Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-08T11:52:01.289Z Has data issue: false hasContentIssue false
  • English
  • Français

Higher-Order Approximations to the Distributions of Fit Indexes Under Fixed Alternatives in Structural Equation Models

Published online by Cambridge University Press:  01 January 2025

Haruhiko Ogasawara
Haruhiko Ogasawara*
Affiliation:
Otaru University of Commerce
*
Requests for reprints should be sent to Haruhiko Ogasawara, Department of Information and Management Science, Otaru University of Commerce, 3-5-21, Midori, Otaru 047-8501, Japan. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Higher-order approximations to the distributions of fit indexes for structural equation models under fixed alternative hypotheses are obtained in nonnormal samples as well as normal ones. The fit indexes include the normal-theory likelihood ratio chi-square statistic for a posited model, the corresponding statistic for the baseline model of uncorrelated observed variables, and various fit indexes as functions of these two statistics. The approximations are given by the Edgeworth expansions for the distributions of the fit indexes under arbitrary distributions. Numerical examples in normal and nonnormal samples with the asymptotic and simulated distributions of the fit indexes show the relative inappropriateness of the normal-theory approximation using noncentral chi-square distributions. A simulation for the confidence intervals of the fit indexes based on the normal-theory Studentized estimators under normality with a small sample size indicates an advantage for the approximation by the Cornish–Fisher expansion over those by the noncentral chi-square distribution and the asymptotic normality.

Keywords

fixed alternativesfit indexesnonnormal distributionsnoncentral chi-square distributionsEdgeworth expansionstructural equation modelingRMSEA
Type
Original Paper
Information
Psychometrika , Volume 72 , Issue 2 , June 2007 , pp. 227 - 243
Copyright
Copyright © 2007 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

The author is indebted to the reviewers for their comments and suggestions, which have led to the improvement of the previous versions of this paper. This work was partially supported by Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science, and Technology.

References

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. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107, 238–246.CrossRefGoogle ScholarPubMed
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
Bentler, P.M., & Dijkstra, T. (1985). Efficient estimation via linearization in structural models. In Krishnaiah, P.R. (Ed.), Multivariate analysis (Vol. VI, pp. 9–12). New York: Elsevier.Google Scholar
Bock, R.D., Bargmann, R.E. (1966). Analysis of covariance structures. Psychometrika, 31, 507–534.CrossRefGoogle ScholarPubMed
Bollen, K.A. (1986). Sample size and Bentler and Bonett’s nonnormed fit index. Psychometrika, 51, 375–377.CrossRefGoogle Scholar
Bollen, K.A. (1989). A new incremental fit index for general equation models. Sociological Methods & Research, 17, 303–316.CrossRefGoogle Scholar
Bollen, K.A., & Stine, R.A. (1993). Bootstrapping goodness-of-fit measures in structural equation models. In Bollen, K.A. & Long, J.S. (Eds.), Testing structural equation models (pp. 111–135). Newbury Park, CA: Sage.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., & Arminger, G. (1995). Specification and estimation of mean- and covariance-structure models. In Arminger, G., Clogg, C.C. & Sobel, M.E. (Eds.), Handbook of statistical modeling for the social and behavioral sciences (pp. 185–241). New York: Plenum Press.CrossRefGoogle Scholar
Browne, M.W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In Bollen, K.A. & Long, J.S. (Eds.), Testing structural equation models (pp. 136–162). Newbury Park, CA: Sage.Google Scholar
Curran, P.J., Bollen, K.A., Paxton, P., Kirby, J., Chen, F. (2002). The noncentral chi-square distribution in misspecified structural equation models: Finite sample results from a Monte Carlo simulation. Multivariate Behavioral Research, 37, 1–36.CrossRefGoogle ScholarPubMed
Hall, P. (1992). The bootstrap and Edgeworth expansion, New York: Springer-Verlag. Corrected printing, 1997.CrossRefGoogle Scholar
Johnson, N.L., & Kotz, S. (1970). Continuous univariate distributions (Vol. 1). New York: Wiley.Google Scholar
Jöreskog, K.G., Sörbom, D. (1981). LISREL V: Analysis of linear structural relations by the method of maximum likelihood, Chicago: International Educational Services.Google Scholar
Kaplan, E.L. (1952). Tensor notation and the sampling cumulants of k-statistics. Biometrika, 39, 319–323.Google Scholar
Maiti, S.S., Mukherjee, B.N. (1990). A note on distributional properties of the Jöreskog-Sörbom fit indices. Psychometrika, 55, 721–726.CrossRefGoogle Scholar
Marsh, H.W., Balla, J.R., McDonald, R.P. (1988). Goodness-of-fit indices in confirmatory factor analysis: Effects of sample size. Psychological Bulletin, 103, 391–411.CrossRefGoogle Scholar
McDonald, R.P. (1989). An index of goodness-of-fit based on noncentrality. Journal of Classification, 6, 97–103.CrossRefGoogle Scholar
McDonald, R.P., Marsh, H.W. (1990). Choosing a multivariate model: Noncentrality and goodness of fit. Psychological Bulletin, 107, 247–255.CrossRefGoogle Scholar
Meijer, E. (1998). Structural equation models for nonnormal data, Leiden: DSWO Press.Google Scholar
Nagahara, Y. (2004). A method of simulating multivariate nonnormal distributions by the Pearson distribution system and estimation. Computational Statistics and Data Analysis, 47, 1–29.CrossRefGoogle Scholar
Ogasawara, H. (2001). Approximations to the distributions of fit indexes for misspecified structural equation models. Structural Equation Modeling, 8, 556–574.CrossRefGoogle Scholar
Ogasawara, H. (2003). A note on the approximate distributions of fit indexes for misspecified structural equation models. In Yanai, H., Okada, A., Shigemasu, K., Kano, Y., & Meulman, J.J. (Eds.), New developments in psychometrics: Proceedings of the International Meeting of the Psychometric Society IMPS2001, Osaka, Japan, July 15–19, 2001 (pp. 103–108). Tokyo: Springer-Verlag.CrossRefGoogle Scholar
Ogasawara, H. (2004a). Higher-order estimation error in factor analysis and structural equation modeling under nonnormality. In Kano, Y. (Organizer), Proceedings of the 83rd symposium of behaviormetrics: Factor analysis centennial symposium (pp. 39–43). Nakanoshima Center (Osaka University), Osaka, Japan.Google Scholar
Ogasawara, H. (2004b). Asymptotic expansion in factor analysis and structural equation modeling under nonnormality/normality. Proceedings of the 72nd annual meeting of the Japan Statistical Society (pp. 101–102 with an error in Corollary 2A corrected at the presentation). Fuji University, Hanamaki, Japan.Google Scholar
Ogasawara, H. (2007a). Higher-order estimation error in structural equation modeling. Economic Review (Otaru University of Commerce), 47(4). http://www.res. otaru-uc.ac.jp/~hogasa/Google Scholar
Ogasawara, H. (2007b). Asymptotic expansion of the distributions of the estimators in factor analysis under nonnormality. British Journal of Mathematical and Statistical Psychology, in press.CrossRefGoogle ScholarPubMed
Olsson, U.H., Foss, T., Breivik, E. (2004). Two equivalent discrepancy functions for maximum likelihood estimation: Do their statistics follow a non-central chi-square distribution under model misspecification?. Sociological Methods and Research, 32, 453–500.CrossRefGoogle Scholar
Ramberg, J.S., Schmeiser, B.W. (1974). An approximate method for generating asymmetric random variables. Communications of the Association for Computing Machinery, 17, 78–82.CrossRefGoogle Scholar
Satorra, A. (1989). Alternative test criteria in covariance structure analysis: A unified approach. Psychometrika, 54, 131–151.CrossRefGoogle 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
Steiger, J.H. (1989). EzPATH: A supplementary module for SYSTAT and SYGRAPH. Evanstone, IL: SYSTAT.Google Scholar
Steiger, J.H. (1990). Structural model evaluation and modification: An interval estimation approach. Multivariate Behavioral Research, 25, 173–180.CrossRefGoogle ScholarPubMed
Steiger, J.H. (1998). A note on multiple sample extensions of the RMSEA fit index. Structural Equation Modeling, 5, 411–419.CrossRefGoogle Scholar
Steiger, J.H., & Lind, J.C. (1980). Statistically based tests for the number of common factors. Paper presented at the annual Spring Meeting of the Psychometric Society, Iowa City, IA.Google Scholar
Tanaka, J.S., Huba, G.J. (1985). A fit index for covariance structural models under arbitrary GLS estimation. British Journal of Mathematical and Statistical Psychology, 38, 197–201.CrossRefGoogle Scholar
Tanaka, J.S., Huba, G.J. (1989). A general coefficient of determination for covariance structure models under arbitrary GLS estimation. British Journal of Mathematical and Statistical Psychology, 42, 233–239.CrossRefGoogle Scholar
Tucker, L.R., Lewis, C. (1973). A reliability coefficient for maximum likelihood factor analysis. Psychometrika, 38, 1–10.CrossRefGoogle Scholar
Wansbeek, T., & Meijer, E. (2000). Measurement error and latent variables in econometrics. Amsterdam: North-Holland.Google Scholar
Weng, L.J., Cheng, C.P. (1997). Why might relative fit indices differ between estimators?. Structural Equation Modeling, 4, 121–128.CrossRefGoogle Scholar
Yuan, K.-H. (2005). Fit indices versus test statistics. Multivariate Behavioral Research, 40, 115–148.CrossRefGoogle ScholarPubMed
Zhou, X.-H., Gao, S. (2000). One-sided confidence intervals for means of positively skewed distributions. American Statistician, 54, 100–104.CrossRefGoogle Scholar