  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.
