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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Apostol, T. M. (1957).Mathematical Analysis. New York: Addison-Wesley.
Barnett, S. (1979).Matrix methods for engineers and scientists. London: McGraw-Hill.
Bentler, P. M. (1980). Multivariate analysis with latent variables: Causal modelling.Annual Review of Psychology, 31, 419–456.
Bentler, P. M. (1982).Theory and implementation of EQSs, a structural equations program (Tech. Rep.). Los Angeles: University of California.
Bentler, P. M., & Bonett, D. G. (1980), Significance tests and goodness of fit in the analysis of covariance structures.Psychological Bulletin, 88, 588–606.
Boomsma, A. (1983).On the robustness of LISREL (maximum likelihood estimation) against small sample size and non-normality. Amsterdam: Sociometric Research foundation.
Browne, M. W. (1982). Covariance structures. In D. M. Hawkins (Ed.),Topics in applied multivariate analysis (pp. 72–141). Cambridge: Cambridge University Press.
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.
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.
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.
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.
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.
Hardy, G. H. (1960).A course of Pure Mathematics (10th ed.). Cambridge: Cambridge University Press.
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.
Jöreskog, K. G. (1970). A general method for analysis of covariance structures.Biometrika, 57, 239–251.
Jöreskog, K. G. (1981). Analysis of covariance structures.Scandinavian Journal of Statistics, 8, 65–92.
Jöreskog, K. G., & Sörbom, D. (1981). LISREL V: Analysis of linear structural relationships by maximum likelihood and least squares methods. Chicago: International Educational Services.
Kendall, M. G., & Stuart A. (1961).The Advanced Theory of Statistics, Vol. 2. London: Charles Griffing.
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.
Lehmann, E. L. (1958). Significance level and power.Ann. Math. Stat., 29, 1167–1176.
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.
Saris, W. E., & Stronkhorst, L. H. (1984).Causal Modelling in Nonexperimental Research. Amsterdam: Sociometric Research Foundation.
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.
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.
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.
Silvey, S. D. (1959). The lagrangian multiplier test.Ann. Math. Stat., 30, 389–407.
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.
Tucker, L. R., & Lewis, C. (1973). A reliability coefficient for maximum likelihood factor analysis.Psychometrika, 38, 1–10.
Wald, A. (1943). Tests of statistical hypotheses concerning several parameters when the number of observations is large.Trans. Amer. Math. Soc., 54, 426–482.
White, H. (1982). Maximum likelihood estimation of misspecified models,Econometrica, 50, 1–25.
Author information
Authors and Affiliations
Additional information
This research was made possible by a grant from the Dutch Organization for Advancement of Pure Research (ZWO). The authors also like to acknowledge the helpful comments and suggestions from the editor and anonymous reviewers.
Rights and permissions
About this article
Cite this article
Satorra, A., Saris, W.E. Power of the likelihood ratio test in covariance structure analysis. Psychometrika 50, 83–90 (1985). https://doi.org/10.1007/BF02294150
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02294150