Generalization of the Lord-Wingersky Algorithm to Computing the Distribution of Summed Test Scores Based on Real-Number Item Scores

Abstract

With known item response theory (IRT) item parameters, Lord and Wingersky provided a recursive algorithm for computing the conditional frequency distribution of number-correct test scores, given proficiency. This article presents a generalized algorithm for computing the conditional distribution of summed test scores involving real-number item scores. The generalized algorithm is distinct from the Lord-Wingersky algorithm in that it explicitly incorporates the task of figuring out all possible unique real-number test scores in each recursion. Some applications of the generalized recursive algorithm, such as IRT test score reliability estimation and IRT proficiency estimation based on summed test scores, are illustrated with a short test by varying scoring schemes for its items.