What is the difference between mixed-effects modelling in the RStan and lme4 packages? I've recently begun running some multilevel/hierarchical models. Initially I was using rstan/rstanarm, but then switched to the lme4 package.
Is the difference between these two packages only in the use of Bayesian priors (as in rstan) or not, as in lme4?
 A: lme4 is fully frequentist, while rstanarm is fully Bayesian. That means there are more differences than just whether a prior is used. For example:


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*rstanarm reports marginal medians of the posterior density for each parameter, while lme4 reports maximum likelihood estimates (approximately analogous to the maximum a posteriori (MAP) estimator, or mode of the posterior distribution, given uninformative priors - but see this CV answer for discussion of why this is a loose analogy)

*rstanarm reports posterior intervals based on quantiles of the marginal posterior distribution (not the more classical highest posterior density intervals), lme4 reports Wald standard errors or likelihood profile confidence intervals

*diagnostics and convergence checking procedures are radically different.


For what it's worth, 


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*brms, also based on Stan, implements a broad class of GLMMs (somewhat broader than rstanarm, I think)

*MCMCglmm implements a broad class of Bayesian mixed models (based on older MCMC approaches rather than Hamiltonian MC)

*the blme package implements a partly Bayesian approach to mixed models that allows for weakly or strongly informative priors, but reports MAP estimates (it builds on lme4's technology)

*the R-INLA package (not on CRAN) uses integrated nested Laplace approximations; it also allows priors and returns MAP estimates

