Evaluating goodness-of-fit indexes for testing measurement invariance

Measurement invariance is usually tested using Multigroup Confirmatory Factor Analysis, which examines the change in the goodness-of-fit index (GFI) when cross-group constraints are imposed on a measurement model. Although many studies have examined the properties of GFI as indicators of overall model fit for single-group data, there have been none to date that examine how GFIs change when between-group constraints are added to a measurement model. The lack of a consensus about what constitutes significant GFI differences places limits on measurement invariance testing. We examine 20 GFIs based on the minimum fit function. A simulation under the two-group situation was used to examine changes in the GFIs (DeltaGFIs) when invariance constraints were added. Based on the results, we recommend using Deltacomparative fit index, DeltaGamma hat, and DeltaMcDonald's Noncentrality Index to evaluate measurement invariance. These three DeltaGFIs are independent of both model complexity and sample size, and are not correlated with the overall fit measures. We propose critical values of these DeltaGFIs that indicate measurement invariance.