Coefficient [beta] as Extension of KR-21 Reliability for Summed and Scaled Scores for Polytomously-Scored Tests

KR-21 reliability and its extension (coefficient [alpha]) gives the reliability estimate of test scores under the assumption of tau-equivalent forms. KR-21 reliability gives the reliability estimate for summed scores for dichotomous items when items are randomly sampled from an infinite pool of similar items (randomly parallel forms). The article extends KR-21 to coefficient [beta] to estimate reliability of summed scores for test with any item scoring pattern under the same assumption of KR-21. Also, the article presents coefficient G[beta] to estimate this reliability for nonlinearly scaled scores for any type of items. The article shows that coefficient [beta] is slightly different from the classical index of reliability. Results show that coefficient [beta] is not equal to the classical index of reliability for summed scores and scaled scores. Moreover, results using real data on psychological instruments reveal that different score scales yield different coefficient G[beta] reliability estimates that are less than coefficient G[alpha] by a function related to form-to-form differences in averages.