How to reconcile that I hate structural equation models, but I love measurement error models and multilevel regressions, even though these are special cases of structural equation models?

Andy Dorsey writes:

I’m a graduate student in psychology. I’m trying to figure out what seems to me to be a paradox: One issue you’ve talked about in the past is how you don’t like structural equation modeling (e.g., your blog post here). However, you have also talked about the problems with noisy measures and measurement error (e.g., your papers here and here).

Here’s my confusion: Isn’t the whole point of structural equation modeling to have a measurement model that accounts for measurement error? So isn’t structural equation modeling actually addressing the measurement problem you’ve lamented?

The bottom line is that I really want to address measurement error (via a measurement model) because I’m convinced that it will improve my statistical inferences. I just don’t know how to do that if structural equation modeling is a bad idea.

My reply:

I do like latent variables. Indeed, when we work with models that don’t have latent variables, we can interpret these as measurement-error models where the errors have zero variance.

And I have no problem with structural equation modeling in the general sense of modeling observed data conditional on an underlying structure.

My problem with structural equation modeling as it is used in social science is that the connections between the latent variables are just too open-ended. Consider the example on the second page of this article.

So, yes, I like measurement-error models and multilevel regressions, and mathematically these are particular examples of structural equation models. But I think that when researchers talk about structural equation models, they’re usually talking about big multivariate models that purport to untangle all sorts of direct and indirect effects from data alone, and I don’t think this is possible. For further discussion of these issues, see Sections 19.7 and B.9 of Regression and Other Stories.

One other thing: I think they should be called “structural models” or “stochastic structural models.” The word “equation” in the name doesn’t seem quite right to me, because the whole point of these models is that they’re not equating the measurement with the structure. The models allow error, so I don’t think of them as equations.

P.S. Zad’s cat, above, is dreaming of latent variables.