The fully Bayesian estimation via the use of Markov chain Monte Carlo (MCMC) techniques has become popular for estimating item response theory (IRT) models. The current development of MCMC includes two major algorithms: Gibbs sampling and the No-U-Turn sampler (NUTS). While the former has been used with fitting various IRT models, the latter is relatively new, calling for the research to compare it with other algorithms. The purpose of the present study is to evaluate the performances of these two emerging MCMC algorithms in estimating two two-parameter logistic (2PL) IRT models, namely, the 2PL unidimensional model and the 2PL multi-unidimensional model under various test situations. Through investigating the accuracy and bias in estimating the model parameters given different test lengths, sample sizes, prior specifications, and/or correlations for these models, the key motivation is to provide researchers and practitioners with general guidelines when it comes to estimating a UIRT model and a multi-unidimensional IRT model. The results from the present study suggest that NUTS is equally effective as Gibbs sampling at parameter estimation under most conditions for the 2PL IRT models. Findings also shed light on the use of the two MCMC algorithms with more complex IRT models.

In this report we give a brief introduction to Item Response Theory models and multilevel models. The general assumptions of two classical Item Response Theory, 1PL and 2PL models are discussed. We follow the discussion by introducing a multilevel level framework for these two Item Response Theory Models. We explain Bock and Aitkin's (1981) work to estimate item parameters for these two models. Finally we illustrate these models with a LSAT exam data and two statistical softwares; R project and Stata. / text