I am hoping someone can clarify how parameter estimates for structural equation models (SEM) are usually interpreted in practice.
By that I mean, suppose we have an SEM of the form
$$ y = \mu + \Lambda \omega + \epsilon $$
$$ \eta = \Gamma \xi + \delta$$
where $\omega = [\eta, \xi]^T$ are latent variables, and $\epsilon, \delta$ are random noise terms.
So we obtain estimates for the coefficients $\lambda_{ij}$ and $\gamma_{ij}$ that comprise the matrices $\Lambda$, $\Gamma$ respectively, and want to make inferences on the relationships between the observed variables $y_i$, the latent explanatory variables $\xi_i$, and the outcome latent variables $\eta_i$.
So the structural equation is straightforward and can be interpretted as with any regression model. Eg. $\gamma_{ij} > 0$ means $\xi_j$ has a positive effect on $\eta_i$.
Where I'm confused is the measurement equation. If we interpret it as we normally do with regression models, we would describe the effect the unobserved latent variables $\omega$ have on the observed variables $y$, which doesn't really make sense to me, since $\omega$ is unobserved?
Since the measurement equation is intended to group observed variables into unobserved latent variables, would it make more sense to instead talk about correlation?
For example, if $\lambda_{ij} > 0$ we might say that $y_i$ and $\omega_j$ are positively correlated. Does that make sense?
Basically, I just want to know how to interpret the measurement equation in a way that makes sense given the explanatory variables are not observed. Hopefully I've explained my confusion clearly.