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 the non-normality of the latent variable distribution of the unidimensional latent trait model for binary data. The test utilizes the pairwise maximum likelihood estimator for the parameters of the latent trait model, which assumes normality of the latent variable, and the maximum likelihood estimator obtained under a semi-non-parametric 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 compared with other test statistics available in the literature for testing latent variable distribution fit and an overall goodness-of-fit test statistic. Additionally, three information criteria are used to select the best-fitted model. 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. The proposed test statistics are used in three data sets.