The model-size effect on traditional and modified tests of covariance structures

Structural Equation Modeling

Volumn 14, Issue 3, 2007, Pages 361-390

  Herzog, Walter ^a,c   Boomsma, Anne ^b   Reinecke, Sven ^a  

^a UNIVERSITY OF ST GALLEN   (Switzerland)

^b UNIVERSITY OF GRONINGEN   (Netherlands)

^c Institute of Marketing and Retailing   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

DEGREES OF FREEDOM (MECHANICS); MAXIMUM LIKELIHOOD; MECHANICS; NORMAL DISTRIBUTION; STATISTICAL TESTS;

BARTLETT CORRECTIONS; CHI-SQUARE TESTS; COVARIANCE STRUCTURES; EMPIRICAL FINDINGS; MAXIMUM LIKELIHOOD RATIOS; MULTI-VARIATE NORMAL DISTRIBUTIONS; STRUCTURAL EQUATION MODELS; TYPE I ERROR RATE;

STATISTICS;

EID: 34548066774     PISSN: 10705511     EISSN: None     Source Type: Journal    
DOI: 10.1080/10705510701301602     Document Type: Article

References (63)

1. Agresti, A., & Coull, B. A. (1998). Approximate is better than "exact" for interval estimation of binomial proportions. The American Statistician, 52, 119-126.
2. Anderson, T. W. (1958). An introduction to multivariate statistical analysis. New York: Wiley.
3. Asparouhov, T. (2005). Sampling weights in latent variable modeling. Structural Equation Modeling, 12, 411-434.
4. Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proceedings of the Royal Society of London, Series A, 160, 268-282.
5. Bartlett, M. S. (1950). Tests of significance in factor analysis. British Journal of Psychology (Statistical Section), 3, 77-85.
6. 2 approximations. Journal of the Royal Statistical Society, Series B, 16, 296-298.
7. Bentler, P. M. (1995). EQS structural equations programmanual. Encino, CA: Multivariate Software.
8. Bender, P. M., & Yuan, K.-H. (1999). Structural equation modeling with small samples: Test statistics. Multivariate Behavioral Research, 34, 181-197.
9. Bimbaum, Z. W. (1952). Numerical tabulation of the distribution of Kolmogorov's statistic for finite sample size. Journal of the American Statistical Association, 47, 425-441.
10. Bollen, K. A. (1989). Structural equations with latent variables. New York: Wiley.
11. 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 1, pp. 149-173). Amsterdam: North-Holland.
12. Boomsma, A. (1983). On the robustness of LISREL (maximum likelihood estimation) against small sample size and non-normality. Unpublished doctoral dissertation, University of Groningen, The Netherlands.
13. Boomsma, A. (1985). Nonconvergence, improper solutions, and starting values in LISREL maximum likelihood estimation. Psychometrika, 50, 229-242.
14. Boomsma, A., & Hoogland, J. J. (2001). The robustness of LISREL modeling revisited. In R. Cudeck, S. du Toit, & D. Sörbom (Eds.), Structural equation modeling: Present and future: A Festschrift in honor of Karl Jöreskog (pp. 139-168). Chicago: Scientific Software International.
15. Box, G. E. P. (1949). A general distribution theory for a class of likelihood criteria. Biometrika, 36, 317-346.
16. Box, G. E. P. (1954). Some theorems on quadratic forms applied in the study of analysis of variance problems: I. Effect of inequality of variance in the one-way classification. Annals of Mathematical Statistics, 25, 290-302.
17. Browne, M. W. ( 1982). Covariance structures. In D. M. Hawkins (Ed.), Topics in applied multivariate analysis (pp. 72-142). Cambridge, UK: Cambridge University Press.
18. Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62-83.
19. Browne, M. W., & Cudeck, R. (1992). Alternative ways of assessing model fit. Sociological Methods & Research, 21, 230-258.
20. 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.
21. 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, I, 16-29.
22. Fouladi, R. T. (1999, April). Model fit in covariance structure analysis under small sample conditions-Modifiedmaximum likelihood and asymptotically distribution free generalized least squares procedures. Paper presented at the annual meeting of the American Educational Research Association, Montreal, Canada.
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. Gnanadesikan, R. (1977). Methods for statistical data analysis of multivariate observations. New York: Wiley.
25. Hau, K.-T., & Marsh, H. W. (2004). The use of item parcels in structural equation modelling: Nonnormal data and small sample sizes. British Journal of Mathematical and Statistical Psychology, 57, 327-351.
26. Herzog, W., & Boomsma, A. (2006, June). Finite sample corrections for RMSEA estimation. Paper presented at the annual meeting of the Psychometric Society, Montreal, Canada.
27. Hoogland, J. J. (1999). The robustness of estimation methods for covariance structure analysis. Unpublished doctoral dissertation, University of Groningen, The Netherlands.
28. Hu, L. T., & Bentier, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6, 1-55.
29. Hu, L. T., Bentler, P. M., & Kano, Y. (1992). Can test statistics in covariance structure analysis be trusted? Psychological Bulletin, 112, 351-362.
30. Jackson, D. L. (2003). Revisiting sample size and number of parameter estimates: Some support for the N:q hypothesis. Structural Equation Modeling, 10, 128-141.
31. Jöreskog, K. G. (1993). Testing structural equation models. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 294-316). Newbury Park, CA: Sage.
32. Jöreskog, K. G. (2005, May). Structural equation modeling with LISREL 8.7. Symposium conducted at The Gleacher Center (in cooperation with K. A. Bollen), Chicago.
33. Jöreskog, K. G., Sörbom, D., Du Toit, S., & Du Toit, M. (2001). LISREL 8: New statistical features (3rd rev. ed.). Chicago: Scientific Software International.
34. Kaplan, D. (1988). The impact of specification error on the estimation, testing, and improvement of structural equation models. Multivariate Behavioral Research, 23, 69-86.
35. Kenny, D. A., & McCoach, D. B. (2003). Effect of the number of variables on measures of fit in structural equation modeling. Structural Equation Modeling, 10, 333-351.
36. Kensuke, O., Takahiro, H., & Kazuo, S. (2005, July). Bartlett correction of likelihood ratio statistics in structural equation modeling. Poster presented at the annual meeting of the Psychometric Society, Tilburg, The Netherlands.
37. Lawley, D. N. (1956). A general method for approximating to the distribution of likelihood ratio criteria. Biometrika, 43, 295-303.
38. Lee, S.-Y., & Song, X.-Y. (2004). Evaluation of the Bayesian and maximum likelihood approaches in analyzing structural equation models with small sample sizes. Multivariate Behavioral Research, 39, 653-686.
39. Marsh, H. W., Hau, K.T., Baila, J. R., & Grayson, D. (1998). Is more ever too much? The number of indicators per factor in confirmatory factor analysis. Multivariate Behavioral Research, 33, 181-220.
40. Marsh, H. W., Hau, K.-T., & Wen, Z. L. (2004). In search of golden rules: Comment on hypothesistesting approaches to setting cutoff values for fit indexes and danger in overgeneralizing Hu and Bender's 1999 findings. Structural Equation Modeling, 11, 320-341.
41. Massey, F. J., Jr. (1951). The Kolmogorov-Smirnov test for goodness of fit. Journal of the American Statistical Association, 46, 68-78.
42. Muthén, B. O. (1993). Goodness of fit with categorical and other nonnormal variables. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 205-234). Newbury Park, CA: Sage.
43. Muthén, B. O. (2004). Mplus: Statistical analysis with latent variables: Technical appendices. Los Angeles: Muthén & Muthén.
44. Muthén, B. O., & Satorra, A. (1995). Technical aspects of Muthén's LISCOMP approach to estimation of latent variable relations with a comprehensive measurement model. Psychometrika, 60, 489-503.
45. Muthén, L. K., & Muthén, B. O. (2004). Mplus user's guide (3rd ed.). Los Angeles: Muthén & Muthén.
46. Nevitt, J., & Hancock, G. R. (2001). Performance of bootstrapping approaches to model test statistics and parameter standard error estimation in structural equation modeling. Structural Equation Modeling, 8, 353-377.
47. Nevitt, J., & Hancock, G. R. (2004). Evaluating small sample approaches for model test statistics in structural equation modeling. Multivariate Behavioral Research, 39, 439-478.
48. Neyman, J., & Pearson, E. S. (1928). On the use and interpretation of certain test criteria for purposes of statistical inference: Part I. Biometrika, 20A, 175-240.
49. Ogasawara, H. (2001). Approximations to the distributions of fit indexes for misspecified structural equation models. Structural Equation Modeling, 8, 556-574.
50. Ramaswami, S. N., & Singh, J. (2003). Antecedents and consequences of merit pay fairness for industrial salespeople. Journal of Marketing, 67, 46-66.
51. Saris, W. E., Den Ronden, J., & Satorra, A. (1987). Testing structural equation models. In P. Cuttance & J. Ecob (Eds.), Structural modeling by example: Applications in educational, sociological, and behavioral research (pp. 202-220). Cambridge, UK: Cambridge University Press.
52. Satorra, A., & Bentler, P. M. (1988). Scaling corrections for statistics in covariance structure analysis (UCLA Statistics Series, No. 2). Los Angeles: University of California, Department of Psychology.
53. 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 variable analysis: Applications for developmental research (pp. 399-419). Thousand Oaks, CA: Sage.
54. Satorra, A., & Bentler, P. M. (2001 ). A scaled difference chi-square test statistic for moment structure analysis. Psychometrika, 66, 507-514.
55. Satterthwaite, F. E. (1941). Synthesis of variance. Psychometrika, 6, 309-316.
56. Satterthwaite, F. E. (1946). An approximate distribution of estimates of variance components. Biometrics Bulletin, 2, 110-114.
57. Scargle, J. D. (2000). Publication bias: The "file drawer" problem in scientific inference. Journal of Scientific Exploration, 14, 91-106.
58. Scheines, R., Hoijtink, H., & Boomsma, A. (1999). Bayesian estimation and testing of structural equation models. Psychometrika, 64, 37-52.
59. Sörbom, D. (1989). Model modification. Psychometrika, 54, 371-384.
60. Swain, A. J. (1975). Analysis of parametric structures for variance matrices. Unpublished doctoral dissertation, University of Adelaide, Australia.
61. Venables, W. N., & Smith, D. M. (2005). An introduction to R (Version 2.1.1). Retrieved August 15, 2005 from http://www.r-project.org.
62. Wakaki, H., Eguchi, S., & Fujikoshi, Y. (1990). A class of tests for a general covariance structure. Journal of Multivariate Analysis, 32, 313-325.
63. Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses. Annals of Mathematical Statistics, 9, 60-62.