Abstract
This paper studies changes of standard errors (SE) of the normal-distribution-based maximum likelihood estimates (MLE) for confirmatory factor models as model parameters vary. Using logical analysis, simplified formulas and numerical verification, monotonic relationships between SEs and factor loadings as well as unique variances are found. Conditions under which monotonic relationships do not exist are also identified. Such functional relationships allow researchers to better understand the problem when significant factor loading estimates are expected but not obtained, and vice versa. What will affect the likelihood for Heywood cases (negative unique variance estimates) is also explicit through these relationships. Empirical findings in the literature are discussed using the obtained results.
Similar content being viewed by others
References
Allen, M.J., & Yen, W.M. (1979). Introduction to measurement theory. Monterey: Brooks-Cole.
Anderson, T.W., & Amemiya, Y. (1988). The asymptotic normal distribution of estimators in factor analysis under general conditions. Annals of Statistics, 16, 759–771.
Anderson, J.C., & Gerbing, D.W. (1984). The effect of sampling error on convergence, improper solutions, and goodness-of-fit indices for maximum likelihood confirmatory factor analysis. Psychometrika, 49, 155–173.
Bentler, P.M. (1983). Some contributions to efficient statistics for structural models: Specification and estimation of moment structures. Psychometrika, 48, 493–517.
Bentler, P.M. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107, 238–246.
Boomsma, A. (1982). The robustness of LISREL against small sample sizes in factor analysis models. In K.G. Jöreskog, & H. Wold (Eds.), Systems under indirect observation: causality, structure, prediction (pp. 149–173). Amsterdam: North-Holland (Part I).
Boomsma, A. (1985). Nonconvergence, improper solutions, and starting values in LISREL maximum likelihood estimation. Psychometrika, 50, 229–242.
Browne, M.W. (1984). Asymptotic distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83.
Cudeck, R., & O’Dell, L.L. (1994). Applications of standard error estimates in unrestricted factor analysis: Significance tests for factor loadings and correlations. Psychological Bulletin, 115, 475–487.
Gerbing, D.W., & Anderson, J.C. (1987). Improper solutions in the analysis of covariance structures: Their interpretability and a comparison of alternate respecifications. Psychometrika, 52, 99–111.
Holzinger, K.J., & Swineford, F. (1939). University of Chicago: supplementary educational monographs: No. 48. A Study in factor analysis: The stability of a bi-factor solution.
Jennrich, R.I. (1974). Simplified formulae for SEs in maximum likelihood factor analysis. British Journal of Mathematical and Statistical Psychology, 27, 122–131.
Jöreskog, K.G. (1969). A general approach to confirmatory maximum likelihood factor analysis. Psychometrika, 34, 183–202.
Kano, Y. (1998). Improper solutions in exploratory factor analysis: Causes and treatments. In A. Rizzi, M. Vichi, & H. Bock (Eds.), Advances in data sciences and classification (pp. 375–382). Berlin: Springer.
Kano, Y., & Shapiro, A. (1987). On asymptotic variance of uniqueness estimators in factor analysis. South African Statistical Journal, 21, 131–136.
Kano, Y., Bentler, P.M., & Mooijaart, A. (1993). Additional information and precision of estimators in multivariate structural models. In K. Matusita, T. Hayakawa et al. (Eds.), Statistical sciences and data analysis: Proceedings of the 3rd Pacific area statistical conference (pp. 187–196). Zeist: VSP International.
Lawley, D.N., & Maxwell, A.E. (1971). Factor analysis as a statistical method (2nd ed.). New York: American Elsevier.
Maxwell, A.E. (1959). Statistical methods in factor analysis. Psychological Bulletin, 56, 228–235.
Ogasawara, H. (2003a). Oblique factors and components with independent clusters. Psychometrika, 68, 299–321.
Ogasawara, H. (2003b). A supplementary note on the paper “Oblique factors and components with independent clusters”. Economic Review, Otaru University of Commerce, 53, 55–66.
Satorra, A., & Bentler, P.M. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In A. von Eye, & C.C. Clogg (Eds.), Latent variables analysis: applications for developmental research (pp. 399–419). Newbury Park: Sage.
Shapiro, A., & Browne, M. (1987). Analysis of covariance structures under elliptical distributions. Journal of the American Statistical Association, 82, 1092–1097.
Spearman, C. (1904). “General intelligence” objectively determined and measured. American Journal of Psychology, 15, 201–293.
Spearman, C., & Holzinger, K.J. (1924). The sampling error in the theory of two factors. British Journal of Psychology, 15, 17–19.
Thurstone, L.L. (1947). Multiple factor analysis. Chicago: University of Chicago Press.
van Driel, O.P. (1978). On various causes of improper solutions in maximum likelihood factor analysis. Psychometrika, 43, 225–243.
Wishart, J. (1928). Sampling errors in the theory of two factors. British Journal of Psychology, 19, 180–187.
Yuan, K.-H., & Bentler, P.M. (1999). On asymptotic distributions of normal theory MLE in covariance structure analysis under some nonnormal distributions. Statistics & Probability Letters, 42, 107–113.
Yuan, K.-H., & Bentler, P.M. (2000). On equivariance and invariance of standard errors in three exploratory factor models. Psychometrika, 65, 121–133.
Yuan, K.-H., & Hayashi, K. (2006). Standard errors in covariance structure models: asymptotics versus bootstrap. British Journal of Mathematical and Statistical Psychology, 59, 397–417.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research was supported by Grants DA00017 and DA01070 from the National Institute on Drug Abuse.
Rights and permissions
About this article
Cite this article
Yuan, KH., Cheng, Y. & Zhang, W. Determinants of Standard Errors of MLEs in Confirmatory Factor Analysis. Psychometrika 75, 633–648 (2010). https://doi.org/10.1007/s11336-010-9169-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11336-010-9169-1