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Asymptotic Biases in Exploratory Factor Analysis and Structural Equation Modeling

Published online by Cambridge University Press:  01 January 2025

Haruhiko Ogasawara
Haruhiko Ogasawara*
Affiliation:
Otaru University of Commerce
*
Requests for reprints should be sent to Haruhiko Ogasawaxa, Department of Information and Management Science, Otaru University of Commerce, 3-5-21, Midori, Otaru 047-8501 JAPAN. E-mail: hogasa@res.otaru-uc.ac.jp
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Abstract

Formulas for the asymptotic biases of the parameter estimates in structural equation models are provided in the case of the Wishart maximum likelihood estimation for normally and nonnormally distributed variables. When multivariate normality is satisfied, considerable simplification is obtained for the models of unstandardized variables. Formulas for the models of standardized variables are also provided. Numerical examples with Monte Carlo simulations in factor analysis show the accuracy of the formulas and suggest the asymptotic robustness of the asymptotic biases with normality assumption against nonnormal data. Some relationships between the asymptotic biases and other asymptotic values are discussed.

Keywords

Asymptotic biasesstructural equation modelingasymptotic variancesnonnormal dataasymptotic robustnessfactor analysis

Information

Type
Theory And Methods
Information
Psychometrika , Volume 69 , Issue 2 , June 2004, pp. 235 - 256
Copyright
Copyright © 2004 The Psychometric Society

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Footnotes

The author is indebted to the editor and anonymous reviewers for their comments, corrections, and suggestions on this paper, and to Yutaka Kano for discussion on biases.

References

Anderson, T.W. (1984). An Introduction to Multivariate Statistical Analysis 2nd ed., New York: WileyGoogle Scholar
Anderson, T.W. (1987). Multivariate linear relations. In Pukkila, T. & Puntanen, S. (Eds.), Proceedings of the Second International Conference in Statistics (pp. 9–36). Tampere, Finland: University of TampereGoogle Scholar
Anderson, T.W. (1989). Linear latent variable models and covariance structures. Journal of Econometrics, 41, 91–119CrossRefGoogle Scholar
Anderson, T.W., & Amemiya, Y. (1988). The asymptotic normal distribution of estimators in factor analysis under general conditions. The Annals of Statistics, 16, 759–771CrossRefGoogle Scholar
Archer, C.O., & Jennrich, R.I. (1973). Standard errors for rotated factor loadings. Psychometrika, 38, 581–592CrossRefGoogle Scholar
Bentler, P.M., & Dijkstra, T. (1985). Efficient estimation via linearization in structural models. In Krishnaiah, P.R. (Eds.), Multivariate Analysis—VI (pp. 9–42). New York: ElsevierGoogle Scholar
Browne, M.W. (1974). Generalized least squares estimators in the analysis of covariance structures. In Aigner, D.J. & Goldberger, A.S. (Eds.), Latent Variables in Socioeconomic Models (pp. 205–226). Amsterdam: North HollandGoogle Scholar
Browne, M.W. (1987). Robustness in statistical inference in factor analysis and related models. Biometrika, 74, 375–384CrossRefGoogle Scholar
Browne, M.W., & Shapiro, A. (1986). The asymptotic covariance matrix of sample correlation coefficients under general conditions. Linear Algebra and Its Applications, 82, 169–176CrossRefGoogle Scholar
Browne, M.W., & Shapiro, A. (1988). Robustness of normal theory methods in the analysis of linear latent variable models. British Journal of Mathematical and Statistical Psychology, 41, 193–208CrossRefGoogle Scholar
Emmett, W.G. (1949). Factor analysis by Lawley's method of maximum likelihood. British Journal of Psychology, Statistical Section, 2, 90–97CrossRefGoogle Scholar
Harman, H.H. (1976). Modern Factor Analysis 3rd ed., Chicago, IL: University of Chicago PressGoogle Scholar
Hayashi, K., & Yung, Y.F. (1999). Standard errors for the class of orthomax-rotated factor loadings: Some matrix results. Psychometrika, 64, 451–460CrossRefGoogle Scholar
Ichikawa, M., & Konishi, S. (2002). Asymptotic expansions and bootstrap approximations in factor analysis. Journal of Multivariate Analysis, 81, 47–66CrossRefGoogle Scholar
Ihara, M. (1985). Asymptotic bias of estimators of the uniqueness in factor analysis. Mathematica Japonica, 30, 885–889Google Scholar
Jennrich, R.I. (1973). Standard errors for obliquely rotated factor loadings. Psychometrika, 38, 593–604CrossRefGoogle Scholar
Jennrich, R.I. (1974). Simplified formulae for standard errors in maximum likelihood factor analysis. British Journal of Mathematical and Statistical Psychology, 27, 122–131CrossRefGoogle Scholar
Jöreskog, K.G. (1969). A general approach to confirmatory maximum likelihood factor analysis. Psychometrika, 34, 183–202CrossRefGoogle Scholar
Jöreskog, K.G. (1978). Structural analysis of covariance and correlation matrices. Psychometrika, 43, 443–477CrossRefGoogle Scholar
Jöreskog, K.G., Sörbom, D., du Toit, S., & du Toit, M. (1999). LISREL 8: New Statistical Features. Chicago, IL: Scientific SoftwareGoogle Scholar
Kaiser, H.F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika, 23, 187–200CrossRefGoogle Scholar
Lawley, D.N., & Maxwell, A.E. (1971). Factor Analysis as a Statistical Method 2nd ed., London: ButterworthsGoogle Scholar
Magnus, J.R., & Neudecker, H. (1999). Matrix Differential Calculus with Applications in Statistics and Econometrics revised ed., New York: WileyGoogle Scholar
Nel, D.G. (1985). A matrix derivation of the asymptotic covariance matrix of sample correlation coefficients. Linear Algebra and Its Applications, 67, 137–145CrossRefGoogle Scholar
Neudecker, H. (1996). The asymptotic variance matrices of the sample correlation matrix in elliptical and normal situations and their proportionality. Linear Algebra and Its Applications, 237–238, 127–132CrossRefGoogle Scholar
Neudecker, H., & Wesselman, A.M. (1990). The asymptotic variance matrix of the sample correlation matrix. Linear Algebra and Its Applications, 127, 589–599CrossRefGoogle Scholar
Ogasawara, H. (1998). Standard errors for rotation matrices with an application to the promax solution. British Journal of Mathematical and Statistical Psychology, 51, 163–178CrossRefGoogle Scholar
Ogasawara, H. (1999). Standard errors for procrustes solutions. Japanese Psychological Research, 41, 121–130CrossRefGoogle Scholar
Ogasawara, H. (2000). On the standard errors of rotated factor loadings with weights for observed variables. Behaviormetrika, 27, 1–14CrossRefGoogle Scholar
Ogasawara, H. (2000). Standard errors for the Harris-Kaiser Case II orthoblique solution. Behaviormetrika, 27, 89–103CrossRefGoogle Scholar
Ogasawara, H. (2002). Asymptotic standard errors of estimated standard errors in structural equation modeling. British Journal of Mathematical and Statistical Psychology, 55, 213–229CrossRefGoogle Scholar
Olkin, I., & Pratt, J.W. (1958). Unbiased estimation of certain correlation coefficients. Annals of Mathematical Statistics, 29, 201–211CrossRefGoogle Scholar
Satorra, A. (1989). Alternative test criteria in covariance structure analysis: A unified approach. Psychometrika, 54, 131–151CrossRefGoogle Scholar
Satorra, A. (2002). Asymptotic robustness in multiple group linear-latent variable models. Econometric Theory, 18, 297–312CrossRefGoogle Scholar
Satorra, A., & Bentler, P.M. (1990). Model conditions for asymptotic robustness in the analysis of linear relations. Computational Statistics and Data Analysis, 10, 235–249CrossRefGoogle Scholar
Shapiro, A. (1983). Asymptotic distribution theory in the analysis of covariance structures (A unified approach). South African Statistical Journal, 17, 33–81Google Scholar
Silvey, S.D. (1975). Statistical Inference. London: Chapman & HallGoogle Scholar
Siotani, M., Hayakawa, T., & Fujikoshi, Y. (1985). Modern Multivariate Statistical Analysis: A Graduate Course and Handbook. Columbus, OH: American Sciences PressGoogle 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, 208–215CrossRefGoogle Scholar
Yuan, K.-H., & Bentler, P.M. (2000). On equivalence and invariance of standard errors in three exploratory factor models. Psychometrika, 65, 121–133CrossRefGoogle Scholar
Yung, Y.-F., & Hayashi, K. (2001). A computationally efficient method for obtaining standard error estimates for the promax and related solutions. British Journal of Mathematical and Statistical Psychology, 54, 125–138CrossRefGoogle ScholarPubMed