| Since the development of structural equation models (SEMs) in early 1970s there has been a tremendous growth in their uses in behavioral, educational, and social research. At present, there are more than ten packages in standard structural equation modeling to cope with the needs in various fields. Because there is a strong demand for some sophisticated models for more complicated theories and data structures, a number of useful generalizations have been proposed. The most important representatives are models with mixed continuous and polytomous variables, nonlinear models, multi-sample models, mixture models and models with incomplete data. Owing to the complexity of these models, the underlying statistical inferences are highly non-trivial. Thanks to the recent powerful tools in statistical computing, a number of important methods on estimation have been developed. Model selection (we use this term to include hypothesis testing and model comparison) is a very important topic beyond estimation. For example, very often investigators are required to identify a number of plausible models and choose the best one among them, and to compare different structures in cross cultural studies, etc. However, existing methods on hypothesis testing cannot be applied to the above mentioned important models, and moreover they are based on significance tests that have a number of key disadvantages. To overcome their difficulties and deficiencies, we will develop a Bayesian approach in this thesis for model selection that can be applied effectively to these complicated models. The statistic will be based on the Bayes factor. Efficient algorithms for computing the Bayes factors for various models will be developed using recent powerful tools in statistical computing; namely, the bridge sampling, the path sampling and some Markov chain Monte Carlo methods. Statistical properties of the proposed approach will be established and demonstrated. |
