“Coefficient Alpha and the Internal Structure of Tests”

Lee Cronbach's 1951 Psychometrika article “Coefficient Alpha and the Internal Structure of Tests” established coefficient alpha as the preeminent estimate of internal consistency reliability. Cronbach demonstrated that coefficient alpha is the mean of all split-half reliability coefficients and discussed the manner in which coefficient alpha should be interpreted. Specifically, alpha estimates the correlation between two randomly parallel tests administered at the same time and drawn from a universe of items like those in the original test. Further, Cronbach showed that alpha does not require the assumption that items be unidimensional. In his reflections 50 years later, Cronbach described how coefficient alpha fits within generalizability theory, which may be employed to obtain more informative explanations of test score variance.

Concerns about the accuracy of test scores are commonly addressed by computing reliability coefficients. An internal consistency reliability coefficient, which may be obtained from a single test administration, estimates the consistency of scores on repeated test administrations taking place at the same time (i.e., no changes in examinees from one test to the next). Split-half reliability coefficients, which estimate internal consistency reliability, were established as a standard of practice for much of the early 20th century, but such coefficients are not unique because they depend on particular splits of items into half tests. Cronbach presented coefficient alpha as an alternative method for estimating internal consistency reliability. Alpha is computed as follows:

where k is the number of items, s2i is the variance of scores on item i, and s2t is the variance of total test scores. As demonstrated by Cronbach, alpha is the mean of all possible split-half coefficients for a test. Alpha is generally applicable for studying measurement consistency whenever data include multiple observations of individuals (e.g., item scores, ratings from multiple judges, stability of performance over multiple trials). Cronbach showed that the well-known Kuder–Richardson formula 20 (KR-20), which preceded alpha, was a special case of alpha when items are scored dichotomously.

One sort of internal consistency reliability coefficient, the coefficient of precision, estimates the correlation between a test and a hypothetical replicated administration of the same test when no changes in the examinees have occurred. In contrast, Cronbach explained that alpha, which estimates the coefficient of equivalence, reflects the correlation between two different k-item tests randomly drawn (without replacement) from a universe of items like those in the test and administered simultaneously. Since the correlation of a test with itself would be higher than the correlation between different tests drawn randomly from a pool, alpha provides a lower bound for the coefficient of precision. Note that alpha (and other internal consistency reliability coefficients) provides no information about variation in test scores that could occur if repeated testings were separated in time. Thus, some have argued that such coefficients overstate reliability.

Cronbach dismissed the notion that alpha requires the assumption of item unidimensionality (i.e., all items measure the same aspect of individual differences). Instead, alpha provides an estimate (lower bound) of the proportion of variance in test scores attributable to all common factors accounting for item responses. Thus, alpha can reasonably be applied to tests typically administered in educational settings and that comprise items that call on several skills or aspects of understanding in different combinations across items. Coefficient alpha, then, climaxed 50 years of work on correlational conceptions of reliability begun by Charles Spearman.

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