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Research Articles

40-Year Old Unbiased Distribution Free Estimator Reliably Improves SEM Statistics for Nonnormal Data

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Pages 872-887 | Received 13 Jan 2022, Accepted 05 Apr 2022, Published online: 01 Jun 2022

References

  • Anderson, T. W., & Darling, D. A. (1952). Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes. The Annals of Mathematical Statistics, 23, 193–212. https://doi.org/10.1214/aoms/1177729437
  • Anderson, T. W., & Darling, D. A. (1954). A test of goodness of fit. Journal of the American Statistical Association, 49, 765–769. https://doi.org/10.1080/01621459.1954.10501232
  • Asparouhov, T., & Muthén, B. (2010). Simple second order chi-square correction. Mplus Technical Appendix. http://statmodel.com/download/WLSMV_new_chi21.pdf
  • Bartlett, M. S. (1951). The effect of standardization on a χ2 approximation in factor analysis. Biometrika, 38, 337–344. https://doi.org/10.2307/2332580
  • Bentler, P. (2006). EQS 6 structural equations program manual. Multivariate Software.
  • Bentler, P. M. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107, 238–246. https://doi.org/10.1037/0033-2909.107.2.238
  • Bradley, J. V. (1978). Robustness? British Journal of Mathematical and Statistical Psychology, 31, 144–152. https://doi.org/10.1111/j.2044-8317.1978.tb00581.x
  • Browne, M. W. (1974). Generalized least squares estimators in the analysis of covariance structures. South African Statistical Journal, 8, 1–24.
  • Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83. https://doi.org/10.1111/j.2044-8317.1984.tb00789.x
  • Cain, M. K., Zhang, Z., & Yuan, K.-H. (2017). Univariate and multivariate skewness and kurtosis for measuring nonnormality: Prevalence, influence and estimation. Behavior Research Methods, 49, 1716–1735.
  • Chan, W., Yung, Y.-F., & Bentler, P. M. (1995). A note on using and unbiased weight matrix in the ADF test statistic. Multivariate Behavioral Research, 30, 453–459.
  • Chou, C.-P., Bentler, P. M., & Satorra, A. (1991). Scaled test statistics and robust standard errors for non-normal data in covariance structure analysis: A monte carlo study. British Journal of Mathematical and Statistical Psychology, 44, 347–357. https://doi.org/10.1111/j.2044-8317.1991.tb00966.x
  • Chun, S. Y., Browne, M. W., & Shapiro, A. (2018). Modified distribution-free goodness-of-fit test statistic. Psychometrika, 83, 48–66.
  • 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.
  • 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. [Database] https://doi.org/10.1037/1082-989X.1.1.16
  • Du, H., & Bentler, P. M. (2021). Distributionally weighted least squares in structural equation modeling. Psychological Methods. https://doi.org/10.1037/met0000388
  • Du, H., Bentler, P. M., & Rosseel, Y. (2022). Distributionally-weighted least squares in growth curve modeling. Structural Equation Modeling: A Multidisciplinary Journal, 29, 1–22. https://doi.org/10.1080/10705511.2021.1931870
  • Foldnes, N., & Grønneberg, S. (2018). Approximating test statistics using eigenvalue block averaging. Structural Equation Modeling: A Multidisciplinary Journal, 25, 101–114. https://doi.org/10.1080/10705511.2017.1373021
  • Foldnes, N., & Grønneberg, S. (2021). Non-normal data simulation using piecewise linear transforms. Structural Equation Modeling: A Multidisciplinary Journal, 29, 36–46.
  • Foldnes, N., & Olsson, U. H. (2015). Correcting too much or too little? The performance of three chi-square corrections. Multivariate Behavioral Research, 50, 533–543. https://doi.org/10.1080/00273171.2015.1036964
  • Foldnes, N., & Olsson, U. H. (2016). A simple simulation technique for nonnormal data with prespecified skewness, kurtosis, and covariance matrix. Multivariate Behavioral Research, 51, 207–219.
  • Fouladi, R. T. (2000). Performance of modified test statistics in covariance and correlation structure analysis under conditions of multivariate nonnormality. Structural Equation Modeling: A Multidisciplinary Journal, 7, 356–410. https://doi.org/10.1207/S15328007SEM0703_2
  • Grønneberg, S., & Foldnes, N. (2017). Covariance model simulation using regular vines. Psychometrika, 82, 1035–1051.
  • Grønneberg, S., & Foldnes, N. (2019). Testing model fit by bootstrap selection. Structural Equation Modeling: A Multidisciplinary Journal, 26, 182–190. https://doi.org/10.1080/10705511.2018.1503543
  • Hayakawa, K. (2019). Corrected goodness-of-fit test in covariance structure analysis. Psychological Methods, 24, 371–389. https://doi.org/10.1037/met0000180
  • Himeno, T., & Yamada, T. (2014). Estimations for some functions of covariance matrix in high dimension under non-normality and its applications. Journal of Multivariate Analysis, 130, 27–44. https://doi.org/10.1016/j.jmva.2014.04.020
  • Holzinger, K. J., & Swineford, F. (1939). A study in factor analysis: The stability of a bi-factor solution (Supplementary Educational Monograph No. 48). University of Chicago Press.
  • Hu, L-t., Bentler, P. M., & Kano, Y. (1992). Can test statistics in covariance structure analysis be trusted? Psychological Bulletin, 112, 351–362.
  • Jiang, G., & Yuan, K.-H. (2017). Four new corrected statistics for SEM with small samples and nonnormally distributed data. Structural Equation Modeling: A Multidisciplinary Journal, 24, 479–494. https://doi.org/10.1080/10705511.2016.1277726
  • Jöreskog, K. G. (1969). A general approach to confirmatory maximum likelihood factor analysis. Psychometrika, 34, 183–202. https://doi.org/10.1007/BF02289343
  • Koning, R. H., Neudecker, H., & Wansbeek, T. (1992). Unbiased estimation of fourth-order matrix moments. Linear Algebra and Its Applications, 160, 163–174. https://doi.org/10.1016/0024-3795(92)90445-G
  • Lee, S.-Y., & Jennrich, R. (1979). A study of algorithms for covariance structure analysis with specific comparisons using factor analysis. Psychometrika, 44, 99–113. https://doi.org/10.1007/BF02293789
  • Lin, J., & Bentler, P. M. (2012). A third moment adjusted test statistic for small sample factor analysis. Multivariate Behavioral Research, 47, 448–462.
  • Liu, H., Tang, Y., & Zhang, H. H. (2009). A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. Computational Statistics & Data Analysis, 53, 853–856. https://doi.org/10.1016/j.csda.2008.11.025
  • Mair, P., Satorra, A., & Bentler, P. M. (2012). Generating nonnormal multivariate data using copulas: Applications to sem. Multivariate Behavioral Research, 47, 547–565.
  • Marcoulides, K. M., Foldnes, N., & Grønneberg, S. (2020). Assessing model fit in structural equation modeling using appropriate test statistics. Structural Equation Modeling: A Multidisciplinary Journal, 27, 369–379. https://doi.org/10.1080/10705511.2019.1647785
  • Muthén, B., & Kaplan, D. (1985). A comparison of some methodologies for the factor analysis of non-normal likert variables. British Journal of Mathematical and Statistical Psychology, 38, 171–189. https://doi.org/10.1111/j.2044-8317.1985.tb00832.x
  • Nevitt, J., & Hancock, G. R. (2004). Evaluating small sample approaches for model test statistics in structural equation modeling. Multivariate Behavioral Research, 39, 439–478. https://doi.org/10.1207/S15327906MBR3903_3
  • Qu, W., Liu, H., & Zhang, Z. (2020). A method of generating multivariate non-normal random numbers with desired multivariate skewness and kurtosis. Behavior Research Methods, 52, 939–946.
  • Rosseel, Y. (2012). Lavaan: An R package for structural equation modeling and more. Version 0.5–12 (beta). Journal of Statistical Software, 48, 1–36. https://doi.org/10.18637/jss.v048.i02
  • Satorra, A., & Bentler, P. M. (1986). Some robustness properties of goodness of fit statistics in covariance structure analysis. In American Statistical Association: Proceedings of the business and economic statistics section (pp. 549–554). American Statistical Association.
  • Satorra, A., & Bentler, P. M. (1988). Scaling corrections for chi-square statistics in covariance structure analysis. In American Statistical Association 1988: Proceedings of business and economics sections (pp. 308–313). American Statistical Association.
  • 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. 339–419). Sage.
  • Savalei, V. (2014). Understanding robust corrections in structural equation modeling. Structural Equation Modeling: A Multidisciplinary Journal, 21, 149–160. https://doi.org/10.1080/10705511.2013.824793
  • Savalei, V., & Rhemtulla, M. (2013). The performance of robust test statistics with categorical data. British Journal of Mathematical and Statistical Psychology, 66, 201–223. https://doi.org/10.1111/j.2044-8317.2012.02049.x
  • Srivastava, M. S., Yanagihara, H., & Kubokawa, T. (2014). Tests for covariance matrices in high dimension with less sample size. Journal of Multivariate Analysis, 130, 289–309. https://doi.org/10.1016/j.jmva.2014.06.003
  • Steiger, J. H. (2016). Notes on the steiger–lind (1980) handout. Structural Equation Modeling: A Multidisciplinary Journal, 23, 777–781. https://doi.org/10.1080/10705511.2016.1217487
  • Stephens, M. A. (1974). Edf statistics for goodness of fit and some comparisons. Journal of the American Statistical Association, 69, 730–737. https://doi.org/10.1080/01621459.1974.10480196
  • Swain, A. J. (1975). Analysis of parametric structures for variance matrices [Unpublished doctoral dissertation]. University of Adelaide, Department of Statistics.
  • Tong, X., & Bentler, P. M. (2013). Evaluation of a new mean scaled and moment adjusted test statistic for sem. Structural Equation Modeling : A Multidisciplinary Journal, 20, 148–156.
  • Vale, C. D., & Maurelli, V. A. (1983). Simulating multivariate nonnormal distributions. Psychometrika, 48, 465–471. https://doi.org/10.1007/BF02293687
  • Wu, H. (2018). Approximations to the distribution of a test statistic in covariance structure analysis: A comprehensive study. The British Journal of Mathematical and Statistical Psychology, 71, 334–362. https://doi.org/10.1111/bmsp.12123
  • Wu, H., & Lin, J. (2016). A scaled f distribution as an approximation to the distribution of test statistics in covariance structure analysis. Structural Equation Modeling: A Multidisciplinary Journal, 23, 409–421. https://doi.org/10.1080/10705511.2015.1057733
  • Yang, M., & Yuan, K.-H. (2019). Optimizing ridge generalized least squares for structural equation modeling. Structural Equation Modeling: A Multidisciplinary Journal, 26, 24–38. https://doi.org/10.1080/10705511.2018.1479853
  • Yuan, K.-H., & Bentler, P. M. (1997). Improving parameter tests in covariance structure analysis. Computational Statistics & Data Analysis, 26, 177–198. https://doi.org/10.1016/S0167-9473(97)00025-X
  • Yuan, K.-H., & Bentler, P. M. (1998). Normal theory based test statistics in structural equation modelling. British Journal of Mathematical and Statistical Psychology, 51, 289–309. https://doi.org/10.1111/j.2044-8317.1998.tb00682.x
  • Yuan, K.-H., & Bentler, P. M. (1999). On normal theory and associated test statistics in covariance structure analysis under two classes of nonnormal distributions. Statistica Sinica, 9, 831–853.
  • Yuan, K.-H., & Chan, W. (2016). Structural equation modeling with unknown population distributions: Ridge generalized least squares. Structural Equation Modeling: A Multidisciplinary Journal, 23, 163–179. https://doi.org/10.1080/10705511.2015.1077335
  • Yuan, K.-H., Tian, Y., & Yanagihara, H. (2015). Empirical correction to the likelihood ratio statistic for structural equation modeling with many variables. Psychometrika, 80, 379–405. https://doi.org/10.1007/s11336-013-9386-5
  • Yuan, K.-H., Yang, M., & Jiang, G. (2017). Empirically corrected rescaled statistics for SEM with small n and large p. Multivariate Behavioral Research, 52, 673–698. https://doi.org/10.1080/00273171.2017.1354759
  • Yung, Y.-F., & Bentler, P. M. (1994). Bootstrap-corrected ADF test statistics in covariance structure analysis. British Journal of Mathematical and Statistical Psychology, 47, 63–84. https://doi.org/10.1111/j.2044-8317.1994.tb01025.x

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