Cronbach's alpha: Information from Answers.com

Cronbach's $α$ (alpha) is a statistic. It is commonly used as an estimator of the internal consistency reliability of a psychometric test score for a sample of examinees. It was first named as alpha by Lee Cronbach in 1951, as he had intended to continue with further coefficients. It can be viewed as an extension of the Kuder-Richardson Formula 20 (KR-20), which is the equivalent for dichotomous items. Alpha is not robust against missing data.

Although this article describes the use of $α$ in psychology, the statistic can be used in any discipline e.g. social sciences, nursing, etc. Similarly, the use of the word items in this article could be replaced with variables.

1 Definition
2 Cronbach's alpha and internal consistency
3 Cronbach's alpha in Generalizability theory
4 Cronbach's alpha and the intra-class correlation
5 Cronbach's alpha and factor analysis
6 See also
7 References

Definition

Cronbach's $α$ is defined as

$\alpha = { { {N} \over{N-1} } \left(1 - {{\sum_{i=1}^N \sigma^{2}_{Y_i}}\over{\sigma^{2}_{X}}}\right) }$

where $N$ is the number of components (items or testlets), $\sigma^{2}_{X}$ is the variance of the observed total test scores for the current sample of persons, and $\sigma^{2}_{Y_i}$ is the variance of component i for the current sample of persons.

Alternatively, the standardized Cronbach's $α$ can also be defined as

$\alpha = {N\cdot\bar c \over (\bar v + (N-1)\cdot\bar c)}$

where N is the number of components (items or testlets), $\bar v$ equals the average variance for the current sample of persons and $\bar c$ is the average of all covariances between the components across the current sample of persons.

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:

$\rho_{XX}= { {\sigma^2_T}\over{\sigma_X^2} }$

Alpha 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 alpha and internal consistency

Cronbach's alpha will generally increase as the intercorrelations among test items increase, and is thus known as an internal consistency estimate of reliability of test scores. Because intercorrelations among test items are maximized when all items measure the same construct, Cronbach's alpha indirectly indicates the degree to which a set of items measures a single unidimensional latent construct. Thus, alpha is most appropriately used when the items measure different substantive areas within a single construct.

When a test has a multidimensional latent structure, Cronbach's alpha (and other internal consistency estimates of reliability) will frequently be low. Thus, alpha is inappropriate for estimating the reliability of an intentionally heterogeneous instrument (such as screening devices like biodata or the original MMPI).

Alpha treats any covariance among items as true-score variance, even if items covary for spurious reasons. For exampe, alpha 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.

Cronbach's alpha in Generalizability theory

Cronbach and others generalized some basic assumptions of classical test theory in their Generalizability Theory. If this theory is applied to test construction, then it is assumed that the items that constitute the test are a random sample from a larger universe of items. The expected score of a person in the universe is called the universe score, analogous to a true score. The generalizability is defined analogously as the variance of the universe scores divided by the variance of the observable scores, analogous to the concept of reliability in classical test theory. In this theory, Cronbach's alpha is an unbiased estimate of the generalizability. For this to be true the assumptions of essential $τ$ -equivalence or parallelness are not needed. Consequently, Cronbach's alpha can be viewed as a measure of how well the sum score on the selected items capture the expected score in the entire domain, even if that domain is heterogeneous.

Cronbach's alpha and the intra-class correlation

Cronbach's alpha is said to be equal to the stepped-up consistency version of the Intra-class correlation coefficient, which is commonly used in observational studies. But this is only conditionally true. In terms of variance components, this condition is, for item sampling: if and only if the value of the item (rater, in the case of rating) variance component equals zero. If this variance component is negative, alpha will be greater than the stepped-up intra-class correlation coefficient; if this variance component is positive, alpha underestimates this stepped-up intra-class correlation coefficient.

Cronbach's alpha and factor analysis

As stated in the section about its relation with classical test theory, Cronbach's alpha has a theoretical relation with factor analysis. There is also a more empirical relation: Selecting items such that they optimize Cronbach's alpha will often result in a test that is homogeneous in that they (very roughly) approximately satisfy a factor analysis with one common factor. The reason for this is that Cronbach's alpha increases with the average correlation between items, so optimization of it tends to select items that have correlations of similar size with most other items. It should be stressed that, although unidimensionality (i.e. fit to the one-factor model) is a necessary condition for alpha to be an unbiased estimator of reliability, the value of alpha is not related to the factorial homogeneity. The reason is that the value of alpha depends on the size of the average inter-item covariance, while unidimensionality depends on the pattern of the inter-item covariances.

References

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297-334.
Bland JM, Altman DG. Statistics notes: Cronbach's alpha. BMJ 1997;314:572.
Allen, M.J., & Yen, W. M. (2002). Introduction to Measurement Theory. Long Grove, IL: Waveland Press.
Cronbach, Lee J., and Richard J. Shavelson. (2004). My Current Thoughts on Coefficient Alpha and Successor Procedures. Educational and Psychological Measurement 64, no. 3 (June 1): 391-418. doi:10.1177/0013164404266386.

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Cronbach's alpha

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