Stratified coefficients of reliability and their sampling behavior under nonnormality

Stratified versions of coefficients for reliability are defined as extensions of the unstratified coefficients given by Guttman and Cronbach: Lambda 3 (Alpha), Lambda 2 and Lambda 6. One of the stratified coefficients is already available as stratified alpha. Among the four stratified coefficients dealt with in this article, two coefficients are for a stratified test with set (stratum)-specific true scores while the other two are for a stratified test with errors correlated within each set. Conditions of some coefficients being equal to population reliability are shown. For the sampling behavior, asymptotic distributions of the sample coefficients with and without stratification are derived using the asymptotic expansions under nonnormality with simulations for confirmation