Abstract
Cronbach’s alpha remains very important as a measure of internal consistency in the social sciences. The Spearman-Brown formula indicates that as the number of items goes to infinity, the reliability of the composite eventually approaches one. Under proper conditions, as the lower bound of the reliability the coefficient alpha also keeps increasing with the number of items. Hayashi et al. (On coefficient alpha in high-dimensions. In: Wiberg M, Molenaar D, Gonzalez J, Bockenholt U, Kim J-S (eds) Quantitative psychology: the 85th annual meeting of the psychometric society, 2020. Springer, New York, pp 127–139, 2021) showed that under the assumption of a one-factor model, the phenomenon of the coefficient alpha approaching one as the number of items increases is closely related to the closeness between factor-analysis (FA) loadings and principal-component-analysis (PCA) loadings, and also the factor score and the principal component agreeing with each other. In this work, their partial results are extended to the case with a multi-factor model, with some extra assumptions. The new results offer another way to characterize the relationship between FA and PCA with respect to the coefficient alpha under more general conditions.
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References
Bentler, P. M., & de Leeuw, J. (2011). Factor analysis via component analysis. Psychometrika, 76, 461–470. https://doi.org/10.1007/s11336-011-9217-5
Bentler, P.M. & Kano, Y. (1990). On the equivalence of factors and components. Multivariate Behavioral Research, 25, 67–74. https://doi.org/10.1207/s15327906mbr2501_8
Brown, W. (1910). Some experimental results in the correlation of mental ability. British Journal of Psychology, 3, 271–295.
Cronbach, L. I. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297–334. https://doi.org/10.1007/BF02310555
Guttman, L. (1956). Best possible systematic estimates of communalities. Psychometrika, 21, 273–285. https://doi.org/10.1007/BF02289137
Harville, D. A. (1997). Matrix algebra from a statistician’s perspective. Springer.
Hayashi, K., & Kamata, A. (2005). A note of the estimator of the alpha coefficient for standardized variables under normality. Psychometrika, 70, 579–586. https://doi.org/10.1007/s11336-001-0888-1
Hayashi, K., Yuan, K.-H., & Sato, R. (2021). On coefficient alpha in high-dimensions. In M. Wiberg, D. Molenaar, J. Gonzalez, U. Bockenholt, & J.-S. Kim (Eds.), Quantitative psychology: The 85th annual meeting of the psychometric society, 2020 (pp. 127–139). Springer.
Jolliffe, I. T. (2002). Principal component analysis (2nd ed.). Springer.
Kijnen, W. P. (2006). Convergence of estimates of unique variances in factor analysis, based on the inverse sample covariance matrix. Psychometrika, 71, 193–199. https://doi.org/10.1007/s11336-000-1142-9
Lawley, D. N., & Maxwell, A. E. (1971). Factor analysis as a statistical method (2nd ed.). American Elsevier.
Press, S. J. (2003). Subjective and objective Bayesian statistics (2nd ed.). Wiley.
Schneeweiss, H. (1997). Factors and principal components in the near spherical case. Multivariate Behavioral Research, 32, 375–401. https://doi.org/10.1207/s15327906mbr3204_4
Schneeweiss, H., & Mathes, H. (1995). Factor analysis and principal components. Journal of Multivariate Analysis, 55, 105–124. https://doi.org/10.1006/jmva.1995.1069
Sijtsma, K. (2009). On the use, the misuse, and the very limited usefulness of Cronbach’s alpha. Psychometrika, 74, 107–120. https://doi.org/10.1007/s11336-008-9101-0
Spearman, C. C. (1910). Correlation calculated from faulty data. British Journal of Psychology, 3, 271–295. https://doi.org/10.1111/j.2044-8295.1910.tb00206.x
Yuan, K.-H., & Bentler, P. M. (2002). On robustness of the normal-theory based asymptotic distributions of three reliability coefficient estimates. Psychometrika, 67, 251–259. https://doi.org/10.1007/BF02294845
Zhang, Z., & Yuan, K.-H. (2016). Robust coefficients alpha and omega and confidence intervals with outlying observations and missing data: Methods and software. Educational and Psychological Measurement, 76, 387–411. https://doi.org/10.1177/0013164415594658
Acknowledgments
The authors would like to thank Dr. Dylan Molenaar for his careful review of the manuscript. This work was supported by a grant from the Department of Education (R305D210023), and by a grant from the Natural Science Foundation of China (31971029). However, the contents of the study do not necessarily represent the policy of the funding agencies, and you should not assume endorsement by the Federal Government.
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Hayashi, K., Yuan, KH. (2023). On the Relationship Between Coefficient Alpha and Closeness Between Factors and Principal Components for the Multi-factor Model. In: Wiberg, M., Molenaar, D., González, J., Kim, JS., Hwang, H. (eds) Quantitative Psychology. IMPS 2022. Springer Proceedings in Mathematics & Statistics, vol 422. Springer, Cham. https://doi.org/10.1007/978-3-031-27781-8_16
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