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When comparing data from different countries, time points, or groups, we run into at least two problems. First, we want to avoid large measurement artifacts that lead to erroneous substantive conclusions [1, 2]. For example, when comparing Finnish to Columbian survey answers, we may want to account for any differences in exuberance. Second, we want to ignore the–likely plentiful–small measurement artifacts whose effect on substantive conclusions is negligible [3, 4]. For example, when comparing Finns in 2002 with Finns in 2004 on an income question, most of the differences found are likely to be substantive; we would not want to spend an inordinate amount of time and modeling power on identifying all the small measurement differences between these already highly comparable groups. Tests for the presence or absence of measurement differences are typically called measurement invariance tests …