• 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.

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

• \(\hat{\pmb{\Sigma}}\): 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

• Inspiration and headache are independent after conditioning on attending IMPS