IRT Linking and Standard Errors of Linking Coefficients Under the Reparameterized Nominal Response Model

Abstract

Under item response theory (IRT), common-item linking methods are used to develop a common ability scale between two test forms administered to examinee groups from different populations. The nominal response (NR) model, proposed first by Bock, was reparameterized by Thissen and his colleagues. This paper presents three types of IRT linking methods, the direct least squares (DLS), mean/least squares (MLS), and item category response function (ICRF) methods, for the reparameterized NR model and investigates their performance through computer simulations. The presentation assumes that the highest response categories are specified for all linking items. This paper also presents analytic formulas for computing the asymptotic standard errors (SEs) of linking coefficient estimates for the three methods. Important findings were obtained from a simulation study. Overall, the ICRF method outperformed the DLS and MLS methods in linking accuracy, and the DLS and MLS methods performed almost equally. The linking coefficients for the DLS and MLS methods should be estimated using item parameter estimates only for the highest response categories whereas those for the ICRF method should be estimated by the criterion function that is defined using all ICRFs across linking items. The analytic formulas for the asymptotic SEs worked properly for the three linking methods, and the SEs were, approximately, inversely proportional to the square root of the sample size.