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Alpha, FACTT, and Beyond

  • Revisiting Cronbach’s Alpha
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Sijtsma and Pfadt (Psychometrika, 2021) provide a wide-ranging defense for the use of coefficient alpha. Alpha is practical and useful when its limitations are acceptable. This paper discusses several methodologies for reliability, some new here, that go beyond alpha and were not emphasized by Sijtsma and Pfadt. Bentler’s (Psychometrika 33:335–345, 1968. combined factor analysis (FA) and classical test theory (CTT) model. FACTT provides a key conceptual foundation.

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  1. This notation is different from S&P. For simplicity, we assume that the variables are linearly independent with means of zero. S&P also recommend the use of estimated factor scores \(y_{w} ={w}'x\) for some weight vector w, a topic previously developed in Bentler (1968) but not discussed here.

  2. The p observed variables x are dependent variables, while the 2p variables \(\tau \) and \(\varepsilon \) are independent variables in the sense of Bentler and Weeks (1980).

  3. Here and elsewhere, if the variables involved are multivariate normally distributed, uncorrelated implies independent.

  4. Or, \(\tau \) is signal, while \(\varepsilon \) is noise (e.g., Cronbach & Gleser, 1964). S&P’s Eq. (1) uses a different conceptualization (Lord & Novick, 1968).

  5. S&P remind us that \(\Sigma _{\varepsilon } =\Delta _{\varepsilon } \) “underlies the lower bound theorem.”

  6. It is also an insurance policy against possibly subjective decisions, since \(\alpha =p^{2}\bar{{\sigma }}_{ij} /\sigma _{y}^{2} \) (S&P’s Eq. 16) depends only on data: number of parts (items) p, average covariance \(\bar{{\sigma }}_{ij} \) of parts, and \(\sigma _{y}^{2} \), the sum of all variances and covariances of parts. No additional parameter estimation or modeling decisions are needed.

  7. This is 40\(+\) years before the 2009 paper cited by S&P, the same year as Lord & Novick (1968).

  8. The FACTT variance composition is illustrated with a Venn diagram in Bentler (2017).

  9. Or internal consistency reliability, and no doubt misleadingly shortened to “reliability” on occasion. S&P state “Thus, in Bentler’s conception, internal consistency refers to unidimensionality operationalized by a common factor.” Actually, a 1-factor model is not assumed in either (2) or (3), although it is not disallowed.

  10. As did Heise and Bohrnstedt (1970).

  11. This is the greatest lower bound (glb) to reliability if \(\Delta _{u}\) contains non-negative variances, though \(\hat{{\Delta }}_{u} \) may contain negative estimates. The possibly larger—and more famous—glb forces \(\hat{{\Delta }}_{u} \) to have non-negative elements; it is discussed further below.

  12. Though at the time Bentler wrote, Jöreskog (1969) had not yet published on CFA.

  13. This recommendation is a bit strange, since internal consistency coefficients (3) and chains of lower bounds do not require \(\Sigma _{c} \) to be rank 1. For example, the glb does not require any specification for number of factors.

  14. Though any bias will be trivial with very large datasets as exist for some internet samples or national testing programs.

  15. The optimization problem also has been called constrained minimum trace factor analysis (ten Berge, Snijders, & Zegers, 1981).

  16. They also provided indirectly corrected versions of these bias-corrected coefficients based on the degree of reliability underestimation by \(\alpha \).

  17. S&P emphasize that “reliability values are dependent on the triplet test, group, and procedure.” They also point out the “misconception…that each particular test allegedly has only one reliability.”

  18. In the context of multiple factors, an application of this partition is to take the covariate-free \(\tau ^{(\bot Z)}\) as that part of the true score due to one or more relevant factors, with the covariate-dependent part as \(\tau ^{(Z)}=\tau -\tau ^{(\bot Z)}\).

  19. S&P are explicit in this assumption in their Eq. (4), stating “…measurement error covaries 0 with any other variable Y, not necessarily a test score, in which E is not included” (S&P’s Y is not this paper’s y). Note, however, that a reviewer of Bentler (2017) explicitly rejected this idea (see 2017, Footnote 5).

  20. A topic with its own methodological issues (e.g., Arruda & Bentler, 2017; Du & Bentler, in press; Jalal & Bentler, 2018; Kim, Reise, & Bentler, 2018; Yuan & Bentler, 2017).


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Correspondence to Peter M. Bentler.

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Bentler, P.M. Alpha, FACTT, and Beyond. Psychometrika 86, 861–868 (2021).

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