Missing responses generally exist in educational and psychological assessments. The statistical inference will lead to serious deviation if the missing responses are not properly modeled in the framework of non-ignorable missing mechanism. In this current study, it is studied whether the different missing mechanism (ignorable missing and non-ignorable missing) models are appropriate to analyze the missing response data from the perspective of parameter estimation and model assessment. In addition, a highly effective Bayesian sampling algorithm based on auxiliary variables is used to estimate the complex models. Compared with the traditional marginal likelihood method and other Bayesian algorithms, the advantages of the new algorithm are discussed in detail. Based on the Markov Chain Monte carlo samples from the posterior distributions, the deviance information criterion (DIC) and the logarithm of the pseudomarignal likelihood (LPML) are employed to compare the different missing mechanism models. Four simulation studies are conducted and a detailed analysis of PISA science data is carried out to further illustrate the proposed methodology.