Structural Equation Modeling (SEM): Formula, Example and How to Report SEM Results
Structural equation modeling (SEM), also spelled structural equation modelling and often searched as SEM statistica (statistics) in Italian and Spanish, is a multivariate analysis that tests a network of relationships among observed and latent variables in one model. It joins factor analysis (the measurement model) with path analysis and regression (the structural model). This guide gives the SEM formula, a worked example, the two-step approach, fit guidelines, remedies for non-normal data, and a complete template for SEM reporting.
1. What is structural equation modeling?
Constructs such as motivation, anxiety or well-being cannot be observed directly, so we measure them through items that contain error. SEM lets you model these latent variables, separate true score from measurement error, test direct, indirect and total effects, and judge the global fit of the whole theory. A structural equation modeling analysis therefore answers two questions: does the model fit, and are the hypothesized paths significant?
| Element | Symbol | Meaning |
|---|---|---|
| Observed variable | Rectangle | Measured item or scale score (X1, Y1) |
| Latent variable | Ellipse | Construct inferred from indicators (ξ, η) |
| Path | Single-headed arrow | Hypothesized effect (β, γ) |
| Covariance | Double-headed arrow | Correlation without direction |
| Loading | Latent → item arrow | How strongly an item reflects its factor (λ) |
| Residual | Small arrow | Unexplained variance (δ, ε, ζ) |
2. Structural equation modeling formula
A full SEM has two sets of equations plus a fitting function that compares the model-implied covariance matrix Σ(θ) with the sample covariance matrix S.
| Part | Equation | Where |
|---|---|---|
| Measurement (exogenous) | x = Λₓ ξ + δ | Λₓ = loadings, ξ = latent predictors, δ = error |
| Measurement (endogenous) | y = Λᵧ η + ε | η = latent outcomes, ε = error |
| Structural | η = Bη + Γξ + ζ | B = effects among η, Γ = effects of ξ on η, ζ = disturbance |
| Model-implied covariance | Σ = Σ(θ) | θ = free parameters |
| ML fit function | FML = ln|Σ| + tr(SΣ⁻¹) − ln|S| − p | p = number of observed variables |
| Chi-square | χ² = (N − 1) · FML | N = sample size |
| Degrees of freedom | df = p(p + 1)/2 − q | q = free parameters |
| Indirect effect | a × b | Product of the two mediation paths |
3. Structural equation modeling in practice: the two-step approach
Anderson and Gerbing (1988) recommended a two-step approach that is still the standard recommendation. First fit a confirmatory factor analysis (CFA) to confirm that the items define their factors. Only then add the structural paths. If you skip step 1, you cannot tell whether poor fit comes from the measurement or from the theory.
- Specify and identify the model (df ≥ 0, one loading or variance fixed per latent variable).
- Check data: missing values, outliers, normality, sample size (200+ or 10–20 cases per parameter).
- Step 1: CFA. Loadings ≥ .50, acceptable fit.
- Step 2: structural model. Add paths, compare fit with the CFA.
- Test indirect effects with bootstrapping.
- Report everything in the order shown in section 8.
4. Structural equation modeling example
A researcher surveys N = 412 students to test whether study habits raise achievement through self-efficacy (Figure 1). The CFA fits well, so the structural model is estimated next.
| Path | β | SE | p | 95% CI |
|---|---|---|---|---|
| Study habits → Self-efficacy (a) | .52 | .05 | < .001 | [.42, .62] |
| Self-efficacy → Achievement (b) | .41 | .06 | < .001 | [.29, .53] |
| Study habits → Achievement (c′) | .18 | .07 | .010 | [.04, .32] |
| Indirect effect (a × b) | .21 | .04 | < .001 | [.13, .30] |
The indirect confidence interval excludes zero and the direct path stays significant, so self-efficacy partially mediates the effect.
5. Structural equation modeling vs path analysis
| Method | Latent variables | Several outcomes | Measurement error | Global fit |
|---|---|---|---|---|
| Multiple regression | No | No | Ignored | No |
| Path analysis | No | Yes | Ignored | Yes |
| CFA | Yes | — | Modeled | Yes |
| Full SEM | Yes | Yes | Modeled | Yes |
Use path analysis when every variable is a single observed score. Choose SEM when constructs are measured by several items and you want to correct for measurement error.
6. Structural equation modelling: guidelines for determining model fit
No single index proves a model is correct. Report several, as discussed by Hu and Bentler (1999) and Hooper et al. (2008). Treat cut-offs as guidelines rather than strict rules; Hu and Bentler themselves noted that no single value works in every condition.
| Index | Good | Acceptable | Note |
|---|---|---|---|
| χ² (df, p) | p > .05 | — | Over-rejects with large N |
| χ²/df | < 2 | < 3 | Rule of thumb only |
| CFI / TLI | ≥ .95 | ≥ .90 | Compares with a null model |
| RMSEA (90% CI) | ≤ .05 | ≤ .08 | Report the interval |
| SRMR | ≤ .05 | ≤ .08 | Average residual |
| AIC / BIC | Lower is better; only for comparing models on the same data | ||
7. Structural equation models with non-normal variables: problems and remedies
Standard ML assumes multivariate normality. With skewed or ordinal data, χ² is inflated and standard errors are too small (West, Finch & Curran, 1995).
| Problem | Remedy |
|---|---|
| Skewed or kurtotic continuous items (|skew| > 2, |kurtosis| > 7) | Robust ML (MLR) with Satorra-Bentler or Yuan-Bentler corrections |
| Non-normal indirect effects | Bootstrap confidence intervals, 5,000 resamples (Preacher & Hayes, 2008) |
| Ordinal items (Likert, fewer than 5 categories) | WLSMV with polychoric correlations |
| Missing data | Full-information ML or multiple imputation |
| Outliers | Check Mahalanobis distance and run with and without them |
8. How to report SEM results (APA7)
Good SEM reporting lets readers reproduce and judge your model. Use this checklist, then adapt the template.
| Report | What to include |
|---|---|
| Sample and data | N, missing data and handling, normality checks |
| Software and estimator | For example lavaan (Rosseel, 2012), MLR, WLSMV, bootstrap resamples |
| Model specification | Theory, path diagram, constraints, identification |
| Fit statistics | χ²(df), p, CFI, TLI, RMSEA with 90% CI, SRMR |
| Parameter estimates | Standardized loadings and paths, SE, p, 95% CI |
| Reliability and validity | Composite reliability, AVE, omega or alpha |
| Indirect effects | Estimate and bootstrap CI |
| Modifications | Any changes and their theoretical justification |
StatFas produces APA7-ready tables for these items, which solves the most time-consuming part of SEM reporting.
9. Related reporting: Wilcoxon signed-rank test and PCA
If your study also uses non-parametric or dimension-reduction methods, here is how to report them.
Principal component analysis is often used to screen items before a CFA. Try the free principal component analysis calculator to view eigenvalues, loadings and the scree plot.
10. FAQ
How do I report SEM results?
State the model, estimator and software, then give χ²(df), p, CFI, TLI, RMSEA with 90% CI, SRMR, standardized estimates with confidence intervals and bootstrapped indirect effects.
What is the difference between SEM and path analysis?
Path analysis uses only observed variables and ignores measurement error. SEM adds latent variables through a measurement model.
What is the two-step approach?
Test the measurement model with CFA first, then add structural paths (Anderson & Gerbing, 1988).
What if my data are not normal?
Use MLR, bootstrapping or WLSMV for ordinal items (Table 6).
Further reading and free tools
- lavaan, the free R package for SEM and CFA: lavaan.ugent.be (Rosseel, 2012).
- JASP offers point-and-click SEM in a free desktop app: jasp-stats.org.
- Run the preliminary steps free on StatFas: PCA calculator and the PCA guide.