This discussion paper argues that both the use of Cronbach’s alpha as a reliability estimate and as a measure of internal consistency suffer from major problems. First, alpha always has a value, which cannot be equal to the test score’s reliability given the interitem covariance matrix and the usual assumptions about measurement error. Second, in practice, alpha is used more often as a measure of the test’s internal consistency than as an estimate of reliability. However, it can be shown easily that alpha is unrelated to the internal structure of the test. It is further discussed that statistics based on a single test administration do not convey much information about the accuracy of individuals’ test performance. The paper ends with a list of conclusions about the usefulness of alpha.

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On the Use, the Misuse, and the Very Limited Usefulness of Cronbach’s Alpha

Coefficient alpha is the most popular measure of reliability (and certainly of internal consistency reliability) reported in psychological research. This is noteworthy given the numerous deficiencies of coefficient alpha documented in the psychometric literature. This mismatch between theory and practice appears to arise partly because users of psychological scales are unfamiliar with the psychometric literature on coefficient alpha and partly because alternatives to alpha are not widely known. We present a brief review of the psychometric literature on coefficient alpha, followed by a practical alternative in the form of coefficient omega. To facilitate the shift from alpha to omega, we also present a brief guide to the calculation of point and interval estimates of omega using a free, open source software environment.

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From alpha to omega: a practical solution to the pervasive problem of internal consistency estimation

Bifactor latent structures were introduced over 70 years ago, but only recently has bifactor modeling been rediscovered as an effective approach to modeling construct-relevant multidimensionality in a set of ordered categorical item responses. I begin by describing the Schmid-Leiman bifactor procedure (Schmid & Leiman, 1957), and highlight its relations with correlated-factors and second-order exploratory factor models. After describing limitations of the Schmid-Leiman, two newer methods of exploratory bifactor modeling are considered, namely, analytic bifactor (Jennrich & Bentler, 2011) and target bifactor rotations (Reise, Moore, & Maydeu-Olivares, 2011). In section two, I discuss limited and full-information estimation approaches to confirmatory bifactor models that have emerged from the item response theory and factor analysis traditions, respectively. Comparison of the confirmatory bifactor model to alternative nested confirmatory models and establishing parameter invariance for the general factor also are discussed. In the final section, important applications of bifactor models are reviewed. These applications demonstrate that bifactor modeling potentially provides a solid foundation for conceptualizing psychological constructs, constructing measures, and evaluating a measure's psychometric properties. However, some applications of the bifactor model may be limited due to its restrictive assumptions.

/pdf/the-rediscovery-of-bifactor-measurement-models-1bluokfjre.pdf

The Rediscovery of Bifactor Measurement Models

There are three fundamental problems in Sijtsma (Psychometrika, 2008): (1) contrary to the name, the glb is not the greatest lower bound of reliability but rather is systematically less than ωt (McDonald, Test theory: A unified treatment, Erlbaum, Hillsdale, 1999), (2) we agree with Sijtsma that when considering how well a test measures one concept, α is not appropriate, but recommend ωt rather than the glb, and (3) the end user needs procedures that are readily available in open source software.

Coefficients Alpha, Beta, Omega, and the glb: Comments on Sijtsma

Empirical studies in psychology commonly report Cronbach's alpha as a measure of internal consistency reliability despite the fact that many methodological studies have shown that Cronbach's alpha is riddled with problems stemming from unrealistic assumptions. In many circumstances, violating these assumptions yields estimates of reliability that are too small, making measures look less reliable than they actually are. Although methodological critiques of Cronbach's alpha are being cited with increasing frequency in empirical studies, in this tutorial we discuss how the trend is not necessarily improving methodology used in the literature. That is, many studies continue to use Cronbach's alpha without regard for its assumptions or merely cite methodological articles advising against its use to rationalize unfavorable Cronbach's alpha estimates. This tutorial first provides evidence that recommendations against Cronbach's alpha have not appreciably changed how empirical studies report reliability. Then, we summarize the drawbacks of Cronbach's alpha conceptually without relying on mathematical or simulation-based arguments so that these arguments are accessible to a broad audience. We continue by discussing several alternative measures that make less rigid assumptions which provide justifiably higher estimates of reliability compared to Cronbach's alpha. We conclude with empirical examples to illustrate advantages of alternative measures of reliability including omega total, Revelle's omega total, the greatest lower bound, and Coefficient H. A detailed software appendix is also provided to help researchers implement alternative methods. (PsycINFO Database Record

Thanks coefficient alpha, we'll take it from here.

We make theoretical comparisons among five coefficients—Cronbach’s α, Revelle’s β, McDonald’s ω
 h
 , and two alternative conceptualizations of reliability. Though many end users and psychometricians alike may not distinguish among these five coefficients, we demonstrate formally their nonequivalence. Specifically, whereas there are conditions under which α, β, and ω
 h
 are equivalent to each other and to one of the two conceptualizations of reliability considered here, we show that equality with this conceptualization of reliability and between α and ω
 h
 holds only under a highly restrictive set of conditions and that the conditions under which β equals ω
 h
 are only somewhat more general. The nonequivalence of α, β, and ω
 h
 suggests that important information about the psychometric properties of a scale may be missing when scale developers and users only report α as is almost always the case

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Wen-Wen Li

Papers

CRONBACH'S α, REVELLE'S β, AND MCDONALD'S ωH : THEIR RELATIONS WITH EACH OTHER AND TWO ALTERNATIVE CONCEPTUALIZATIONS OF RELIABILITY

Vesicle transport related protein Synaptotagmin-1 mediates paraquat transport to antagonize paraquat toxicity.