A Definition of a Heywood Case in Item Response Theory Based on Fisher Information

Heywood cases and other improper solutions occur frequently in latent variable models, e.g., factor analysis, item response theory, latent class analysis, multilevel models, or structural equation models, all of which are models with response variables taken from an exponential family. They have important consequences for scoring with the latent variable model and are indicative of issues in a model, such as poor identification or model misspecification. In the context of the 2PL and 3PL models in IRT, they are more frequently known as Guttman items and are identified by having a discrimination parameter that is deemed excessively large. Other IRT models, such as the newer asymmetric item response theory (AsymIRT) or polytomous IRT models often have parameters that are not easy to interpret directly, so scanning parameter estimates are not necessarily indicative of the presence of problematic values. The graphical examination of the IRF can be useful but is necessarily subjective and highly dependent on choices of graphical defaults. We propose using the derivatives of the IRF, item Fisher information functions, and our proposed Item Fraction of Total Information (IFTI) decomposition metric to bypass the parameters, allowing for the more concrete and consistent identification of Heywood cases. We illustrate the approach by using empirical examples by using AsymIRT and nominal response models.