Apply Structural Equation Modeling (SEM) to test hypothesized causal structures by combining measurement models (CFA) and structural models (path analysis). Use this skill when the user needs to validate latent constructs, test mediation or moderation paths, assess model fit with CFI/TLI/RMSEA/SRMR, or when they ask 'do these variables form a causal chain', 'how do I test my theoretical model', or 'is my measurement model valid'.
Structural Equation Modeling (SEM) simultaneously estimates measurement models (how observed indicators map to latent constructs) and structural models (directional paths among constructs). It integrates confirmatory factor analysis with path analysis to test whether empirical data are consistent with a hypothesized theoretical structure.
IRON LAW: SEM does NOT prove causation — it tests whether data is CONSISTENT
with a hypothesized causal structure. Good fit does NOT mean the model is
correct; it means the model cannot be rejected.
Key assumptions:
Define latent constructs and their observed indicators. Run CFA to confirm factor loadings, assess convergent validity (AVE ≥ 0.50), and discriminant validity.
Evaluate fit indices: CFI ≥ 0.90, TLI ≥ 0.90, RMSEA ≤ 0.08, SRMR ≤ 0.08. Examine modification indices cautiously — only respecify with theoretical justification.
Add directional paths among latent constructs based on theory. Estimate path coefficients and their significance. Compare nested models using chi-square difference test.
Report standardized path coefficients, R² for endogenous constructs, and overall fit. Discuss indirect effects if mediation is hypothesized. See references/estimation.md for mathematical notation and estimation details.
## SEM Analysis: [Study Title]
### Measurement Model (CFA)
| Construct | Indicator | Std. Loading | AVE | CR |
|-----------|-----------|-------------|-----|-----|
| [name] | [item] | x.xx | x.xx | x.xx |
### Model Fit
| Index | Value | Threshold | Assessment |
|-------|-------|-----------|------------|
| CFI | x.xx | ≥ 0.90 | [pass/fail] |
| TLI | x.xx | ≥ 0.90 | [pass/fail] |
| RMSEA | x.xx | ≤ 0.08 | [pass/fail] |
| SRMR | x.xx | ≤ 0.08 | [pass/fail] |
### Structural Paths
| Path | Std. β | S.E. | p-value | Supported? |
|------|--------|------|---------|------------|
| X → M | x.xx | x.xx | x.xx | [Yes/No] |
### Key Findings
- [Interpretation of results]
### Limitations
- [Note any assumption violations]