  46. van der Burg, E., de Leeuw, J., & Verdegaal, R. (1988). Homogeneity analysis withk sets of variables,Psychometrika, 53, 177–197.Google Scholar
  47. van der Heijden, P. G. M., & de Leeuw, J. (1985). Correspondence analysis used complementary to loglinear analysis.Psychometrika, 50, 429–447.Google Scholar
  48. 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.Google Scholar
  49. van Rijckevorsel, J. L. A. (1987).The application of horseshoes and fuzzy coding in multiple correspondence analysis. Leiden: DSWO-Press.Google Scholar
  50. van Rijckevorsel, J. L. A., & de Leeuw, J. (Eds.). (1988).Progress in component and correspondence analysis. New York: Wiley.Google Scholar
  51. Winsberg, S., & Ramsay, J. O. (1980). Monotonic transformations to additivity using splines.Biometrika, 67, 669–674.Google Scholar
  52. Young, F. W. (1981). Quantitative analysis of qualitative data.Psychometrika, 46, 357–388.Google Scholar

Copyright information

© The Psychometric Society 1988

Authors and Affiliations

  • Jan de Leeuw
    • 1
  1. 1.Department of Psychology and MathematicsUniversity of California, Los AngelesLos Angeles

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