This document discusses structural equation modeling (SEM) and its applications. SEM allows analyzing relationships between independent and dependent variables that can be continuous or discrete. It involves two main components: a measurement model using confirmatory factor analysis to represent unobserved latent variables, and a structural model to represent paths between variables. Model examination involves assessing fit indices, factor loadings, errors, and regression coefficients. Mediation can be examined using SEM by analyzing indirect paths from an independent variable to a dependent variable through a mediator variable. Bootstrapping provides a more accurate method than the Sobel test for estimating standard errors and confidence intervals in mediation models.

1. STRUCTURAL
EQUATIONS
MODELING (SEM)
Wei-Jiun, Shen Ph. D.

2. Measurement model
 Confirmatory factor analysis (CFA)
x1
x3
x2
δ1
δ 2
δ 3
ξ1
λ11
λ21
λ31
δ[delta] ; λ[lambda]; ξ[xi]
x1= λ11 ξ1+ δ1
x2= λ21 ξ1+ δ2
x3= λ31 ξ1+ δ3

3. After CFA…
 The goal of SEM is to analyze a set of
relationships between one or more IVs, either
continuous or discrete, and one or more DVs,
either continuous or discrete.
 Model examination
 Observed variable only
 Latent variable

4. Graphical representation for SEM
x1
x2
x3
ξ belief η behavior
y1
y2
y3
δ１ ε１
λx1
γ
ζ
ε２
ε３
δ２
δ３
Error
Exogenous
observed
variable
Factor loading
Exogenous
latent variable
Structural parameter
Endogenous
latent variable
Factor loading
Endogenous
observed
variable
Structural
model
measurement
model
Error
λx2
λx3
λy1
λy2
λy3
measurement
model

5. Model examination
 Fit index
 Absolute fit index
 Comparative fit index
 Parsimonious fit index
 Residual
 Coefficients
 Loading
 Error
 Correlation/Regression coefficient

6. PRACTICE

7. 個人表現與團隊表現
 史蒂分∙咖哩的助攻及得分與好鬥者隊的團隊表
現是否有關？ 。

8. 個人表現與團隊表現
 以路徑分析檢驗助攻、得分對團隊表現的預測情
形。分析結果顯示助攻能顯著正向預測團隊表現
(β= .44, p < .001)，得分則不會影響團隊表現
(β= .01, p = .777)。

9. MEDIATION
EXAMINATION
APPLYING
BOOTSTRAPPING

10. Mediation
 SEM
 Observed variables
X Y
M
βa βb
βc

11. Mediation
 SEM
 Observed and latent variables
X Y
M
βa βb
βc

12. Mediation examination
 B-K
 Regression coefficients
Visual examine?
 Sobel test
 Test the difference between change of coefficient
Biased SE?
 Bootstrap
 More accurate estimation

13. Bootstrap
 Resampling technique
 Applied to mediation model
 SE
 Confidence interval

14. Confidence interval
 Interval estimation
𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝐶𝐼 = 𝑀 𝑥 ± 𝑡∗
× 𝑆𝐸 𝑥
95% 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝐶𝐼 = 𝑀 𝑥 ± 1.96 × 𝑆𝐸 𝑥

15. Why bootstrap provides more accurate SE?
 Standard error
 The standard deviation of the sampling distribution
of a statistic
𝑆𝐸 𝑥 =
𝑠
𝑛
Representative sample

16. PRACTICE