Fit Bayesian generalized (non-)linear multivariate multilevel models using Stan for full Bayesian inference. 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 distributions can be predicted in order to perform distributional regression. Prior specifications are flexible and explicitly encourage users to apply prior distributions that actually reflect their beliefs. In addition, model fit can easily be assessed and compared with posterior predictive checks and leave-one-out cross-validation.
brm(formula, data, family = gaussian(), prior = NULL, autocor = NULL,
cov_ranef = NULL, sample_prior = c("no", "yes", "only"), sparse = FALSE,
knots = NULL, stanvars = NULL, stan_funs = NULL, fit = NA,
save_ranef = TRUE, save_mevars = FALSE, save_all_pars = FALSE,
inits = "random", chains = 4, iter = 2000, warmup = floor(iter/2),
thin = 1, cores = getOption("mc.cores", 1L), control = NULL,
algorithm = c("sampling", "meanfield", "fullrank"),
future = getOption("future", FALSE), silent = TRUE, seed = NA,
save_model = NULL, save_dso = TRUE, ...)An object of class
formula,
brmsformula, or mvbrmsformula
(or one that can be coerced to that classes):
A symbolic description of the model to be fitted.
The details of model specification are explained in
brmsformula.
An object of class data.frame
(or one that can be coerced to that class)
containing data of all variables used in the model.
A description of the response distribution and link function
to be used in the model. This can be a family function,
a call to a family function or a character string naming the family.
Every family function has a link argument allowing to specify
the link function to be applied on the response variable.
If not specified, default links are used.
For details of supported families see
brmsfamily.
By default, a linear gaussian model is applied.
In multivariate models, family might also be a list of families.
An optional cor_brms object describing
the correlation structure within the response variable
(i.e., the 'autocorrelation').
See the documentation of cor_brms for a description
of the available correlation structures. Defaults to NULL,
corresponding to no correlations.
In multivariate models, autocor might also be a list
of autocorrelation structures.
A list of matrices that are proportional to the
(within) covariance structure of the group-level effects.
The names of the matrices should correspond to columns
in data that are used as grouping factors.
All levels of the grouping factor should appear as rownames
of the corresponding matrix. This argument can be used,
among others to model pedigrees and phylogenetic effects.
See vignette("brms_phylogenetics") for more details.
Indicate if samples from all specified
proper priors should be drawn additionally to the posterior samples
(defaults to "no"). Among others, these samples can be used
to calculate Bayes factors for point hypotheses.
If set to "only", samples are drawn solely from
the priors ignoring the likelihood. In this case,
all parameters must have proper priors.
Logical; indicates whether the population-level
design matrices should be treated as sparse (defaults to FALSE).
For design matrices with many zeros, this can considerably
reduce required memory. Sampling speed is currently not
improved or even slightly decreased.
Optional list containing user specified knot values to be
used for basis construction of smoothing terms.
See gamm for more details.
An optional stanvars object generated
by function stanvar to define additional variables
in the data block of Stan.
An optional character string containing self-defined Stan functions, which will be included in the functions block of the generated Stan code.
An instance of S3 class brmsfit derived from a previous fit;
defaults to NA.
If fit is of class brmsfit, the compiled model associated
with the fitted result is re-used and all arguments
modifying the model code or data are ignored.
It is not recommended to use this argument directly, but to call
the update method, instead.
A flag to indicate if group-level effects
for each level of the grouping factor(s) should be saved
(default is TRUE). Set to FALSE to save memory.
The argument has no impact on the model fitting itself.
A flag to indicate if samples
of latent noise-free variables obtained by using me
and mi terms should be saved (default is FALSE).
Saving these samples allows to better use methods such as
predict with the latent variables but
leads to very large R objects even for models
of moderate size and complexity.
A flag to indicate if samples from
all variables defined in Stan's parameters block
should be saved (default is FALSE). Saving these samples
is required in order to apply the methods bridge_sampler,
bayes_factor, and post_prob.
Either "random" or "0".
If inits is "random" (the default),
Stan will randomly generate initial values for parameters.
If it is "0", all parameters are initialized to zero.
This option is sometimes useful for certain families,
as it happens that default ("random") inits cause samples
to be essentially constant.
Generally, setting inits = "0" is worth a try, if chains do not behave well.
Alternatively, inits can be a list of lists containing
the initial values, or a function (or function name) generating initial values.
The latter options are mainly implemented for internal testing.
Number of Markov chains (defaults to 4).
Number of total iterations per chain (including warmup; defaults to 2000).
A positive integer specifying number of warmup (aka burnin) iterations.
This also specifies the number of iterations used for stepsize adaptation,
so warmup samples should not be used for inference. The number of warmup should not
be larger than iter and the default is iter/2.
Thinning rate. Must be a positive integer.
Set thin > 1 to save memory and computation time if iter is large.
Number of cores to use when executing the chains in parallel,
which defaults to 1 but we recommend setting the mc.cores option
to be as many processors as the hardware and RAM allow (up to the number of chains).
For non-Windows OS in non-interactive R sessions, forking is used
instead of PSOCK clusters.
A named list of parameters to control the sampler's behavior.
It defaults to NULL so all the default values are used.
The most important control parameters are discussed in the 'Details'
section below. For a comprehensive overview see stan.
Character string indicating the estimation approach to use.
Can be "sampling" for MCMC (the default), "meanfield" for
variational inference with independent normal distributions, or
"fullrank" for variational inference with a multivariate normal
distribution.
logical; If TRUE (the default), most of the
informational messages of compiler and sampler are suppressed.
The actual sampling progress is still printed.
Set refresh = 0 to turn this off as well.
The seed for random number generation to make results
reproducible. If NA (the default), Stan will set
the seed randomly.
Either NULL or a character string.
In the latter case, the model code is
saved in a file named after the string supplied in save_model,
which may also contain the full path where to save the file.
If only a name is given, the file is saved in the current working directory.
Logical, defaulting to TRUE, indicating whether
the dynamic shared object (DSO) compiled from the C++ code for the model
will be saved or not. If TRUE, we can draw samples from the same
model in another R session using the saved DSO
(i.e., without compiling the C++ code again).
Further arguments to be passed to Stan.
An object of class brmsfit, which contains the posterior samples along
with many other useful information about the model.
Use methods(class = "brmsfit") for an overview on available methods.
Fit a generalized (non-)linear multivariate multilevel model
via full Bayesian inference using Stan. A general overview is provided
in the vignettes vignette("brms_overview") and
vignette("brms_multilevel"). For a full list of available
vignettes see vignette(package = "brms").
Formula syntax of brms models
Details of the formula syntax applied in brms
can be found in brmsformula.
Families and link functions
Details of families supported by brms
can be found in brmsfamily.
Prior distributions
Priors should be specified using the
set_prior function.
Its documentation contains detailed information
on how to correctly specify priors. To find out on
which parameters or parameter classes priors can be defined,
use get_prior.
Default priors are chosen to be non or very weakly informative
so that their influence on the results will be negligible and
you usually don't have to worry about them.
However, after getting more familiar with Bayesian statistics,
I recommend you to start thinking about reasonable informative
priors for your model parameters: Nearly always, there is at least some
prior information available that can be used to improve your inference.
Adjusting the sampling behavior of Stan
In addition to choosing the number of iterations, warmup samples,
and chains, users can control the behavior of the NUTS sampler,
by using the control argument.
The most important reason to use control is to decrease
(or eliminate at best) the number of divergent transitions
that cause a bias in the obtained posterior samples.
Whenever you see the warning
"There were x divergent transitions after warmup."
you should really think about increasing adapt_delta.
To do this, write control = list(adapt_delta = <x>),
where <x> should usually be value between 0.8
(current default) and 1. Increasing adapt_delta
will slow down the sampler but will decrease the number of
divergent transitions threatening the validity of your
posterior samples.
Another problem arises when the depth of the tree being evaluated
in each iteration is exceeded. This is less common than having
divergent transitions, but may also bias the posterior samples.
When it happens, Stan will throw out a warning suggesting
to increase max_treedepth, which can be accomplished by
writing control = list(max_treedepth = <x>) with a positive
integer <x> that should usually be larger than the current
default of 10. For more details on the control argument
see stan.
Paul-Christian Buerkner (2017). brms: An R Package for Bayesian Multilevel Models Using Stan. Journal of Statistical Software, 80(1), 1-28. doi:10.18637/jss.v080.i01
Paul-Christian Buerkner (in review). Advanced Bayesian Multilevel Modeling with the R Package brms. arXiv preprint.
# NOT RUN {
# Poisson regression for the number of seizures in epileptic patients
# using student_t priors for population-level effects
# and half cauchy priors for standard deviations of group-level effects
bprior1 <- prior(student_t(5,0,10), class = b) +
prior(cauchy(0,2), class = sd)
fit1 <- brm(count ~ log_Age_c + log_Base4_c * Trt + (1|patient),
data = epilepsy, family = poisson(), prior = bprior1)
# generate a summary of the results
summary(fit1)
# plot the MCMC chains as well as the posterior distributions
plot(fit1, ask = FALSE)
# predict responses based on the fitted model
head(predict(fit1))
# plot marginal effects for each predictor
plot(marginal_effects(fit1), ask = FALSE)
# investigate model fit
loo(fit1)
pp_check(fit1)
# Ordinal regression modeling patient's rating of inhaler instructions
# category specific effects are estimated for variable 'treat'
fit2 <- brm(rating ~ period + carry + cs(treat),
data = inhaler, family = sratio("logit"),
prior = set_prior("normal(0,5)"), chains = 2)
summary(fit2)
plot(fit2, ask = FALSE)
WAIC(fit2)
# Survival regression modeling the time between the first
# and second recurrence of an infection in kidney patients.
fit3 <- brm(time | cens(censored) ~ age * sex + disease + (1|patient),
data = kidney, family = lognormal())
summary(fit3)
plot(fit3, ask = FALSE)
plot(marginal_effects(fit3), ask = FALSE)
# Probit regression using the binomial family
ntrials <- sample(1:10, 100, TRUE)
success <- rbinom(100, size = ntrials, prob = 0.4)
x <- rnorm(100)
data4 <- data.frame(ntrials, success, x)
fit4 <- brm(success | trials(ntrials) ~ x, data = data4,
family = binomial("probit"))
summary(fit4)
# Simple non-linear gaussian model
x <- rnorm(100)
y <- rnorm(100, mean = 2 - 1.5^x, sd = 1)
data5 <- data.frame(x, y)
bprior5 <- prior(normal(0, 2), nlpar = a1) +
prior(normal(0, 2), nlpar = a2)
fit5 <- brm(bf(y ~ a1 - a2^x, a1 + a2 ~ 1, nl = TRUE),
data = data5, prior = bprior5)
summary(fit5)
plot(marginal_effects(fit5), ask = FALSE)
# Normal model with heterogeneous variances
data_het <- data.frame(
y = c(rnorm(50), rnorm(50, 1, 2)),
x = factor(rep(c("a", "b"), each = 50))
)
fit6 <- brm(bf(y ~ x, sigma ~ 0 + x), data = data_het)
summary(fit6)
plot(fit6)
marginal_effects(fit6)
# extract estimated residual SDs of both groups
sigmas <- exp(posterior_samples(fit6, "^b_sigma_"))
ggplot(stack(sigmas), aes(values)) +
geom_density(aes(fill = ind))
# Quantile regression predicting the 25%-quantile
fit7 <- brm(bf(y ~ x, quantile = 0.25), data = data_het,
family = asym_laplace())
summary(fit7)
marginal_effects(fit7)
# use the future package for more flexible parallelization
library(future)
plan(multiprocess)
fit7 <- update(fit7, future = TRUE)
# }
# NOT RUN {
# }
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