The generalized Hausman test for detecting non-normality in the latent variable distribution of the two-parameter IRT model

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.