Ensuring Positiveness of the Scaled Diﬀerence Chi-square Test Statistic
- Author(s): Satorra, Albert
- Bentler, P M
- et al.
A scaled diﬀerence test statistic T_tildad that can be computed from standard software of structural equation models (SEM) by hand calculations was proposed in Satorra and Bentler (2001). The statistic T_tildad is asymptotically equivalent to the scaled diﬀerence test statistic T_hatd introduced in Satorra (2000), which requires more involved computations beyond standard output of SEM software. The test statistic T_tildad 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 T_hatd that avoids negative chi-square values.