Heywood cases in unidimensional factor models and item response models for binary data

Heywood cases are known from linear factor analysis literature as variables with communalities larger than 1.00, and in present day factor models, the problem also shows in negative residual variances. For binary data, ordinal factor models can be applied with either delta parameterization or theta parametrization. The former is more common than the latter and can yield Heywood cases when limited information estimation is used. The same problem shows up as nonconvergence cases in theta parameterized factor models and as extremely large discriminations in item response theory (IRT) models. In this study, we explain why the same problem appears in different forms depending on the method of analysis. We first discuss this issue using equations and then illustrate our conclusions using a small simulation study, where all three methods, delta and theta parameterized ordinal factor models (with estimation based on polychoric correlations) and an IRT model (with full information estimation), are used to analyze the same datasets. We also compared the performances of the WLS, WLSMV, and ULS estimators for the ordinal factor models. Finally, we analyze real data with the same three approaches. The results of the simulation study and the analysis of real data confirm the theoretical conclusions.