Abstract
Cronbach’s coefficient alpha remains very important as a measure of internal consistency. The well-known Spearman-Brown formula indicates that as the number of items (i.e., the dimension) goes to infinity, the coefficient alpha eventually approaches one. In this work, we show that under the assumption of a one-factor model, not necessarily with parallel items, the phenomenon of the coefficient alpha approaching one is closely related to four different phenomena: (1) the closeness between factor-analysis (FA) loadings and principal-component-analysis (PCA) loadings, (2) the inverse of the population covariance matrix of the manifest variables becoming a diagonal matrix, (3) the communalities between FA and PCA approaching each other, and (4) the factor score and the principal component agreeing with each other. These results allow us to characterize the relationship between FA and PCA with respect to the coefficient alpha.
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Acknowledgments
We are thankful to Dr. Jorge Andres Gonzalez Burgos for his valuable comments on an earlier version of the article. This work was supported by Grant 31971029 from the Natural Science Foundation of China.
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Hayashi, K., Yuan, KH., Sato, R. (2021). On the Coefficient Alpha in High-Dimensions. In: Wiberg, M., Molenaar, D., González, J., Böckenholt, U., Kim, JS. (eds) Quantitative Psychology. IMPS 2020. Springer Proceedings in Mathematics & Statistics, vol 353. Springer, Cham. https://doi.org/10.1007/978-3-030-74772-5_12
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