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Factor Analysis for Non-Normal Variables

Published online by Cambridge University Press:  01 January 2025

Ab Mooijaart
Ab Mooijaart*
Affiliation:
Leiden University
*
Requests for reprints should be sent to Ab Mooijaart, Department of Psychology, Leiden University, Hooigracht 15, 2312 KM LEIDEN, THE NETHERLANDS.
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Abstract

Factor analysis for nonnormally distributed variables is discussed in this paper. The main difference between our approach and more traditional approaches is that not only second order cross-products (like covariances) are utilized, but also higher order cross-products. It turns out that under some conditions the parameters (factor loadings) can be uniquely determined. Two estimation procedures will be discussed. One method gives Best Generalized Least Squares (BGLS) estimates, but is computationally very heavy, in particular for large data sets. The other method is a least squares method which is computationally less heavy. In one example the two methods will be compared by using the bootstrap method. In another example real life data are analyzed.

Keywords

factor analysisnonnormalitythird order cross-productsunique factor loadingsBGLS estimatesbootstrap

Information

Type
Original Paper
Information
Psychometrika , Volume 50 , Issue 3 , September 1985 , pp. 323 - 342
Copyright
Copyright © 1985 The Psychometric Society

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Footnotes

This paper has partly been written while the author was a visiting scholar at the Department of Psychology, University of California, Los Angeles. He wants to thank Peter Bentler who made this stay at UCLA possible and for his valuable contributions to this paper. This research was supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O) under number R56-150 and by USPHS Grant DA01070.

References

Aitchison, J. R., Brown, J. A. C. (1957). The Lognormal Distribution, London: Cambridge University Press.Google Scholar
Barankin, E., Gurland, J. (1951). On asymptotically normal efficient estimators: 1. University of California Publications in Statistics, 1(6), 86–130.Google Scholar
Bentler, P. M. (1983). Simultaneous equation systems as moment structure models: With an introduction to latent variable models. Journal of Econometrics, 22, 13–42.CrossRefGoogle Scholar
Bentler, P. M. (1983). Some contributions to efficient statistics in structural models: Specification and estimation of moment structures. Psychometrika, 48, 493–517.CrossRefGoogle Scholar
Bentler, P. M., Lee, S. Y. (1975). Some extensions of matrix calculus. General Systems, 20, 145–150.Google Scholar
Bentler, P. M., Lee, S. Y. (1978). Matrix derivatives with chain rule and rules for simple, Hadamard, and Kronecker products. Journal of Mathematical Psychology, 17, 255–262.CrossRefGoogle Scholar
Boomsma, A. (1983). On the robustness of LISREL (maximum likelihood estimation) against small sample size and non-normality, Groningen: University of Groningen.Google Scholar
Browne, M. W. (1974). Generalized least squares estimators in the analysis of covariance structures. South African Statistical Journal, 8, 1–24.Google Scholar
Browne, M. W. (1982). Covariance structures. In Hawkins, D. M. (Eds.), Topics in Applied Multivariate Analysis (pp. 72–141). London: Cambridge University Press.CrossRefGoogle Scholar
Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83.CrossRefGoogle ScholarPubMed
Cook, M. B. (1951). Bivariatek-statistics and cumulants of their joint sampling distribution. Biometrika, 38, 179–195.CrossRefGoogle ScholarPubMed
Cramér, H. (1946). Mathematical Methods of Statistics, Princeton, N J: Princeton University Press.Google Scholar
de Leeuw, J. (1983). Models and methods for the analysis of correlation coefficients. Journal of Econometrics, 22, 113–137.CrossRefGoogle Scholar
Dunn, J. E. (1973). A note on a sufficiency condition for uniqueness of a restricted factor matrix. Psychometrika, 38, 141–143.CrossRefGoogle Scholar
Du Toit, S. H. C. (1979). The Analysis of growth curves, Pretoria: University of South Africa.Google Scholar
Efron, B. (1979). The 1977 Rietz Lecture: Bootstrap methods: another look at the Jack-knife. The Annals of Statistics, 7, 1–26.Google Scholar
Ferguson, T. S. (1958). A method of generating best asymptotically normal estimates with application to the estimation of bacterial densities. Annals of Mathematical Statistics, 29, 1046–1062.CrossRefGoogle Scholar
Graham, A. (1981). Kronecker products and matrix calculus with applications, Chichester: Ells Horwood.Google Scholar
Howe, W. G. (1955). Some contributions to factor analysis (Report No. ORNL-1919), Oak Ridge, TN: Oak Ridge National Laboratory.CrossRefGoogle Scholar
Hsu, P. L. (1949). The limiting distribution of functions of sample means and application to testing hypotheses. Proceedings of the First Berkeley Symposium in Mathematical Statistics and Probability (pp. 359–402). Berkeley.Google Scholar
Isserlis, D. Sc. (1916). On certain probable errors and correlation coefficients of multiple frequency distributions with skew regression. Biometrika, 11, 185–190.CrossRefGoogle Scholar
Jennrich, R. I. (1978). Rotational equivalence of factor loading matrices with specified values. Psychometrika, 43, 421–426.CrossRefGoogle Scholar
Johnson, N. L., Kotz, S. (1970). Continuous Univariate Distribution—I, Boston: Houghton Mifflin.Google Scholar
Jöreskog, K. G. (1967). Some contributions to maximum likelihood factor analysis. Psychometrika, 32, 443–482.CrossRefGoogle Scholar
Jöreskog, K. G., Goldberger, A. S. (1972). Factor analysis by generalized least squares. Psychometrika, 37, 243–260.CrossRefGoogle Scholar
Jöreskog, K. G., Sörbom, D. (1979). Advances in factor analysis and structural equation models, Cambridge, MA.: Abt..Google Scholar
Jöreskog, K. G., Sörbom, D. (1981). Lisrel V Users Guide, Chicago: International Educational Resources.Google Scholar
Kagan, A. M., Linnik, Y. V., Rao, C. R. (1973). Characterization problems in mathematical statistics, New York: John Wiley & Sons.Google Scholar
Kaplan, E. L. (1952). Tensor notation and the sampling cumulants ofk-statistics. Biometrika, 39, 319–323.Google Scholar
Kendall, M. G., Stuart, A. (1977). The advanced theory of statistics, Volume 1, New York: Mac-Millan.Google Scholar
Lawley, D. N. (1940). The estimation of factor loadings by the method of maximum likelihood. Proceedings of the Royal Society of Edinburgh, 60, 64–82.CrossRefGoogle Scholar
Layard, M. W. J. (1972). Large sample tests for the equality of two covariance matrices. Annals of Mathematical Statistics, 43, 123–141.CrossRefGoogle Scholar
Layard, M. W. J. (1974). A Monte Carlo comparison of test of equality of covariance matrices. Biometrika, 16, 461–465.Google Scholar
McDonald, R. P., Swaminathan, H. (1973). A simple matrix calculus with applications to multivariate analysis. General Systems, 18, 37–54.Google Scholar
Mooijaart, A. (1978). Latent Structure Models, Leiden: Leiden University.Google Scholar
Pearson, K., Filon, L. N. G. (1898). Mathematical contributions to the theory of evolution: IV. On the probable error of frequency constants and on the influence of random selection of variation and correlation. Philosophical Transactions of the Royal Society of London, 191, 229–311.CrossRefGoogle Scholar
Rao, C. R. (1966). Characterization of the distribution of random variables in linear structural relations. Sankhya, 28, 251–260.Google Scholar
Reiersøl, O. (1950). Identifiability of a linear relation between variables which are subject to error. Econometrika, 18, 375–389.CrossRefGoogle Scholar
Steiger, J. H., Browne, M. W. (1984). The comparison of interdependent correlations between optimal linear composities. Psychometrika, 49, 11–24.CrossRefGoogle Scholar
Steiger, J. H., Hakstian, A. R. (1982). The asymptotic distribution of elements of a correlation matrix: Theory and Application. British Journal of Mathematical and Statistical Psychology, 35, 208–215.CrossRefGoogle Scholar
Steiger, J. H., Hakstian, A. R. (1983). A historical note on the asymptotic distribution of correlations. British Journal of Mathematical and Statistical Psychology, 36, 157–157.CrossRefGoogle Scholar
Swain, A. J. (1975). A class of factor analysis estimation procedures with common asymptotic sampling properties. Psychometrika, 40, 315–335.CrossRefGoogle Scholar
Van Putten, C. M., Commandeur, J. J. F. (1984). Structuur van leerkrachtuitspraken over sociaal-emotionele leerlingkenmerken in IBO, LBO en VBO/MLK [Structure of statements of teachers about social-emotional characteristics of pupils at schools for individual vocational education], Leiden: Rijksuniversiteit Leiden.Google Scholar