A statistical test to assess the non - normality of the latent variable distribution

This paper presents the generalized Hausman test to detect non-normality of the latent variable distribution in unidimensional Item Response Theory (IRT) models for binary data. The test is based on the estimators resulting from the two- parameter IRT model, that assumes normality of the latent variable, and the semi- nonparametric IRT model, that assumes a more flexible latent variable distribution. The performance of the test is evaluated through a simulation study, including the cases where the latent variable is generated from a skew-normal and mixture of nor- mals. The results highlight the good performance of the test when the latent variable is generated from a mixture of normals and from a skew-normal only with many items and large sample sizes.