Part II: On the Use, the Misuse, and the Very Limited Usefulness of Cronbach’s Alpha: Discussing Lower Bounds and Correlated Errors
Authors: Sijtsma Klaas , Pfadt Julius M. 
Published: August 2021

Prior to discussing and challenging two criticisms on coefficient <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}, the well-known lower bound to test-score reliability, we discuss classical test theory and the theory of coefficient <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}. The first criticism expressed in the psychometrics literature is that coefficient <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} is only useful when the model of essential <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}-equivalence is consistent with the item-score data. Because this model is highly restrictive, coefficient <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} incorrectly ignores correlated errors. If correlated errors would enter the computation of coefficient <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}, theoretical values of coefficient <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} could be greater than the test-score reliability. Because quality measures that are systematically too high are undesirable, critics dismiss coefficient <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}. We argue that introducing correlated errors is inconsistent with the derivation of the lower bound theorem and that the properties of coefficient <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} remain intact when data contain correlated errors.