Projection predictive variable selection has been shown to be stable and able to find small models with good predictive performance
Our projpred package supported before just generalized linear models. @AlejandroCatalina did great work in extending the methods to
handle also GLMMs and GAMMs and running many experiments to show that the extended method really works.
The reference can be defined with R formula syntax for GLMMs and GAMMs, forward search through the projected models finds a simpler model structure that has similar predictive performance as the full model, and we get a gnew formula.
The benefit of finding automatically a simpler model structure and formula is is that the modeller can focus more quickly on what is relevant. The forward search (or any search through the model space) is usually associated with overfitting. The projection predictive approach avoids the overfitting by examining only the projected submodels and not fitting each model independently to the data.
To be able to support R formula syntax for GLMMs and GAMMs, @AlejandroCatalina refactored the projpred. The new code supporting GLMMs and GAMMs is not yet in CRAN, but is available in master branch at https://github.com/stan-dev/projpred. See also a vignette showing examples how to use projpred for GLMMs
EDIT: correct link for the vignette