Satorra-Bentler adjusted chi-square

• For data not of multivariate normal distribuiton, ML is not appropriate. Some suggest use asymptotic (large-sample) distribution-free (ADF) estimation method. However, Satorra and Bentler aruge that it may be more appropriate to correct the test statistic rather than to use different estimation methods. Satorra-Bentler developed the SB chi-squre statistic, which incorporates a scaling correction for the chi-square statistic when distributional assumptions are violated. This is the approach taken by EQS in the treatment of non-normal data. In contrast to LISREL, EQS uses an estimation method that assumes the data are multivariate normal but bases evaluation of model fit on a test statistic that has been corrected to take non-normality into account.
• S-B chi-square incorporates a scaling correction for the chi-square statistic when distributional assumptions are violated. Its computation takes into account the model, the estimation method, and the sample curtosis values. The S-B chi-square has been shoen to more closely approximate chi-square than the uncorrected test statistic, to have robust standard errors, and to perform as well, or bettern than the usual asymptotically distribution-free methods generally recommended for nonnormal multivariate data. However, now, the S-B chi-square is not yet available fro multigroup analysis in EQS.
• Bias levels in standard errors were reduced when the Satorra–Bentler (1988) rescaling correction was applied to nonnormal, ordered categorical data.
• Amos does not report the Satorra-Bentler adjusted chi square.
• Mplus and EQS can compute a Satorra-Bentler chi-square
• Satorra Bentler Chi square is good tool for correcting standard errors under non- normality data
• SPSS macros for getting the Satorra-Bentler adjusted chi-square are now available at http://rubens.its.unimelb.edu.au/~dudgeon They are limited to single sample models without any equality constraints being imposed.
• The chi-square fit index is also very sensitive to violations of the assumption of multivariate normality. When this assumption is known to be violated, the researcher may prefer Satorra-Bentler scaled chi-square, which adjusts model chi-square for non-normality.

• Satorra, A., & Bentler, P. M. (2001). A scaled difference chi-square test statistic for moment structure analysis. Psychometrika, 66(4), 507-514.