Fixed and Free Parameters
In CFA, each indicator $y_j$ loads on a pre-specified subset of factors. The implied covariance matrix is:
\[\boldsymbol{\Sigma}(\boldsymbol{\theta}) = \boldsymbol{\Lambda}\boldsymbol{\Phi}\boldsymbol{\Lambda}^\top + \boldsymbol{\Theta}\]where $\boldsymbol{\Phi}$ is the factor correlation (or covariance) matrix and $\boldsymbol{\Theta} = \text{diag}(\psi_1, \ldots, \psi_p)$ contains unique variances. Free parameters are estimated; fixed parameters (usually 0 for cross-loadings) are held at their specified values.
Identification
Each factor requires at least one scale-setting constraint: either fix one loading per factor to 1 (marker indicator) or fix each factor variance to 1. A sufficient condition for local identification is the two-indicator rule (each factor has $\ge 2$ indicators with no cross-loadings) combined with a positive-definite $\boldsymbol{\Phi}$.
Model Fit and Modification
The chi-squared test statistic $\chi^2 = (N-1)F_{\min}$ has $df = p^* - q$ degrees of freedom. Because $\chi^2$ is sensitive to sample size, approximate indices are preferred:
| Index | Acceptable | Good |
|---|---|---|
| CFI | $\ge 0.90$ | $\ge 0.95$ |
| RMSEA | $\le 0.08$ | $\le 0.06$ |
| SRMR | $\le 0.10$ | $\le 0.08$ |
Modification indices (MI) estimate the $\chi^2$ drop if a fixed parameter is freed. Freeing parameters post-hoc requires cross-validation to avoid capitalizing on chance.