Reconceptualization of Coefficient Alpha Reliability for Test Summed and Scaled Scores

Coefficient alpha reliability persists as the most common reliability coefficient reported in research. The assumptions for its use are, however, not well-understood. The current paper challenges the commonly used expressions of coefficient alpha and argues that while these expressions are correct when estimating reliability for summed scores, they are not appropriate to extend coefficient alpha to correctly estimate the reliability for nonlinearly transformed scaled scores such as percentile ranks and stanines. The current paper reconceptualizes coefficient alpha as a complement of the ratio of two unbiased estimates of the summed score variance. These include conditional summed score variance assuming uncorrelated item scores (gives the error score variance) and unconditional summed score variance incorporating intercorrelated item scores (gives the observed score variance). Using this reconceptualization, a new equation of coefficient generalized alpha is introduced for scaled scores. Coefficient alpha is a special case of this new equation since the latter reduces to coefficinet alpha if the scaled scores are the summed scores themselves. Two applications (cognitive and psychological assessments) are used to compare the performance (estimation and bootstrap confidence interval) of the reliability coefficients for different scaled scores. Results support the new equation of coefficient generalized alpha and compare it to coefficient generalized beta for parallel test forms. Coefficient generalized alpha produced different reliability values, which were larger than coefficient generalized beta for different scaled scores.