# Interpreting Structural Equation Model Estimates?

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.

Broadly, structural equation models are causal models, and causal assumptions are laden in the path coefficients and the absence of estimated paths. A truly agnostic interpretation of factor loadings is that a one unit change in the level of the latent variable is associated with a $$\lambda_j$$ change in the expected value of the indicator. Typically, though, if we want to interpret latent variable models as measurement models, we need to invoke a causal interpretation of the factor loading and claim that a change in the latent variable causes a change in the expected value of the indicator. The validity of the causal claim is subject to critique, but that is what the process of validating measurement models and instruments is designed to protect against.