A Bayesian Nonparametric Model for Test Equating

Abstract

We introduce a Bayesian nonparametric model for test score equating, which can be applied to any of the major equating designs. It provides a flexible model for the continuized distribution of test scores, by means of a mixture of beta distributions with an unknown number of mixture components. Also, the model can be specified to account for dependence between score distributions from the tests to be equated. This dependence can be accounted for even under an equivalent-groups design, where typically the questionable assumption of independence is made. Moreover, unlike the current methods of observed score equating, the Bayesian nonparametric model provides symmetric equating and always equates scores that fall within the correct range of test scores. Given data of observed test scores, an application of Bayes’ theorem provides a means to infer the posterior distribution of the equating function, including the 95% credible interval of the equated score of the posterior distribution. Thus, the Bayesian model fully accounts for the uncertainty in the equated scores, for any sample size. In contrast, current approaches to test score equating only provide large-sample approximations to estimate the confidence interval of the equated score. This Bayesian equating model is illustrated through the analysis of two data sets.