Flexible Bayesian modelling in dichotomous item response theory using mixtures of skewed item curves.

Most item response theory (IRT) models for dichotomous responses are based on probit or logit link functions which assume a symmetric relationship between the probability of a correct response and the latent traits of individuals taking a test. This assumption restricts the use of those models to the case in which all items behave symmetrically. On the other hand, asymmetric models proposed in the literature impose that all the items in a test behave asymmetrically. This assumption is inappropriate for great majority of tests which are, in general, composed of both symmetric and asymmetric items. Furthermore, a straightforward extension of the existing models in the literature would require a prior selection of the items' symmetry/asymmetry status. This paper proposes a Bayesian IRT model that accounts for symmetric and asymmetric items in a flexible but parsimonious way. That is achieved by assigning a finite mixture prior to the skewness parameter, with one of the mixture components being a point mass at zero. This allows for analyses under both model selection and model averaging approaches. Asymmetric item curves are designed through the centred skew normal distribution, which has a particularly appealing parametrization in terms of parameter interpretation and computational efficiency. An efficient Markov chain Monte Carlo algorithm is proposed to perform Bayesian inference and its performance is investigated in some simulated examples. Finally, the proposed methodology is applied to a data set from a large-scale educational exam in Brazil.