Scale reliability evaluation for a-priori clustered data

According to the classical measurement theory, the reliability of the relationship between a latent variable describing a true measure and its corresponding manifest proxies can be assessed through the Cronbach’s alpha index. The Cronbach’s alpha index can be used 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, after having estimated the loading coefficients, e.g. by means of a structural equation model with latent variables (SEM-LV). Let us assume the existence of an a-priori segmentation, based upon a categorical variable Z. We want to test the reliability of the construct over all the groups. This corresponds to the null joint hypothesis that the loadings are equal within each group as well as they do not vary among groups. Otherwise different measurement models need to be defined over groups. A test for measuring group differences in reliability is presented in Raykov (2002), basing on differences of loading estimates in a SEM-LV framework. We consider a formulation of the Cronbach’s a coefficient according to the decomposition of pairwise covariances in a clustered framework.