Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-07T16:29:57.583Z Has data issue: false hasContentIssue false
  • English
  • Français

Power of the Likelihood Ratio Test in Covariance Structure Analysis

Published online by Cambridge University Press:  01 January 2025

Albert Satorra  and
Willem E. Saris
Albert Satorra*
Affiliation:
Department of Statistics and Econometrics, University of Barcelona
Willem E. Saris
Affiliation:
Department of Methods and Techniques, University of Amsterdam
*
Requests for reprints should be sent to Albert Satorra, Department of Statistics and Econometrics, Faculty of Economics, University of Barcelona, Avinguda Diagonal s/n, Barcelona-08034, Spain.
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

covariance structure analysismaximum likelihood estimationlikelihood ratio testpower of the testlocal alternativesnoncentral Chi-squarenoncentrality parameterMonte Carlo experiment
Type
Original Paper
Information
Psychometrika , Volume 50 , Issue 1 , March 1985 , pp. 83 - 90
Copyright
Copyright © 1985 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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.

References

Apostol, T. M. (1957). Mathematical Analysis, New York: Addison-Wesley.Google Scholar
Barnett, S. (1979). Matrix methods for engineers and scientists, London: McGraw-Hill.Google Scholar
Bentler, P. M. (1980). Multivariate analysis with latent variables: Causal modelling. Annual Review of Psychology, 31, 419–456.CrossRefGoogle Scholar
Bentler, P. M. (1982). Theory and implementation of EQSs, a structural equations program, Los Angeles: University of California.Google Scholar
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
Boomsma, A. (1983). On the robustness of LISREL (maximum likelihood estimation) against small sample size and non-normality, Amsterdam: Sociometric Research foundation.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
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.CrossRefGoogle Scholar
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
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.CrossRefGoogle Scholar
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
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.CrossRefGoogle Scholar
Hardy, G. H. (1960). A course of Pure Mathematics 10th ed.,, Cambridge: Cambridge University Press.Google Scholar
Haynam, G. E., Govindarajulu, Z., & Leone, F. C. (1973). Tables of the cumulative Chi-square distribution. In Harter, H. L., Owen, D. B. (Eds.), Selected Tables in Mathematical Statistics, Providence, R. I.: American Mathematical Society.Google Scholar
Jöreskog, K. G. (1970). A general method for analysis of covariance structures. Biometrika, 57, 239–251.CrossRefGoogle Scholar
Jöreskog, K. G. (1981). Analysis of covariance structures. Scandinavian Journal of Statistics, 8, 65–92.Google Scholar
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
Kendall, M. G., & Stuart, A. (1961). The Advanced Theory of Statistics, Vol. 2, London: Charles Griffing.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
Lehmann, E. L. (1958). Significance level and power. Ann. Math. Stat., 29, 1167–1176.CrossRefGoogle Scholar
Saris, W. E., de Pijper, W. M., & Zegwaart, P. (1979). Detection of specification errors in linear structural equation models. In Schuessler, K. (Eds.), Sociological Methodology 1979 (pp. 151–171). San Francisco: Jossey-Bass.Google Scholar
Saris, W. E., & Stronkhorst, L. H. (1984). Causal Modelling in Nonexperimental Research, Amsterdam: Sociometric Research Foundation.Google Scholar
Saris, W. E., den Ronden, J., & Satorra, A. (in press). Testing Structural Equation Models. In Cuttance, P. F. & Ecob, J. R. (Eds.), Structural Modeling. Cambridge: Cambridge University Press.Google Scholar
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 Middendrop, C. P., Niemöller, B. & Saris, W. E. (Eds.), Socimetric Research 1982 (pp. 127–190). Amsterdam: Sociometric Research Foundation.Google Scholar
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
Silvey, S. D. (1959). The lagrangian multiplier test. Ann. Math. Stat., 30, 389–407.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Tucker, L. R. & Lewis, C. (1973). A reliability coefficient for maximum likelihood factor analysis. Psychometrika, 38, 1–10.CrossRefGoogle Scholar
Wald, A. (1943). Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans. Amer. Math. Soc., 54, 426–482.CrossRefGoogle Scholar
White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50, 1–25.CrossRefGoogle Scholar