Some of my friends/colleagues have recently taken an interest in structural equation modelling, and I have been having to field an increasing number of questions about SEM. Often times, these questions are about how to interpret the meaning of the estimates of various measurement model parameters (i.e., factor loadings, residual variances, and intercepts), and why it is important for these values to be roughly equivalent between groups (i.e., establishing measurement invariance) before comparing groups on estimated structural parameters (i.e., variances, covariances, and means).
I feel as though I have a pretty good handle on providing accessible definitions of factor loadings and residual variances, and accessible explanations for why it might be important for these estimated parameters to be equivalent between groups before comparing those groups on structural parameters. But for some reason, I've felt like a similarly accessible definition and explanation of intercepts has eluded me.
So, my question is: how best to accessibly explain what an intercept is, and explain why it might be important for intercepts to be invariant between groups before comparing groups' latent means?
For example: a factor loading represents the estimated direction and strength of association between an observed variable and a latent variable. In other words, a factor loading represents how central that observed variable is to the manifestation of its associated latent variable. When comparing groups' structural parameters, it's important to ensure that they are invariant across groups, because otherwise it suggests that the same observed variables aren't equally important to both groups' understanding of a given latent variable--the latent variable means something different to each group.
An intercept is the expected value of a given observed variable when its associated latent variable is equal to zero...what is the In other words... and it's important to ensure they are invariant, because... portions of the explanation of an interpret (in the SEM context)?