Abstract
The normal theory maximum likelihood and asymptotically distribution free methods are commonly used in covariance structure practice. When the number of observed variables is too large, neither method may give reliable inference due to bad condition numbers or unstable solutions. The main existing solution to the problem of high dimension is to build a model based on marginal variables. This practice is inefficient because the omitted variables may still contain valuable information regarding the structural model. In this paper, we propose a simple method of averaging proper variables which have similar factor structures in a confirmatory factor model. The effects of averaging variables on estimators and tests are investigated. Conditions on the relative errors of the measured variables are given that verify when a model based on averaged variables can give better estimators and tests than one based on omitted variables. Our method is compared to the method of variable selection based on mean square error of predicted factor scores. Some aspects related to averaging, such as improving the normality of observed variables, are also discussed.
Article PDF
Similar content being viewed by others
Change history
13 February 2021
A Correction to this paper has been published: https://doi.org/10.1007/s41237-020-00126-4
References
Anderson, J.C. and Gerbing, D.W. (1982). Some methods for respecifying measurement models to obtain unidimensional construct measurement. Journal of Marketing Research, 19, 453–460.
Anderson, J.C. and Gerbing, D.W. (1988). Structural equation modeling in practice: A review and recommended two-step approach. Psychological Bulletin, 103, 411–423.
Bentler, P.M. and Wu, E.J.C. (1995). EQS for Windows (Macintosh) User’s Guide. Encino, CA: Multivariate Software.
Hattie, J. (1985). Methodology review: Assessing unidimensionality of tests and items. Applied Psychological Measurement, 9, 139–164.
Hunter, J.E. and Gerbing, D.W. (1982). Unidimensional measurement, second-order factor analysis, and causal models. In B.M. Staw and L.L. Cummings (eds.), Research in organizational behavior (Vol. 4, pp. 267–299). Greenwich, CT: JAI Press.
Jolliffe, I.T. (1972). Discarding variables in a principal component analysis I. Artificial data. Applied Statistics, 21, 160–173.
Jolliffe, I.T. (1973). Discarding variables in a principal component analysis II. Real data. Applied Statistics, 22, 21–31.
Jöreskog, K.G. (1971). Statistical analysis of sets of congeneric tests. Psychometrika, 36, 109–133.
Jöreskog, K.G. and Sörbom, D. (1993). New Features in LISREL 8. Chicago: Scientific Software.
Jöreskog, K.G. and Yang, F. (1996). Nonlinear structural equation models: The Kenny-Judd model with interaction effects. In G.A. Marcoulides and R.E. Schumacker (Eds.), Advanced Structural Equation Modeling: Issues and Techniques (pp. 57–88). Mahwah, NJ: Lawrence Erlbaum Associates.
Kano, Y. (1986). A condition for the regression predictor to be consistent in a single common factor model. British Journal of Mathematical and Statistical Psychology, 39, 221–227.
Kenny, D.A. and Judd, C.M. (1984). Estimating the nonlinear and interactive effects of latent variables. Psychological Bulletin, 96, 201–210.
Khatri, C.G. (1966). A note on a MANOVA model applied to problems in growth curves. Annals of the Institute of Statistical Mathematics, 18, 75–86.
Magnus, J.R. and Neudecker, H. (1988). Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley, New York.
Tanaka, Y. (1983). Some criteria for variable selection in factor analysis. Behaviormetrika, 13, 31–45.
Tanaka, Y. and Kodake, K. (1981). A method of variable selection in factor analysis and its numerical investigation. Behaviormetrika, 10, 49–61
Yanai, H. (1980). A proposition of generalized method for forward selection of variables. Behaviormetrika, 7, 31–45.
Author information
Authors and Affiliations
Additional information
The original online version of this article was revised due to the retrospective open access order.
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Yuan, KH., Bentler, P.M. & Kano, Y. On Averaging Variables in a Confirmatory Factor Analysis Model. Behaviormetrika 24, 71–83 (1997). https://doi.org/10.2333/bhmk.24.71
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.2333/bhmk.24.71