Flexible Bayesian modelling in dichotomous item response theory using mixtures of skewed item curves
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Most Item Response Theory (IRT) models for dichotomous responses are based on probit or logit link functions which assume a symmetric relationship between the responses and the latent traits of individuals submitted to a test. Such an assumption restricts the use of such models to situations in which all items have symmetric behavior. Similar constraint is imposed by the asymmetric models proposed in the literature as it is required that all items have an asymmetric behavior. Such assumptions are inappropriate for great part of the tests which, in general, are composed by both symmetric and asymmetric items. Furthermore, a straightforward extension of the existing models in the literature of 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 though parsimonious way. That is achieved by assigning a point-mass mixture prior to the skewness parameter of the item, allowing for an analysis under the model selection or model averaging approaches. Asymmetric item curves are design through the centred skew normal distribution which has a particularly appealing parametrisation in terms of parameter interpretation and computational efficiency. An efficient MCMC 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.
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