SEM 2 Symposium 2017

• Both models imply a variance-covariance matrix $$\hat{\pmb{\Sigma}}$$, aimed to closely resemble the sample variance-covariance matrix $$\pmb{S}$$ with positive degrees of freedom.

## Structural Equation Modeling

$\hat{\pmb{\Sigma}} = \pmb{\Lambda} \left( \pmb{I} - \pmb{B} \right)^{-1} \pmb{\Psi} \left( \pmb{I} - \pmb{B} \right)^{-1\top} \pmb{\Lambda}^{\top} + \pmb{\Theta}$

• $$\hat{\pmb{\Sigma}}$$: model implied variance-covariance matrix
• $$\pmb{\Lambda}$$: matrix containing factor loadings
• $$\pmb{B}$$: matrix containing structural relationships between latents
• $$\pmb{\Psi}$$: variance-covariance matrix of latents or latent residuals
• $$\pmb{\Theta}$$: matrix containing residual variances and covariances
• Usually diagonal!

\begin{aligned} \hat{\pmb{\Sigma}} &= \pmb{\Lambda} \pmb{\Psi} \pmb{\Lambda}^{\top} + \pmb{\Theta} \\ \begin{bmatrix} 2 & 1 & 1 & 1 \\ 1 & 2 & 1 & 1 \\ 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 2 \end{bmatrix} &= \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} \begin{bmatrix} 1 \end{bmatrix} \begin{bmatrix} 1 & 1 & 1 & 1 \end{bmatrix} + \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \end{aligned}

• Degrees of freedom: 2