Abstract
It is very important to choose appropriate variables to be analyzed in multivariate analysis when there are many observed variables such as those in a questionnaire. What is actually done in scale construction with factor analysis is nothing but variable selection.
In this paper, we take several goodness-of-fit statistics as measures of variable selection and develop backward elimination and forward selection procedures in exploratory factor analysis. Once factor analysis is done for a certain numberp of observed variables (thep-variable model is labeled the current model), simple formulas for predicted fit measures such as chi-square, GFI, CFI, IFI and RMSEA, developed in the field of the structural equation modeling, are provided for all models obtained by adding an external variable (so that the number of variables isp + 1) and for those by deleting an internal variable (so that the number isp − 1), provided that the number of factors is held constant.
A programSEFA (Stepwise variable selection in Exploratory Factor Analysis) is developed to actually obtain a list of the fit measures for all such models. The list is very useful in determining which variable should be dropped from the current model to improve the fit of the current model. It is also useful in finding a suitable variable that may be added to the current model. A model with more appropriate variables makes more stable inference in general.
The criteria traditionally often used for variable selection is magnitude of communalities. This criteria gives a different choice of variables and does not improve fit of the model in most cases.
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References
Akaike, H. (1987). Factor analysis and AIC.Psychometrika, 52, 317–332.
Anderson, T. W., & Rubin, H. (1956). Statistical inference in factor analysis.Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability (Vol. 5, pp. 111–150). Berkeley: University of California Press.
Arbuckle, J. L. (1995).AMOS 3.5. Chicago: Smallwaters.
Bentler, P. M. (1990). Comparative fit indexes in structural models.Psychological Bulletin, 107, 238–246.
Bentler, P. M. (1995).EQS structural equations program manual. Los Angeles: Multivariate Software.
Bollen, K. A. (1989a).Structural equations with latent variables. New York: Wiley.
Bollen, K. A. (1989b). A new incremental fit index in general structural equation models.Sociological Methods and Research, 17, 303–316.
Browne, M. W. (1982). Covariance structures. In D. M. Hawkins (Ed.),Topics in applied multivariate analysis (pp. 72–141). Cambridge: Cambridge University Press.
Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures.British Journal of Mathematical and Statistical Psychology, 37, 62–83.
Browne, M. W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In K. A. Bollen & J. S. Long (Eds.),Testing structural equation models (pp. 137–162). Newbury Park: SAGE Publications.
Browne, M. W., Cudeck, R., Tateneni, K., & Mels, G. (1998).CEFA: Comprehensive exploratory factor analysis. Columbus, OH: The Ohio State University, Department of Psychology.
Buse, A. (1982). The likelihood ratio, Wald, and Lagrange multiplier tests: An expository note.The American Statistician, 36, 153–157.
Clarke, M. R. B. (1970). A rapidly convergent method for maximum likelihood factor analysis.British Journal of Mathematical and Statistical Psychology, 23, 43–52.
Cudeck, R. (1989). Noniterative factor analysis estimators, with algorithms for subset and instrumental variable selection.Journal of Educational Statistics, 16, 35–52.
Davis, F. B. (1944). Fundamental factors of comprehension in reading.Psychometrika, 9, 185–197.
Gorsuch, R. L. (1988). Exploratory factor analysis. In J. R. Nesselroade & R. B. Cattell (Eds.),Handbook of multivariate experimental psychology (2nd ed., pp. 231–258.) New York and London: Plenum Press.
Harada, A., & Kano, Y. (1998).SEFA (Stepwise Exploratory Factor Analysis) program manual (Technical Report of Statistical Group DATA, DATA98-04). Osaka: Osaka University, Faculty of Human Sciences.
Harman, H. H. (1976).Modern factor analysis (3rd ed.) Chicago and London: The University of Chicago Press.
Ihara, M., & Kano, Y. (1992). Asymptotic equivalence of uniqueness estimators in marginal and conditional factor analysis models.Statistics & Probability Letters, 14, 337–341.
Jöreskog, K. G. (1967). Some contributions to maximum likelihood factor analysis.Psychometrika, 34, 183–202.
Jöreskog, K. G. (1978). Structural analysis of covariance and correlation matrices.Psychometrika, 43, 443–477.
Jöreskog, K. G., & Sörbom, D. (1981).LISREL 5: User's guide. Chicago, IL.
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., Bentler, P. M., & Mooijaart, A. (1993). Additional information and precision of estimators in multivariate analysis. In K. Matusita, M. L. Puri & T. Hayakawa (Eds.),Statistical sciences and data analysis: Proceedings of the third pacific area statistical conference (pp. 187–196). Zeist, The Netherlands: VSP International Science Publisher.
Kano, Y., & Ihara, M. (1994). Identification of inconsistent variates in factor analysis.Psychometrika, 59, 5–20.
Lawley, D. N., & Maxwell, A. E. (1971).Factor analysis as a statistical method (2nd ed.). London: Butterworths.
Magnus, J. R., & Neudecker, H. (1988).Matrix differential calculus with applications in statistics and econometrics. New York: Wiley.
Maxwell, A. E. (1961). Recent trends in factor analysis.Journal of the Royal Statistical Society, Series A, 124, 49–59.
Mulaik, S. A. (1976). Comments on “The measurement factorial indeterminacy”.Psychometrika, 41, 249–262.
Mulaik, S. A., & McDonald, R. P. (1978). The effect of additional variables on factor indeterminacy in models with a single common factor.Psychometrika, 43, 177–192.
Rao, C. R. (1973).Linear statistical inference and its application (2nd ed.) New York: Wiley.
SAS (1989). Cary, NC: SAS Institute.
Sato, M. (1987). Pragmatic treatment of improper solutions in factor analysis.Annals of Institute of Statistical Mathematics, 39, 443–455.
SPSS (1977). Chicago: SPSS.
Steiger, J. H. (1979). Factor indeterminacy in the 1930's and the 1970's: Some interesting parallels.Psychometrika, 44, 157–167.
Tanaka, Y. (1983). Some criteria for variable selection in factor analysis.Behaviormetrika, 13, 31–45.
van Driel, O. P. (1978). On various causes of improper solutions in maximum likelihood factor analysis.Psychometrika, 43, 225–243.
West, R. W., Ogden, R. T., & Rossini, J. A. (1998). Statistical tools on the World Wide Web.The American Statistician, 52, 257–262.
Yanai, H. (1980). A proposition of generalized method for forward selection of variables.Behaviormetrika, 7, 95–107.
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The URL of the programSEFA is http://koko15.hus.osaka-u.ac.jp/~harada/factor/stepwise/.
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Kano, Y., Harada, A. Stepwise variable selection in factor analysis. Psychometrika 65, 7–22 (2000). https://doi.org/10.1007/BF02294182
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DOI: https://doi.org/10.1007/BF02294182