Asymptotic Standard Errors of IRT Linking Coefficients for the Bifactor Extension of the 3PL Model

Abstract

The bifactor extension of the three-parameter logistic (B3PL) model has been used in applications of multidimensional item response theory (IRT) such as test equating and vertical scaling. Developing a common multidimensional IRT scale (that is, multidimensional coordinate system) is critical in those applications. Three common-item scale linking methods, the direct least squares (DLS), mean/least squares (MLS), and item response function (IRF) methods, for the B3PL model have been found to be effective in developing a common multidimensional IRT ability scale between two test forms to be linked. In this paper, the asymptotic standard errors (SEs) of IRT linking coefficients estimated by the DLS, MLS, and IRF methods are derived assuming that the B3PL model holds and the asymptotic variance-covariance matrix of item parameter estimates from separate calibrations is available. The delta method is used for the derivations. Computer simulations which investigate the accuracy of the derivations under various conditions are given, showing that the derivations are reasonably accurate when sufficiently large samples are used and that in general the SEs of the IRF method are smaller than those of the DLS and MLS methods. The simulation results also suggest that the SEs of linking coefficient estimates are, approximately, inversely proportional to the square root of the sample size when two test forms are administered to the same number of examinees.