| Structural equation models (SEMs) are commonly used in behavioral, educational, medical, and social sciences. Lots of software, such as EQS, LISREL, MPlus, and WinBUGS, can be used for the analysis of SEMs. Also many methods have been developed to analyze SEMs. One popular method is the Bayesian approach. An important issue in the Bayesian analysis of SEMs is model selection. In the literature, Bayes factor and deviance information criterion (DIC) are commonly used statistics for Bayesian model selection. However, as commented in Chen et al. (2004), Bayes factor relies on posterior model probabilities, in which proper prior distributions are needed. And specifying prior distributions for all models under consideration is usually a challenging task, in particular when the model space is large. In addition, it is well known that Bayes factor and posterior model probability are generally sensitive to the choice of the prior distributions of the parameters. Furthermore the computational burden of Bayes factor is heavy. Alternatively, criterion-based methods are attractive in the sense that they do not require proper prior distributions in general, and the computation is quite simple. One of commonly used criterion-based methods is DIC, which however assumes the posterior mean to be a good estimator. For some models like the mixture SEMs, WinBUGS does not provide the DIC values. Moreover, if the difference in DIC values is small, only reporting the model with the smallest DIC value may be misleading. In this thesis, motivated by the above limitations of the Bayes factor and DIC, a Bayesian model selection criterion called the Lv measure is considered. It is a combination of the posterior predictive variance and bias, and can be viewed as a Bayesian goodness-of-fit statistic. The calibration distribution of the Lv measure, defined as the prior predictive distribution of the difference between the Lv measures of the candidate model and the criterion minimizing model, is discussed to help understanding the Lv measure in detail. The computation of the Lv measure is quite simple, and the performance is satisfactory. Thus, it is an attractive model selection statistic. In this thesis, the application of the Lv measure to various kinds of SEMs will be studied, and some illustrative examples will be conducted to evaluate the performance of the Lv measure for model selection of SEMs. To compare different model selection methods, Bayes factor and DIC will also be computed. Moreover, different prior inputs and sample sizes are considered to check the impact of the prior information and sample size on the performance of the Lv measure. In this thesis, when the performances of two models are similar, the simpler one is selected. |
