A Direct Comparison Approach for Testing Measurement Invariance

TLDR

Measurement equivalence/invariance (ME/I) is a prerequisite for meaningful cross‑group survey comparisons. This teaching note demonstrates how bias‑corrected bootstrap confidence intervals can be used to test ME/I, offering an alternative to the likelihood ratio test, ΔCFI rules, and modification index approaches. The BC bootstrap confidence‑interval method allows item‑level ME/I testing with a single model, avoiding the need for separate estimations required by LRT and ΔCFI methods. Using this approach simplifies the search for invariant items, extends factor‑ratio and list‑and‑delete tests to scalar invariance, and facilitates comparison of uniqueness variances, factor variances, covariances, and latent means across groups.

Abstract

Measurement equivalence/invariance (ME/I) is a condition that should be met before meaningful comparisons of survey results across groups can be made. As an alternative to the likelihood ratio test (LRT), the change in comparative fit index (ΔCFI) rules of thumb, and the modification index (MI), this teaching note demonstrates the procedures for establishing bias-corrected (BC) bootstrap confidence intervals for testing ME/I. Unlike the LRT and ΔCFI methods, which need a different model estimation per item, the BC bootstrap confidence intervals approach can examine item-level ME/I tests using a single model. This method greatly simplifies the search for an invariant item as the reference indicator in the factor-ratio test. Also demonstrated here is how the factor-ratio test and the list-and-delete method can be extended from the metric invariance test to the scalar invariance test. Finally, the BC bootstrap confidence interval procedures for comparing uniqueness variances, factor variances, factor covariances, and latent means across groups are shown.