sem-guide

Structural Equation Modeling Guide

Build, estimate, and evaluate structural equation models (SEM) with latent variables using Python (semopy) and R (lavaan), including confirmatory factor analysis and path analysis.

What Is SEM?

Structural Equation Modeling is a multivariate statistical framework that combines factor analysis and path analysis to test complex theoretical models involving:

• Observed (manifest) variables: Directly measured (e.g., survey items, test scores)
• Latent (unobserved) variables: Theoretical constructs measured indirectly through observed indicators (e.g., "motivation," "intelligence")
• Structural paths: Directional relationships between variables (regression-like)
• Measurement model: How latent variables relate to their indicators (CFA)
• Structural model: How latent variables relate to each other (path analysis)

SEM Components

Component	Description	Diagram Symbol
Observed variable	Measured directly	Rectangle
Latent variable	Inferred from indicators	Oval/circle
Regression path	Directional relationship	Single-headed arrow
Covariance	Non-directional association	Double-headed arrow
Error/residual	Unexplained variance	Small circle with arrow

Step 1: Confirmatory Factor Analysis (CFA)

CFA tests whether observed variables load onto hypothesized latent factors.

In R (lavaan)

library(lavaan)

# Define the measurement model
# =~ means "is measured by"
cfa_model <- '
  # Latent variable definitions
  Motivation =~ mot1 + mot2 + mot3 + mot4
  SelfEfficacy =~ se1 + se2 + se3
  Performance =~ perf1 + perf2 + perf3 + perf4

  # Covariances between latent variables (estimated by default in CFA)
'

# Fit the model
fit <- cfa(cfa_model, data = mydata, estimator = "MLR")

# View results
summary(fit, fit.measures = TRUE, standardized = TRUE)

# Key output to examine:
# - Factor loadings (standardized > 0.5 is desirable)
# - Model fit indices (see table below)
# - Modification indices (for model improvement)
modindices(fit, sort = TRUE, minimum.value = 10)