Latent variable models (LVMs), as useful multivariate techniques, have attracted significant attention from various fields, including the behavioral, educational, social-psychological, and medical sciences. In the analysis of LVMs, most existing statistical methods and software have been developed under the normal assumption of response variables. While some recent developments can partially address the non-normality of data, they are still problematic in dealing with highly non-normal data. Moreover, the presence of incomplete data, such as missing data and censoring data, is a practical issue in substantive research. Simply ignoring incomplete data or wrongly managing incomplete data might seriously distort statistical influence results. In this thesis, we develop a Bayesian P-spline approach, coupled with Markov chain Monte Carlo (MCMC) methods, to analyze transformation LVMs with highly non-normal and incomplete data. Different types of incomplete data, such as missing completely at random data, missing at random data, nonignorable missing data, as well as censored data, are discussed in the context of transformation LVMs. The deviance information criterion is proposed to conduct model comparison and select an appropriate missing mechanism. The empirical performance of the proposed methodologies is examined via many simulation studies. Applications to a study concerning people's job satisfaction, home life, and work attitude, as well as a study on cardiovascular diseases for type 2 diabetic patients in Hong Kong are presented.
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