Thinking Clearly About Correlations and Causation (2018) [pdf]
dacemirror.sci-hub.tw
dacemirror.sci-hub.tw
The approach outlined in the paper is close to [Directed Acyclic Graphs](https://en.wikipedia.org/wiki/Directed_acyclic_graph) and Pearl's approach to causality (see [this comment](https://news.ycombinator.com/item?id=24506593) for more info).
The basic issue with these approaches is that the structural model needs to be well specified in order to get to an unbiased estimate of the causal effect that you want to estimate. By well-specified, I mean that you have measured all possible variables impacting the relationship of interest and that you have specified the correct relationships among the variables (no controls for mediators or no colliders, as the paper outlines). If you believe that, SEM can give you unbiased estimates of the coefficient of interest.
On the other hand, if the model is misspecified, coefficients will be biased and it is difficult to figure out where the bias is. Bollen (1989) pretty much says that any coefficient could be biased in a misspecified model.
(Micro) Economists have addressed this issues by moving away from a regression framework and towards experimental and quasi-experimental methods. The same methods are not really used outside of microeconomics, as far as I can tell.
SEM is a little bit more common (at least with developmental psychologists that I know of) and that framework doesn't really work well with even basic quasi-experimental methods (e.g., diff-in-diff).
this is a common critique of linear SEMs -- some ML researchers refer to a broader class of "structural causal models (SCMs)", which are basically SEMs where the relationships may be drawn from broader function classes.
Non-linear models are not common in social sciences so that's probably why I never used them. There are a few variables that are usually quadratic (like the effect of age on wages). Other than that, linear relationships are good enough most of the time.
The only real application of non-linear model that I have seen are generalized structural equation models (GSEM). These allow for the use of link functions in SEM (logits, logs, poisson, exponentials, etc.) and are the multiple equations analogue of generalized linear models.
I am not familiar with structural causal models (SCMs), but a quick google search shows that these are a non-parametric version of SEM based on Baysian estimation and are a generalization of Baysian network. They sound cool but I don't think that they will become mainstream in psychology research anytime soon.
This complicates experimental designs also, for what it's worth, in that it can be difficult to determine what a manipulation is acting on, and whether it's acting on what you intend it to.
This is in part why there's such a huge interest in mediation modeling, although you're back to the misspecification issue. It's difficult to know what you don't know.
Person A: I used to think correlation implied causation, then I took a statistics class. Now I don't.
Person B: Sounds like that class helped.
Person A: Well, maybe...