Model Components

A standard SEM has two parts. The measurement model maps observed indicators $\mathbf{y}$ to latent variables $\boldsymbol{\eta}$:

\[\mathbf{y} = \boldsymbol{\Lambda}\boldsymbol{\eta} + \boldsymbol{\varepsilon}\]

The structural model specifies directional paths among the latent variables:

\[\boldsymbol{\eta} = \mathbf{B}\boldsymbol{\eta} + \boldsymbol{\Gamma}\boldsymbol{\xi} + \boldsymbol{\zeta}\]

where $\mathbf{B}$ contains endogenous-to-endogenous paths, $\boldsymbol{\Gamma}$ contains exogenous effects, and $\boldsymbol{\zeta}$ is the structural disturbance.

Model Identification

A model is identified if every free parameter has a unique solution given $\boldsymbol{\Sigma}$. The necessary (but not sufficient) condition is $df = p^* - q \ge 0$, where $p^* = p(p+1)/2$ unique covariance elements and $q$ is the number of free parameters. Each latent variable requires a scale-setting constraint (fixed loading or fixed variance).

Fit Indices

Index	Formula / Benchmark
$\chi^2$	$\chi^2 = (N-1)F_{\min}$; non-significant desirable
CFI	$\ge 0.95$ indicates good fit
RMSEA	$\le 0.06$ good; $\le 0.08$ acceptable
SRMR	$\le 0.08$ acceptable

Estimation proceeds by minimizing the maximum likelihood discrepancy function:

\[F_{\text{ML}} = \log|\boldsymbol{\Sigma}(\boldsymbol{\theta})| + \text{tr}[\mathbf{S}\boldsymbol{\Sigma}^{-1}(\boldsymbol{\theta})] - \log|\mathbf{S}| - p\]

Modification indices indicate the expected drop in $\chi^2$ per freed parameter, guiding model respecification.