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Structural Equation Modeling

Structural Equation Modeling. Estimates and Tests in SEM. 資料來源 : 結構方程式 LISREL 的理論技術與應用 2003 邱皓政. 外衍觀測變數.  觀測變數. 內衍觀測變數. 外衍變數 exogenous variables (X).  潛在變數. 內衍潛在變數. 外衍潛在變數. 內衍變數 endogenous variables (Y). 外衍觀測變數.  觀測變數. 內衍觀測變數. Terminology.

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Structural Equation Modeling

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  1. Structural Equation Modeling Estimates and Tests in SEM

  2. 資料來源:結構方程式 LISREL 的理論技術與應用 2003邱皓政

  3. 外衍觀測變數  觀測變數 內衍觀測變數 外衍變數 exogenous variables (X) 潛在變數 內衍潛在變數 外衍潛在變數 內衍變數 endogenous variables (Y) 外衍觀測變數  觀測變數 內衍觀測變數 Terminology 資料來源:結構方程式 LISREL 的理論技術與應用 2003邱皓政

  4.              y y      x x Parameter in Structural and Measurement Model ■ 結構模型方程式 y ■ 變項測量模型方程式 x ■ 變項測量模型方程式 資料來源:結構方程式 LISREL 的理論技術與應用 2003邱皓政

  5. 資料來源:結構方程式 LISREL 的理論技術與應用 2003邱皓政

  6. Statistical Theory of Covariance 資料來源:結構方程式 LISREL 的理論技術與應用 2003邱皓政

  7. Generating the Covariance Matrix 資料來源:結構方程式 LISREL 的理論技術與應用 2003邱皓政

  8. Variance Covariance 資料來源:結構方程式 LISREL 的理論技術與應用 2003邱皓政

  9. cov(v1,v1) cov(v2,v1) cov(v3,v1) cov(v4,v1) cov(v5,v1) cov(v6,v1) cov(v1,v2) cov(v2,v2) cov(v3,v2) cov(v4,v2) cov(v5,v2) cov(v6,v2) Σ= cov(v1,v3) cov(v2,v3) cov(v3,v3) cov(v4,v3) cov(v5,v3) cov(v6,v3) cov(v1,v4) cov(v2,v4) cov(v3,v4) cov(v4,v4) cov(v5,v4) cov(v6,v4) cov(v1,v5) cov(v2,v5) cov(v3,v5) cov(v4,v5) cov(v5,v5) cov(v6,v5) cov(v1,v6) cov(v2,v6) cov(v3,v6) cov(v4,v6) cov(v5,v6) cov(v6,v6) Estimation of Parameters in Covariance Matrix Variables: v1 v2 v3 v4 v5 v6 p: number of variables in covariance matrix p(p+1)/2: non-redundant elements in the sample covariance matrix q: number of free parameters q<=p(p+1)/2 T- rule

  10. Simultaneous equations x+y=5 ---------- (1) 2x+y=8----------(2) x+2y=9----------(3) • Only (1)  (c, 5-c) • (1) and (2)  (3,2) • (1) , (2) , (3)  no exact solution under-identified just identified over-identified

  11. Smallest absolute difference from the constants (2.273,3.273) (2.333,3.333) Smallest squared difference (3,2) Y criterion X

  12. Example :T rule Variables: v1 v2 v3 v4 v5 v6 Simultaneous Equations Cov(v1,v1)=λ12+θ1 Cov(v3,v3)= λ32+θ3 Cov(v1,v2)=λ1λ2 Cov(v3,v4)= λ3λ6φ21 Cov(v1,v3)= λ1λ3 Cov(v3,v5)= λ3λ5φ21 Cov(v1,v4)= λ1λ4φ21 Cov(v3,v6)= λ3λ6φ21 Cov(v1,v5)=λ1λ5φ21 Cov(v4,v4)= λ42+θ4 Cov(v1,v6)=λ1λ6φ21 Cov(v4,v5)= λ4λ5 Cov(v2,v2)= λ22+θ2 Cov(v4,v6)= λ4λ6 Cov(v2,v3)= λ2λ3 Cov(v5,v5)= λ52+θ5 Cov(v2,v4)= λ2λ4φ21 Cov(v5,v6)= λ5λ6 Cov(v2,v5)= λ2λ5φ21 Cov(v6,v6)= λ62+θ6 Cov(v2,v6)= λ2λ6φ21 6(6+1)/2 =15 13 Estimate parameter λ1,λ2λ3λ4λ5λ6φ21θ1 θ2 θ3 θ4 θ5 θ6 13<=6(6+1)/2  T rule

  13. Estimation methods • Different estimation methods will yield different results for estimates and model tests. (Chapter 5 ) • Normal distribution assumption • ML:maximum likelihood • GLS: generalized least squares • ADF: asymptotic distribution free method --- non-normal distribution most commonly used

  14. Fitting Functions • The criterion selected for parameter estimation is AKA the discrepancy function. • F=F(S, Σ(hat θ))

  15. Maximum likelihood • 觀察數據都是從母體中抽取得到的資料，而所抽出的樣本必須是所有可能樣本中被選擇的機率的最大者，若能符合此一假設，估計的參數即能反應母體的參數。 • FML=log| Σ(θ)| + Trace[Σ(θ)-1S] – log|S| - p p : number of measured variable if Σ(θ)=S then Trace[Σ(θ)-1S] =Trace(I)=p  FML=0

  16. Generalized Least Squares • 使用差異平方和的概念，只是在計算每一個差異值時，同時計算了一個特定的權數用以整合個別的比較值。 資料來源:結構方程式 LISREL 的理論技術與應用 2003邱皓政

  17. Asymptotic Distribution Free • 一種無須常態假設為基礎的參數估計法，由於不需考慮常態分配的問題，因此稱為分配自由（free）。利用W-1權數，來消除多變量常態假設的影響。 資料來源:結構方程式 LISREL 的理論技術與應用 2003邱皓政

  18. estimated Population (sample) Goodness-of-fit Similarity Σ( hat θ) ~ Σ H0: Σ(θ)= Σ

  19. Testing(1) T=Min(F)*(N-1) ~ X2 distribution • q parameters p(p+1)/2 equations • df=(q-p(p+1)/2 ) >0  testable model • H0: Σ(θ)= Σ  significance  reject H0 • Large sample assumption

  20. Testing(2) • Scaled test (non-normal distribution) • SCALED T=c-1T • T : function of standard goodness-of-fit X2

  21. Practical problems • Covariance matrix is not positive definite– linear dependency among observed variables. • Non-convergence – iterative process is terminated before the predetermined criteria have been satisfied.

  22. Thank you The End

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