Nonparametric Bayesian modelling for item response

Abstract

Item response theory is widely used in standardized testing to model the relationship between test takers’ unobserved ability levels and their responses to items on the test. Item characteristic curves give the probability of a correct response to an item as a function of ability and are most often modelled with logistic curves. In this paper we demonstrate how to model the item characteristic curve with nonparametric Bayesian methods through the use of Dirichlet process priors and present a complementary model in which the ability distribution is modelled nonparametrically while the item characteristic curves are logistic. We compare the nonparametric models with the two-parameter logistic Bayesian model on data from an exam in an introductory statistics course. We find that the nonparametric curve model produces significantly different item characteristic curves for a few of the items and that the corresponding ability estimates also change substantially for some individuals.