Finally! A valid test of configural invariance using permutation in multigroup CFA

In multigroup factor analysis, configural measurement invariance is accepted as tenable when researchers either (a) fail to reject the null hypothesis of exact fit using a χ² test or (b) conclude that a model fits approximately well enough, according to one or more alternative fit indices (AFIs). These criteria fail for two reasons. First, the test of perfect fit confounds model fit with group equivalence, so rejecting the null hypothesis of perfect fit does not imply that the null hypothesis of configural invariance should be rejected. Second, treating common rules of thumb as critical values for judging approximate fit yields inconsistent results across conditions because fixed cutoffs ignore sampling variability of AFIs. As a solution, we propose replacing χ² and fixed AFI cutoffs with permutation tests. Iterative permutation of group assignment yields an empirical distribution of any fit measure under the null hypothesis of invariance. Simulations show the permutation test of configural invariance controls Type I error rates better than χ² or AFIs when a model has parsimony error (i.e., negligible misspecification) but the factor structure is equivalent across groups (i.e., the null hypothesis is true).