Scale reliability evaluation for a-priori clustered data

According to the classical measurement theory, the reliability of a set of indicators related to a latent variable describing a true measure can be assessed through the Cronbach’s α index. The Cronbach’s α index can be used for τ-equivalent measures and for parallel measures and represents a lower bound for the reliability value in presence of congeneric measures, for which the assessment can properly be made only ex post, once the loading coefficients have been estimated, e.g. by means of a structural equation model with latent variables.Once assumed the existence of an a-priori segmentation based upon a categorical variable Z, we test whether the construct is reliable all over the groups. In this case the measurement model is the same across groups, which means that loadings are equal within each group as well as they do not vary across groups. A formulation of the Cronbach’s α coefficient is considered according to the decomposition of pairwise covariances in a clustered framework, and a test procedure assessing the possible presence of congeneric measures in a multigroup framework is proposed.