Hostname: page-component-7f64f4797f-zzvtc Total loading time: 0 Render date: 2025-11-04T00:41:54.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Alternative Test Criteria in Covariance Structure Analysis: A Unified Approach

Published online by Cambridge University Press:  01 January 2025

Albert Satorra
Albert Satorra*
Affiliation:
University of Barcelona
*
Request for reprints should be sent to Albert Satorra, Department of Statistics and Econometrics, Faculty of Economics, University of Barcelona, Avinguda Diagonal 690, Barcelona-08034, SPAIN.
Get access Rights & Permissions [Opens in a new window]

Abstract

In the context of covariance structure analysis, a unified approach to the asymptotic theory of alternative test criteria for testing parametric restrictions is provided. The discussion develops within a general framework that distinguishes whether or not the fitting function is asymptotically optimal, and allows the null and alternative hypothesis to be only approximations of the true model. Also, the equivalent of the information matrix, and the asymptotic covariance matrix of the vector of summary statistics, are allowed to be singular. When the fitting function is not asymptotically optimal, test statistics which have asymptotically a chi-square distribution are developed as a natural generalization of more classical ones. Issues relevant for power analysis, and the asymptotic theory of a testing related statistic, are also investigated.

Keywords

covariance structure analysistest statisticsasymptotic equivalencenoncentral chi-squarepower of the test

Information

Type
Original Paper
Information
Psychometrika , Volume 54 , Issue 1 , March 1989 , pp. 131 - 151
Copyright
Copyright © 1989 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.)

Article purchase

Temporarily unavailable

Footnotes

This research has been supported by the U.S.-Spanish Joint Committee for Cultural and Educational Cooperation, grant number V-B.854020. The author wishes to express his gratitude to P. M. Bentler who provided very helpful suggestions and research facilities—with an stimulating working environment—at the University of California, Los Angeles, where this work was undertaken. Thanks are also due to W. E. Saris who provided very valuable comments to earlier versions of this paper. Finally, it has also to be acknowledged the editor's and reviewers suggestions which led to substantial improvements of this article.

References

Aitchison, J. (1962). Large-sample restricted parametric tests. Journal of the Royal Statistical Society, Series B, 24, 234–250.CrossRefGoogle Scholar
Bentler, P. M. (1983). Simultaneous equations as moment structure models: With an introduction to latent variable models. Journal of Econometrics, 22, 13–42.CrossRefGoogle Scholar
Bentler, P. M. (1983). Some contributions to efficient statistics in structural models: specification and estimation of moment structures. Psychometrika, 48, 493–517.CrossRefGoogle Scholar
Bentler, P. M. (1985). Theory and implementation of EQS, a structural equations program, Los Angeles: BMDP Statistical Software.Google Scholar
Bentler, P. M. (1986). Lagrange multiplier and Wald tests for EQS and EQS/PC, Los Angeles: BMDP Statistical Software.Google Scholar
Bentler, P. M., & Chou, C. P. (1986). Statistics for parameter expansion and constraction in structural models. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco.Google Scholar
Bentler, P. M., Dijkstra, T. (1985). Efficient estimation via linearization in structural models. In Kirshnaiah, P. R. (Eds.), Multivariate analysis VI (pp. 9–42). Amsterdam: North-Holland.Google Scholar
Browne, M. W. (1974). Generalized least squares estimates in the analysis of covariance structures. South African Statistical Journal, 8, 1–24.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
Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83.CrossRefGoogle ScholarPubMed
Browne, M. W. (1987). Robustness of statistical inference in factor analysis and related models. Biometrika, 74, 375–84.CrossRefGoogle Scholar
Burguete, J. F., Gallant, A. R., Souza, G. (1982). On unification of the asymptotic theory of nonlinear econometric models. Econometric Reviews, 1, 151–190.CrossRefGoogle Scholar
Buse, A. (1982). The likelihood ratio, Wald, and Lagrange multiplier tests: An expository note. The American Statistician, 36, 153–157.Google Scholar
Chou, C. P., & Bentler, P. M. (1987). Model modification in covariance structure modeling: A comparison among likelihood ratio, lagrange multiplier and Wald tests. Paper presented at the Annual Meeting of the American Educational Research Association, Washington, D. C.Google Scholar
Cox, D. R., Hinkley, D. V. (1974). Theoretical statistics, London: Chapman & Hall.CrossRefGoogle Scholar
Dijkstra, T. (1983). Some comments on maximum likelihood and partial least squares methods. Journal of Econometrics, 22, 67–90.CrossRefGoogle Scholar
Engle, R. F. (1984). Wald, likelihood ratio, and Lagrange multiplier test in econometrics. In Griliches, Z., Intriligator, M. (Eds.), Handbook of Econometrics (pp. 775–825). Amsterdam: North-Holland.CrossRefGoogle Scholar
Ferguson, T. S. (1958). A method of generating best asymptotically normal estimates with application to the estimation of bacterial densities. Annals of Mathematical Statistics, 29, 1046–1062.CrossRefGoogle Scholar
Jennings, D. E. (1986). Judging inference adequacy in logistic regression. Journal of the American Statistical Association, 81, 471–476.CrossRefGoogle Scholar
Jöreskog, K. G. (1970). A general method for the analysis of covariance structures. Biometrika, 57, 239–251.CrossRefGoogle Scholar
Jöreskog, K. G. (1977). Structural equation models in the social sciences: Specification, estimation and testing. In Krishnaiah, P. R. (Eds.), Application of statistics (pp. 265–287). Amsterdam: North-Holland.Google Scholar
Jöreskog, K. G. (1981). Analysis of covariance structures. Scandinavian Journal of Statistics, 8, 65–92.Google Scholar
Jöreskog, K. G., Sörbom, D. (1984). LISREL VI: Analysis of linear structural relationships by maximum likelihood, instrumental variables and least squares methods, Mooresville: Scientific Software.Google Scholar
Lee, S.-Y. (1985). On testing functional constraints in structural equation models. Biometrika, 72, 125–131.CrossRefGoogle Scholar
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
Lee, S.-Y., Tsui, K. L. (1982). Covariance structure analysis in several populations. Psychometrika, 47, 297–308.CrossRefGoogle Scholar
Luijben, T., Boomsma, A., Molenaar, I. W. (1987). Modification of factor analysis models in covariance structure analysis: A monte study, Groningen: University of Groningen, Heymans Bulletins Psychologische Instituten.Google Scholar
Lustbader, E. D., Moolgavkar, S. H., Venzon, D. J. (1984). Tests of the null hypothesis in case-control studies. Biometrics, 40, 1017–1024.CrossRefGoogle ScholarPubMed
Rao, C. R. (1947). Large sample tests of statistical hypothesis concerning several parameters with applications to problems of estimation. Proceedings Cambridge Philosophy Society, 44, 50–57.Google Scholar
Rao, C. R. (1965). Linear statistical inference and its applications, New York: John Wiley.Google Scholar
Rao, C. R., Mitra, S. K. (1971). Generalized inverse of matrices and its applications, New York: John Wiley.Google Scholar
Saris, W. E., Satorra, A., Sörbom, D. (1987). The detection and correction of specification errors in structural equation models. In Clogg, C. C. (Eds.), Sociological methodology 1987 (pp. 105–129). Washington, D. C.: American Sociological Association.Google Scholar
Saris, W. E., Stronkhorst, L. H. (1984). Causal modeling in nonexperimental research: An introduction to the LISREL approach, Amsterdam: Sociometric Research Foundation.Google Scholar
Satorra, A., Bentler, P. M. (1988). Scaling corrections for statistics on covariance structure analysis, Los Angeles: University of California.Google Scholar
Satorra, A., Bentler, P. M. (1988). Model conditions for asymptotic robustness in the analysis of linear relations, Los Angeles: University of California.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 Middendorp, C. P., Niemöller, B., Saris, W. E. (Eds.), Sociometric Research 1982 (pp. 127–190). Amsterdam: Sociometric Research Foundation.Google Scholar
Satorra, A., Saris, W. E. (1985). Power of the likelihood ratio test in covariance structure analysis. Psychometrika, 50, 83–90.CrossRefGoogle Scholar
Satorra, A., Saris, W. E., & de Pijper, W. M. (1987). A comparison of several approximations to the power of the likelihood ratio test in covariance structure analysis. American Statistical Association 1987 Proceedings, Business and Economic Statistics Section (pp. 393–398).Google Scholar
Serfling, R. J. (1980). Approximation theorems in mathematical statistics, New York: Wiley.CrossRefGoogle Scholar
Searle, S. R. (1982). Matrix algebra useful for statistics, New York: Wiley.Google Scholar
Shapiro, A. (1983). Asymptotic distribution theory in the analysis of covariance structures (a unified approach). South African Statistical Journal, 17, 33–81.Google Scholar
Shapiro, A. (1984). A note on the consistency of estimates in the analysis of moment structures. British Journal of Mathematical and Statistical Psychology, 37, 84–88.CrossRefGoogle Scholar
Shapiro, A. (1985). Asymptotic equivalence of minimum discrepancy function estimators to G.L.S. estimators. South African Statistical Journal, 19, 73–81.Google Scholar
Shapiro, A. (1985). Second-order derivatives of extremal-value functions and optimality conditions for semi-infinite programs. Mathematics of Operations Research, 10, 207–219.CrossRefGoogle Scholar
Shapiro, A. (1986). Asymptotic theory of overparameterized structural models. Journal of the American Statistical Association, 81, 142–149.CrossRefGoogle Scholar
Shapiro, A. (1987). Robustness properties of the MDF analysis of moment structures. South African Statistical Journal, 21, 39–62.Google Scholar
Silvey, S. D. (1959). The Lagrangian multiplier test. Annals of Mathematical Statistics, 30, 389–407.CrossRefGoogle Scholar
Sörbom, D. (1986). Model modification, Uppsala: University of Uppsala.Google Scholar
Steiger, J. H., Shapiro, A., Browne, M. W. (1985). On the multivariate asymptotic distribution of sequential chi-square statistics. Psychometrika, 50, 253–263.CrossRefGoogle Scholar
van Praag, R. M. S., Dijkstra, T. K., Van Velzen, J. (1985). Least-squares theory based on general distributional assumptions with an application to the incomplete observations problem. Psychometrika, 50, 25–36.CrossRefGoogle Scholar
van Praag, R.M.S., de Leeuw, J., Kloek, T. (1986). The population-sample decomposition approach to multivariate estimation methods. Applied Stochastic Models and Data Analysis, 2, 99–119.CrossRefGoogle Scholar
Wald, A. (1943). Tests of statistical hypothesis concerning several parameters when the number of observations is large. Transactions of the American Mathematical Society, 54, 426–482.CrossRefGoogle Scholar
White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50, 1–25.CrossRefGoogle Scholar
Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses. Annals of Mathematical Statistics, 9, 60–62.CrossRefGoogle Scholar