Stat 542: Item Response Theory Modeling Using The Extended Rank Likelihood

Researchers commonly apply item response theory (IRT) models to binary and ordinal items in order to measure an underlying trait using a collection of observed responses to test questions or items. For binary items, a common IRT model is the two parameter logistic (2PL) form. Assume that we have observed p item responses for n individuals where a correct responses is designated with a ’1’ and an incorrect response a ’0’. Let the matrix Y = {Yij : i = 1, . . . , n; j = 1, . . . , p} denote the set of binary item responses. The 2PL IRT form for the probability of a correct response on item j for participant i denoted by Yij = 1 is the following,