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 (see http://mc-stan.org/). The formula syntax is very similar to that of the package lme4 to provide a familiar and simple interface for performing regression analyses. A wide range of distributions and link functions are supported, allowing users to fit -- among others -- linear, robust linear, count data, survival, response times, ordinal, zero-inflated, hurdle, 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, meta-analytic standard errors, and quite a few more. In addition, all parameters of the response distribution can be predicted in order to perform distributional regression. Multivariate models (i.e. models with multiple response variables) can be fitted, as well. Prior specifications are flexible and explicitly encourage users to apply prior distributions that actually reflect their beliefs. Model fit can easily be assessed and compared with posterior predictive checks and leave-one-out cross-validation. A list of blog posts about brms can be found at https://paul-buerkner.github.io/blog/old-brms-blogposts/.
library(brms)As a simple example, we use poisson regression to model the seizure counts in epileptic patients to investigate whether the treatment (represented by variable Trt) can reduce the seizure counts. Two group-level intercepts are incorporated to account for the variance between patients as well as for the residual variance.
fit <- brm(count ~ log_Age_c + log_Base4_c * Trt + (1|patient) + (1|obs),
data = epilepsy, family = poisson())
#> Compiling the C++ model
#> Start samplingThe results (i.e. posterior samples) can be investigated using
summary(fit, waic = TRUE)
#> Family: poisson
#> Links: mu = log
#> Formula: count ~ log_Age_c + log_Base4_c * Trt + (1 | patient) + (1 | obs)
#> Data: epilepsy (Number of observations: 236)
#> Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
#> total post-warmup samples = 4000
#> ICs: LOO = NA; WAIC = 1145.29; R2 = NA
#>
#> Group-Level Effects:
#> ~obs (Number of levels: 236)
#> Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
#> sd(Intercept) 0.37 0.04 0.29 0.46 1641 1.00
#>
#> ~patient (Number of levels: 59)
#> Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
#> sd(Intercept) 0.51 0.07 0.38 0.66 2021 1.00
#>
#> Population-Level Effects:
#> Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
#> Intercept 1.74 0.11 1.52 1.96 3010 1.00
#> log_Age_c 0.48 0.37 -0.29 1.17 2474 1.00
#> log_Base4_c 0.88 0.14 0.60 1.15 2353 1.00
#> Trt1 -0.34 0.16 -0.65 -0.03 2801 1.00
#> log_Base4_c:Trt1 0.35 0.22 -0.07 0.79 2646 1.00
#>
#> 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).On the top of the output, some general information on the model is given, such as family, formula, number of iterations and chains, as well as the WAIC, which is an information criterion for Bayesian models. Next, group-level effects are displayed seperately for each grouping factor in terms of standard deviations and (in case of more than one group-level effect per grouping factor; not displayed here) correlations between group-level effects. On the bottom of the output, population-level effects are displayed. If incorporated, autocorrelation effects and family specific parameters (e.g., the residual 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 distribution as well as two-sided 95% credible intervals ('l-95% CI' and 'u-95% CI') based on quantiles. The last two values ('Eff.Sample' and 'Rhat') provide information on how well the algorithm could estimate the posterior distribution of this parameter. If 'Rhat' is considerably greater than 1, the algorithm has not yet converged and it is necessary to run more iterations and / or set stronger priors.
To visually investigate the chains as well as the posterior distributions, you can use
plot(fit) An even more detailed investigation can be achieved by applying the shinystan package:
launch_shinystan(fit) There are several methods to compute and visualize model predictions. Suppose that we want to predict responses (i.e. seizure counts) of a person in the treatment group (Trt = 1) and in the control group (Trt = 0) with average age and average number of previous seizures. Than we can use
newdata <- data.frame(Trt = c(0, 1), log_Age_c = 0, log_Base4_c = 0)
predict(fit, newdata = newdata, allow_new_levels = TRUE, probs = c(0.05, 0.95))
#> Estimate Est.Error 5%ile 95%ile
#> [1,] 6.92075 5.530291 1 17
#> [2,] 4.92100 4.059295 0 12We need to set allow_new_levels = TRUE because we want to predict responses of a person that was not present in the data used to fit the model. While the predict method returns predictions of the responses, the fitted method returns predictions of the regression line.
fitted(fit, newdata = newdata, allow_new_levels = TRUE, probs = c(0.05, 0.95))
#> Estimate Est.Error 5%ile 95%ile
#> [1,] 6.913456 4.718377 1.990409 15.98126
#> [2,] 4.931857 3.343603 1.445182 11.38421Both methods return the same etimate (up to random error), while the latter has smaller variance, because the uncertainty in the regression line is smaller than the uncertainty in each response. If we want to predict values of the original data, we can just leave the newdata argument empty.
A related feature is the computation and visualization of marginal effects, which can help in better understanding the influence of the predictors on the response.
plot(marginal_effects(fit, probs = c(0.05, 0.95)))For a complete list of methods to apply on brms models see
methods(class = "brmsfit")
#> [1] add_ic add_loo add_waic
#> [4] as.array as.data.frame as.matrix
#> [7] as.mcmc bayes_factor bayes_R2
#> [10] bridge_sampler coef control_params
#> [13] expose_functions family fitted
#> [16] fixef formula getCall
#> [19] hypothesis kfold launch_shinystan
#> [22] log_lik log_posterior logLik
#> [25] loo LOO loo_linpred
#> [28] loo_predict loo_predictive_interval marginal_effects
#> [31] marginal_smooths model.frame neff_ratio
#> [34] ngrps nobs nsamples
#> [37] nuts_params pairs parnames
#> [40] plot post_prob posterior_interval
#> [43] posterior_linpred posterior_predict posterior_samples
#> [46] posterior_summary pp_average pp_check
#> [49] pp_mixture predict predictive_error
#> [52] print prior_samples prior_summary
#> [55] ranef residuals rhat
#> [58] stancode standata stanplot
#> [61] summary update VarCorr
#> [64] vcov waic WAIC
#> see '?methods' for accessing help and source codeDetails on formula syntax, families and link functions, as well as prior distributions can be found on the help page of the brm function:
help("brm") More instructions on how to use brms are given in the package's main vignettes.
vignette("brms_overview")
vignette("brms_multilevel")To install the latest release version from CRAN use
install.packages("brms")The current developmental version can be downloaded from github via
if (!require("devtools")) {
install.packages("devtools")
}
devtools::install_github("paul-buerkner/brms", dependencies = TRUE)Because brms is based on Stan, a C++ compiler is required. The program Rtools (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 the prerequisites section on https://github.com/stan-dev/rstan/wiki/RStan-Getting-Started.
Questions can be asked in the google group brms-users. To propose a new feature or report a bug, please open an issue on github. Of course, you can always write me an email (paul.buerkner@gmail.com).
If you have already fitted a model, just apply the stancode method on the fitted model object. If you just want to generate the Stan code without any model fitting, use the make_stancode function.
When you fit your model for the first time with brms, there is currently no way to avoid compilation. However, if you have already fitted your model and want to run it again, for instance with more samples, you can do this without recompilation by using the update method. For more details see
help("update.brmsfit")Specification of non-linear or distributional models requires multiple formulae. In brms, the function brmsformula (or short bf) is used to combine all formulae into one object, which can then be passed to the formula argument of brm. More help is given in
help("brmsformula")For a detailed discussion of some examples see
vignette("brms_nonlinear")vignette("brms_distreg")rstanarm is an R package similar to brms that also allows to fit regression models using Stan for the backend estimation. Contrary to brms, rstanarm comes with precompiled code to save the compilation time (and the need for a C++ compiler) when fitting a model. However, as brms generates its Stan code on the fly, it offers much more flexibility in model specification than rstanarm. Also, multilevel models are currently fitted a bit more efficiently in brms. For a detailed comparison of brms with other common R packages implementing multilevel models, see
vignette("brms_overview")