Using ΔGoodness-of-Fit Indexes in Assessing Mean Structure Invariance

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Identity, Critical Agency, and Engineering: An Affective Model for Predicting Engineering as a Career Choice (2016, methodology): “Cutoff values of 0.01 were used for the delta CFI tests (Fan & Sivo, 2009).”
A Bifactor Exploratory Structural Equation Modeling Framework for the Identification of Distinct Sources of Construct-Relevant Psychometric Multidimensionality (2016, background): “…descriptive guidelines for the comparison of nested invariance models has been validated in CFA for tests of weak, strong, and strict measurement invariance (Chen, 2007; Cheung & Rensvold, 2002), they appear to be of questionable efficacy for tests of latent mean invariance (Fan & Sivo, 2009).”
Measurement Invariance Conventions and Reporting: The State of the Art and Future Directions for Psychological Research (2016, methodology): “Fan and Sivo (2009) questioned the use of change in alternative fit indices (e.g., ΔCFI, ΔRMSEA) because the performance of these tests was highly related to model size (e.g., tests were more sensitive in small models and much less sensitive as model size grew) in their Monte Carlo simulation.”