# 結構方程式模型

(由比較適合指數跳轉過嚟)

## 基本概念

### 迴歸分析

${\displaystyle DV=a+b(IV)}$（線性迴歸[英 5]）；
${\displaystyle DV=a+b(IV)+c(IV)^{2}}$（多項式迴歸[英 6]）；

... 等等。原則上，如果有個方法可以由過往數據嗰度搵出 ${\displaystyle a}$ 同埋 ${\displaystyle b}$參數嘅數值，第時就可以靠條式大致上用 IV 嘅值估計 DV 嘅值；統計學同機械學習等嘅領域上有好多演算法可以用嚟搵出呢啲參數嘅數值[7]

### 因素分析

1. 想像家陣手上個數據庫有若干個被觀察咗[英 8]隨機變數 ${\displaystyle x_{1},x_{2},...,x_{p}}$，而呢柞變數嘅平均值係 ${\displaystyle \mu _{1},\mu _{2},...,\mu _{p}}$
2. 想像有 ${\displaystyle k}$冇被觀察到[英 9]隱藏變數[英 10] ${\displaystyle F_{j}}$${\displaystyle j\in 1,...,k}$（呢柞 ${\displaystyle F_{j}}$ 係所謂嘅因素）[註 1]
3. 喺做因素分析前，${\displaystyle F_{j}}$ 嘅數值係未知，而因素分析嘅目的就係要搵出以下呢啲式當中嘅參數：
${\displaystyle x_{i}-\mu _{i}=l_{i1}F_{1}+\cdots +l_{ik}F_{k}+\varepsilon _{i}}$；當中
• ${\displaystyle i\in 1,...,p}$
• ${\displaystyle l_{ij}}$ 係參數；
• ${\displaystyle \varepsilon _{i}}$誤差，平均值係 0，而變異數係一個有限數值，唔同 ${\displaystyle i}$${\displaystyle \varepsilon _{i}}$ 變異數數值可以唔同[12]

## 建立模型

### 事前數據處理

${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle X_{3}}$ ${\displaystyle X_{4}}$
${\displaystyle X_{1}}$ ${\displaystyle 10}$ ${\displaystyle 6}$ ${\displaystyle 6}$ ${\displaystyle 6}$
${\displaystyle X_{2}}$ ${\displaystyle 6}$ ${\displaystyle 11}$ ${\displaystyle 6}$ ${\displaystyle 6}$
${\displaystyle X_{3}}$ ${\displaystyle 6}$ ${\displaystyle 6}$ ${\displaystyle 12}$ ${\displaystyle 6}$
${\displaystyle X_{4}}$ ${\displaystyle 6}$ ${\displaystyle 6}$ ${\displaystyle 6}$ ${\displaystyle 13}$

${\displaystyle \operatorname {cov} (X,Y)=\operatorname {E} {{\big [}(X-\operatorname {E} [X])(Y-\operatorname {E} [Y]){\big ]}}}$，當中
• ${\displaystyle \operatorname {cov} (X,Y)}$${\displaystyle x}$${\displaystyle y}$ 呢兩個變數之間嘅協方差；
• ${\displaystyle X}$ 係第 ${\displaystyle i}$ 個個案嘅 ${\displaystyle x}$ 數值；
• ${\displaystyle Y}$ 係第 ${\displaystyle i}$ 個個案嘅 ${\displaystyle y}$ 數值；
• ${\displaystyle \operatorname {E} [X]}$ 係啲個案喺 ${\displaystyle x}$ 上嘅平均值
• ${\displaystyle \operatorname {E} [Y]}$ 係啲個案喺 ${\displaystyle y}$ 上嘅平均值。

### 估計模型參數

${\displaystyle {\text{Pr}}(X|\theta )}$ 可以表達成[20]

${\displaystyle {\text{Pr}}(x_{1}\cap x_{2}\cap ...\cap x_{n}|\theta )}$ [註 4]

#### 最佳化

1. 睇吓自己身處嗰點周圍每個方向有幾斜，
2. 揀最能夠令自己向上爬嗰一個方向，移去嗰個方向，
3. 重複，直至某啲條件達到（例如 ${\displaystyle {\text{Pr}}(X|\theta )}$ 超過咗某個特定數值）為止。

#### 結構模型

${\displaystyle x_{i}-\mu _{i}=l_{i1}F_{1}+\cdots +l_{ik}F_{k}+\varepsilon _{i}}$

## 模型評估

### 適合度嘅指標

• 卡方檢定[英 29]，符號係 χ2 [註 6]：呢種做法將「個模型係正確嘅」當做 ${\displaystyle H_{0}}$虛無假說），並且攞「個模型嘅協方差矩陣」同「實際觀察到嘅協方差矩陣」做卡方檢定（下面係變數有三個咁多嘅情況）－如果 χ2 愈大，就表示兩個矩陣之間嘅差異愈大，研究者就愈有理由相信個模型係錯嘅[29][30]。卡方檢定可以話係 SEM 適合度嘅基礎－RMSEA 同 CFI 都係由卡方檢定計出嚟嘅[31][32]

${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle X_{3}}$
${\displaystyle X_{1}}$ ${\displaystyle 5.88}$ . .
${\displaystyle X_{2}}$ ${\displaystyle 2.88}$ ${\displaystyle 7.16}$ .
${\displaystyle X_{3}}$ ${\displaystyle 3.12}$ ${\displaystyle 4.44}$ ${\displaystyle 8.14}$

${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle X_{3}}$
${\displaystyle X_{1}}$ ${\displaystyle 6.42}$ . .
${\displaystyle X_{2}}$ ${\displaystyle 1.24}$ ${\displaystyle 8.25}$ .
${\displaystyle X_{3}}$ ${\displaystyle 4.26}$ ${\displaystyle 2.75}$ ${\displaystyle 7.99}$
• 近似值根均方誤差[英 30]，英文簡稱 RMSEA：一個數值愈低愈好嘅適合度指標；RMSEA 最細嘅可能數值係 0，而一般認為，RMSEA 數值喺 0.1 或者以上嘅話個模型嘅適合度就算係低到唔可以接受[33]
• 標準化根均殘差[英 31]，英文簡稱 SRMR：另一個數值愈低愈好嘅適合度指標；一般認為，SRMR 嘅數值最好係喺 0.1 以下[2]，亦都有統計學家主張 SRMR 數值要喺 0.08 以下個模型先可以算係有充足嘅適合度[29]
• 比較適合指數[英 32]，英文簡稱 CFI：一個主要反映數據當中嘅統計相關嘅大細嘅適合度指標，所以數值係愈高愈好；一般嚟講，CFI 嘅數值過到 0.95，個模型就算係可以接受[29]

... 等等。

## 多組分析

### 量度不變特性

• ${\displaystyle c_{1}}$ 表示由喺美國得到嘅數據造出嚟嘅 ${\displaystyle c}$ 量度模型、
• ${\displaystyle c_{2}}$ 表示由喺歐洲得到嘅數據造出嚟嘅 ${\displaystyle c}$ 量度模型、同埋
• ${\displaystyle c_{3}}$ 表示由喺日本得到嘅數據造出嚟嘅 ${\displaystyle c}$ 量度模型、

### 約束分析

• 首先俾個程式完全自由噉建立一個結構模型先，得出一個 χ2 值；
• 跟住佢作出約束，要求個程式喺「假設兩組嘅嗰一個 ${\displaystyle \beta _{xy}}$ 數值一樣」嘅情況下建立一個結構模型，又得出一個 χ2 值，
• 如果呢個 χ2 值明顯大過[註 7]打前嗰個，就表示「假設兩組嘅嗰一個 ${\displaystyle \beta _{xy}}$ 數值一樣」會搞到個結構模型嘅適合度明顯變差－噉個分析者就有理由相信「兩組嘅嗰一個 ${\displaystyle \beta _{xy}}$ 數值有差異」，可以作出「『屬實驗組定對照組』呢個變數會對 ${\displaystyle X}$${\displaystyle Y}$ 之間嘅關係有調節效應」嘅推論[38]

## 統計功效

${\displaystyle p=P{\big (}}$成功拒絕 ${\displaystyle H_{0}\mid H_{1}}$ 係真 ${\displaystyle {\big )}}$

• 大過 200 [42]
• 係要估計嘅參數數量嘅最少 10 倍[43]、而且
• 係變數數量嘅最少 10 倍[44]

## 軟件

• LISREL
• R 程式語言當中都有軟件包做結構方程式模型。
• SPSS 當中嘅軟件包「AMOS」專係整嚟做結構方程式模型嘅。

## 文獻

• Bowen, N. K., & Guo, S. (2011). Structural equation modeling. Oxford University Press.
• Elliott, M. R. (2003). Causality and how to model it (PDF). BT technology journal, 21(2), 120-125.
• Kaplan, D. (2008). Structural Equation Modeling: Foundations and Extensions (2nd ed.). SAGE. ISBN 978-1412916240.
• Kline, Rex (2011). Principles and Practice of Structural Equation Modeling (3rd ed.). Guilford. ISBN 978-1-60623-876-9.
• Werner C., & Schermelleh-Engel, K. (2010). Deciding Between Competing Models: Chi-Square Difference Tests. Introduction to Structural Equation Modeling with LISREL.

• Bagozzi, Richard P; Yi, Youjae (2011). "Specification, evaluation, and interpretation of structural equation models". Journal of the Academy of Marketing Science. 40 (1): 8–34.
• Hox, J. J., & Bechger, T. M. (1998). An introduction to structural equation modeling. Family Science Review, 11, 354-373.
• Hu, Li‐tze; Bentler, Peter M (1999). "Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives". Structural Equation Modeling: A Multidisciplinary Journal. 6: 1-55.
• MacCallum, Robert; Austin, James (2000). "Applications of Structural Equation Modeling in Psychological Research". Annual Review of Psychology. 51: 201-226.
• Quintana, Stephen M.; Maxwell, Scott E. (1999). "Implications of Recent Developments in Structural Equation Modeling for Counseling Psychology". The Counseling Psychologist. 27 (4): 485-527.
• Tarka, P. (2018). An overview of structural equation modeling: its beginnings, historical development, usefulness and controversies in the social sciences (PDF). Quality & quantity, 52(1), 313-354.
• Wulff, J. N., Sajons, G. B., Pogrebna, G., Lonati, S., Bastardoz, N., Banks, G. C., & Antonakis, J. (2023). Common methodological mistakes. The Leadership Quarterly, 34(1), 101677，當中其中幾段有講到啲人做 SEM 分析嗰陣嘅常犯錯誤。

## 註釋

1. ${\displaystyle a\in b}$」意思係「${\displaystyle a}$${\displaystyle b}$ 呢個入面」。
2. 可以睇吓迴歸分析
3. 又有做法會事先將數據庫入面每個數據嘅數值減以相應嘅平均值，噉 ${\displaystyle \operatorname {cov} (X,Y)}$ 就會簡化變成 ${\displaystyle \operatorname {E} {{\big [}XY{\big ]}}}$
4. 喺實際應用上，考慮咁多極細嘅數值可能會引起算術下溢（指要處理嘅數值細過部電腦能夠表示嘅最細值），所以「要點樣計 ${\displaystyle {\text{Pr}}(X|\theta )}$」呢家嘢有一定嘅學問。
5. 順帶一提，通徑分析基本上就係冇量度模型、齋用觀察到嘅變數嚟建立結構模型嘅 SEM。
6. 卡方檢定喺樣本過要估計嘅參數數量（${\displaystyle df>0}$）嗰陣先會有用，而且當樣本大得滯嗰陣，卡方檢定會變得靠唔住。
7. 點樣算係「明顯大過」都有學問。

## 詞彙

1. regression analysis
2. continuous
3. independent variable；IV
4. dependent variable；DV
5. linear regression
6. polynomial regression
7. factor analysis
8. observed
9. unobserved
10. latent variable
11. factor structure
12. intelligence
14. measurement model
15. structural model
16. iid
17. covariance matrix
18. measurement model
19. maximum likelihood estimation
20. probability function
21. optimization
23. structural model
24. confirmatory factor analysis
25. exploratory factor analysis
27. goodness of fit / model fit
28. fit indices
29. Chi-squared test
30. Root Mean Square Error of Approximation，RMSEA
31. Standardized Root Mean Residual，SRMR
32. Comparative Fit Index，CFI
33. modification indices
34. multigroup analysis
35. moderation effect
36. measurement invariance
37. constraint
38. statistical power

## 引咗

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