Noncentral chi-square versus normal distributions in describing the likelihood ratio statistic: The univariate case and its multivariate implication

Multivariate Behavioral Research

Volumn 43, Issue 1, 2008, Pages 109-136

  Yuan, Ke Hai ^a  

^a UNIVERSITY OF NOTRE DAME   (United States)

Author keywords

[No Author keywords available]

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EID: 41149114218     PISSN: 00273171     EISSN: None     Source Type: Journal    
DOI: 10.1080/00273170701836729     Document Type: Article

References (42)

1. Bentler, P. M. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107, 238-246.
2. Bender, P. M., & Dijkstra, T. K. (1985). Efficient estimation via linearization in structural models. In P. R. Krishnaiah (Ed.), Multivariate analysis VI (pp. 9-42). Amsterdam: North-Holland.
3. Bollen, K. A. (1989). Structural equations with latent variables. New York: Wiley.
4. Bollen, K. A. (2002). Latent variables in psychology and the social sciences. Annual Review of Psychology, 53, 605-634.
5. Boomsma, A. (2000). Reporting on structural equation analyses. Structural Equation Modeling, 7, 461-483.
6. Browne, M. W. (1984). Asymptotic distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62-83.
7. Browne, M. W., & Arminger, G. (1995). Specification and estimation of mean and covariance structure models. In G. Arminger, C. C. Clogg, & M. E. Sobel (Eds.), Handbook of statistical modeling for the social and behavioral sciences (pp. 185-249). New York: Plenum.
8. Browne, M. W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 136-162). Newbury Park, CA: Sage.
9. Cohen, J. J. (1977). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
10. Ferguson, T. S. (1996). A course in large sample theory. London: Chapman & Hall.
11. Hancock, G. R. (2001). Effect size, power, and sample size determination for structured means modeling and MIMIC approaches to between-groups hypothesis testing of means on a single latent construct. Psychometrika, 66, 373-388.
12. Holzinger, K. J., & Swineford, F. (1939). A study in factor analysis: The stability of a bi-factor solution (Monograph No. 48). Chicago: University of Chicago Supplementary Educational Monographs.
13. Howell, D. C. (2007). Statistical methods for psychology (6th ed.). Belmont, CA: Thomson Wadsworth.
14. Kano, Y. (2001). Structural equation modeling for experimental, data. In R. Cudeck, S.H.C. du Toit, & D. Sörbom. (Eds.), Structural equation modeling: Present and future (pp. 381-402). Lincolnwood, IL: Scientific Software International.
15. Kim, K. (2005). The relationship among fit indexes, power, and sample size in structural equation modeling. Structural Equation Modeling, 12, 368-390.
16. Li, L., & Bentler, P. M. (2006). Robust statistical tests for evaluating the hypothesis of close fit of misspecified mean and covariance structural models. UCLA Statistics Electronic Publications (http://prepri.nts.stat.ucla.edu/).
17. MacCallum, R. C., & Austin, J. T. (2000). Applications of structural, equation modeling in psychological research. Annual Review of Psychology, 51, 201-226.
18. MacCallum, R. C., Browne, M. W., & Sugawara, H. M. (1996). Power analysis and determination of sample size for covariance structure modeling. Psychological Methods, 1, 130-149.
19. Olkin, I., & Siotani, M. (1976). Asymptotic distribution of functions of a correlation matrix. Essays in probability and statistics (pp. 235-251). Tokyo: Shinko Tsusho.
20. Olsson U. H., Foss T., & Breivik E. (2004). Two equivalent discrepancy functions for maximum likelihood estimation: Do their test statistics follow a non-central chi-square distribution under model misspecification? Sociological Methods & Research, 32, 453-500.
21. Satorra, A. (1989). Alternative test criteria in covariance structure analysis: A unified approach. Psychometrika, 54, 131-151.
22. Satorra, A., & Saris, W. (1985). Power of the likelihood ratio test in covariance structure analysis. Psychometrika, 50, 83-90.
23. Satorra, A., Saris, W. E., & de Pijper W. M. (1991). A comparison of several approximations to the power function of the likelihood ratio test in covariance structure analysis. Statistica Neerlandica, 45, 173-185.
24. Shapiro, A. (1983). Asymptotic distribution theory in the analysis of covariance structures (a unified approach). South African Statistical Journal, 17, 33-81.
25. Shapiro, A., & Browne, M. (1987). Analysis of covariance structures under elliptical distributions. Journal of the American Statistical Association, 82, 1092-1097.
26. Snedecor, G. W., & Cochran, W. G. (1989). Statistical methods (8th ed.). Ames: Iowa State University Press.
27. Steiger, J. H., & Lind, J. M. (1980, June). Statistically based tests for the number of common factors. Paper presented at the annual meeting of the Psychometric Society, Iowa City, IA.
28. Steiger, J. H., Shapiro, A., & Browne, M. W. (1985). On the multivariate asymptotic distribution of sequential chi-square statistics. Psychometrika, 50, 253-264.
29. Sugiura, N. (1969). Asymptotic expansions of the distributions of the likelihood ratio criteria for covariance matrix. Annals of Mathematical Statistics, 40, 2051-2063.
30. Sugiura, N., & Fujikoshi, Y. (1969). Asymptotic expansions of the non-null distributions of the like-lihood ratio criteria for multivariate linear hypothesis and independence. Annals of Mathematical Statistics, 40, 942-952.
31. van der Vaart, A. W. (1998). Asymptotic statistics. Cambridge, UK: Cambridge University Press.
32. Vuong, Q. H. (1989). Likelihood ratio tests for model selection and nonnested hypotheses. Econometrica, 57, 307-333.
33. Wald, A. (1943). Tests of statistical hypotheses concerning several, parameters when the number of observations is large. Transactions of the American Mathematical Society, 54, 426-482.
34. Yanagihara, H., Tonda, T., & Matsumoto, C. (2005). The effects of nonnormality on asymptotic distributions of some likelihood ratio criteria for testing covariance structures under normal assumption. Journal of Multivariate Analysis, 96, 237-264.
35. Yuan, K.-H., & Bender, P. M. (2006). Mean comparison: Manifest variable versus latent variable. Psychometrika, 71, 139-159.
36. Yuan, K.-H., & Bentler, P. M. (2007). Structural equation modeling. In C. R. Rao & S. Sinharay (Eds.), Handbook of statistics 25: Psychometrics (pp. 297-358). Amsterdam: North-Holland.
37. Yuan, K.-H., Bentler, P. M., & Chan, W. (2004). Structural equation modeling with heavy tailed distributions. Psychometrika, 69, 421-436.
38. Yuan, K.-H., Bentler, P. M., & Zhang, W. (2005). The effect of skewness and kurtosis on mean and covariance structure analysis: The univariate case and its multivariate implication. Sociological Methods & Research, 34, 249-258.
39. Yuan, K.-H., & Chan, W. (2005). On nonequivalence of several procedures of structural equation modeling. Psychometrika, 70, 791-798.
40. Yuan, K.-H., & Hayashi, K. (2003). Bootstrap approach to inference and power analysis based on three statistics for covariance structure models. British Journal of Mathematical and Statistical Psychology, 56, 93-110.
41. Yuan, K.-H., Hayashi, K., & Bentler, P. M. (2007). Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses. Journal of Multivariate Analysis, 98, 1262-1282.
42. Yuan, K.-H., & Marshall, L. L. (2004). A new measure of misfit for covariance structure models. Behaviormetrika, 31, 67-90.