Structural Equation Modeling (SEM)

SEM combines a measurement model (latent constructs measured by multiple indicators) with a structural model (causal paths among latent constructs). The most comprehensive multivariate method in the module.

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Step 1 — Define the measurement model

For each latent variable, specify its observed indicators (minimum 2 per construct).

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Step 2 — Define the structural model

Specify causal paths between latent variables.

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Step 3 — Provide data

Observations x observed indicators (numeric). Supports Excel Import, Sample Data Generator, and Manual Entry.

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Step 4 — Results

Measurement model:

• Factor loadings (standardised) with SE and significance
• Reliability per construct

Structural model:

• Path coefficients (standardised) with SE, z-values, p-values
• R-squared per endogenous latent variable

Model fit indices:

Index	Good fit
Chi-square	p > 0.05
CFI	>= 0.95
TLI	>= 0.95
RMSEA	<= 0.05 (close), <= 0.08 (reasonable)
SRMR	<= 0.08
GFI/AGFI	>= 0.90

Also: AIC, BIC for model comparison.

Factor scores — estimated per subject.

Path diagram — full model visualisation.

A UniversalChatBot is available for discussion.

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Statistical methods used

Model-implied covariance:

Sigma = Lambda * (I-B)^-1 * Psi * ((I-B)^-1)' * Lambda' + Theta

Chi-square: chi2 = (n-1) * F_ML(S, Sigma) with df = p(p+1)/2 - free_params.

H0: model fits (Sigma = S in population). Non-significant p desired.

CFI: 1 - max(chi2_model - df_model, 0) / max(chi2_null - df_null, 0)

RMSEA: sqrt(max(chi2 - df, 0) / (df * (n-1)))

SRMR: Standardised root mean square of residuals between S and Sigma.

Best practice: report multiple indices for convergent evidence of fit (Hu & Bentler, 1999).

Sample size guidance: at least 200 observations or 10-20 per estimated parameter.