Principles and Practice of Scaled Difference Chi-Square Testing

We highlight critical conceptual and statistical issues and how to resolve them in conducting Satorra–Bentler (SB) scaled difference chi-square tests. Concerning the original (Satorra & Bentler, 2001 Satorra, A. and Bentler, P. M. 2001. A scaled difference chi-square test statistic for moment structure analysis. Psychometrika, 66: 507–514. [Crossref], [Web of Science ®] , [Google Scholar]) and new (Satorra & Bentler, 2010 Satorra, A. and Bentler, P. M. 2010. Ensuring positiveness of the scaled chi-square test statistic. Psychometrika, 75: 243–248. [Crossref], [Web of Science ®] , [Google Scholar]) scaled difference tests, a fundamental difference exists in how to compute properly a model's scaling correction factor (c), depending on the particular structural equation modeling software used. Because of how LISREL 8 defines the SB scaled chi-square, LISREL users should compute c for each model by dividing the model's normal theory weighted least-squares (NTWLS) chi-square by its SB chi-square, to recover c accurately with both tests. EQS and Mplus users, in contrast, should divide the model's maximum likelihood (ML) chi-square by its SB chi-square to recover c. Because ML estimation does not minimize the NTWLS chi-square, however, it can produce a negative difference in nested NTWLS chi-square values. Thus, we recommend the standard practice of testing the scaled difference in ML chi-square values for models M ₁ and M ₀ (after properly recovering c for each model), to avoid an inadmissible test numerator. We illustrate the difference in computations across software programs for the original and new scaled tests and provide LISREL, EQS, and Mplus syntax in both single- and multiple-group form for specifying the model M ₁₀ that is involved in the new test.