Part II: On the Use, the Misuse, and the Very Limited Usefulness of Cronbach’s Alpha: Discussing Lower Bounds and Correlated Errors

Abstract

Prior to discussing and challenging two criticisms on coefficient α, the well-known lower bound to test-score reliability, we discuss classical test theory and the theory of coefficient α. The first criticism expressed in the psychometrics literature is that coefficient α is only useful when the model of essential τ-equivalence is consistent with the item-score data. Because this model is highly restrictive, coefficient α is smaller than test-score reliability and one should not use it. We argue that lower bounds are useful when they assess product quality features, such as a test-score’s reliability. The second criticism expressed is that coefficient α incorrectly ignores correlated errors. If correlated errors would enter the computation of coefficient α, theoretical values of coefficient α could be greater than the test-score reliability. Because quality measures that are systematically too high are undesirable, critics dismiss coefficient α. We argue that introducing correlated errors is inconsistent with the derivation of the lower bound theorem and that the properties of coefficient α remain intact when data contain correlated errors.