Scale Linking Under a Mixture of Multidimensional IRT Models for Mixed-Format Tests

Under the context of the common-item nonequivalent groups design, item response theory (IRT) scale linking methods have been presented for each of the multidimensional extensions of the three-parameter logistic (M3PL), graded response (MGR), generalized partial credit (MGPC), and nominal response (MNR) models. The common-item multidimensional IRT (MIRT) linking methods that have been proved effective for single-format tests include the direct least squares (DLS), item category response function (ICRF), and test response function (TRF) methods. This paper extends the three MIRT linking methods to any mixture of the exploratory M3PL, MGR, MGPC, and MNR models for mixed-format tests. Each linking method extended attempts to find the linking coefficients in the rotation matrix and translation vector that constitute a linear transformation between two multidimensional ability scales. Simulations are conducted to investigate the performance of the three linking methods under various linking conditions. The results indicated that overall, the ICRF method performed best in estimating the rotation matrix and translation vector of the linear transformation between the new and true simulation scales. In not a few conditions, the DLS method resulted in the second best linking solutions. The TRF method tended to estimate the rotation matrix well but the translation vector poorly and thus showed the worst performance among the linking methods in some conditions.