Slice-Gibbs sampling algorithm for estimating the parameters of a multilevel item response model

In a fully Bayesian framework, a novel slice-Gibbs algorithm is developed to estimate a multilevel item response theory (IRT) model. The advantage of this algorithm is that it can recover parameters well based on various types of prior distributions of the item parameters, including informative and non-informative priors. In contrast to the traditional Metropolis-Hastings (M-H) within Gibbs algorithm, the slice-Gibbs algorithm is faster and more efficient, due to its drawing the sample with acceptance probability as one, rather than tuning the proposal distributions to achieve the reasonable acceptance probabilities, especially for the logistic model without conjugate distribution. In addition, based on the Markov chain Monte Carlo (MCMC) output, two model assessment methods are investigated concerning the goodness of fit between models. The information criterion method on the basis of marginal likelihood is proposed to assess the different structural multilevel models, and the cross-validation method is used to evaluate the overall multilevel IRT models. The feasibility and effectiveness of the slice-Gibbs algorithm are investigated in simulation studies. An application using a real data involving students' mathematics test achievements is reported.