結構方程模式(Structural Equation Modeling, SEM) 是一個在社會學與心理學中極為廣用的分析方法，用以探討觀測值與潛在因素間其各自相依關係的一個重要工具，因此眾多的套裝軟體相繼而生。然而若資料型態不符合SEM 模型的假設，便不能進行SEM 的分析研究，儘管擁有與SEM 模型類似的模型構念。因此本文先基於SEM 模型的貝氏分析，提出一具時間序列的SEM 模型，使其也能充分的應用在時間序列的資料中。另外也對此模型加以提升改進，加入相關性的元素，讓模型更能滿足較多樣化的資料。此外，針對縱貫性資料之SEM模型分析，我們將採用隨機效果 (random effect) 的概念於SEM 模型中，使其充分的解釋資料既有的相關性。在實例分析中，具時間序列的SEM 模型對歐洲、亞洲以及美洲實際股價指數進行配適與預測，皆獲得不錯的解釋及準確的預測結果，另一方面，具縱貫性的SEM 模型對於我們所考慮的青少年左右眼資料之分析，亦有著相當良好的配適能力及解釋程度。最後，我們考慮在 SEM 模型之貝氏分析中，加入混合先驗分佈 (Mixture prior)來完成模型選擇問題，並對模型中之反應變數及隨機變數進行篩選，藉由剔除模型中不必要的關連性，來降低模型之參數個數，挑選出一具解釋力且較簡化之SEM模型。 Structural equation models (SEM) have been extensively used in behavioral, social, and psychological research to model relations between latent variables and observations. Most softwares for fitting SEM rely on frequentist methods. However, traditional models and softwares are not appropriate to analyze dependent observations such as the time-series data and multidimensional longitudinal data. We first introduce a structural equation model with time series feature via Bayesian approach with the aid of the Markov chain Monte Carlo method. Bayesian inference as well as prediction of the proposed time series structural equation model will be developed which can also reveal certain unobserved relationship among the observations. The approach is successfully employed using real Asian, American and European stock return data. Moreover, we consider to deal with the multidimensional longitudinal myopia data with correlation between both eyes. Myopia is becoming a significant public health problem, affecting more and more people. Motivated by the increase in the number of people affected by this problem, the primary focus is to utilize mathematical methods to gain further insight into their relationship with myopia. Accordingly, utilizing multidimensional longitudinal myopia data with correlation between both eyes, a Bayesian structural equation model including random effects is developed. Four observed factors, including intraocular pressure, anterior chamber depth, lens thickness and axial length, are considered. The results indicate that the genetic effect has much greater influence on myopia than the environmental effects. We also consider to apply the model selection problem with mixture prior to the SEMs. To put the reasonable mixture prior on the specific parameter which describes the doubting relationship in the SEMs, the model posterior probability can be computed via the MCMC iterations and viewed as a Bayesian model selection criterion. An advantage of the method using mixture priors is that it can automatically identify the predictors having non-zero fixed effect coefficients or non-zero random effects variance in the MCMC procedure. Specifically, we will focus on the multidimensional longitudinal myopia data to reduce the dimensionality of the parameter space and to select the simpler model.
