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Regularized generalized linear mixed models and generalized additive mixed models are exactly what I need. I'm an R user, so it looks like bGAMM and maybe glmmLASSO are possibilities outside of doing my own implementation (which I don't have time to do for this project). My question is whether these packages can handle reasonably large numbers of observations (up to 150K) and random effects (over 500) along with, say, up to 1000 parameters. I'll be using a beefy Amazon instance for this use case.

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Thanks to Fabian Scheipl, developer of package::spikeSlabGAM among other things, for pointing me toward mboost, which is more scalable than spikeSlabGAM, GMMBoost, and glmmLASSO, and includes all of the model specifications of these packages as subsets. It is more scalable because it does not use MCMC. So it is good for use cases when what you are looking for are point estimates, and you've got large-ish data.

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