On the Quality of Test Statistics in Covariance Structure Analysis: Caveat Emptor

  • Peter M. Bentler
Part of the Perspectives on Individual Differences book series (PIDF)


Structural equation modeling, and its important special cases of covariance structure analysis and confirmatory factor analysis, has become an important tool for testing theories with nonexperimental data (see Bentler, 1986; Bollen, 1989a; Loehlin, 1987). Some of the useful properties of structural modeling methods in theory testing are: the requirement for explicitness of a theory, so that it can be represented in path diagram form; the emphasis on a distinction between the variables of true interest, typically constructs or latent variables, and particular operationalizations of these; the capability to distinguish between direct, indirect, and total effects of certain variables on others; the ability to provide a statistical evaluation of the adequacy of the theory as a whole; the availability of tests to compare competing, nested models for their relative adequacy; the possibility of isolating potential problems with a theory via tests on missing parameters; and so on. Clearly, structural modeling has provided a useful methodology for theory testing in situations where more traditional or alternative methods may not work.


Normal Theory Elliptical Theory Covariance Structure Analysis Covariance Structure Model Maximum Likelihood Factor Analysis 
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  1. Amemiya, Y. (1985). On the goodness-of-fit tests for linear structural relationships. Technical Report No. 10. Stanford University.Google Scholar
  2. Amemiya, Y. & Anderson, T. W. (1990). Asymptotic chi-square tests for a large class of factor analysis models. The Annals of Statistics, 18, 1453–1463.CrossRefGoogle Scholar
  3. Anderson, T. W. & Amemiya, Y. (1988). The asymptotic normal distribution of estimators in factor analysis under general conditions. The Annals of Statistics, 16, 759–771.CrossRefGoogle Scholar
  4. Anderson, T. W., & Rubin, H. (1956). Statistical inference in factor analysis. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 5, 111–150.Google Scholar
  5. Arminger, G., & Küsters, U. (1989). Construction principles for latent trait models. In C. C. Clogg (Ed.), Sociological methodology 1989 (pp. 369–393 ). Oxford: Basil Blackwell.Google Scholar
  6. Bentler, P. M. (1983). Some contributions to efficient statistics for structural models: Specification and estimation of moment structures. Psychometrika, 48, 493–517.CrossRefGoogle Scholar
  7. Bentler, P. M. (1986). Structural modeling and Psychometrika: An historical perspective on growth and achievements. Psychometrika, 51, 35–51.CrossRefGoogle Scholar
  8. Bentler, P. M. (1989). EQS structural equations program manual. Los Angeles: BMDP Statistical Software.Google Scholar
  9. Bentler, P. M. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107, 283–246.CrossRefGoogle Scholar
  10. Bentler, P. M., & Berkane, M. (1986). The greatest lower bound to the elliptical theory kurtosis parameter. Biometrika, 73, 240–241.CrossRefGoogle Scholar
  11. Bentler, P. M., Berkane, M., & Kano, Y. (1991). Covariance structure analysis under a simple kurtosis model. In E. M. Keramidas (Ed.), Computing science and statistics (pp. 463–465 ). Fairfax Station, VA: Interface Foundation of North America.Google Scholar
  12. Bentler, P. M., & Bonett, D. G. (1980). Significance tests and goodness of fit in the analysis of covariance structures. Psychological Bulletin, 88, 588–606.CrossRefGoogle Scholar
  13. Bentler, P. M., & Dijkstra, T. (1985). Efficient estimation via linearization in structural models. In P. R. Krishnaiah (Ed.), Multivariate analysis VI (pp. 9–42 ). Amsterdam: North-Holland.Google Scholar
  14. Bentler, P. M., Lee, S.-Y., & Weng, J. (1987). Multiple population covariance structure analysis under arbitrary distribution theory. Communications in Statistics-Theory, 16, 1951–1964.CrossRefGoogle Scholar
  15. Bollen, K. A. (1989a). Structural equations with latent variables. New York: Wiley.Google Scholar
  16. Bollen, K. A. (1989b). A new incremental fit index for general structural equation models. Sociological Methods and Research, 17, 303–316.CrossRefGoogle Scholar
  17. Boomsma, A. (1983). On the robustness of LISREL (maximum likelihood estimation) against small sample size and nonnormality. Ph.D. Thesis, University of Groningen.Google Scholar
  18. Browne, M. W. (1974). Generalized least squares estimators in the analysis of covariance structures. South African Statistical Journal, 8, 1–24.Google Scholar
  19. Browne, M. W. (1982). Covariance structures. In D. M. Hawkins (Ed.), Topic in Applied Multivariate Analysis (pp. 72–141 ). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  20. Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83.PubMedCrossRefGoogle Scholar
  21. Browne, M. W. (1985). Robustness of normal theory tests of fit of factor analysis and related models against nonnormally distributed common factors. Paper presented at the Fourth European Meeting of the Psychometric Society and the Classification Societies, Cambridge.Google Scholar
  22. Browne, M. W. (1987). Robustness of statistical inference in factor analysis and related models. Biometrika, 74, 375–384.CrossRefGoogle Scholar
  23. Browne, M. W. & 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
  24. Chamberlain, G. (1982). Multivariate regression models for panel data. Journal of Econometrics, 18, 5–46.CrossRefGoogle Scholar
  25. Chou, C.-P., Bentler, P. M., & Satorra, A. (1991). Scaled test statistics and robust standard errors for non-normal data in covariance structure analysis: A Monte Carlo study.Google Scholar
  26. British Journal of Mathematical and Statistical Psychology, 44,347–357.Google Scholar
  27. Collins, L. M., Cliff, N., McCormick, D. J., & Zatkin, J. L. (1986). Factor recovery in binary data sets: A simulation. Multivariate Behavioral Research, 21, 377–391.CrossRefGoogle Scholar
  28. Cuttance, P. (1987). Issues and problems in the application of structural equation models. In P. Cuttance and R. Ecob (Eds.), Structural modeling by example (pp. 241–279 ). Cambridge: Cambridge University Press, 1987.Google Scholar
  29. Harlow, L. L. (1985). Behavior of some elliptical theory estimators with non-normal data in a covariance structure framework: A Monte Carlo Study. Unpublished Ph. D. dissertation, University of California, Los Angeles.Google Scholar
  30. Harlow, L. L., Chou, C. P., & Bentler, P. M. (1986). Performance of chi-square statistic with ML, ADF, and elliptical estimators for covariance structures. Paper presented at the Annual Meeting, Psychometric Society, Toronto, Canada.Google Scholar
  31. Hu, L., Bentler, P. M., & Kano, Y. (1992). On test statistics in covariance structure analysis be trusted? Psychological Bulletin, 112, 351–362.PubMedCrossRefGoogle Scholar
  32. Jöreskog, K.G. (1967). Some contributions to maximum likelihood factor analysis. Psychometrika, 32, 443–482.CrossRefGoogle Scholar
  33. Jöreskog, K. G. (1969). A general approach to confirmatory maximum likelihood factor analysis. Psychometrika, 34, 183–202.CrossRefGoogle Scholar
  34. Jöreskog, K. G., & Sörbom, D. (1988). LISREL 7, A guide to the program and applications. Chicago: SPSS.Google Scholar
  35. Kano, Y. (1990). A simple adjustment of the normal theory inference for a wide class of distribution in linear latent variate models. Technical Report, University of Osaka Prefecture.Google Scholar
  36. Kano, Y., Berkane, M., & Bentler, P. M. (1990). Covariance structure analysis with heterogeneous kurtosis parameters. Biometrika, 77, 575–585.CrossRefGoogle Scholar
  37. Lawley, D. N. (1940). The estimation of factor loadings by the method of maximum likelihood. Proceedings of the Royal Society of Edinburgh, 60, 64–82.Google Scholar
  38. Lawley, D. N., & Maxwell, A. E. (1971). Factor analysis as a statistical method. London: ButterworthsGoogle Scholar
  39. 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
  40. Lee, S.-Y., Poon, W.-Y., & Bentler, P. M. (1990). A three-stage estimation procedure for structural equation models with polytomous variables. Psychometrika, 55, 45–51.CrossRefGoogle Scholar
  41. Lee, S.-Y., Poon, W.-Y., & Bentler, P. M. (1991). Some theoretical and empirical problems with LISCOMP. Technical Report, The Chinese University of Hong Kong.Google Scholar
  42. Lee, S.-Y., Poon, W.-Y., & Bentler, P. M. (1992). Structural equation models with continuous and polytomous variables. Psychometrika, 57, 89–105.CrossRefGoogle Scholar
  43. Linn, R. L. (1968). A Monte Carlo approach to the number of factors problem. Psychometrika, 33, 37–72.PubMedCrossRefGoogle Scholar
  44. Loehlin, J. C. (1987). Latent variable models: An introduction to factor, path, and structural analysis. Hillsdale, NJ: Erlbaum.Google Scholar
  45. Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin, 105, 156–166.CrossRefGoogle Scholar
  46. Mooijaart, A., & Bentler, P. M. (1991). Robustness of normal theory statistics in structural equation models. Statistica Neerlandica, 45, 159–171.CrossRefGoogle Scholar
  47. Mosier, C. I. (1939). Determining a simple structure when loadings for certain tests are known. Psychometrika, 4, 149–162.Google Scholar
  48. Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49, 115–132.CrossRefGoogle Scholar
  49. Muthén, B. (1987). LISCOMP: Analysis of linear statistical equations using a comprehensive measurement model. Mooresville, IN: Scientific Software.Google Scholar
  50. Muthén, B. (1989). Tobit factor analysis. British Journal of Mathematical and Statistical Psychology, 42, 241–250.CrossRefGoogle Scholar
  51. Muthén, B. & Kaplan, D. (1985). A comparison of some methodologies for the factor analysis of non-normal Likert variables. British Journal of Mathematical and Statistical Psychology, 38, 171–189.CrossRefGoogle Scholar
  52. Muthén, B. & Kaplan, D. (1992). A comparison of some methodologies for the factor analysis of nonnormal Likert variables: A note on the size of the model. British Journal of Mathematical and Statistical Psychology, 45, 19–30.CrossRefGoogle Scholar
  53. Satorra, A. (1989). Alternative test criteria in covariance structure analysis: A unified approach. Psychometrika, 54, 131–151.CrossRefGoogle Scholar
  54. Satorra, A., & Bentler, P. M. (1988a). Scaling corrections for chi-square statistics in covariance structure analysis. Proceedings of the American Statistical Association, 308–313.Google Scholar
  55. Satorra, A., & Bentler, P. M. (1988b). Scaling corrections for statistics in covariance structure analysis. Los Angeles: UCLA Statistics Series #2.Google Scholar
  56. Satorra, A., & Bentler, P. M. (1990). Model conditions for asymptotic robustness in the analysis of linear relations. Computational Statistics and Data Analysis, 10, 235–249.CrossRefGoogle Scholar
  57. Satorra, A., & Bentler, P. M. (1991). Goodness-of-fit test under IV estimation: Asymptotic robustness of a NT test statistic. In R. Gutiérrez and M. J. Valderrama (Eds.), Applied Stochastic Models and Data Analysis (pp. 555–567 ). Singapore: World Scientific Publishing Co.Google Scholar
  58. Shapiro, A. (1987). Robustness properties of the MDF analysis of moment structures. South African Statistical Journal, 21, 39–62.Google Scholar
  59. Shapiro, A. & Browne, M. (1987). Analysis of covariance structures under elliptical distributions. Journal of the American Statistical Association, 82, 1092–1097.CrossRefGoogle Scholar
  60. Tanaka, J. S. (1984). Some results on the estimation of covariance structure models. Ph.D. Thesis, University of California, Los Angeles.Google Scholar
  61. Tanaka, J. S. (1987). “How big is big enough?”: Sample size and goodness of fit in structural equation models with latent variables. Child Development, 58, 134–146.Google Scholar
  62. Tanaka, J. S., & Huba, G. J. (1985). A fit index for covariance structure models under arbitrary GLS estimation. British Journal of Mathematical and Statistical Psychology, 38, 197–201.CrossRefGoogle Scholar
  63. Flicker, L. R., & Lewis, C. (1973). A reliability coefficient for maximum likelihood factor analysis. Psychometrika, 38, 1–10.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Peter M. Bentler
    • 1
  1. 1.Department of PsychologyUniversity of California, Los AngelesLos AngelesUSA

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