Abstract
Robust schemes in regression are adapted to mean and covariance structure analysis, providing an iteratively reweighted least squares approach to robust structural equation modeling. Each case is properly weighted according to its distance, based on first and second order moments, from the structural model. A simple weighting function is adopted because of its flexibility with changing dimensions. The weight matrix is obtained from an adaptive way of using residuals. Test statistic and standard error estimators are given, based on iteratively reweighted least squares. The method reduces to a standard distribution-free methodology if all cases are equally weighted. Examples demonstrate the value of the robust procedure.
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References
Ammann, L. P. (1989). Robust principal components.Communications in Statistics: Simulation and Computation, 18, 857–874.
Bentler, P. M. (1995).EQS structural equations program manual. Encino, CA: Multivariate Software.
Bentler, P. M., & Dudgeon, P. (1996). Covariance structure analysis: Statistical practice, theory, directions.Annual Review of Psychology, 47, 563–592.
Berkane, M., & Bentler, P. M. (1988). Estimation of contamination parameters and identification of outliers in multivariate data.Sociological Methods & Research, 17, 55–64.
Birch, J. B., & Myers, R. H. (1982). Robust analysis of covariance.Biometrics, 38, 699–713.
Bollen, K. A. (1987). Outliers and improper solutions: A confirmatory factor analysis example.Sociological Methods & Research, 15, 375–384.
Bollen, K. A. (1989).Structural equations with latent variables. New York: Wiley.
Bollen, K. A., & Arminger, G. (1991). Observational residuals in factor analysis and structural equation models.Sociological methodology, 21, 235–262.
Breckler, S. J. (1990). Application of covariance structure modeling in psychology: Cause for concern?Psychological Bulletin, 107, 260–273.
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). Asymptotic distribution-free methods for the analysis of covariance structures.British Journal of Mathematical and Statistical Psychology, 37, 62–83.
Cadigan, N. G. (1995). Local influence in structural equation models.Structural Equation Modeling, 2, 13–30.
Campbell, N. A. (1980). Robust procedures in multivariate analysis I: Robust covariance estimation.Applied Statistics, 29, 231–237.
Campbell, N. A. (1982). Robust procedures in multivariate analysis II: Robust canonical variate analysis.Applied Statistics, 31, 1–8.
Carroll, R. J. (1979). On estimating variances of robust estimators when the errors are asymmetric.Journal of the American Statistical Association, 74, 674–679.
Curran, P. S. (1994).The robustness of confirmatory factor analysis to model misspecification and violations of normality. Unpublished doctoral dissertation, Arizona State University.
Curran, P. S., West, S. G., & Finch, J. F. (1996). The robustness of test statistics to nonnormality and specification error in confirmatory factor analysis.Psychological Methods, 1, 16–29.
Devlin, S. J., Gnanadesikan, R., & Kettenring, J. R. (1981). Robust estimation of dispersion matrices and principal components.Journal of the American Statistical Association, 76, 354–362.
Gabriel, K. R., & Odoroff, L. (1984). Resistant lower rank approximation of matrices. In E. Diday, M. Jambu, L. Lebart, J. Pages, & R. Tomassone (Eds.),Data analysis and informatics III (pp. 23–30). Amsterdam: North-Holland.
Green, P. J. (1984). Iteratively reweighted least squares for maximum likelihood estimation, and some robust and resistent alternatives (with discussion).Journal of the Royal Statistical Society, Series B, 46, 149–192.
Gross, A. M. (1977). Confidence intervals for bisquare regression estimates.Journal of the American Statistical Association, 72, 341–354.
Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., & Stahel, W. A. (1986).Robust statistics: The approach based on influence functions. New York: Wiley.
Heiser, W. J. (1987). Correspondence analysis with least absolute residuals.Computational Statistics & Data Analysis, 5, 337–356.
Hoaglin, D. C., Mosteller, F., & Tukey, J. W. (1983).Understanding robust and exploratory data analysis. New York: Wiley.
Holland, P. W., & Welsch, R. E. (1977). Robust regression using iteratively reweighted least-squares.Communications in Statistics-Theory and Methods, Series A 6, 813–827.
Holzinger, K. J., & Swineford, F. (1939).A study in factor analysis: The stability of a bi-factor solution (Supplementary Educational Monographs No. 48). Chicago: University of Chicago.
Hu, L., Bentler, P. M., & Kano, Y. (1992). Can test statistics in covariance structure analysis be trusted?Psychological Bulletin, 112, 351–362.
Huba, G. J., & Harlow, L. L. (1987). Robust structural equation models: Implications for developmental psychology.Child Development, 58, 147–166.
Huber, P. J. (1964). Robust estimation of a location parameter.Annals of Mathematical Statistics, 35, 73–101.
Huber, P. J. (1973). Robust regression: Asymptotics, conjectures and Monte Carlo.Annals of Statistics, 1, 799–821.
Huber, P. J. (1981).Robust statistics. New York: Wiley.
Jöreskog, K. G. (1969). A general approach to confirmatory maximum likelihood factor analysis.Psychometrika, 34, 183–202.
Jöreskog, K. G. (1977). Structural equation models in the social sciences: Specification, estimation and testing. In P. R. Krishnaiah (Ed.),Applications of statistics (pp. 265–287). Amsterdam: North Holland.
Jöreskog, K. G., & Sörbom, D. (1993).LISREL 8 user's reference guide. Chicago: Scientific Software International.
Kharin, Y. S. (1996). Robustness in discriminant analysis. In H. Rieder (Ed.),Robust statistics, data analysis, and computer intensive methods (pp. 225–234). New York: Springer.
Lange, K. L., Little, R. J. A., & Taylor, J. M. G. (1989). Robust statistical modeling using thet distribution.Journal of the American Statistical Association, 84, 881–896.
Lee, S.-Y., & Jennrich, R. I. (1979). A study of algorithms for covariance structure analysis with specific comparisons using factor analysis.Psychometrika, 44, 99–114.
Lee, S.-Y., & Wang, S. J. (1996). Sensitivity analysis of structural equation models.Psychometrika, 61, 93–108.
Mardia, K. V., Kent, J. T., & Bibby, J. M. (1979).Multivariate analysis. New York: Academic Press.
Maronna, R. A. (1976). Robust M-estimators of multivariate location and scatter.Annals of Statistics, 4, 51–67.
Newcomb, M. D., & Bentler, P. M. (1988).Consequences of adolescent drug use: Impact on the lives of young adults. Beverly Hills: Sage Publications.
Rousseeuw, P. J., & Leroy, A. M. (1987).Robust regression and outlier detection. New York: Wiley.
Rousseeuw, P. J., & Van Zomeren, B. C. (1990). Unmasking multivariate outliers and leverage points.Journal of the American Statistical Association, 85, 633–639.
Rubin, D. B. (1983). Iteratively reweighted least squares. In N. L. Johnson & S. Kotz (Eds.)Encyclopedia of statistical sciences, Volume 4 (pp. 272–275). New York: Wiley.
Stein, J. A., Newcomb, M. D., & Bentler, P. M. (1996). Initiation and maintenance of tobacco smoking: Changing personality correlates in adolescence and young adulthood.Journal of Applied Social Psychology, 26, 160–187.
Tanaka, Y., Watadani, S., & Moon, S. H. (1991). Influence in covariance structure analysis: with an application to confirmatory factor analysis.Communication in Statistics-Theory and Method, 20, 3805–3821.
Tyler, D. E. (1983). Robustness and efficiency properties of scatter matrices.Biometrika, 70, 411–420.
Verboon, P., & Heiser, W. J. (1994). Resistant lower rank approximation of matrices by iterative majorization.Computational Statistics & Data Analysis, 18, 457–467.
Wilcox, R. R. (1997).Introduction to robust estimation and hypothesis testing. San Diego: Academic Press.
Yuan, K.-H., & Bentler, P. M. (1997a). Mean and covariance structure analysis: Theoretical and practical improvements.Journal of the American Statistical Association, 92, 767–774.
Yuan, K.-H., & Bentler, P. M. (1997b). Improving parameter tests in covariance structure analysis.Computational Statistics & Data Analysis, 26, 177–198.
Yuan, K.-H., & Bentler, P. M. (1998a). Robust mean and covariance structure analysis.British Journal of Mathematical and Statistical Psychology, 51, 63–88.
Yuan, K.-H., & Bentler, P. M. (1998b). Structural equation modeling with robust covariances.Sociological Methodology, 28, 363–396.
Yuan, K.-H., & Bentler, P. M. (1998c). Normal theory based test statistics in structural equation modeling.British Journal of Mathematical and Statistical Psychology, 51, 289–309.
Yung, Y. F. (1997). Finite mixtures in confirmatory factor-analysis models.Psychometrika, 62, 297–330.
Yung, F.-Y., & Bentler, P. M. (1996). Bootstrapping techniques in analysis of mean and covariance structures. In G. A. Marcoulides & R. E. Schumacker (Eds.)Advanced structural equation modeling techniques (pp. 195–226). New Jersey: Erlbaum.
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The authors acknowledge the constructive comments of three referees and the Editor that lead to an improved version of the paper. This work was supported by National Institute on Drug Abuse Grants DA01070 and DA00017 and by the University of North Texas Faculty Research Grant Program.
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Yuan, KH., Bentler, P.M. Robust mean and covariance structure analysis through iteratively reweighted least squares. Psychometrika 65, 43–58 (2000). https://doi.org/10.1007/BF02294185
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DOI: https://doi.org/10.1007/BF02294185