Introduction to Structural Equation Modeling (SEM) with Latent Variables

Structural Equation Modeling (SEM) is a powerful statistical technique for testing and estimating causal relationships among observed and latent (unobserved) variables. It combines factor analysis (to model latent variables from observed indicators) and path analysis (to model relationships between latents). SEM is commonly implemented in software like lavaan (R), Mplus, AMOS (SPSS), or LISREL.

To conduct SEM with latent variables:

• Define your theoretical model based on prior research (e.g., constructs like "intelligence" or "job satisfaction" as latents).


• Specify the measurement model using Confirmatory Factor Analysis (CFA) to link observed variables (indicators) to latents.


• Specify the structural model to define paths (regressions) between latents.


• Estimate the model (e.g., via maximum likelihood), assess fit, and interpret.

Step 1: Specify the Measurement Model (CFA)

The measurement model defines how observed variables (indicators) load onto latent variables. It's essentially CFA, confirming that your indicators reliably measure the intended constructs. Key components:

• Latent variables (factors): Unobserved constructs (e.g., η₁ = Job Demands).


• Observed indicators: Measured variables (e.g., x₁ to x₃ for demands).


• Factor loadings (λ): Regressions from latents to indicators (assume λ > 0 for positive relationships).


• Measurement errors (θ or ψ): Unique variances/errors for indicators (diagonal matrix, uncorrelated unless specified).


• No direct paths between indicators (all go through latents).


• Assumptions: Indicators are continuous, multivariate normal; no cross-loadings unless theory supports them.

Example Specification

• Latent Job Demands (η₁) measured by 3 indicators: Workload (x₁), Time Pressure (x₂), Role Conflict (x₃).


• Latent Job Resources (η₂) measured by 3 indicators: Autonomy (x₄), Support (x₅), Feedback (x₆).


• Latent Burnout (η₃) measured by 3 indicators: Exhaustion (y₁), Cynicism (y₂), Inefficacy (y₃). (Note: Exogenous latents use x; endogenous use y.)

• Measurement for exogenous latents: x = Λₓ ξ + δ

• Measurement for endogenous latents: y = Λᵧ η + ε

Explicit equations:

η₁ = λ₁₁ x₁ + λ₁₂ x₂ + λ₁₃ x₃ + θ₁  (Job Demands)
η₂ = λ₂₁ x₄ + λ₂₂ x₅ + λ₂₃ x₆ + θ₂  (Job Resources)
η₃ = λ₃₁ y₁ + λ₃₂ y₂ + λ₃₃ y₃ + θ₃  (Burnout)

• Fix one loading per latent to 1 for identification (e.g., λ₁₁ = 1).


• Allow correlations between exogenous latents (Φ matrix, e.g., cov(η₁, η₂)).


• Errors (Θ) are diagonal.

library(lavaan)
model.cfa <- '
  # Measurement model
  Demands =~ x1 + x2 + x3
  Resources =~ x4 + x5 + x6
  Burnout =~ y1 + y2 + y3
  
  # Correlations between exogenous latents
  Demands ~~ Resources
'
fit.cfa <- cfa(model.cfa, data = your_data)
summary(fit.cfa, fit.measures = TRUE)

• Fit assessment: Use χ², CFI (>0.95 good), RMSEA (<0.06 good), SRMR (<0.08 good). Modify if needed (e.g., allow correlated errors).

Step 2: Specify the Structural Model (Paths Between Latent Variables)

The structural model extends CFA by adding regressions (paths, β or γ coefficients) between latents, explaining how exogenous latents predict endogenous ones. It tests hypotheses like "Higher demands increase burnout."

• Exogenous latents (ξ): Independent (e.g., Demands, Resources); can correlate.


• Endogenous latents (η): Dependent (e.g., Burnout); predicted by others.


• Structural equations: η = B η + Γ ξ + ζ (B = paths among η; Γ = paths from ξ to η; ζ = disturbance).


• Disturbances (ζ or ψ): Residual variances for endogenous latents (uncorrelated with predictors).

Example Specification

Structural equations:

η₃ = β₃₁ η₁ + β₃₂ η₂ + ζ₃  (Burnout regressed on Demands and Resources)

• β₃₁ > 0 (positive path); β₃₂ < 0 (negative path).


• No recursive loops (B matrix must be solvable; no cycles).

• Measurement as above.


• Structural: η = Γ ξ + ζ (Here, ξ = [η₁, η₂], η = [η₃], Γ = [β₃₁, β₃₂]).

model.sem <- '
  # Measurement model (same as CFA)
  Demands =~ x1 + x2 + x3
  Resources =~ x4 + x5 + x6
  Burnout =~ y1 + y2 + y3
  
  # Structural paths
  Burnout ~ Demands + Resources
  
  # Correlations (from CFA)
  Demands ~~ Resources
'
fit.sem <- sem(model.sem, data = your_data)
summary(fit.sem, standardized = TRUE, fit.measures = TRUE)

• ~ denotes regression (path); ~~ denotes covariance.


• Use standardized = TRUE for β coefficients (effect sizes).

Identification and Estimation

• Identification: Ensure 3+ indicators per latent; fix loadings/variances; no under-identification (check with software warnings). Degrees of freedom = (observed vars choose 2) - parameters.


• Estimation: Maximum likelihood (ML) for normal data; robust ML or bootstrapping for non-normal.


• Sample size: At least 200; rule of thumb 10:1 (subjects:parameters).

Step 3: Assessment and Interpretation

• Overall fit: Same indices as CFA; compare CFA vs. SEM χ² difference for path significance.


• Path significance: t-tests (p < 0.05); inspect β (standardized paths, |β| > 0.2 meaningful).


• R²: Explained variance in endogenous latents (e.g., R² for Burnout).


• Modifications: Use MI (modification indices) sparingly; theory-driven.


• Indirect effects: Add if mediation (e.g., Resources → Demands → Burnout via =~ and ~).


• Common issues: Multicollinearity (high correlations >0.85); Heywood cases (negative variances → fix bounds).

Next Steps

• Collect/prepare data: Continuous indicators, handle missingness (FIML in lavaan).


• Run in software: Start with CFA, then SEM. If categorical indicators, use WLSMV estimator.


• Example data: Use lavaan datasets like HolzingerSwineford1939 for practice.


• Resources: Read Bollen (1989) Structural Equations with Latent Variables or Byrne's software guides.