I'm looking for a simple concrete example of Confirmatory Factor Analysis. What are the precise steps?
What I (think) understand so far:
Get data (random sample).
Bring in a theory along the line of
This unobserved variable $\eta_1$ is correlated to the observed variables $x,y,z$ and not to $v, w$, $\eta_1$ is correlated to $v, w, z$ (and not $x,y$).
The unobserved variables are chosen because theory predicts they cause the others and are called latent or underlying factors.
Compute the covariance matrix of $x,y,z,v,w$.
Fit a linear models... (but the $\eta$'s are the dependent variable (but still unbserved) so how can we know the coefficients on the observed variables (which are called loadings $\lambda$)?)
Compute (how?) the implied covariance matrix and compare to the actual covariance matrix use some kind of a statistic to estimate how likely the observed difference between that matrices is due to random chance.