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brms

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 (see https://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 response distributions are supported, allowing users to fit – among others – linear, robust linear, 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 response distribution can be predicted in order to perform distributional regression. Multivariate models (i.e., models with multiple response variables) can be fit, 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, cross-validation, and Bayes factors.

Resources

• Introduction to brms (Journal of Statistical Software)
• Advanced multilevel modeling with brms (The R Journal)
• Website (Website of brms with documentation and vignettes)
• Blog posts (List of blog posts about brms)
• Ask a question (Stan Forums on Discourse)
• 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 seizure counts in epileptic patients to investigate whether the treatment (represented by variable Trt) can reduce the seizure counts and whether 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-level intercept is incorporated to account for the resulting dependency in the data.

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
#> 
#> Multilevel Hyperparameters:
#> ~patient (Number of levels: 59) 
#>               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sd(Intercept)     0.59      0.07     0.46     0.74 1.01      566     1356
#> 
#> Regression Coefficients:
#>            Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept      1.78      0.12     1.55     2.01 1.00      771     1595
#> zAge           0.09      0.09    -0.08     0.27 1.00      590     1302
#> zBase          0.71      0.12     0.47     0.96 1.00      848     1258
#> Trt1          -0.27      0.16    -0.60     0.05 1.01      749     1172
#> zBase:Trt1     0.05      0.17    -0.30     0.38 1.00      833     1335
#> 
#> 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 is given, such as family, formula, number of iterations and chains. Next, group-level effects are displayed separately 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 (i.e. regression coefficients) 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. We see that the coefficient of Trt is negative with a zero overlapping 95%-CI. This indicates that, on average, the treatment may reduce seizure counts by some amount but the evidence based on the data and applied model is not very strong and still insufficient by standard decision rules. Further, we find little evidence that the treatment effect varies with the baseline number of seizures.

The last three values (‘ESS_bulk’, ‘ESS_tail’, 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, we can use the plot method. If we just want to see results of the regression coefficients of Trt and zBase, we go for

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

<img src="man/figures/README-plot-1.png" width="60%" style="display: block; margin: auto;" />

A more detailed investigation can be performed by running launch_shinystan(fit1). To better understand the relationship of the predictors with the response, I recommend the conditional_effects method:

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

<img src="man/figures/README-conditional_effects-1.png" width="60%" style="display: block; margin: auto;" />

This method uses some prediction functionality behind the scenes, which can also be called directly. 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), zAge = 0, zBase = 0)
predict(fit1, newdata = newdata, re_formula = NA)
#>      Estimate Est.Error Q2.5 Q97.5
#> [1,]  5.91200  2.494857    2    11
#> [2,]  4.57325  2.166058    1     9

We need to set re_formula = NA in order not to condition of the group-level effects. While the predict method returns predictions of the responses, the fitted method returns predictions of the regression line.

fitted(fit1, newdata = newdata, re_formula = NA)
#>      Estimate Est.Error     Q2.5    Q97.5
#> [1,] 5.945276 0.7075160 4.696257 7.450011
#> [2,] 4.540081 0.5343471 3.579757 5.665132

Both methods return the same estimate (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.

Suppose, we want to investigate whether there is overdispersion in the model, that is residual variation not accounted for by the response distribution. For this purpose, we include a second group-level intercept 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 approximate leave-one-out (LOO) cross-validation.

loo(fit1, fit2)
#> Output of model 'fit1':
#> 
#> Computed from 4000 by 236 log-likelihood matrix.
#> 
#>          Estimate   SE
#> elpd_loo   -671.7 36.6
#> p_loo        94.3 14.2
#> looic      1343.4 73.2
#> ------
#> MCSE of elpd_loo is NA.
#> MCSE and ESS estimates assume MCMC draws (r_eff in [0.4, 2.0]).
#> 
#> Pareto k diagnostic values:
#>                          Count Pct.    Min. ESS
#> (-Inf, 0.7]   (good)     228   96.6%   157     
#>    (0.7, 1]   (bad)        7    3.0%   <NA>    
#>    (1, Inf)   (very bad)   1    0.4%   <NA>    
#> See help('pareto-k-diagnostic') for details.
#> 
#> Output of model 'fit2':
#> 
#> Computed from 4000 by 236 log-likelihood matrix.
#> 
#>          Estimate   SE
#> elpd_loo   -596.8 14.0
#> p_loo       109.7  7.2
#> looic      1193.6 28.1
#> ------
#> MCSE of elpd_loo is NA.
#> MCSE and ESS estimates assume MCMC draws (r_eff in [0.4, 1.7]).
#> 
#> Pareto k diagnostic values:
#>                          Count Pct.    Min. ESS
#> (-Inf, 0.7]   (good)     172   72.9%   83      
#>    (0.7, 1]   (bad)       56   23.7%   <NA>    
#>    (1, Inf)   (very bad)   8    3.4%   <NA>    
#> See help('pareto-k-diagnostic') for details.
#> 
#> Model comparisons:
#>      elpd_diff se_diff
#> fit2   0.0       0.0  
#> fit1 -74.9      27.2

The loo output when comparin