The extensions of the logit/probit regressions discussed here are multivariate logit/probit models, which still have binary outcomes, the probit version is straightforward because it only requires a multivariate normal that is implemented in Stan (and is pretty common). The logit version requires a multivariate logistic instead, but since it’s less common it’s not implemented in Stan and less straightforward.
Those are distinct from multilogit (softmax) regression, which has multiple outcome categories but doesn’t use a multivariate distribution, so you can see this and the previous as different extensions from the basic logistic/probit regression that will be suitable to different kinds of data.
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I’ve never heard of SEM being inadequate for prediction! Although I’m far from an expert on that kind of model. Do you have any references I could check out?
As far as I understand, it is a controversial topic. Some say SEM models are descriptive only and cannot be used for out of sample prediction and some have already developed some methods (link).
Lavaan itself does not seem to have a function for out of sample predication (link1 & link2)
I am not an expert in the field but this is what I have found based on some quick research.
Interesting! I tend to believe that you can predict using SEM when you account for noisy indicators (so, always when your SEM is well constructed).
I guess the big question is what is being forecasted. If it’s the latent variable, then it’s useful only as input for other models. If it’s questionnaire responses, then I guess you’ll inevitably observe a lot of noise. Anyways, interesting stuff!