The Relationship Between Unstandardized and Standardized Alpha, True Reliability, and the Underlying Measurement Model

Abstract

Popular computer programs print 2 versions of Cronbach's alpha: unstandardized alpha, α_Σ, based on the covariance matrix, and standardized alpha, α _R , based on the correlation matrix. Sources that accurately describe the theoretical distinction between the 2 coefficients are lacking, which can lead to the misconception that the differences between α _R and α_Σ are unimportant and to the temptation to report the larger coefficient. We explore the relationship between α _R and α_Σ and the reliability of the standardized and unstandardized composite under 3 popular measurement models; we clarify the theoretical meaning of each coefficient and conclude that researchers should choose an appropriate reliability coefficient based on theoretical considerations. We also illustrate that α _R and α_Σ estimate the reliability of different composite scores, and in most cases cannot be substituted for one another.

Notes

The variance of the composite can be obtained by either computing the composite and taking its variance, or by summing up all the elements in the covariance matrix Σ.

Throughout the article, we assume that the test's items are measured continuously or at least using a sufficient number of categories so that the true score model holds approximately at the item level (five to seven categories is usually deemed enough). For the case of binary items or items with few response options, although one can compute a reliability coefficient from the tetrachoric or polychoric correlation matrix, this coefficient estimates the reliability of a composite made of the underlying continuous responses to the items, not of the observed categorical items. Thus, this is not the estimate of interest. To model the observed categorical items appropriately requires switching to the item response theory framework, where the concept of reliability is replaced with the concept of information.

If the total scale score is standardized after raw item scores have been added, this standardization does not change the scale's reliability, and unstandardized alpha can still be used.

We are assuming no correlated errors. The important case in which the errors are correlated is discussed by Raykov (1998a, 2001).

All three measurement models discussed in this article allow for different intercepts among the items; however, item intercepts are omitted for simplicity and because they do not affect any of the reliability computations.

The form of this matrix is obtained by applying the rules of covariance algebra to find the variances of individual items and simplify the covariances between X_i =T+E_i and X_j =T+E_j (e.g., Bollen, 1989).