A Latent-Trait Based Reliability Estimate and Upper Bound

An estimate and an upper-bound estimate for the reliability of a test composed of binary items is derived from the multidimensional latent trait theory proposed by Bock and Aitkin (1981). The estimate derived here is similar to internal consistency estimates (such as coefficient alpha) in that it is a function of the correlations among test items; however, it is not a lowerbound estimate as are all other similar methods. An upper bound to reliability that is less than unity does not exist in the context of classical test theory. The richer theoretical background provided by Bock and Aitkin's latent trait model has allowed the development of an index (called δ here) that is always greater-than or equal-to the reliability coefficient for a test (and is less-than or equal-to one). The upper bound estimate of reliability has practical uses—one of which makes use of the “greatest lower bound”.