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Structural Equation Modeling: Applications Using Mplus by Xiaoqian Wang, Jichuan Wang

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5.1 Multi-Group CFA Model

The multi-group CFA model (Sörbom, 1974; Hayduk, 1987; Bollen, 1989a) is often used to test factorial invariance of measurement scales. According to Jöreskog (1971b), to test factorial invariance using multi-group CFA, one should first conduct an omnibus test of equality of the observed indicator variance/covariance matrix across groups. If the omnibus test demonstrates invariance of observed variances/covariances, then no more invariance tests are needed. However, the necessity of the global omnibus test is debatable, and there is no consensus in practice of multi-group CFA modeling in this regard. Studies have found some inconsistencies in the omnibus invariance test and model parameter invariance test. For example, even though the omnibus test results show invariance of the observed variances/covariances, some measurement parameters (e.g., factor loadings and item intercepts) and structural parameters (e.g., factor variances/covariances, and factor means) may be still noninvariant across groups, and vice versa (Byrne, 2006). We prefer not to conduct the omnibus test, but focus on tests of measurement invariance and structural invariance in this section.

Conceptually speaking, factorial invariance of a scale consists of two different kinds of invariance: measurement invariance and structural invariance. The former involves invariance of patterns of factor loadings, values of factor loadings, observed indicator/item intercepts, and error variances; and ...

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