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A scaled difference test statistic Formula: see text that can be computed from standard software of structural equation models (SEM) by hand calculations was proposed in Satorra and Bentler (2001). The statistic Formula: see text is asymptotically equivalent to the scaled difference test statistic T̄(d) introduced in Satorra (2000), which requires more involved computations beyond standard output of SEM software. The test statistic Formula: see text has been widely used in practice, but in some applications it is negative due to negativity of its associated scaling correction. Using the implicit function theorem, this note develops an improved scaling correction leading to a new scaled difference statistic T̄(d) that avoids negative chi-square values.
Satorra et al. (Fri,) studied this question.