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Identification of Inconsistent Variates in Factor Analysis

Published online by Cambridge University Press:  01 January 2025

Yutaka Kano  and
Masamori Ihara
Yutaka Kano*
Affiliation:
University of Osaka Prefecture
Masamori Ihara
Affiliation:
Osaka Electro-Communication University
*
Requests for reprints should be sent to Yutaka Kano, Department of Mathematical Sciences, College of Engineering, University of Osaka Prefecture, Sakai 593, Osaka, JAPAN
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Abstract

When some of observed variates do not conform to the model under consideration, they will have a serious effect on the results of statistical analysis. In factor analysis the model with inconsistent variates may result in improper solutions. In this article a useful method for identifying a variate as inconsistent is proposed in factor analysis. The procedure is based on the likelihood principle. Several statistical properties such as the effect of misspecified hypotheses, the problem of multiple comparisons, and robustness to violation of distributional assumptions are investigated. The procedure is illustrated by some examples.

Keywords

likelihood ratio testmultiple comparisonrobustnessvariable selection
Type
Original Paper
Information
Psychometrika , Volume 59 , Issue 1 , March 1994 , pp. 5 - 20
Copyright
Copyright © 1994 The Psychometric Society

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References

Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, AC-19, 716–723.CrossRefGoogle Scholar
Akaike, H. (1987). Factor analysis and AIC. Psychometrika, 52, 317–332.CrossRefGoogle Scholar
Amemiya, Y., Anderson, T. W. (1990). Asymptotic chi-square tests for a large class of factor analysis models (Tech. Rep. No. 13, Stanford Univ., 1985). The Annals of Statistics, 18, 1453–1463.CrossRefGoogle Scholar
Anderson, T. W. (1989). Linear latent variable models and covariance structures. Journal of Econometrics, 41, 91–119.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. (1989). EQS structural equations program manual, Los Angeles: BMDP Statistical Software.Google Scholar
Boomsma, A. (1985). Nonconvergence, improper solutions, and starting values in LISREL maximum likelihood estimation. Psychometrika, 50, 229–242.CrossRefGoogle 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). Cambridge: 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
Browne, M., Shapiro, A. (1988). Robustness of normal theory methods in the analysis of linear latent variate models. British Journal of Mathematical and Statistical Psychology, 41, 193–208.CrossRefGoogle Scholar
Davis, F. B. (1944). Fundamental factors of comprehension in reading. Psychometrika, 9, 185–197.CrossRefGoogle Scholar
Fujikoshi, Y. (1985). Selection of variables in two-group discriminant analysis by error rate and Akaike's information criteria. Journal of Multivariate Analysis, 17, 27–37.CrossRefGoogle Scholar
Hochberg, Y., Tamhane, A. C. (1987). Multiple comparison procedures, New York: Wiley.CrossRefGoogle Scholar
Ihara, M., & Kano, Y. (1991, July). Variable selection in factor analysis. Paper presented at the 7th European Meeting of the Psychometric Society. Trier, Germany.Google Scholar
Ihara, M., Okamoto, M. (1985). Experimental comparison of least-squares and maximum likelihood methods in factor analysis. Statistics & Probability Letters, 6, 287–293.CrossRefGoogle Scholar
Jolliffe, I. T. (1986). Principal component analysis, New York: Springer-Verlag.CrossRefGoogle Scholar
Jöreskog, K. G. (1967). Some contributions to maximum likelihood factor analysis. Psychometrika, 32, 443–482.CrossRefGoogle Scholar
Jöreskog, K. G., Sörbom, M. (1988). LISREL 7: A guide to the program and applications, Chicago: SPSS.Google Scholar
Kano, Y. (1990). Noniterative estimation and the choice of the number of factors in exploratory factor analysis. Psychometrika, 55, 277–291.CrossRefGoogle Scholar
Kano, Y. (1993). Asymptotic properties of statistical inference based on Fisher consistent estimators in the analysis of covariance structures. In Haagen, K., Bartholomew, D. J., Deistler, M. (Eds.), Proceedings of the international workshop on statistical modelling and latent variables, Amsterdam: Elsevier Science Publisher.Google Scholar
Lawley, D. N., Maxwell, A. E. (1971). Factor analysis as a statistical method 2nd ed.,, London: Butterworths.Google Scholar
Lee, S. Y. (1980). Estimation of covariance structure models with parameters subject to functional restraints. Psychometrika, 45, 309–324.CrossRefGoogle Scholar
Lee, S. Y., Bentler, P. M. (1980). Some asymptotic properties of constrained generalized least squares estimation in covariance structure models. South African Statistical Journal, 14, 121–136.Google Scholar
Lee, S. Y., Jennrich, R. I. (1979). A study of algorithms for covariance structure analysis with specific comparisons using factor analysis. Psychometrika, 44, 99–113.CrossRefGoogle Scholar
Magnus, J. R., Neudecker, H. (1988). Matrix differential calculus with applications in statistics and econometrics, New York: Wiley.Google Scholar
Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 57, 519–530.CrossRefGoogle Scholar
Mardia, K. V. (1974). Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies. Sankhyā, Series B, 36, 115–128.Google Scholar
Martin, J. K., McDonald, R. P. (1975). Bayesian estimation in unrestricted factor analysis: A treatment for Heywood cases. Psychometrika, 40, 505–517.CrossRefGoogle Scholar
Maxwell, A. E. (1961). Recent trends in factor analysis. Journal of the Royal Statistical Society, Series A, 124, 49–59.CrossRefGoogle Scholar
Mooijaart, A., Bentler, P. M. (1991). Robustness of normal theory statistics in structural equation models. Statistica Neerlandica, 45, 159–171.CrossRefGoogle Scholar
Muirhead, R. J., Waternaux, C. M. (1980). Asymptotic distributions in canonical analysis and other multivariate procedures for nonnormal populations. Biometrika, 67, 31–43.CrossRefGoogle Scholar
Okamoto, M., Ihara, M. (1984). Partial Gauss-Newton algorithm for least-squares and maximum likelihood methods in factor analysis. Journal of the Japan Statistical Society, 14, 137–144.Google Scholar
Rao, C. R. (1955). Estimation and tests of significance in factor analysis. Psychometrika, 20, 93–111.CrossRefGoogle Scholar
Rao, C. R. (1973). Linear statistical inference and its application 2nd ed.,, New York: Wiley.CrossRefGoogle Scholar
Rindskopf, D. M. (1983). Parameterizing inequality constraints on unique variances in linear structural equation models. Psychometrika, 48, 73–83.CrossRefGoogle Scholar
Ryan, T. A. (1959). Multiple comparisons in psychological research. Psychological Bulletin, 56, 26–47.CrossRefGoogle ScholarPubMed
Ryan, T. A. (1960). Significance tests for multiple comparison of proportions, variances, and other statistics. Psychological Bulletin, 57, 318–328.CrossRefGoogle ScholarPubMed
SAS Institute (1988). SAS/STAT: User's guide 4th ed.,, Cary, NC: Author.Google Scholar
Sato, M. (1987). Pragmatic treatment of improper solutions in factor analysis. Annals of the Institute of Statistical Mathematics, Part B, 39, 443–455.CrossRefGoogle Scholar
Satorra, A. (1989). Alternative test criteria in covariance structure analysis. Psychometrika, 54, 131–151.CrossRefGoogle Scholar
Shapiro, A., Browne, M. (1987). Analysis of covariance structures under elliptical distributions. Journal of the American Statistical Association, 82, 1092–1097.CrossRefGoogle Scholar
Tanaka, Y. (1983). Some criteria for variable selection in factor analysis. Behaviormetrika, 13, 31–45.CrossRefGoogle Scholar
van Driel, O. P. (1978). On various causes of improper solutions in maximum likelihood factor analysis. Psychometrika, 43, 225–243.CrossRefGoogle Scholar
Yanai, H. (1980). A proposition of generalized method for forward selection of variables. Behaviormetrika, 7, 95–107.CrossRefGoogle Scholar