For unidimensional item response theory (UIRT) models, three linking methods, which are the separate, concurrent, and fixed parameter calibration methods, have been developed and widely used in applications such as vertical scaling, differential item functioning, computerized adaptive testing (CAT), and equating. By contrast, even though a few studies have compared the separate and concurrent calibration methods for full multidimensional IRT (MIRT) models or applied the concurrent calibration method to vertical scaling using the bifactor model, no study has yet provided technical descriptions of the concurrent and fixed parameter calibration methods for any MIRT models. Thus, the purpose of this dissertation was to extend the concurrent and fixed parameter calibration methods for UIRT models to the two-tier item factor analysis model. In addition, the relative performance of the separate, concurrent, and fixed parameter calibration methods was compared in terms of the recovery of item parameters and accuracy of IRT observed score equating using both real and simulated datasets.
The separate, concurrent, and fixed parameter calibration methods well recovered the item parameters, with the concurrent calibration method performing slightly better than the other two linking methods. Despite the comparable performance of the three linking methods in terms of the recovery of item parameters, however, some discrepancy was observed between the IRT observed score equating results obtained with the three linking methods. In general, the concurrent calibration method provided equating results with the smallest equating error, whereas the separate calibration method provided equating results with the largest equating error due to the largest standard error of equating. The performance of the fixed parameter calibration method depended on the proportion of common items. When the proportion was , the fixed parameter calibration method provided more biased equating results than the concurrent calibration method because of the underestimated specific slope parameters. However, when the proportion of common items was 40%, the fixed parameter calibration method worked as well as the concurrent calibration method.