Atlas › Validate Constructs › Structural Equation Modeling (SEM)

Structural Equation Modeling (SEM)

Combines measurement models with structural paths among latent constructs — testing whole theoretical models, measurement error included.

Validate ConstructsMultivariate also known as: Covariance-based SEM, CB-SEM, LISREL-type modeling

✓ When to use

Testing a network of hypothesized relations among latent constructs (e.g., leadership → empowerment → engagement → performance).
Latent mediation/moderation with measurement error properly handled.
Comparing competing theoretical models on the same data.
Multi-group questions: does the model hold for both genders? (measurement invariance, then structural comparison).

✗ When NOT to use

Small samples (< ~200) with complex models — estimates and fit are unstable; consider PLS-SEM.
Exploratory, prediction-oriented work — PLS-SEM or machine learning.
Weak theory: SEM confirms/compares specified models; it does not find models for you.
Formative constructs poorly accommodated by the reflective CB-SEM default.

Data requirements

Dependent / outcome variable	Multiple latent constructs, each with 3+ reflective indicators, plus specified directional paths.
Independent / grouping variable	—
Design	One sample (or groups for invariance testing); temporal separation strengthens causal ordering.
Sample size guidance	Guidelines vary: N ≥ 200, or 5–10 cases per estimated parameter; power analysis (e.g., via simulation or MacCallum RMSEA approach) is best.

Assumptions

All CFA assumptions for the measurement portion.
Correct structural specification — omitted paths/confounders bias estimates.
Multivariate normality for ML (robust corrections available).
Identification: enough indicators/constraints for a unique solution.
Causal interpretation requires design support, not just arrows.

Hypotheses

H₀ — Model-implied covariances equal population covariances (global); each path coefficient is zero (local).

H₁ — Misfit exists (global); paths differ from zero (local).

The concept

SEM stitches together two layers: a measurement layer (CFA — items reflect latents) and a structural layer (regressions among latents). Because relations are estimated between error-purged latent variables, structural coefficients are corrected for the attenuation that plagues scale-score regressions.

Best practice is Anderson–Gerbing two-step: validate the measurement model first, then fix it and test structural variants. Global fit uses the same battery as CFA; local results are standardized path coefficients with significance, plus indirect effects via bootstrapping for mediation chains. Model comparison (nested Δχ², or AIC/BIC) against plausible rivals — including reversed-path models — is what gives SEM its evidential bite. Equivalent models exist for almost any SEM; good fit never proves the causal story.

Worked example

A three-wave study (N = 385) tests: transformational leadership (T1) → psychological empowerment (T2) → work engagement (T3), with a direct path leadership → engagement.

Result: fit χ²(163) = 322.8, CFI = .95, RMSEA = .050, SRMR = .046. Paths: leadership → empowerment β = .54***; empowerment → engagement β = .43***; direct β = .16*. Bootstrapped indirect effect = .23, 95% CI [.16, .31] — empowerment substantially transmits the effect.

How to run it

library(lavaan)
model <- '
  # measurement
  lead    =~ l1 + l2 + l3 + l4
  empower =~ e1 + e2 + e3 + e4
  engage  =~ g1 + g2 + g3 + g4
  # structural
  empower ~ a*lead
  engage  ~ b*empower + c*lead
  # indirect effect
  ab := a*b
'
fit <- sem(model, data = df, se = "bootstrap", bootstrap = 5000)
summary(fit, fit.measures = TRUE, standardized = TRUE)
parameterEstimates(fit, boot.ci.type = "perc")  # CI for ab

import semopy

model = """
lead    =~ l1 + l2 + l3 + l4
empower =~ e1 + e2 + e3 + e4
engage  =~ g1 + g2 + g3 + g4
empower ~ lead
engage  ~ empower + lead
"""
m = semopy.Model(model)
m.fit(df)
print(m.inspect(std_est=True))
print(semopy.calc_stats(m))

Interpreting the output

Two-step reporting: measurement fit first, structural results second.
Standardized β per path with significance; R² for each endogenous construct.
Indirect effects with bootstrap CIs for mediation claims.
Model comparisons against rivals; acknowledge equivalent-model ambiguity.
Causal wording only as strong as the design (waves, controls, theory).

APA-style reporting

The hypothesized structural model fit the data well, χ²(163) = 322.8, p < .001, CFI = .95, RMSEA = .050, SRMR = .046. Transformational leadership predicted empowerment (β = .54, p < .001), which predicted engagement (β = .43, p < .001); the bootstrapped indirect effect was significant, β = .23, 95% CI [.16, .31], alongside a smaller direct effect (β = .16, p = .022). The model explained 41% of variance in engagement.

Common mistakes

Skipping measurement validation and interpreting paths from a misfitting model.
Treating good fit as proof of causality (equivalent models!).
Small-sample complex models with proudly reported but unstable estimates.
Trimming/adding paths post hoc without disclosure.
Ignoring measurement invariance before multi-group comparisons.

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