Coefficient α, although ubiquitous in the research literature, is frequently criticized for being a poor estimate of test reliability. In this note, we consider the range of α and prove that it has no lower bound (i.e., α ∈ ( - ∞, 1]). While outlining our proofs, we present algorithms for generating data sets that will yield any fixed value of α in its range. We also prove that for some data sets-even those with appreciable item correlations-α is undefined. Although α is a putative estimate of the correlation between parallel forms, it is not a correlation as α can assume any value below-1 (and α values below 0 are nonsensical reliability estimates). In the online supplemental materials, we provide R code for replicating our empirical findings and for generating data sets with user-defined α values. We hope that researchers will use this code to better understand the limitations of α as an index of scale reliability. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

中文翻译：

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系数 α 的数学界限是什么？

系数 α 虽然在研究文献中无处不在，但经常被批评为对测试可靠性的估计不佳。在这篇文章中，我们考虑了 α 的范围并证明它没有下界（即 α ∈ ( - ∞, 1]）。在概述我们的证明时，我们提出了生成数据集的算法，这些数据集将产生任何固定值α 在其范围内。我们还证明对于某些数据集 - 即使是那些具有明显项目相关性的数据集 - α 是未定义的。虽然 α 是平行形式之间相关性的推定估计，但它不是相关性，因为 α 可以假设以下任何值-1（低于 0 的 α 值是无意义的可靠性估计）。在在线补充材料中，我们提供 R 代码来复制我们的经验发现和生成具有用户定义的 α 值的数据集。我们希望研究人员将使用此代码来更好地理解 α 作为量表可靠性指标的局限性。（PsycInfo 数据库记录 (c) 2023 APA，保留所有权利）。