A Bayesian skew mixture item response model

Abstract

Under the Item Response Theory, the two most common link functions used to model dichotomous data are the symmetric probit and logit. However, some authors have emphasized that these symmetric links do not always provide the best t for some data sets. To overcome this issue, asymmetric links have been proposed. This work aims at introducing a exible Item Response Model able to accommodate both symmetric and asymmetric link. The c.d.f. of a centered skew normal distribution is assumed as the link function and, additionally, we consider a nite mixture of Beta distributions and a point mass distribution at zero to describe the uncertainty about the skewness parameter, so not all items need to be assumed asymmetric a priori. Therefore, the proposed model embraces symmetric and asymmetric normal models in one also performing an intrinsic model selection. We o er the full condition distribution of ability, discrimination and dificulty parameters. We also introduce efficient algorithms to sample from the posterior distributions.