Advanced Bayesian Multilevel Modeling with the R Package brms

The brms package allows R users to easily specify a wide range of Bayesian single-level and multilevel models, which are ﬁtted with the probabilistic programming language Stan behind the scenes. Several response distributions are supported, of which all parameters (e.g., location, scale, and shape) can be predicted at the same time thus allowing for distributional regression. Non-linear relationships may be speciﬁed using non-linear predictor terms or semi-parametric approaches such as splines or Gaussian processes. To make all of these modeling options possible in a multilevel framework, brms provides an intuitive and powerful formula syntax, which extends the well known formula syntax of lme4 . The purpose of the present paper is to introduce this syntax in detail and to demonstrate its usefulness with four examples, each showing other relevant aspects of the syntax.


Introduction
Multilevel models (MLMs) offer great flexibility for researchers across sciences (Brown and Prescott 2015;Demidenko 2013;Gelman and Hill 2006;Pinheiro and Bates 2006). They allow modeling of data measured on different levels at the same time -for instance data of students nested within classes and schools -thus taking complex dependency structures into account. It is not surprising that many packages for R have been developed to fit MLMs. Usually, however, the functionality of these implementations is limited insofar as it is only possible to predict the mean of the response distribution. Other parameters of the response distribution, such as the residual standard deviation in linear models, are assumed constant across observations, which may be violated in many applications. Accordingly, it is desirable to allow for prediction of all response parameters at the same time. Models doing exactly that are often referred to as distributional models or more verbosely models for location, scale and shape (Rigby and Stasinopoulos 2005). Another limitation of basic MLMs is that they only allow for linear predictor terms. While linear predictor terms already offer good flexibility, they are of limited use when relationships are inherently non-linear. Such non-linearity can be handled in at least two ways: (1) by fully specifying a non-linear predictor term with corresponding parameters each of which can be predicted using MLMs (Lindstrom and Bates 1990), or (2) estimating the form of the non-linear relationship on the fly using splines (Wood 2004) or Gaussian processes (Rasmussen and Williams 2006). The former are often simply The purpose of the present article is to provide an introduction of the advanced multilevel formula syntax implemented in brms, which allows to fit a wide and growing range of nonlinear distributional multilevel models. A general overview of the package is already given in Bürkner (2017). Accordingly, the present article focuses on more recent developments. We begin by explaining the underlying structure of distributional models. Next, the formula syntax of lme4 and its extensions implemented in brms are explained. Four examples that demonstrate the use of the new syntax are discussed in detail. Afterwards, the functionality of brms is compared with that of rstanarm (Stan Development Team 2017a) and MCMCglmm (Hadfield 2010). We end by describing future plans for extending the package.

Model description
The core of models implemented in brms is the prediction of the response y through predicting all parameters θ p of the response distribution D, which is also called the model family in many R packages. We write y i ∼ D(θ 1i , θ 2i , ...) to stress the dependency on the i th observation. Every parameter θ p may be regressed on its own predictor term η p transformed by the inverse link function f p that is θ pi = f p (η pi ) 2 . Such models are typically refered to as distributional models 3 . Details about the parameterization of each family are given in vignette("brms_families").
Suppressing the index p for simplicity, a predictor term η can generally be written as In this equation, β and u are the coefficients at population-level and group-level respectively and X, Z are the corresponding design matrices. The terms s k (x k ) symbolize optional smooth functions of unspecified form based on covariates x k fitted via splines (see Wood (2011) for the underlying implementation in the mgcv package) or Gaussian processes (Williams and Rasmussen 1996). The response y as well as X, Z, and x k make up the data, whereas β, u, and the smooth functions s k are the model parameters being estimated. The coefficients β and u may be more commonly known as fixed and random effects, but I avoid theses terms following the recommendations of Gelman and Hill (2006). Details about prior distributions of β and u can be found in Bürkner (2017) and under help("set_prior").
As an alternative to the strictly additive formulation described above, predictor terms may also have any form specifiable in Stan. We call it a non-linear predictor and write The structure of the function f is given by the user, c r are known or observed covariates, and φ s are non-linear parameters each having its own linear predictor term η φs of the form specified above. In fact, we should think of non-linear parameters as placeholders for linear predictor terms rather than as parameters themselves. A frequentist implementation of such models, which inspired the non-linear syntax in brms, can be found in the nlme package (Pinheiro, Bates, DebRoy, Sarkar, and R Core Team 2016).

Extended multilevel formula syntax
The formula syntax applied in brms builds upon the syntax of the R package lme4 (Bates et al. 2015). First, we will briefly explain the lme4 syntax used to specify multilevel models and then introduce certain extensions that allow to specify much more complicated models in brms. An lme4 formula has the general form response~pterms + (gterms | group) The pterms part contains the population-level effects that are assumed to be the same across obervations. The gterms part contains so called group-level effects that are assumed to vary accross grouping variables specified in group. Multiple grouping factors each with multiple group-level effects are possible. Usually, group contains only a single variable name pointing to a factor, but you may also use g1:g2 or g1/g2, if both g1 and g2 are suitable grouping factors. The : operator creates a new grouping factor that consists of the combined levels of g1 and g2 (you could think of this as pasting the levels of both factors together). The / operator indicates nested grouping structures and expands one grouping factor into two or more when using multiple / within one term. If, for instance, you write (1 | g1/g2), it will be expanded to (1 | g1) + (1 | g1:g2). Instead of | you may use || in grouping terms to prevent group-level correlations from being modeled. This may be useful in particular when modeling so many group-level effects that convergence of the fitting algorithms becomes an issue due to model complexity. One limitation of the || operator in lme4 is that it only splits up terms so that columns of the design matrix originating from the same term are still modeled as correlated (e.g., when coding a categorical predictor; see the mixed function of the afex package by Singmann, Bolker, and Westfall (2015) for a way to avoid this behavior).
While intuitive and visually appealing, the classic lme4 syntax is not flexible enough to allow for specifying the more complex models supported by brms. In non-linear or distributional models, for instance, multiple parameters are predicted, each having their own population and group-level effects. Hence, multiple formulas are necessary to specify such models 4 . Then, however, specifying group-level effects of the same grouping factor to be correlated across formulas becomes complicated. The solution implemented in brms (and currently unique to it) is to expand the | operator into |<ID>|, where <ID> can be any value. Group-level terms with the same ID will then be modeled as correlated if they share same grouping factor(s) 5 . For instance, if the terms (x1|ID|g1) and (x2|ID|g1) appear somewhere in the same or different formulas passed to brms, they will be modeled as correlated.
Further extensions of the classical lme4 syntax refer to the group part. It is rather limited in its flexibility since only variable names combined by : or / are supported. We propose two extensions of this syntax: Firstly, group can generally be split up in its terms so that, say, (1 | g1 + g2) is expanded to (1 | g1) + (1 | g2). This is fully consistent with the way / is handled so it provides a natural generalization to the existing syntax. Secondly, there are some special grouping structures that cannot be expressed by simply combining grouping variables. For instance, multi-membership models cannot be expressed this way. To overcome this limitation, we propose wrapping terms in group within special functions that allow specifying alternative grouping structures: (gterms | fun(group)). In brms, there are currently two such functions implemented, namely gr for the default behavior and mm for multi-membership terms. To be compatible with the original syntax and to keep formulas short, gr is automatically added internally if none of these functions is specified.
While some non-linear relationships, such as quadratic relationships, can be expressed within the basic R formula syntax, other more complicated ones cannot. For this reason, it is possible in brms to fully specify non-linear predictor terms similar to how it is done in nlme, but fully compatible with the extended multilevel syntax described above. Suppose, for instance, we want to model the non-linear growth curve between y and x with parameters b 1 , b 2 , and b 3 (see Example 3 in this paper for an implementation of this model with real data). Furthermore, we want all three parameters to vary by a grouping variable g and model those group-level effects as correlated. Additionally b 1 should be predicted by a covariate z. We can express this in brms using multiple formulas, one for the non-linear model itself and one per non-linear parameter: The first formula will not be evaluated using standard R formula parsing, but instead taken literally. In contrast, the formulas for the non-linear parameters will be evaluated in the usual way and are compatible with all terms supported by brms. Note that we have used the above described ID-syntax to model group-level effects as correlated across formulas.
There are other syntax extensions implemented in brms that do not directly target grouping terms. Firstly, there are terms formally included in the pterms part that are handled separately. The most prominent examples are smooth terms specified through the s and t2 functions of the mgcv package (Wood 2011). Other examples are category specific effects cs, monotonic effects mo, noise-free effects me, or Gaussian process terms gp. The former is explained in Bürkner (2017), while the latter three are documented in help(brmsformula).
Internally, these terms are extracted from pterms and not included in the construction of the population-level design matrix. Secondly, making use of the fact that | is unused on the left-hand side of ∼ in formula, additional information on the response variable may be specified via

response | aterms~<predictor terms>
The aterms part may contain multiple terms of the form fun(<variable>) separated by + each providing special information on the response variable. This allows among others to weight observations, provide known standard errors for meta-analysis, or model censored or truncated data. As it is not the main topic of the present paper, we refer to help("brmsformula") and help("addition-terms") for more details.
To set up the model formulas and combine them into one object, brms defines the brmsformula (or short bf) function. Its output can then be passed to the parse_bf function, which splits up the formulas in separate parts and prepares them for the generation of design matrices and related data. Other packages may re-use these functions in their own routines making it easier to offer support for the above described multilevel syntax.

Examples
The idea of brms is to provide one unified framework for multilevel regression models in R. As such, the above described formula syntax in all of its variations can be applied in combination with all response distributions supported by brms (currently about 35 response distributions are supported; see help("brmsfamily") and vignette("brms_families") for an overview).
In this section, we will discuss four examples in detail, each focusing on certain aspects of the syntax. They are chosen to provide a broad overview of the modeling options. The first is about the number of fish caught be different groups of people. It does not actually contain any multilevel structure, but helps in understanding how to set up formulas for different model parts. The second example is about housing rents in Munich. We model the data using splines and a distributional regression approach. The third example is about cumulative insurance loss payments across several years, which is fitted using a rather complex non-linear multilevel model. Finally, the fourth example is about the performance of school children, who change school during the year, thus requiring a multi-membership model.
Despite not being covered in the four examples, there are a few more modeling options that we want to briefly describe. First, brms allows fitting so called phylogenetic models. These models are relevant in evolutionary biology when data of many species are analyzed at the same time. Species are not independent as they come from the same phylogenetic tree, implying that different levels of the same grouping-factor (i.e., species) are likely correlated. There is a whole vignette dedicated to this topic, which can be found via vignette("brms_phylogenetics"). Second, there is a canonical way to handle ordinal predictors, without falsely assuming they are either categorical or continuous. We call them monotonic effects and discuss them in vignette("brms_monotonic"). Last but not least, it is possible to account for measurement error in both response and predictor variables. This is often ignored in applied regression modeling (Westfall and Yarkoni 2016), although measurement error is very common in all scientific fields making use of observational data. There is no vignette yet covering this topic, but one will be added in the future. In the meantime, help("brmsformula") is the best place to start learning about such models as well as about other details of the brms formula syntax.

Example 1: Catching fish
An important application of the distributional regression framework of brms are so called zero-inflated and hurdle models. These models are helpful whenever there are more zeros in the response variable than one would naturally expect. Here, we consider an example dealing with the number of fish caught by various groups of people. On the UCLA website (https: //stats.idre.ucla.edu/stata/dae/zero-inflated-poisson-regression), the data are described as follows: "The state wildlife biologists want to model how many fish are being caught by fishermen at a state park. Visitors are asked how long they stayed, how many people were in the group, were there children in the group and how many fish were caught. Some visitors do not fish, but there is no data on whether a person fished or not. Some visitors who did fish did not catch any fish so there are excess zeros in the data because of the people that did not fish." zinb <-read.csv("http://stats.idre.ucla.edu/stat/data/fish.csv") zinb$camper <-factor(zinb$camper, labels = c("no", "yes")) head (  (see Figure 1). In fact, the conditional_effects method turned out to be so powerful in visualizing effects of predictors that I am using it almost as frequently as summary. According to the parameter estimates, larger groups catch more fish, campers catch more fish than noncampers, and groups with more children catch less fish. The zero-inflation probability zi is pretty large with a mean of 41%. Please note that the probability of catching no fish is actually higher than 41%, but parts of this probability are already modeled by the Poisson distribution itself (hence the name zero-inflation). If you want to treat all zeros as originating from a separate process, you can use hurdle models instead (not shown here). Now, we try to additionally predict the zero-inflation probability by the number of children.
The underlying reasoning is that we expect groups with more children to not even try catching fish, since children often lack the patience required for fishing. From a purely statistical perspective, zero-inflated (and hurdle) distributions are a mixture of two processes and predicting both parts of the model is natural and often very reasonable to make full use of the data.

Example 2: Housing rents
In their book about regression modeling, Fahrmeir, Kneib, Lang, and Marx (2013)  Here, we aim at predicting the rent per square meter with the size of the apartment as well as the construction year, while taking the district of Munich into account. As the effect of both predictors on the rent is of unknown non-linear form, we model these variables using a bivariate tensor spline (Wood, Scheipl, and Faraway 2013). The district is accounted for via a varying intercept.

fit_rent1 <-brm(rentsqm~t2(area, yearc) + (1|district), data = rent99, chains = 2, cores = 2)
We fit the model using just two chains (instead of the default of four chains) on two processor cores to reduce the model fitting time for the purpose of the present paper. In general, using the default option of four chains (or more) is recommended. Samples were drawn using sampling (NUTS). For each parameter, Eff.Sample is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat = 1).
For models including splines, the output of summary is not tremendously helpful, but we get at least some information. Firstly, the credible intervals of the standard deviations of the coefficients forming the splines (under 'Smooth Terms') are sufficiently far away from zero to indicate non-linearity in the (combined) effect of area and yearc. Secondly, even after controlling for these predictors, districts still vary with respect to rent per square meter by a sizable amount as visible under 'Group-Level Effects' in the output. To further understand the effect of the predictor, we apply graphical methods:

conditional_effects(fit_rent1, surface = TRUE)
In Figure 2, the conditional effects of both predictors are displayed, while the respective other predictor is fixed at its mean. In Figure 3, the combined effect is shown, clearly demonstrating an interaction between the variables. In particular, housing rents appear to be highest for small and relatively new apartments. In the above example, we only considered the mean of the response distribution to vary by area and yearc, but this my not necessarily reasonable assumption, as the variation of the response might vary with these variables as well. Accordingly, we fit splines and effects of district for both the location and the scale parameter, which is called sigma in Gaussian models.

sigma~t2(area, yearc) + (1|ID1|district)) fit_rent2 <-brm(bform, data = rent99, chains = 2, cores = 2)
If not otherwise specified, sigma is predicted on the log-scale to ensure it is positive no matter how the predictor term looks like. Instead of (1|district) as in the previous model, we now use (1|ID1|district) in both formulas. This results in modeling the varying intercepts of both model parts as correlated (see the description of the ID-syntax above). The group-level part of the summary output looks as follows: As visible from the positive correlation of the intercepts, districts with overall higher rent per square meter have higher variation at the same time. Lastly, we want to turn our attention to the splines. While conditional_effects is used to visualize effects of predictors on the expected response, conditional_smooths is used to show just the spline parts of the model:

conditional_smooths(fit_rent2)
The plot on the left-hand side of Figure 4 resembles the one in Figure 3, but the scale is different since only the spline is plotted. The right-hand side of 4 shows the spline for sigma.
Since we apply the log-link on sigma by default the spline is on the log-scale as well. As visible in the plot, the variation in the rent per square meter is highest for relatively small and old apartments, while the variation is smallest for medium to large apartments build around the 1960s.

Example 3: Insurance loss payments
On his blog, Markus Gesmann predicts the growth of cumulative insurance loss payments over time, originated from different origin years (see http://www.magesblog.com/2015/11/ loss-developments-via-growth-curves-and.html). We will use a slightly simplified version of his model for demonstration purposes here. It looks as follows: The cumulative insurance payments cum will grow over time, and we model this dependency using the variable dev. Further, ult AY is the (to be estimated) ultimate loss of accident each year. It constitutes a non-linear parameter in our framework along with the parameters θ and ω, which are responsible for the growth of the cumulative loss and are for now assumed to be the same across years. We load the data url <-paste0("https://raw.githubusercontent.com/mages/", "diesunddas/master/Data/ClarkTriangle.csv") loss <-read.csv(url) head(loss) AY dev cum 1 1991AY dev cum 1 6 357.848 2 1991AY dev cum 1 18 1124AY dev cum 1 .788 3 1991AY dev cum 1 30 1735AY dev cum 1 .330 4 1991AY dev cum 1 42 2182AY dev cum 1 .708 5 1991AY dev cum 1 54 2745AY dev cum 1 .596 6 1991 and translate the proposed model into a non-linear brms model.
nlform <-bf(cum~ult * (1 -exp(-(dev / theta) nlprior <-c(prior(normal(5000, 1000), nlpar = "ult"), prior(normal (1, 2), nlpar = "omega"), prior (normal(45, 10), nlpar = "theta")) fit_loss1 <-brm(formula = nlform, data = loss, family = gaussian(), prior = nlprior, control = list(adapt_delta = 0.9)) In the above functions calls, quite a few things are going on. The formulas are wrapped in bf to combine them into one object. The first formula specifies the non-linear model. We set argument nl = TRUE so that brms takes this formula literally and instead of using standard R formula parsing. We specify one additional formula per non-linear parameter (a) to clarify what variables are covariates and what are parameters and (b) to specify the predictor term for the parameters. We estimate a group-level effect of accident year (variable AY) for the ultimate loss ult. This also shows nicely how a non-linear parameter is actually a placeholder for a linear predictor, which in the case of ult, contains only a varying intercept for year. Both omega and theta are assumed to be constant across observations so we just fit a population-level intercept.
Priors on population-level effects are required and, for the present model, are actually mandatory to ensure identifiability. Otherwise, we may observe that different Markov chains converge to different parameter regions as multiple posterior distribution are equally plausible. Setting prior distributions is a difficult task especially in non-linear models. It requires some experience and knowledge both about the model that is being fitted and about the data at hand. Additionally, there is more to keep in mind to optimize the sampler's performance: Firstly, using non-or weakly informative priors in non-linear models often leads to problems even if the model is generally identified. For instance, if a zero-centered and reasonably wide prior such as normal(0, 10000) it set on ult, there is little information about theta and omega for samples of ult being close to zero, which may lead to convergence problems. Secondly, Stan works best when parameters are roughly on the same order of magnitude (Stan Development Team 2017b). In the present example, ult is of three orders larger than omega. Still, the sampler seems to work quite well, but this may not be true for other models. One solution is to rescale parameters before model fitting. For instance, for the present example, one could have downscaled ult by replacing it with ult * 1000 and correspondingly the normal(5000, 1000) prior with normal(5, 1).
In the control argument we increase adapt_delta to get rid of a few divergent transitions (cf. Stan Development Team, 2017b;Bürkner, 2017 Samples were drawn using sampling (NUTS). For each parameter, Eff.Sample is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat = 1).

as well as
conditional_effects (fit_loss1) (see Figure 5). We can also visualize the cumulative insurance loss over time separately for each year.
conditions <-data.frame(AY = unique(loss$AY)) rownames(conditions) <-unique(loss$AY) me_year <-conditional_effects(fit_loss1, conditions = conditions, re_formula = NULL, method = "predict") plot(me_year, ncol = 5, points = TRUE) (see Figure 6). It is evident that there is some variation in cumulative loss across accident years, for instance due to natural disasters happening only in certain years. Further, we see that the uncertainty in the predicted cumulative loss is larger for later years with fewer available data points. In the above model, we considered omega and delta to be constant across years, which may not necessarily be true. We can easily investigate this by fitting varying intercepts for all three non-linear parameters also estimating group-level correlation using the above introduced ID syntax. We could have also specified all predictor terms more conveniently within one formula as because the structure of the predictor terms is identical. To compare model fit, we perform leave-one-out cross-validation.  -fit_loss2 -5.15 5.34 1996 1997 1998 1999 2000 1991 1992 1993 1994 1995 30 60  Since smaller values indicate better expected out-of-sample predictions and thus better model fit, the simpler model that only has a varying intercept over parameter ult is preferred. This may not be overly surprising, given that three varying intercepts as well as three group-level correlations are probably overkill for data containing only 55 observations. Nevertheless, it nicely demonstrates how to apply the ID syntax in practice. More examples of non-linear models can be found in vignette("brms_nonlinear").

Example 4: Performance of school children
Suppose that we want to predict the performance of students in the final exams at the end of the year. There are many variables to consider, but one important factor will clearly be school membership. Schools might differ in the ratio of teachers and students, the general quality of teaching, in the cognitive ability of the students they draw, or other factors we are not aware of that induce dependency among students of the same school. Thus, it is advised to apply a multilevel modeling techniques including school membership as a group-level term. Of course, we should account for class membership and other levels of the educational hierarchy as well, but for the purpose of the present example, we will focus on schools only. Usually, accounting for school membership is pretty-straight forward by simply adding a varying intercept to the formula: (1 | school). However, a non-negligible number of students might change schools during the year. This would result in a situation where one student is a member of multiple schools and so we need a multi-membership model. Setting up such a model not only requires information on the different schools students attend during the year, but also the amount of time spend at each school. The latter can be used to weight the influence each school has on its students, since more time attending a school will likely result in greater influence. For now, let us assume that students change schools maximally once a year and spend equal time at each school. We will later see how to relax these assumptions.
Real educational data are usually relatively large and complex so that we simulate our own data for the purpose of this tutorial paper. We simulate 10 schools and 1000 students, with each school having the same expected number of 100 students. We model 10% of students as changing schools. Thus, school variables are identical, but we still have to specify both in order to pass the data appropriately. Incorporating no other predictors into the model for simplicity, a multimembership model is specified as fit_mm <-brm(y~1 + (1 | mm(s1, s2)), data = data_mm) The only new syntax element is that multiple grouping factors (s1 and s2) are wrapped in mm. Everything else remains exactly the same. Note that we did not specify the relative weights of schools for each student and thus, by default, equal weights are assumed. With regard to the assumptions made in the above example, it is unlikely that all children who change schools stay in both schools equally long. To relax this assumption, we have to specify weights. First, we amend the simulated data to contain non-equal weights for students changing schools. For all other students, weighting does of course not matter as they stay in the same school anyway. Incorporating these weights into the model is straight forward.
fit_mm2 <-brm(y~1 + (1 | mm(s1, s2, weights = cbind(w1, w2))), data = data_mm) The summary output is similar to the previous, so we do not show it here. The second assumption that students change schools only once a year, may also easily be relaxed by providing more than two grouping factors, say, mm(s1, s2, s3).

Comparison between packages
Over the years, many R packages have been developed that implement MLMs, each being more or less general in their supported models. In Bürkner (2017) Since then, quite a few new features have been added in particular to brms and rstanarm. Accordingly, in the present paper, I will update these comparisons, but focus on brms, rstanarm, and MCMCglmm as the possibly most important R packages implementing Bayesian MLMs. While brms and rstanarm are both based on the probabilistic programming language Stan, MCMCglmm implements its own custom MCMC algorithm. Modeling options and other important information of these packages are summarized in Table 1 and will be discussed in detail below.
Regarding general model classes, all three packages support the most common types such as linear, count data and certain survival models. Currently, brms and MCMCglmm provide more flexibility when modeling categorical and ordinal data. Additionally, they offer support for zero-inflated and hurdle models to account for access zeros in the data (see Example 1 above). For survival / time-to-event models, rstanarm offers great flexibility via the stan_jm function, which allows for complex association structures between time-to-event data and one or more models of longitudinal covariates (for details see https://cran.r-project.org/ web/packages/rstanarm/vignettes/jm.html). Model classes currently specific to brms are robust linear models using Student's t-distribution (family student) as well as response times models via the exponentially modified Gaussian (family exgaussian) distribution or the Wiener diffusion model (family wiener). The latter allows to simultaneously model dichotomous decisions and their corresponding response times (for a detailed example see http://singmann.org/wiener-model-analysis-with-brms-part-i/).
All three packages offer many additional modeling options, with brms currently having the greatest flexibility (see Table 1 for a summary). Moreover, the packages differ in the general framework, in which they are implemented: brms and MCMCglmm each come with a single model fitting function (brm and MCMCglmm respectively), through which all of their models can be specified. Further, their framework allows to seamlessly combine most modeling options with each other in the same model. In contrast, the approach of rstanarm is to emulate existing functions of other packages. This has the advantage of an easier transition between classical and Bayesian models, since the syntax used to specify models stays the same. However, it comes with the main disadvantage that many modeling options cannot be used in combination within the same model.
Information criteria are available in all three packages. The advantages of WAIC and LOO implemented in brms and rstanarm, are their less restrictive assumptions and that their standard errors can be easily estimated to get a better sense of the uncertainty in the criteria.
Comparing the prior options of the packages, brms offers a little more flexibility than MCM-Cglmm and rstanarm, as virtually any prior distribution can be applied on population-level effects as well as on the standard deviations of group-level effects. In addition, I believe that the way priors are specified in brms is more intuitive as it is directly evident what prior is actually applied. In brms, Bayes factors are available both via Savage-Dickey ratios (Wagenmakers, Lodewyckx, Kuriyal, and Grasman 2010) and bridge-sampling (Gronau and Singmann 2017), while rstanarm allows for the latter option. For a detailed comparison with respect to sampling efficiency, see Bürkner (2017).

Conclusion
The present paper is meant to introduce R users and developers to the extended lme4 formula syntax applied in brms. Only a subset of modeling options were discussed in detail, which ensured the paper was not too broad. For some of the more basic models that brms can fit, see Bürkner (2017). Many more examples can be found in the growing number of vignettes accompanying the package (see vignette(package = "brms") for an overview).
To date, brms is already one of the most flexible R packages when it comes to regression modeling. However, for the future, there are quite a few more features that I am planning to implement (see https://github.com/paul-buerkner/brms/issues for the current list of issues). In addition to smaller, incremental updates, I have two specific features in mind: (1) latent variables estimated via confirmatory factor analysis and (2) missing value imputation. I receive ideas and suggestions from users almost every day -for which I am always grateful -and so the list of features that will be implemented in the proceeding versions of brms will continue to grow.