Relating measurement invariance, cross-level invariance, and multilevel reliability

Data often have a nested, multilevel structure, for example when data are collected from children in classrooms. This kind of data complicate the evaluation of reliability and measurement invariance, because several properties can be evaluated at both the individual level and the cluster level, as well as across levels. For example, cross-level invariance implies equal factor loadings across levels, which is needed to give latent variables at the two levels a similar interpretation. Reliability at a specific level refers to the ratio of true score variance over total variance at that level. This paper aims to shine light on the relation between reliability, cross-level invariance, and strong factorial invariance across clusters in multilevel data. Specifically, we will illustrate how strong factorial invariance across clusters implies cross-level invariance and perfect reliability at the between level in multilevel factor models.