"Rassling the hog": The influence of correlated item error on internal consistency, Classical reliability, and congeneric reliability

The properties of internal consistency (α), classical reliability (ρ), and congeneric reliability (ω) for a composite test with correlated item error are analytically investigated. Possible sources of correlated item error are contextual effects, item bundles, and item models that ignore additional attributes or higher-order attributes. The relation between reliability and internal consistency is determined by the deviance from true-score equivalence. The influence of correlated item error on α, ρ, and ω is conveyed strictly through the total item error covariance. As the total item error covariance increases, ρ and ω decrease, but α increases. The necessary and sufficient condition for α to be a lower bound to ρ and to ω is that the total item error covariance not exceed the deviance from true-score equivalence. Coefficient α will uniformly exceed ρ or ω in true-score equivalent tests with positively correlated item error.