Formal Relations and an Empirical Comparison among the Bi-Factor,the Testlet,and a Second-Order Multidimensional IRT Model

Testlet effects can be taken into account by incorporating specific dimensions in addition to the general dimension into the item response theory model. Three such multidimensional models are described: the bi-factor model, the testlet model, and a second-order model. It is shown how the second-order model is formally equivalent to the testlet model. In turn, both models are constrained bi-factor models. Therefore, the efficient full maximum likelihood estimation method that has been established for the bi-factor model can be modified to estimate the parameters of the two other models. An application on a testlet-based international English assessment indicated that the bi-factor model was the preferred model for this particular data set.