Latent trait models (LTMs) for binary data analysis must address the issue of model determination, which divides into two components: model assessment and model selection. In this thesis, a Bayesian framework is presented for assessing model adequacy and selecting the most plausible model. Simulated and real data sets with different item numbers, degrees of sparseness, sample sizes and factor dimensions are studied to investigate the performance of the proposed procedure. Limited information statistics have been recommended as the goodness-of-fit measures in sparse 2k contingency tables, but the p-values of these test statistics are computationally difficult to obtain. A Bayesian model diagnostic tool, Relative Entropy - Posterior Predictive Model Checking (RE-PPMC), is proposed to assess the global fit for LTMs. This approach utilizes the relative entropy (RE) to resolve possible problems in the original PPMC procedure based on the posterior predictive p-value (PPP-value). Compared with the typical conservatism of PPP-value, the RE value measures the discrepancy effectively. RE-PPMC is much more capable of evaluating model adequacy than parametric bootstrapping. The determination of the number of factors is another unresolved problem in LTMs. Under the Bayesian framework, the criteria used are the marginal posterior probabilities which involve high dimensional integrals. The integrals are usually computationally intractable and hence the Laplace expansion is utilized. However, the unidentifiable parameters involved prohibit the computation of the Hessian matrix in Laplace approximation. A new asymptotic expansion with polar coordinate transformation is thus introduced, separating identifiable and unidentifiable parameters and allowing the marginal posterior to be computed. The simulation study shows that the proposed method always selects the correct number of factors under different conditions, and performs better than Schwarz’s Bayesian information criterion (BIC) and the simulation approach based on Markov chain Monte Carlo (MCMC).