Some Relationships between the Binomial Error Model and Classical Test Theory

Abstract

Straightforward application of the binomial error model defines parallel forms as random samples of items from a large pool of items. Such a model includes form-to-form differences in difficulty as a component of error variance and leads to Kuder-Richardson formula 21 as an estimate of reliability. Such a component is inappropriate if all examinees take the same form of the test. It is demonstrated here that if the form-to-form component is removed from the estimate of average error variance, the binomial model leads to KR 20 as the estimate of reliability. Empirical data are cited which support deductions from the compound binomial error model regarding the trend in the standard error of measurement over the observed score range. A computational formula derived from this model is recommended for two practical purposes: to estimate the standard error for individual examinees or to implement the recommendation in the APA/AERA/NCME Standards (1974) that the standard error of measurement be reported for several score levels.