Detecting the Violation of Factorial Invariance with an Unknown Reference Variable
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A widely used tool for testing measurement invariance is multi-group confirmatory factor analysis (MCFA). Identification of MCFA models is usually done by imposing invariance constraints on parameters of chosen reference variables (RV). If the chosen RVs were not actually invariant, one could draw invalid conclusions regarding the source of noninvariance. How can an invariant RV be selected accurately? To our knowledge, no method is yet available, yet two approaches have been suggested to detect non-invariant (or invariant) items without choosing specific RVs. One is the factor-ratio test (FR-T), and the other is the use of the largest modification index (Max-Mod). These two approaches have yet to be directly compared under the same conditions. To address unsolved problems in partial measurement invariance testing, two studies were conducted. The first aimed to identify a truly invariant RV using the smallest modification index. The second aimed to directly compare the performances of FR-T and the backward approach using the Max-Mod in correctly specifying the source of noninvariance. The second study also proposes a new method?the forward approach facilitated by the bias-corrected bootstrapping confidence intervals. The performances of the three methods was compared in terms of perfect recovery rates, model-level Type I error rates, and model-level Type II error rates. The results of the first study indicated that the Min-Mod successfully identify a truly invariant RV across all conditions. In the second study, overall, the backward approach also showed best performance under 99% confidence level (? = 0.01) in both partial metric invariance (PMI) and partial scalar invariance (PSI) conditions. The performance of the forward approach was comparable with that of the backward approach only in PMI conditions. The factor-ratio test had the poorest performance. Limitations and future directions are also discussed.