A Generalized Definition of Multidimensional Item Response Theory Parameters

In this paper, we generalize the multidimensional discrimination (MDISC) and difficulty (MID) parameters in the Multidimensional 2-Parameter Logistic model to account for non-identity latent covariances and negatively-keyed items. We apply Reckase’s maximum discrimination point method to define them in an algebraic sense in an arbitrary basis, and then defining such basis to be a geometrical representation of the measured construct. This results in three different versions of the parameters: The original one, based on the item parameters solely, one that incorporates the covariance structure of the latent space, and one that uses the correlation structure instead. Importantly, we find that the items should be properly represented in a test space, distinct from the latent space. We also provide a procedure for the geometrical representation of the items in the test space, and apply our results to examples from the literature to get a more accurate representation of the measurement properties of the items. We recommend using the covariance structure version for describing the properties of the parameters, and the correlation structure version for graphical representation. Finally, we discuss the implications of this generalization for other multidimensional IRT models and the parallels of our results in common factor model theory.