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Psychometrika

, Volume 50, Issue 1, pp 83–90 | Cite as

Power of the likelihood ratio test in covariance structure analysis

  • Albert Satorra
  • Willem E. Saris
Article

Abstract

A procedure for computing the power of the likelihood ratio test used in the context of covariance structure analysis is derived. The procedure uses statistics associated with the standard output of the computer programs commonly used and assumes that a specific alternative value of the parameter vector is specified. Using the noncentral Chi-square distribution, the power of the test is approximated by the asymptotic one for a sequence of local alternatives. The procedure is illustrated by an example. A Monte Carlo experiment also shows how good the approximation is for a specific case.

Key words

covariance structure analysis maximum likelihood estimation likelihood ratio test power of the test local alternatives noncentral Chi-square noncentrality parameter Monte Carlo experiment 

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References

  1. Apostol, T. M. (1957).Mathematical Analysis. New York: Addison-Wesley.Google Scholar
  2. Barnett, S. (1979).Matrix methods for engineers and scientists. London: McGraw-Hill.Google Scholar
  3. Bentler, P. M. (1980). Multivariate analysis with latent variables: Causal modelling.Annual Review of Psychology, 31, 419–456.Google Scholar
  4. Bentler, P. M. (1982).Theory and implementation of EQSs, a structural equations program (Tech. Rep.). Los Angeles: University of California.Google Scholar
  5. Bentler, P. M., & Bonett, D. G. (1980), Significance tests and goodness of fit in the analysis of covariance structures.Psychological Bulletin, 88, 588–606.Google Scholar
  6. Boomsma, A. (1983).On the robustness of LISREL (maximum likelihood estimation) against small sample size and non-normality. Amsterdam: Sociometric Research foundation.Google Scholar
  7. Browne, M. W. (1982). Covariance structures. In D. M. Hawkins (Ed.),Topics in applied multivariate analysis (pp. 72–141). Cambridge: Cambridge University Press.Google Scholar
  8. Burguete, J. P., Gallant, A. R., & Souza G. (1982). On unification of the asymptotic theory of nonlinear econometric models.Econometric Reviews, 1(2), 151–190.Google Scholar
  9. Davidson, R.R., & Lever, W. E. (1970). The limiting distribution of the likelihood ratio statistic under a class of local alternatives.Sankhyä, Ser. A32, 209–224.Google Scholar
  10. Feder, P. I. (1968). On the distribution of the log likelihood ratio test statistic when the true parameter is “near” the boundaries of the hypothesis regions.Ann. Math. Stat., 39, 2044–2055.Google Scholar
  11. Foutz, R. V., & Srivastava, R. C. (1977). The performance of the likelihood ratio test when the model is incorrect.Ann. Math. Stat., 5, 1183–1194.Google Scholar
  12. Gallant, A. R., & Holly A. (1980). Statistical inference in an implicit nonlinear, simultaneous equation model in the context of maximum likelihood estimation.Econometrica, 48, 697–720.Google Scholar
  13. Hardy, G. H. (1960).A course of Pure Mathematics (10th ed.). Cambridge: Cambridge University Press.Google Scholar
  14. Haynam, G. E., Govindarajulu, Z., & Leone, F. C. (1973). Tables of the cumulative Chi-square distribution. In H. L. Harter & D. B. Owen (Eds.),Selected Tables in Mathematical Statistics. Providence, R. I.: American Mathematical Society.Google Scholar
  15. Jöreskog, K. G. (1970). A general method for analysis of covariance structures.Biometrika, 57, 239–251.Google Scholar
  16. Jöreskog, K. G. (1981). Analysis of covariance structures.Scandinavian Journal of Statistics, 8, 65–92.Google Scholar
  17. Jöreskog, K. G., & Sörbom, D. (1981). LISREL V: Analysis of linear structural relationships by maximum likelihood and least squares methods. Chicago: International Educational Services.Google Scholar
  18. Kendall, M. G., & Stuart A. (1961).The Advanced Theory of Statistics, Vol. 2. London: Charles Griffing.Google Scholar
  19. 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.Google Scholar
  20. Lehmann, E. L. (1958). Significance level and power.Ann. Math. Stat., 29, 1167–1176.Google Scholar
  21. Saris, W. E., de Pijper, W. M., & Zegwaart, P. (1979). Detection of specification errors in linear structural equation models. In K. Schuessler (Ed.),Sociological Methodology 1979 (pp. 151–171). San Francisco: Jossey-Bass.Google Scholar
  22. Saris, W. E., & Stronkhorst, L. H. (1984).Causal Modelling in Nonexperimental Research. Amsterdam: Sociometric Research Foundation.Google Scholar
  23. Saris, W. E., den Ronden, J., & Satorra, A. (in press). Testing Structural Equation Models. In P. F. Cuttance & J. R. Ecob (Eds.),Structural Modeling. Cambridge: Cambridge University Press.Google Scholar
  24. Satorra, A., & Saris, W. E. (1983). The accuracy of a procedure for calculating the power of the likelihood ratio test as used within the LISREL framework. In C. P. Middendrop, B. Niemöller, & W. E. Saris (Eds.),Socimetric Research 1982 (pp. 127–190). Amsterdam: Sociometric Research Foundation.Google Scholar
  25. Satorra, A., & Saris, W. E., & de Pijper, W. M. (in preparation).Several approximations of the power function of the likelihood ratio test in covariance structure analysis.Google Scholar
  26. Silvey, S. D. (1959). The lagrangian multiplier test.Ann. Math. Stat., 30, 389–407.Google Scholar
  27. Stroud, T. W. F. (1972). Fixed alternatives and Wald's formulation of the noncentral asymptotic behavior of the likelihood ratio statistic.Annals of Math. Stat., 43, 447–454.Google Scholar
  28. Tucker, L. R., & Lewis, C. (1973). A reliability coefficient for maximum likelihood factor analysis.Psychometrika, 38, 1–10.Google Scholar
  29. Wald, A. (1943). Tests of statistical hypotheses concerning several parameters when the number of observations is large.Trans. Amer. Math. Soc., 54, 426–482.Google Scholar
  30. White, H. (1982). Maximum likelihood estimation of misspecified models,Econometrica, 50, 1–25.Google Scholar

Copyright information

© The Psychometric Society 1985

Authors and Affiliations

  • Albert Satorra
    • 2
  • Willem E. Saris
    • 1
  1. 1.Department of Methods and TechniquesUniversity of AmsterdamThe Netherlands
  2. 2.Department of Statistics and Econometrics, Faculty of EconomicsUniversity of BarcelonaBarcelona-08034Spain

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