Assessing Measurement Invariance Across Multiple Groups: When Is Fit Good Enough?

Complex research questions often need large samples to obtain accurate estimates of parameters and adequate power. Combining extant data sets into a large, pooled data set is one way this can be accomplished without expending resources. Measurement invariance (MI) modeling is an established approach to ensure participant scores are on the same scale. There are two major problems when combining independent data sets through MI. First, sample sizes will often be large leading to small differences becoming noninvariant. Second, not all data sets may include the same combination of measures. In this article, we present a method that can deal with both these problems and is user friendly. It is a combination of generating random normal deviates for variables missing completely in combination with assessing model fit using the root mean square error of approximation good enough principle, based on the hypothesis that the difference between groups is not zero but small. We demonstrate the method by examining MI across eight independent data sets and compare the MI decisions of the traditional and good enough approach. Our results show the approach has potential in combining educational data.