Scaling metric measurement invariance models

This paper aims at clarifying the questions regarding the effects of the scaling method on the discrepancy function of the metric measurement invariance model. We provide examples and a formal account showing that neither the choice of the scaling method in general nor the choice of a particular referent indicator affects the value of the discrepancy function. Thus, the test statistic is not affected by the scaling method, either. The results rely on an appropriate application of the scaling restrictions, which can be phrased as a simple rule: "Apply the scaling restriction in one group only!" We develop formulas to calculate the degrees of freedom of χ²-difference tests comparing metric models to the corresponding configural model. Our findings show that it is impossible to test the invariance of the estimated loading of exactly one indicator, because metric MI models aimed at doing so are actually equivalent to the configural model.