| Summary: | A Monte Carlo simulation was conducted to investigate the Type I error rates of several versions of chi-square difference tests for nonnormal data in confirmatory factor analysis (CFA) models. The studied statistics include: 1) the original uncorrected difference test, D, obtained by taking the difference of the ML chi-squares for the respective models; 2) the original robust difference test, DR₁, due to Satorra and Bentler (2001); 3) the recent modification to this test, DR₂, which ensures that the statistic remains positive (Satorra & Bentler, 2010); and 4) a hybrid statistic, DH, proposed by Asparouhov and Muthén (2010), which is equal to DR₁ when DR₁ > 0, and otherwise is equal to DR₁. Types of constraints studied included constraining factor correlations to 0, constraining factor correlations to 1, and constraining factor loadings to equal each other within or across factors. An interesting finding was that the uncorrected test appeared to be robust to nonnormality when the constraint was setting factor correlations to zero. The robust tests performed well and similarly to each other in many conditions. The new strictly positive test, DR₂ exhibited slightly inflated rejection rates in conditions that involved constraining factor loadings, while DR₁ and DH exhibited rejection rates slightly below nominal in conditions that involved constraining factor correlations or factor loadings. While more research is needed on the new strictly positive test, the original robust difference test or the hybrid procedure are tentatively recommended. === Arts, Faculty of === Psychology, Department of === Graduate
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