There’s a lot of great discussion here! I recently posted a review article on one-level models for survey data: (Pseudo) Bayesian Inference for Complex Survey Data
For multi-level models, there is an additional level of complexity if the sampling design is also multi-stage and the sampled units correspond to entities in the model hierarchy. For example sampling schools then students or hospitals then patients. There are competing approaches (are we surprised?!) but they all seem to need the sampling weights for each stage, which is often not available. This challenge is well known in the survey literature, but most approaches are not satisfactory. There’s a lot of potential to taking a Bayesian or (Pseudo-Bayesian) approach. We have a couple pre-prints in this space that might be interesting.
The general approach with perhaps some theory missing
Some more theory for a super simple model
Both papers do review some of the competing methods so take a look!
We’ve been using Stan (of course!) but with multiple weights, we are using custom models and Rstan instead of using packages like brms.