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The assumption of equivalence between measurement-model configurations across groups is typically investigated by evaluating overall fit of the same model simultaneously to multiple samples. However, the null hypothesis (H 0 ) of configural invariance is distinct from the H 0  of overall model fit. Permutation tests of configural invariance yield nominal Type I error rates even when a model does not fit perfectly (Jorgensen et al.,  2017 ,  in press ). When the configural model requires modification, lack of evidence against configural invariance implies that researchers should reconsider their model's structure simultaneously across all groups. Application of multivariate modification indices is therefore proposed to help decide which parameter(s) to free simultaneously in all groups, and I present Monte Carlo simulation results comparing their Type I error control to traditional 1- df  modification indices. I use the Holzinger and Swineford ( 1939 ) data set to illustrate these methods.

Applying Permutation Tests and Multivariate Modification Indices to Configurally Invariant Models That Need Respecification