Cronbach's alpha
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Cronbach's α (alpha) has an important use as a measure of the reliability of a psychometric instrument. It indicates the extent to which a set of test items can be treated as measuring a single latent variable. It was first named as alpha by Cronbach (1951), as he had intended to continue with further instruments. It is the extension of an earlier version, the Kuder-Richardson Formula 20 (often shortened to KR-20), which is the equivalent for dichotomous items, and Guttman (1945) developed the same quantity under the name lambda-2.
Cronbach's α is defined as
,
where N is the number of components (items or testlets), 
is the variance of the observed total test scores, and 
is the variance of component i.
Alpha is an unbiased estimator of reliability when the components are all parallel. Although this assumption may sometimes be met (at least approximately) by testlets, when applied to items it is probably never true, because test developers invariably include items with a range of difficulties (or stimuli that vary in their standing on the latent trait, in the case of personality, attitude or other non-cognitive instruments). When this assumption of parallel components is violated, alpha is not an unbiased estimator of reliability. Instead, it is a lower bound on reliability.
α can take values between negative infinity and 1 (although only positive values make sense). Some professionals, as a rule of thumb, require a reliability of 0.70 or higher (obtained on a substantial sample) before they will use an instrument. Obviously, this rule should be applied with caution when α has been computed from items that systematically violate its assumptions. Further, the appropriate degree of reliability depends upon the use of the instrument, e.g., an instrument designed to be used as part of a battery may be intentionally designed to be as short as possible (and thus somewhat less reliable). Other situations may require extremely precise measures (with very high reliabilities).
Cronbach's α is related conceptually to the Spearman-Brown prediction formula. Both arise from the basic classical test theory result that the reliability of test scores can be expressed as the ratio of the true score and total score (error and true score) variances:
Alpha is most appropriately used when the items measure different substantive areas within a single construct. Conversely, alpha (and other internal consistency estimates of reliability) are inappropriate for estimating the reliability of an intentionally hetrogeneous instrument (such as screening device such as a biodata or the original MMPI). Also, α can be artificially inflated by making scales which consist of superficial changes to the wording within a set of items or by analyzing speeded tests.
Although this description of the use of α is given in terms of psychology, the statistic can be used in any discipline.
[edit] References
- Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297-334.
- Allen, M.J., & Yen, W. M. (2002). Introduction to Measurement Theory. Long Grove, IL: Waveland Press.
