The Effect of Upper and Lower Asymptotes of IRT Models on Computerized Adaptive Testing

Abstract

In this article, the effect of the upper and lower asymptotes in item response theory models on computerized adaptive testing is shown analytically. This is done by deriving the step size between adjacent latent trait estimates under the four-parameter logistic model (4PLM) and two models it subsumes, the usual three-parameter logistic model (3PLM) and the 3PLM with upper asymptote (3PLMU). The authors show analytically that the large effect of the discrimination parameter on the step size holds true for the 4PLM and the two models it subsumes under both the maximum information method and the b-matching method for item selection. Furthermore, the lower asymptote helps reduce the positive bias of ability estimates associated with early guessing, and the upper asymptote helps reduce the negative bias induced by early slipping. Relative step size between modeling versus not modeling the upper or lower asymptote under the maximum Fisher information method (MI) and the b-matching method is also derived. It is also shown analytically why the gain from early guessing is smaller than the loss from early slipping when the lower asymptote is modeled, and vice versa when the upper asymptote is modeled. The benefit to loss ratio is quantified under both the MI and the b-matching method. Implications of the analytical results are discussed.