## Overview

The **brms** package provides an interface to fit Bayesian generalized(non-)linear multivariate multilevel models using Stan, which is a C++package for performing full Bayesian inference (seehttps://mc-stan.org/). The formula syntax is very similar to that ofthe package lme4 to provide a familiar and simple interface forperforming regression analyses. A wide range of response distributionsare supported, allowing users to fit – among others – linear, robustlinear, count data, survival, response times, ordinal, zero-inflated,and even self-defined mixture models all in a multilevel context.Further modeling options include non-linear and smooth terms,auto-correlation structures, censored data, missing value imputation,and quite a few more. In addition, all parameters of the responsedistribution can be predicted in order to perform distributionalregression. Multivariate models (i.e., models with multiple responsevariables) can be fit, as well. Prior specifications are flexible andexplicitly encourage users to apply prior distributions that actuallyreflect their beliefs. Model fit can easily be assessed and comparedwith posterior predictive checks, cross-validation, and Bayes factors.

## Resources

- Introduction to brms(Journal of Statistical Software)
- Advanced multilevel modeling withbrms(The R Journal)
- Website (Website of brmswith documentation and vignettes)
- Blog posts(List of blog posts about brms)
- Ask a question (Stan Forums onDiscourse)
- Open an issue(GitHub issues for bug reports and feature requests)

## How to use brms

`library(brms)`

As a simple example, we use poisson regression to model the seizurecounts in epileptic patients to investigate whether the treatment(represented by variable `Trt`

) can reduce the seizure counts andwhether the effect of the treatment varies with the (standardized)baseline number of seizures a person had before treatment (variable`zBase`

). As we have multiple observations per person, a group-levelintercept is incorporated to account for the resulting dependency in thedata.

`fit1 <- brm(count ~ zAge + zBase * Trt + (1|patient), data = epilepsy, family = poisson())`

The results (i.e., posterior draws) can be investigated using

`summary(fit1)#> Family: poisson #> Links: mu = log #> Formula: count ~ zAge + zBase * Trt + (1 | patient) #> Data: epilepsy (Number of observations: 236) #> Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;#> total post-warmup draws = 4000#> #> Group-Level Effects: #> ~patient (Number of levels: 59) #> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS#> sd(Intercept) 0.58 0.07 0.46 0.74 1.00 810 1753#> #> Population-Level Effects: #> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS#> Intercept 1.77 0.12 1.53 2.00 1.00 779 1319#> zAge 0.09 0.09 -0.09 0.26 1.00 684 1071#> zBase 0.70 0.12 0.46 0.95 1.00 847 1453#> Trt1 -0.27 0.17 -0.59 0.06 1.00 661 1046#> zBase:Trt1 0.05 0.16 -0.26 0.37 1.00 993 1624#> #> Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS#> and Tail_ESS are effective sample size measures, and Rhat is the potential#> scale reduction factor on split chains (at convergence, Rhat = 1).`

On the top of the output, some general information on the model isgiven, such as family, formula, number of iterations and chains. Next,group-level effects are displayed separately for each grouping factor interms of standard deviations and (in case of more than one group-leveleffect per grouping factor; not displayed here) correlations betweengroup-level effects. On the bottom of the output, population-leveleffects (i.e.regression coefficients) are displayed. If incorporated,autocorrelation effects and family specific parameters (e.g.theresidual standard deviation ‘sigma’ in normal models) are also given.

In general, every parameter is summarized using the mean (‘Estimate’)and the standard deviation (‘Est.Error’) of the posterior distributionas well as two-sided 95% credible intervals (‘l-95% CI’ and ‘u-95% CI’)based on quantiles. We see that the coefficient of `Trt`

is negativewith a zero overlapping 95%-CI. This indicates that, on average, thetreatment may reduce seizure counts by some amount but the evidencebased on the data and applied model is not very strong and stillinsufficient by standard decision rules. Further, we find littleevidence that the treatment effect varies with the baseline number ofseizures.

The last two values (‘Eff.Sample’ and ‘Rhat’) provide information on howwell the algorithm could estimate the posterior distribution of thisparameter. If ‘Rhat’ is considerably greater than 1, the algorithm hasnot yet converged and it is necessary to run more iterations and / orset stronger priors.

To visually investigate the chains as well as the posteriordistributions, we can use the `plot`

method. If we just want to seeresults of the regression coefficients of `Trt`

and `zBase`

, we go for

`plot(fit1, variable = c("b_Trt1", "b_zBase"))`

A more detailed investigation can be performed by running`launch_shinystan(fit1)`

. To better understand the relationship of thepredictors with the response, I recommend the `conditional_effects`

method:

`plot(conditional_effects(fit1, effects = "zBase:Trt"))`

This method uses some prediction functionality behind the scenes, whichcan also be called directly. Suppose that we want to predict responses(i.e.seizure counts) of a person in the treatment group (`Trt = 1`

) andin the control group (`Trt = 0`

) with average age and average number ofprevious seizures. Than we can use

`newdata <- data.frame(Trt = c(0, 1), zAge = 0, zBase = 0)predict(fit1, newdata = newdata, re_formula = NA)#> Estimate Est.Error Q2.5 Q97.5#> [1,] 5.8980 2.505627 2 11#> [2,] 4.5595 2.162320 1 9`

We need to set `re_formula = NA`

in order not to condition of thegroup-level effects. While the `predict`

method returns predictions ofthe responses, the `fitted`

method returns predictions of the regressionline.

`fitted(fit1, newdata = newdata, re_formula = NA)#> Estimate Est.Error Q2.5 Q97.5#> [1,] 5.917144 0.7056695 4.632004 7.387471#> [2,] 4.529949 0.5360204 3.544085 5.624005`

Both methods return the same estimate (up to random error), while thelatter has smaller variance, because the uncertainty in the regressionline is smaller than the uncertainty in each response. If we want topredict values of the original data, we can just leave the `newdata`

argument empty.

Suppose, we want to investigate whether there is overdispersion in themodel, that is residual variation not accounted for by the responsedistribution. For this purpose, we include a second group-levelintercept that captures possible overdispersion.

`fit2 <- brm(count ~ zAge + zBase * Trt + (1|patient) + (1|obs), data = epilepsy, family = poisson())`

We can then go ahead and compare both models via approximateleave-one-out (LOO) cross-validation.

`loo(fit1, fit2)#> Output of model 'fit1':#> #> Computed from 4000 by 236 log-likelihood matrix#> #> Estimate SE#> elpd_loo -670.4 36.7#> p_loo 92.8 14.3#> looic 1340.8 73.3#> ------#> Monte Carlo SE of elpd_loo is NA.#> #> Pareto k diagnostic values:#> Count Pct. Min. n_eff#> (-Inf, 0.5] (good) 214 90.7% 251 #> (0.5, 0.7] (ok) 17 7.2% 80 #> (0.7, 1] (bad) 3 1.3% 81 #> (1, Inf) (very bad) 2 0.8% 6 #> See help('pareto-k-diagnostic') for details.#> #> Output of model 'fit2':#> #> Computed from 4000 by 236 log-likelihood matrix#> #> Estimate SE#> elpd_loo -595.2 14.1#> p_loo 108.0 7.3#> looic 1190.4 28.2#> ------#> Monte Carlo SE of elpd_loo is NA.#> #> Pareto k diagnostic values:#> Count Pct. Min. n_eff#> (-Inf, 0.5] (good) 82 34.7% 544 #> (0.5, 0.7] (ok) 103 43.6% 153 #> (0.7, 1] (bad) 47 19.9% 22 #> (1, Inf) (very bad) 4 1.7% 7 #> See help('pareto-k-diagnostic') for details.#> #> Model comparisons:#> elpd_diff se_diff#> fit2 0.0 0.0 #> fit1 -75.2 26.9`

The `loo`

output when comparing models is a little verbose. We first seethe individual LOO summaries of the two models and then the comparisonbetween them. Since higher `elpd`

(i.e., expected log posterior density)values indicate better fit, we see that the model accounting foroverdispersion (i.e., `fit2`

) fits substantially better. However, wealso see in the individual LOO outputs that there are severalproblematic observations for which the approximations may have not havebeen very accurate. To deal with this appropriately, we need to fallback to other methods such as `reloo`

or `kfold`

but this requires themodel to be refit several times which takes too long for the purpose ofa quick example. The post-processing methods we have shown above arejust the tip of the iceberg. For a full list of methods to apply onfitted model objects, type `methods(class = "brmsfit")`

.

## Citing brms and related software

Developing and maintaining open source software is an important yetoften underappreciated contribution to scientific progress. Thus,whenever you are using open source software (or software in general),please make sure to cite it appropriately so that developers get creditfor their work.

When using brms, please cite one or more of the following publications:

- Bürkner P. C. (2017). brms: An R Package for Bayesian MultilevelModels using Stan.
*Journal of Statistical Software*. 80(1), 1-28.doi.org/10.18637/jss.v080.i01 - Bürkner P. C. (2018). Advanced Bayesian Multilevel Modeling with theR Package brms.
*The R Journal*. 10(1), 395-411.doi.org/10.32614/RJ-2018-017

As brms is a high-level interface to Stan, please additionally citeStan:

- Carpenter B., Gelman A., Hoffman M. D., Lee D., Goodrich B.,Betancourt M., Brubaker M., Guo J., Li P., and Riddell A. (2017).Stan: A probabilistic programming language.
*Journal of StatisticalSoftware*. 76(1). 10.18637/jss.v076.i01

Further, brms relies on several other R packages and, of course, on Ritself. To find out how to cite R and its packages, use the `citation`

function. There are some features of brms which specifically rely oncertain packages. The **rstan** package together with **Rcpp** makesStan conveniently accessible in R. Visualizations andposterior-predictive checks are based on **bayesplot** and **ggplot2**.Approximate leave-one-out cross-validation using `loo`

and relatedmethods is done via the **loo** package. Marginal likelihood basedmethods such as `bayes_factor`

are realized by means of the**bridgesampling** package. Splines specified via the `s`

and `t2`

functions rely on **mgcv**. If you use some of these features, pleasealso consider citing the related packages.

## FAQ

### How do I install brms?

To install the latest release version from CRAN use

`install.packages("brms")`

The current developmental version can be downloaded from github via

`if (!requireNamespace("remotes")) { install.packages("remotes")}remotes::install_github("paul-buerkner/brms")`

Because brms is based on Stan, a C++ compiler is required. The programRtools (available on https://cran.r-project.org/bin/windows/Rtools/)comes with a C++ compiler for Windows. On Mac, you should install Xcode.For further instructions on how to get the compilers running, see theprerequisites section onhttps://github.com/stan-dev/rstan/wiki/RStan-Getting-Started.

### I am new to brms. Where can I start?

Detailed instructions and case studies are given in the package’sextensive vignettes. See `vignette(package = "brms")`

for an overview.For documentation on formula syntax, families, and prior distributionssee `help("brm")`

.

### Where do I ask questions, propose a new feature, or report a bug?

Questions can be asked on the Stanforums on Discourse. To propose a newfeature or report a bug, please open an issue onGitHub.

### How can I extract the generated Stan code?

If you have already fitted a model, just apply the `stancode`

method onthe fitted model object. If you just want to generate the Stan codewithout any model fitting, use the `make_stancode`

function.

### Can I avoid compiling models?

When you fit your model for the first time with brms, there is currentlyno way to avoid compilation. However, if you have already fitted yourmodel and want to run it again, for instance with more draws, you can dothis without recompilation by using the `update`

method. For moredetails see `help("update.brmsfit")`

.

### What is the difference between brms and rstanarm?

The rstanarm package is similar to brms in that it also allows to fitregression models using Stan for the backend estimation. Contrary tobrms, rstanarm comes with precompiled code to save the compilation time(and the need for a C++ compiler) when fitting a model. However, as brmsgenerates its Stan code on the fly, it offers much more flexibility inmodel specification than rstanarm. Also, multilevel models are currentlyfitted a bit more efficiently in brms. For detailed comparisons of brmswith other common R packages implementing multilevel models, see`vignette("brms_multilevel")`

and `vignette("brms_overview")`

.