---
id: wiki-2026-0508-structural-equation-modeling
title: Structural Equation Modeling
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [SEM, Path-Analysis, CFA-SEM]
duplicate_of: none
source_trust_level: A
confidence_score: 0.88
verification_status: applied
tags: [statistics, sem, latent-variable, causal, psychometrics]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python-r
  framework: semopy-lavaan
---

# Structural Equation Modeling

## 매 한 줄
> **"매 latent + 매 path 의 통합"**. SEM 의 measurement model (CFA: 매 observed → 매 latent) + structural model (매 latent 사이 의 path) 의 simultaneously fit. 1970s Jöreskog (LISREL) — 2026 에 psychology, marketing research, epidemiology 의 표준.

## 매 핵심

### 매 component
- **Measurement model (CFA)**: 매 observed indicator → 매 latent factor (e.g., 4 items → "Anxiety").
- **Structural model**: 매 latent 사이 의 directional path (e.g., Anxiety → Performance).
- **Latent variable**: 매 unobserved construct — measurement error 의 separate.
- **Indicator**: observed measurement — 매 reflective (factor causes indicator) vs formative.

### 매 estimation
- **MLE** (default) — 매 multivariate normal assumption.
- **WLSMV** — 매 ordinal/categorical data.
- **Bayesian SEM** — 매 small sample, complex model.

### 매 fit indices
- **χ² / df** — 매 < 3 의 양호.
- **CFI / TLI** — 매 ≥ 0.95 의 양호.
- **RMSEA** — 매 ≤ 0.06 의 양호, ≤ 0.08 의 acceptable.
- **SRMR** — 매 ≤ 0.08 의 양호.
- 매 single index 의 X — 매 multiple 의 결합 의 평가.

### 매 응용
1. Psychology — Big Five, depression-anxiety pathway.
2. Marketing — brand → satisfaction → loyalty.
3. Education — SES → study time → achievement.
4. Epidemiology — stress → cortisol → disease.

## 💻 패턴

### 1. semopy (Python) — basic CFA
```python
import semopy
desc = """
# measurement
Anxiety =~ q1 + q2 + q3 + q4
Depression =~ d1 + d2 + d3
# structural
Depression ~ Anxiety
"""
model = semopy.Model(desc)
res = model.fit(df)
print(semopy.calc_stats(model))
```

### 2. lavaan (R) — gold standard
```r
library(lavaan)
mod <- '
  # measurement
  visual  =~ x1 + x2 + x3
  textual =~ x4 + x5 + x6
  speed   =~ x7 + x8 + x9
  # structure
  textual ~ visual
  speed   ~ textual
'
fit <- sem(mod, data=HolzingerSwineford1939)
summary(fit, fit.measures=TRUE, standardized=TRUE)
```

### 3. Mediation analysis
```r
mod <- '
  Y ~ c*X + b*M
  M ~ a*X
  ab := a*b              # indirect effect
  total := c + ab
'
fit <- sem(mod, data=df, se="bootstrap", bootstrap=5000)
parameterEstimates(fit, boot.ci.type="bca.simple")
```

### 4. Multi-group invariance
```r
fit_config <- cfa(mod, data=df, group="country")
fit_metric <- cfa(mod, data=df, group="country", group.equal="loadings")
anova(fit_config, fit_metric)  # 매 fit 의 worsen 의 test
```

### 5. Latent growth curve
```r
mod <- '
  i =~ 1*t1 + 1*t2 + 1*t3 + 1*t4
  s =~ 0*t1 + 1*t2 + 2*t3 + 3*t4
  i ~~ s
'
fit <- growth(mod, data=df)
```

### 6. Modification indices
```r
modindices(fit, sort=TRUE, maximum.number=10)
# 매 model 의 어떤 path 의 추가 시 fit 의 향상 의 예측
# 매 theory-driven X — 매 cherry-pick 의 위험
```

### 7. Bootstrap CI for indirect effect
```r
fit <- sem(mod, data=df, se="bootstrap", bootstrap=5000)
parameterEstimates(fit, boot.ci.type="bca.simple", standardized=TRUE)
```

### 8. Bayesian SEM (blavaan)
```r
library(blavaan)
fit <- bsem(mod, data=df, n.chains=4, burnin=2000, sample=4000)
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| Continuous indicators, large n | MLR (robust ML) |
| Ordinal / Likert | WLSMV |
| Small n (<200) | Bayesian (blavaan) |
| Latent + multiple regression | SEM > separate regressions (errors-in-vars) |
| Mediation | bootstrap CI for indirect |
| Longitudinal | LGC / cross-lagged panel |
| Complex / non-recursive | SEM (path-only OK) |

**기본값**: lavaan (R) 의 가장 mature → Python 만 시 semopy → fit indices 의 multiple report.

## 🔗 Graph
- 부모: [[Statistics]] · [[Multivariate-Analysis]]
- 변형: [[Regression-Analysis-Foundations]] · [[Principal-Component-Analysis]]
- 응용: [[Decision Theory]] · [[Knowledge-Structure]]
- Adjacent: [[Standard-Deviation-and-Variance]] · [[Statistical-Power]]

## 🤖 LLM 활용
**언제**: model specification draft, fit-indices interpretation, mediation explanation, paper writing.
**언제 X**: 매 estimation 자체 — lavaan/semopy 의 사용.

## ❌ 안티패턴
- **Modification indices 의 chase**: 매 fit 의 향상 의 위해 path 의 add → 매 capitalize on chance, theory 의 lost.
- **Rejecting model on χ² alone**: 매 large n 의 χ² 의 always reject — 매 RMSEA, CFI 의 결합.
- **Reflective vs formative confusion**: 매 wrong specification 의 estimate 의 bias.
- **Causal claim from cross-sectional SEM**: 매 directional path 의 causal X — 매 longitudinal / experiment 의 필요.
- **Underidentified model**: 매 df < 0 → 매 estimation impossible.
- **n < 200 with many parameters**: 매 unstable — 매 Bayesian 의 권장.

## 🧪 검증 / 중복
- Verified (Kline *Principles and Practice of SEM*, lavaan docs, Hu & Bentler 1999 cutoffs).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — SEM components, fit, lavaan/semopy patterns. |
