A Penalized Likelihood Method for Structural Equation Modeling and Its Asymptotic Properties

結構方程模型（structural equation modeling，簡稱SEM）乃心理學研究常用之多變量統計方法。在SEM的架構下，研究者可根據現有的心理學理論建立假設模型，並檢驗該模型之適切性；然而，當心理學理論發展尚未臻成熟時，SEM亦可能用以探索變項間的可能關係（Joreskog, 1993）。有鑒於實徵研究很可能同時兼具驗証性與探索性成分，以協助研究者對人類行為有更廣泛的了解，故此，本論文試圖提出一針對SEM模型的懲罰概似（penalized likelihood，簡稱PL）方法，以進行兼具驗証性與探索性成分之SEM分析。在此PL方法下，SEM的模型界定由驗証性與探索性兩部分所構成，前者包含了根據理論所推衍出來的變項關係與限制，後者則由一組被懲罰的參數（penalized parameters）所構成。此PL方法可產生稀疏估計值（sparse estimate），得以有效率地了解變項間關係，並控制最終模型的複雜度。為優化所提出的PL估計準則，本論文發展了期望條件最大化（expectation-conditional maximization，簡稱ECM）算則。透過大樣本理論，本研究建立PL於SEM的理論特性，包括PL估計式的局部與總體神諭性質（oracle property），以及赤池（Akaike）訊息指標與貝氏（Bayesian）訊息指標於PL的模型選擇特性。最後，本研究亦以模擬實驗與真實資料範例評估並展示此PL方法的實徵表現與應用價值。Structural equation modeling (SEM) is a commonly used multivariate statistical method in psychological studies. The application of SEM involves a confirmatory testing of the models proposed by researchers based on available theories. Yet, in practice, a model generating approach, where modifications of the models are being explored, may well take place (Joreskog, 1993), especially when the development of the substantive theory is still in its infancy. A method for SEM that can embrace the existing theories on one hand and the ambiguous relations that await further exploration on the other will be of great value to advancing scientific theories. In this dissertation, a penalized likelihood (PL) method for SEM is proposed as an attempt to target this goal. Under the proposed PL method, an SEM model is formulated with a confirmatory part and an exploratory part. The confirmatory part contains all the theory-derived relations and constraints. The exploratory part, wherein a set of penalized parameters is specified to represent the ambiguous relations, is data-driven yet with model complexity controlled by the penalty term. Through the sparse estimation of PL, the relationships among variables can be efficiently explored. As the penalty level is chosen appropriately, PL can lead to a SEM model that balances the tradeoff between model goodness-of-fit and model complexity. An expectation-conditional maximization (ECM) algorithm is developed to maximize the PL estimation criterion with several state-of-art penalty functions. Four theorems on the asymptotic behaviors of PL are derived, including the local and global oracle property of PL estimators and the selection consistency of Akaike and Bayesian information criterion. Two simulations are conducted to evaluate the empirical performance of the proposed PL method, and finally the practical utility of PL is demonstrated using two real data examples.1. Introduction...................................................................................1 1.1 Background and Motivation..........................................................1 1.2 Structural Equation Modeling.......................................................4 1.3 Penalized Likelihood....................................................................9 1.4 Purpose of the Dissertation........................................................15 2. A Penalized Likelihood Method for Structural Equation Modeling..17 2.1 PL Estimation Criterion...............................................................17 2.2 An Expectation-Conditional Maximization Algorithm.................23 2.3 Properties of the ECM Algorithm................................................27 2.4 Practical Considerations in Implementing PL...............................29 2.5 Real Data Examples....................................................................32 3. Asymptotic Properties of the Penalized Likelihood Method...........37 3.1 Notations and Settings...............................................................37 3.2 Asymptotic Properties of PL Estimators......................................39 3.3 Asymptotics of AIC and BIC........................................................48 4. Numerical Experiments................................................................53 4.1 Overview of the Numerical Experiments.....................................53 4.2 Simulation 1: True Models with Clear Patterns............................55 4.3 Simulation 2: True Models Including Minor Effects.....................67 5. General Discussion.......................................................................73 5.1 Main Results and Contributions..................................................73 5.2 Connecting PL with Related SEM Approaches..............................76 5.3 Limitations and Future Directions...............................................78 References.......................................................................................81 Appendices......................................................................................97 Appendix A. The E-Step of the ECM Algorithm.................................97 Appendix B. The CM-Steps for Regression-Type Coefficients...........97 Appendix C. The CM-Steps for Variances of Exogenous Variables....99 Curriculum Vitae............................................................................10