  This paper introduces the generalized Hausman test as a novel method for
detecting non-normality of the latent variable distribution of unidimensional
Item Response Theory (IRT) models for binary data. The test utilizes the
pairwise maximum likelihood estimator obtained for the parameters of the
classical two-parameter IRT model, which assumes normality of the latent
variable, and the quasi-maximum likelihood estimator obtained under a
semi-nonparametric framework, allowing for a more flexible distribution of the
latent variable. The performance of the generalized Hausman test is evaluated
through a simulation study and it is compared with the likelihood-ratio and the
M2 test statistics. Additionally, various information criteria are computed.
The simulation results show that the generalized Hausman test outperforms the
other tests under most conditions. However, the results obtained from the
information criteria are somewhat contradictory under certain conditions,
suggesting a need for further investigation and interpretation.
