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

Prior to discussing and challenging two criticisms on coefficient α [12pt]{minimal}{amsmath}{wasysym}{amsfonts}{amssymb}{amsbsy}{mathrsfs}{upgreek}{}{-69pt}{document}α{document} , the well-known lower bound to test-score reliability, we discuss classical test theory and the theory of coefficient α [12pt]{minimal}{amsmath}{wasysym}{amsfonts}{amssymb}{amsbsy}{mathrsfs}{upgreek}{}{-69pt}{document}α{document} . The first criticism expressed in the psychometrics literature is that coefficient α [12pt]{minimal}{amsmath}{wasysym}{amsfonts}{amssymb}{amsbsy}{mathrsfs}{upgreek}{}{-69pt}{document}α{document} is only useful when the model of essential τ [12pt]{minimal}{amsmath}{wasysym}{amsfonts}{amssymb}{amsbsy}{mathrsfs}{upgreek}{}{-69pt}{document}τ{document} -equivalence is consistent with the item-score data. Because this model is highly restrictive, coefficient α [12pt]{minimal}{amsmath}{wasysym}{amsfonts}{amssymb}{amsbsy}{mathrsfs}{upgreek}{}{-69pt}{document}α{document} 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 [12pt]{minimal}{amsmath}{wasysym}{amsfonts}{amssymb}{amsbsy}{mathrsfs}{upgreek}{}{-69pt}{document}α{document} incorrectly ignores correlated errors. If correlated errors would enter the computation of coefficient α [12pt]{minimal}{amsmath}{wasysym}{amsfonts}{amssymb}{amsbsy}{mathrsfs}{upgreek}{}{-69pt}{document}α{document} , theoretical values of coefficient α [12pt]{minimal}{amsmath}{wasysym}{amsfonts}{amssymb}{amsbsy}{mathrsfs}{upgreek}{}{-69pt}{document}α{document} could be greater than the test-score reliability. Because quality measures that are systematically too high are undesirable, critics dismiss coefficient α</mml:mi