Bayesian Dependence Structure Analysis for Ordinal Data

This dissertation explores different methods to study the dependence structure among many ordinal variables under the Bayesian framework. Chapter 1 introduces ordinal data analysis methods, and the related literature works are briefly reviewed. An outline of the dissertation is put forward. In Chapter 2, Gaussian copula graphical models with different priors of graphical Lasso, adaptive graphical Lasso, and spike-and-slab Lasso on the precision matrix are assessed and compared. The proposed models are well illustrated via simulations and a real ordinal survey data analysis. In Chapter 3, adaptive spike-and-slab Lasso prior is proposed as an extension of Chapter 2. The developed adaptive spike-and-slab prior yields good results based on the simulation study. An improved simulation setting is utilized in this chapter compared to that in Chapter 2. Thus better guidance is achieved for the dependence structure learning in real data analysis. Chapter 4 applies a Bayesian factor analysis model with Gaussian copula to the TSCC (Trauma Symptom Checklist for Children) ordinal data. The variable structure is investigated with a global-factor-local shrinkage before the factor loading matrix. The results with different numbers of factors are compared, and the corresponding estimates of the covariance matrix are obtained and compared with those obtained using the Gaussian copula graphical models in Chapters 2 and 3. In Chapter 5, a summary of the studies in the previous chapters is presented, and at the same time, we put forward some ideas for future work