The most common effect size when using a multiple-group confirmatory factor analysis approach to measurement invariance is [delta]CFI and [delta]TLI with a cutoff value of 0.01. However, this recommended cutoff value may not be ubiquitously appropriate and may be of limited application for some tests (e.g., measures using dichotomous items or different estimation methods, sample sizes, or model complexity). Moreover, prior cutoff value estimations often have ignored consequences resulting in using measures that more accurately estimate countries' or learners' proficiency for some countries or groups versus others. In this study, we investigate whether the cutoff value proposed by Cheung and Rensvold ([delta]CFI or [delta]TLI > 0.01) is appropriate across educational measurement contexts. Specifically, we investigated the performance of [delta]CFI and [delta]TLI in capturing LOI at the scalar level in dichotomous items within item response theory on groups whose test characteristic curves differed by 0.5. Simulation results showed that the proposed cutoff value of 0.01 in [delta]CFI and [delta]TLI was not appropriate to capture LOI under the study conditions, which may result in the misinterpretation of test results or inaccurate inferences.