Empirical Correction to the Likelihood Ratio Statistic for Structural Equation Modeling with Many Variables

Psychometrika

Volumn 80, Issue 2, 2015, Pages 379-405

  Yuan, Ke Hai ^a   Tian, Yubin ^b   Yanagihara, Hirokazu ^c  

^a UNIVERSITY OF NOTRE DAME   (United States)

^b BEIJING INSTITUTE OF TECHNOLOGY   (China)

^c HIROSHIMA UNIVERSITY   (Japan)

Author keywords

Bartlett correction; Bayesian information criterion; maximum likelihood; type I errors

Indexed keywords

ALGORITHM; CHI SQUARE DISTRIBUTION; HUMAN; METHODOLOGY; PSYCHOMETRY; QUESTIONNAIRE; STATISTICAL MODEL;

ALGORITHMS; CHI-SQUARE DISTRIBUTION; HUMANS; LIKELIHOOD FUNCTIONS; PSYCHOMETRICS; RESEARCH DESIGN; SURVEYS AND QUESTIONNAIRES;

EID: 84930576487     PISSN: 00333123     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11336-013-9386-5     Document Type: Article

References (61)

1. Amemiya, Y., & Anderson, T.W. (1990). Asymptotic chi-square tests for a large class of factor analysis models. The Annals of Statistics, 18, 1453–1463.
2. Anderson, J.C., & Gerbing, D.W. (1984). The effect of sampling error on convergence, improper solutions, and goodness-of-fit indices for maximum likelihood confirmatory factor analysis. Psychometrika, 49, 155–173.
3. Anderson, J.C., & Gerbing, D.W. (1988). Structural equation modeling in practice: a review and recommended two-step approach. Psychological Bulletin, 103, 411–423.
4. Barndorff-Nielsen, O.E., & Hall, P. (1988). On the level-error after Bartlett adjustment of the likelihood ratio statistic. Biometrika, 75, 374–378.
5. Bartlett, M.S. (1937). Properties of sufficiency and statistical tests. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 160, 268–282.
6. 2approximation in factor analysis. Biometrika, 38, 337–344.
7. 2approximations. Journal of the Royal Statistical Society, Series B, 16, 296–298.
8. Bentler, P.M. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107, 238–246.
9. Bentler, P.M. (2008). EQS 6 structural equations program manual. Encino, CA: Multivariate Software.
10. Bentler, P.M., & Bonett, D.G. (1980). Significance tests and goodness of fit in the analysis of covariance structures. Psychological Bulletin, 88, 588–606.
11. Bentler, P.M., & Yuan, K.-H. (1999). Structural equation modeling with small samples: test statistics. Multivariate Behavioral Research, 34, 181–197.
12. Beran, R. (1988). Prepivoting test statistics: a bootstrap view of asymptotic refinements. Journal of the American Statistical Association, 83, 687–697.
13. Bollen, K.A., & Stine, R. (1993). Bootstrapping goodness of fit measures in structural equation models. In K.A. Bollen & J.S. Long (Eds.), Testing structural equation models (pp. 111–135). Newbury Park: Sage.
14. Boomsma, A. (1982). The robustness of LISREL against small sample sizes in factor analysis models. In K.G. Jöreskog & H. Wold (Eds.), Systems under indirect observation: causality, structure, prediction (Part I) (pp. 149–173). Amsterdam: North-Holland.
15. Box, G.E.P. (1949). A general distribution theory for a class of likelihood criteria. Biometrika, 36, 317–346.
16. Browne, M.W., MacCallum, R.C., Kim, C.-T., Andersen, B.L., & Glaser, R. (2002). When fit indices and residuals are incompatible. Psychological Methods, 7, 403–421.
17. Casella, G., & Berger, R.L. (2002). Statistical inference (2nd ed.). Pacific Grove: Duxbury Press.
18. Cox, D.R., & Hinkley, D.V. (1974). Theoretical statistics. Boca Raton: Chapman and Hall.
19. Curran, P.J., West, S.G., & Finch, J.F. (1996). The robustness of test statistics to nonnormality and specification error in confirmatory factor analysis. Psychological Methods, 1, 16–29.
20. Efron, B., & Tibshirani, R.J. (1993). An introduction to the bootstrap. New York: Chapman & Hall.
21. Evans, M., Hastings, N., & Peacock, B. (2000). Statistical distributions (3rd ed.). New York: Wiley.
22. Fan, X., Thompson, B., & Wang, L. (1999). Effects of sample size, estimation methods, and model specification on structural equation modeling fit indexes. Structural Equation Modeling, 6, 56–83.
23. Fouladi, R.T. (2000). Performance of modified test statistics in covariance and correlation structure analysis under conditions of multivariate nonnormality. Structural Equation Modeling, 7, 356–410.
24. Fujikoshi, Y. (2000). Transformations with improved chi-squared approximations. Journal of Multivariate Analysis, 72, 249–263.
25. Geweke, J.F., & Singleton, K.J. (1980). Interpreting the likelihood ratio statistic in factor models when sample size is small. Journal of the American Statistical Association, 75, 133–137.
26. Hall, P. (1992). The bootstrap and Edgeworth expansion. New York: Springer.
27. Herzog, W., & Boomsma, A. (2009). Small-sample robust estimators of noncentrality-based and incremental model fit. Structural Equation Modeling, 16, 1–27.
28. Herzog, W., Boomsma, A., & Reinecke, S. (2007). The model-size effect on traditional and modified tests of covariance structures. Structural Equation Modeling, 14, 361–390.
29. Holzinger, K.J., & Swineford, F. (1939). Supplementary educational monographs: Vol. 48. A study in factor analysis: the stability of a bi-factor solution. Chicago: University of Chicago Press.
30. Hu, L.T., Bentler, P.M., & Kano, Y. (1992). Can test statistics in covariance structure analysis be trusted? Psychological Bulletin, 112, 351–362.
31. Ichikawa, M., & Konishi, S. (1995). Application of the bootstrap methods in factor analysis. Psychometrika, 60, 77–93.
32. Jackson, D.L. (2001). Sample size and number of parameter estimates in maximum likelihood confirmatory factor analysis: a Monte Carlo investigation. Structural Equation Modeling, 8, 205–223.
33. Jöreskog, K.G. (1969). A general approach to confirmatory maximum likelihood factor analysis. Psychometrika, 34, 183–202.
34. Kano, Y. (1992). Robust statistics for test-of-independence and related structural models. Statistics & Probability Letters, 15, 21–26.
35. Lawley, D.N. (1956). A general method for approximating to the distribution of likelihood ratio criteria. Biometrika, 43, 295–303.
36. Lawley, D.N., & Maxwell, A.E. (1971). Factor analysis as a statistical method (2nd ed.). New York: Elsevier.
37. MacCallum, R.C., & Austin, J.T. (2000). Applications of structural equation modeling in psychological research. Annual Review of Psychology, 51, 201–226.
38. Mardia, K.V. (1970). Measure of multivariate skewness and kurtosis with applications. Biometrika, 57, 519–530.
39. McDonald, R.P., & Marsh, H.W. (1990). Choosing a multivariate model: noncentrality and goodness of fit. Psychological Bulletin, 107, 247–255.
40. Mooijaart, A., & Bentler, P.M. (1991). Robustness of normal theory statistics in structural equation models. Statistica Neerlandica, 45, 159–171.
41. Moshagen, M. (2012). The model size effect in SEM: inflated goodness-of-fit statistics are due to the size of the covariance matrix. Structural Equation Modeling, 19, 86–98.
42. Nevitt, J., & Hancock, G.R. (2004). Evaluating small sample approaches for model test statistics in structural equation modeling. Multivariate Behavioral Research, 39, 439–478.
43. Ogasawara, H. (2010). Asymptotic expansions of the null distributions of discrepancy functions for general covariance structures under nonnormality. American Journal of Mathematical and Management Sciences, 30, 385–422.
44. Okada, K., Hoshino, T., & Shigemasu, K. (2007). Bartlett correction of test statistics in structural equation modeling. Japanese Journal of Educational Psychology, 55, 382–392 (in Japanese).
45. Satorra, A., & Bentler, P.M. (1990). Model conditions for asymptotic robustness in the analysis of linear relations. Computational Statistics & Data Analysis, 10, 235–249.
46. Satorra, A., & Bentler, P.M. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In A. von Eye & C.C. Clogg (Eds.), Latent variables analysis: applications for developmental research (pp. 399–419). Thousand Oaks: Sage.
47. Schott, J.R. (2007). A test for the equality of covariance matrices when the dimension is large relative to the sample size. Computational Statistics & Data Analysis, 51, 6535–6542.
48. Sörbom, D. (1989). Model modification. Psychometrika, 54, 371–384.
49. Srivastava, M.S., & Yanagihara, H. (2010). Testing the equality of several covariance matrices with fewer observations than the dimension. Journal of Multivariate Analysis, 101, 1319–1329.
50. Swain, A.J. (1975). Analysis of parametric structures for variance matrices. Doctoral dissertation, University of Adelaide, Australia.
51. Wakaki, H., Eguchi, S., & Fujikoshi, Y. (1990). A class of tests for a general covariance structure. Journal of Multivariate Analysis, 32, 313–325.
52. White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50, 1–25.
53. Yuan, K.-H. (2005). Fit indices versus test statistics. Multivariate Behavioral Research, 40, 115–148.
54. Yuan, K.-H., & Bentler, P.M. (1998). Normal theory based test statistics in structural equation modeling. British Journal of Mathematical & Statistical Psychology, 51, 289–309.
55. Yuan, K.-H., & Bentler, P.M. (1999a). On normal theory and associated test statistics in covariance structure analysis under two classes of nonnormal distributions. Statistica Sinica, 9, 831–853.
56. Yuan, K.-H., & Bentler, P.M. (1999b). F-tests for mean and covariance structure analysis. Journal of Educational and Behavioral Statistics, 24, 225–243.
57. Yuan, K.-H., & Chan, W. (2002). Fitting structural equation models using estimating equations: a model segregation approach. British Journal of Mathematical & Statistical Psychology, 55, 41–62.
58. Yuan, K.-H., & Chan, W. (2008). Structural equation modeling with near singular covariance matrices. Computational Statistics & Data Analysis, 52, 4842–4858.
59. 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 & Statistical Psychology, 56, 93–110.
60. Yung, Y.-F., & Bentler, P.M. (1996). Bootstrapping techniques in analysis of mean and covariance structures. In G.A. Marcoulides & R.E. Schumacker (Eds.), Advanced structural equation modeling: techniques and issues (pp. 195–226). Hillsdale: Erlbaum.
61. Yung, Y.-F., & Chan, W. (1999). Statistical analyses using bootstrapping: concepts and implementation. In R.H. Hoyle (Ed.), Statistical strategies for small sample research (pp. 81–105). Thousand Oaks: Sage.