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Fit Bayesian Generalized (Non-)Linear Multivariate Multilevel Models

Source: R/brm.R
brm.Rd

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, extended-support beta regression, 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.

Usage

brm(
  formula,
  data,
  family = gaussian(),
  prior = NULL,
  autocor = NULL,
  data2 = NULL,
  cov_ranef = NULL,
  sample_prior = "no",
  sparse = NULL,
  knots = NULL,
  drop_unused_levels = TRUE,
  stanvars = NULL,
  stan_funs = NULL,
  fit = NA,
  save_pars = getOption("brms.save_pars", NULL),
  save_ranef = NULL,
  save_mevars = NULL,
  save_all_pars = NULL,
  init = NULL,
  inits = NULL,
  chains = 4,
  iter = getOption("brms.iter", 2000),
  warmup = floor(iter/2),
  thin = 1,
  cores = getOption("mc.cores", 1),
  threads = getOption("brms.threads", NULL),
  opencl = getOption("brms.opencl", NULL),
  normalize = getOption("brms.normalize", TRUE),
  control = NULL,
  algorithm = getOption("brms.algorithm", "sampling"),
  backend = getOption("brms.backend", "rstan"),
  future = getOption("future", FALSE),
  silent = getOption("brms.silent", 1),
  seed = NA,
  save_model = NULL,
  stan_model_args = list(),
  file = NULL,
  file_compress = TRUE,
  file_refit = getOption("brms.file_refit", "never"),
  empty = FALSE,
  rename = TRUE,
  ...
)

Arguments

formula

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.

data

An object of class data.frame (or one that can be coerced to that class) containing data of all variables used in the model.

family

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.

prior

One or more brmsprior objects created by set_prior or related functions and combined using the c method or the + operator. See also default_prior for more help.

autocor

(Deprecated) 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. It is now recommend to specify autocorrelation terms directly within formula. See brmsformula for more details.

data2

A named list of objects containing data, which cannot be passed via argument data. Required for some objects used in autocorrelation structures to specify dependency structures as well as for within-group covariance matrices.

cov_ranef

(Deprecated) 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. It is now recommended to specify those matrices in the formula interface using the gr and related functions. See vignette("brms_phylogenetics") for more details.

sample_prior

Indicate if draws from priors should be drawn additionally to the posterior draws. Options are "no" (the default), "yes", and "only". Among others, these draws can be used to calculate Bayes factors for point hypotheses via hypothesis. Please note that improper priors are not sampled, including the default improper priors used by brm. See set_prior on how to set (proper) priors. Please also note that prior draws for the overall intercept are not obtained by default for technical reasons. See brmsformula how to obtain prior draws for the intercept. If sample_prior is set to "only", draws are drawn solely from the priors ignoring the likelihood, which allows among others to generate draws from the prior predictive distribution. In this case, all parameters must have proper priors.

sparse

(Deprecated) 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. It is now recommended to use the sparse argument of brmsformula and related functions.

knots

Optional list containing user specified knot values to be used for basis construction of smoothing terms. See gamm for more details.

drop_unused_levels

Should unused factors levels in the data be dropped? Defaults to TRUE.

stanvars

An optional stanvars object generated by function stanvar to define additional variables for use in Stan's program blocks.

stan_funs

(Deprecated) An optional character string containing self-defined Stan functions, which will be included in the functions block of the generated Stan code. It is now recommended to use the stanvars argument for this purpose instead.

fit

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.

save_pars

An object generated by save_pars controlling which parameters should be saved in the model. The argument has no impact on the model fitting itself.

save_ranef

(Deprecated) 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.

save_mevars

(Deprecated) A flag to indicate if draws of latent noise-free variables obtained by using me and mi terms should be saved (default is FALSE). Saving these draws 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.

save_all_pars

(Deprecated) A flag to indicate if draws from all variables defined in Stan's parameters block should be saved (default is FALSE). Saving these draws is required in order to apply the methods bridge_sampler, bayes_factor, and post_prob. Can be set globally for the current R session via the "brms.save_pars" option (see options).

init

Initial values for the sampler. If NULL (the default) or "random", Stan will randomly generate initial values for parameters in a reasonable range. If 0, all parameters are initialized to zero on the unconstrained space. This option is sometimes useful for certain families, as it happens that default random initial values cause draws to be essentially constant. Generally, setting init = 0 is worth a try, if chains do not initialize or behave well. Alternatively, init 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 but are available to users if necessary. If specifying initial values using a list or a function then currently the parameter names must correspond to the names used in the generated Stan code (not the names used in R). For more details on specifying initial values you can consult the documentation of the selected backend.

inits

(Deprecated) Alias of init.

chains

Number of Markov chains (defaults to 4).

iter

Number of total iterations per chain (including warmup; defaults to 2000). Can be set globally for the current R session via the "brms.iter" option (see options).

warmup

A positive integer specifying number of warmup (aka burnin) iterations. This also specifies the number of iterations used for stepsize adaptation, so warmup draws should not be used for inference. The number of warmup should not be larger than iter and the default is iter/2.

thin

Thinning rate. Must be a positive integer. Set thin > 1 to save memory and computation time if iter is large.

cores

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.

threads

Number of threads to use in within-chain parallelization. For more control over the threading process, threads may also be a brmsthreads object created by threading. Within-chain parallelization is experimental! We recommend its use only if you are experienced with Stan's reduce_sum function and have a slow running model that cannot be sped up by any other means. Can be set globally for the current R session via the "brms.threads" option (see options).

opencl

The platform and device IDs of the OpenCL device to use for fitting using GPU support. If you don't know the IDs of your OpenCL device, c(0,0) is most likely what you need. For more details, see opencl. Can be set globally for the current R session via the "brms.opencl" option

normalize

Logical. Indicates whether normalization constants should be included in the Stan code (defaults to TRUE). Setting it to FALSE requires Stan version >= 2.25 to work. If FALSE, sampling efficiency may be increased but some post processing functions such as bridge_sampler will not be available. Can be controlled globally for the current R session via the `brms.normalize` option.

control

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.

algorithm

Character string naming the estimation approach to use. Options are "sampling" for MCMC (the default), "meanfield" for variational inference with independent normal distributions, "fullrank" for variational inference with a multivariate normal distribution, "pathfinder" for the pathfinder algorithm, "laplace" for the laplace approximation, or "fixed_param" for sampling from fixed parameter values. Can be set globally for the current R session via the "brms.algorithm" option (see options).

backend

Character string naming the package to use as the backend for fitting the Stan model. Options are "rstan" (the default) or "cmdstanr". Can be set globally for the current R session via the "brms.backend" option (see options). Details on the rstan and cmdstanr packages are available at https://mc-stan.org/rstan/ and https://mc-stan.org/cmdstanr/, respectively. Additionally a "mock" backend is available to make testing brms and packages that depend on it easier. The "mock" backend does not actually do any fitting, it only checks the generated Stan code for correctness and then returns whatever is passed in an additional mock_fit argument as the result of the fit.

future

Logical; If TRUE, the future package is used for parallel execution of the chains and argument cores will be ignored. Can be set globally for the current R session via the "future" option. The execution type is controlled via plan (see the examples section below).

silent

Verbosity level between 0 and 2. If 1 (the default), most of the informational messages of compiler and sampler are suppressed. If 2, even more messages are suppressed. The actual sampling progress is still printed. Set refresh = 0 to turn this off as well. If using backend = "rstan" you can also set open_progress = FALSE to prevent opening additional progress bars. Can be set globally for the current R session via the "brms.silent" option (see options).

seed

The seed for random number generation to make results reproducible. If NA (the default), Stan will set the seed randomly.

save_model

Either NULL or a character string. In the latter case, the model's Stan code is saved via cat in a text file named after the string supplied in save_model.

stan_model_args

A list of further arguments passed to rstan::stan_model for backend = "rstan" or to cmdstanr::cmdstan_model for backend = "cmdstanr", which allows to change how models are compiled.

file

Either NULL or a character string. In the latter case, the fitted model object is saved via saveRDS in a file named after the string supplied in file. The .rds extension is added automatically. If the file already exists, brm will load and return the saved model object instead of refitting the model. Unless you specify the file_refit argument as well, the existing files won't be overwritten, you have to manually remove the file in order to refit and save the model under an existing file name. The file name is stored in the brmsfit object for later usage.

file_compress

Logical or a character string, specifying one of the compression algorithms supported by saveRDS. If the file argument is provided, this compression will be used when saving the fitted model object.

file_refit

Modifies when the fit stored via the file argument is re-used. Can be set globally for the current R session via the "brms.file_refit" option (see options). For "never" (default) the fit is always loaded if it exists and fitting is skipped. For "always" the model is always refitted. If set to "on_change", brms will refit the model if model, data or algorithm as passed to Stan differ from what is stored in the file. This also covers changes in priors, sample_prior, stanvars, covariance structure, etc. If you believe there was a false positive, you can use brmsfit_needs_refit to see why refit is deemed necessary. Refit will not be triggered for changes in additional parameters of the fit (e.g., initial values, number of iterations, control arguments, ...). A known limitation is that a refit will be triggered if within-chain parallelization is switched on/off.

empty

Logical. If TRUE, the Stan model is not created and compiled and the corresponding 'fit' slot of the brmsfit object will be empty. This is useful if you have estimated a brms-created Stan model outside of brms and want to feed it back into the package.

rename

For internal use only.

...

Further arguments passed to Stan. For backend = "rstan" the arguments are passed to sampling or vb. For backend = "cmdstanr" the arguments are passed to the cmdstanr::sample or cmdstanr::variational method.

Value

An object of class brmsfit, which contains the posterior draws along with many other useful information about the model. Use methods(class = "brmsfit") for an overview on available methods.

Details

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 default_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 draws, 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 draws. 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 draws.

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 draws. 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.

Parallelization with multiple CPU cores

brms can make use of multiple CPU cores in parallel to speed up computations in various ways. For efficient use of the available resources it is recommended to only use parallelism to an extend such that the available physical CPUs are not oversubscribed. For example, when you have 8 CPU cores locally available, then you may consider to run 4 chains with 2 threads per chain for best performance if you happen to just run a single model. In case you run a simulation study which requires to run many times a given model, then neither chain nor within-chain parallelization is advisable as the computational resources are already exhausted by the simulation study and any further parallelization beyond the simulation study itself will in fact slow down the overall runtime. Please be aware that for historical reasons the nomenclature of the arguments is possibly confusing. The cores argument refers to running different chains in parallel and the within-chain parallelization will allocate for each chain as many threads as requested. The requested threads therefore increase the use of overall CPUs in a multiplicative way.

For more advanced parallelization (including beyond single model fits), brms also integrates with the future package. Importantly, this enables seamless integration with the mirai parallelization framework through the use of the future.mirai adapter. With mirai local and remote machines can be used in a fully transparent manner to the user. This includes the possibility to use large number of remote machines running in the context of a computer cluster, which are managed with queuing systems. Please refer to the section on distributed computing of mirai::daemons.

References

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 (2018). Advanced Bayesian Multilevel Modeling with the R Package brms. The R Journal. 10(1), 395–411. doi:10.32614/RJ-2018-017

See also

brms, brmsformula, brmsfamily, brmsfit

Author

Paul-Christian Buerkner paul.buerkner@gmail.com

Examples

# \dontrun{
# Poisson regression for the number of seizures in epileptic patients
fit1 <- brm(
  count ~ zBase * Trt + (1|patient),
  data = epilepsy, family = poisson(),
  prior = prior(normal(0, 10), class = b) +
    prior(cauchy(0, 2), class = sd)
)
#> Compiling Stan program...
#> Start sampling
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 3.6e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.36 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1: 
#> Chain 1: 
#> Chain 1: Iteration:    1 / 2000 [  0%]  (Warmup)
#> Chain 1: Iteration:  200 / 2000 [ 10%]  (Warmup)
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#> Chain 1: Iteration: 1000 / 2000 [ 50%]  (Warmup)
#> Chain 1: Iteration: 1001 / 2000 [ 50%]  (Sampling)
#> Chain 1: Iteration: 1200 / 2000 [ 60%]  (Sampling)
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#> Chain 1: Iteration: 1800 / 2000 [ 90%]  (Sampling)
#> Chain 1: Iteration: 2000 / 2000 [100%]  (Sampling)
#> Chain 1: 
#> Chain 1:  Elapsed Time: 2.188 seconds (Warm-up)
#> Chain 1:                1.533 seconds (Sampling)
#> Chain 1:                3.721 seconds (Total)
#> Chain 1: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
#> Chain 2: 
#> Chain 2: Gradient evaluation took 2.6e-05 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.26 seconds.
#> Chain 2: Adjust your expectations accordingly!
#> Chain 2: 
#> Chain 2: 
#> Chain 2: Iteration:    1 / 2000 [  0%]  (Warmup)
#> Chain 2: Iteration:  200 / 2000 [ 10%]  (Warmup)
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#> Chain 2: Iteration: 2000 / 2000 [100%]  (Sampling)
#> Chain 2: 
#> Chain 2:  Elapsed Time: 2.193 seconds (Warm-up)
#> Chain 2:                1.514 seconds (Sampling)
#> Chain 2:                3.707 seconds (Total)
#> Chain 2: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3: 
#> Chain 3: Gradient evaluation took 2.6e-05 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.26 seconds.
#> Chain 3: Adjust your expectations accordingly!
#> Chain 3: 
#> Chain 3: 
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#> Chain 3: Iteration: 2000 / 2000 [100%]  (Sampling)
#> Chain 3: 
#> Chain 3:  Elapsed Time: 1.898 seconds (Warm-up)
#> Chain 3:                1.564 seconds (Sampling)
#> Chain 3:                3.462 seconds (Total)
#> Chain 3: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
#> Chain 4: 
#> Chain 4: Gradient evaluation took 2.6e-05 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.26 seconds.
#> Chain 4: Adjust your expectations accordingly!
#> Chain 4: 
#> Chain 4: 
#> Chain 4: Iteration:    1 / 2000 [  0%]  (Warmup)
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#> Chain 4: 
#> Chain 4:  Elapsed Time: 2.129 seconds (Warm-up)
#> Chain 4:                1.699 seconds (Sampling)
#> Chain 4:                3.828 seconds (Total)
#> Chain 4: 

# generate a summary of the results
summary(fit1)
#>  Family: poisson 
#>   Links: mu = log 
#> Formula: count ~ 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      606     1109
#> 
#> Regression Coefficients:
#>            Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept      1.79      0.12     1.55     2.03 1.00      682     1268
#> zBase          0.70      0.12     0.46     0.95 1.00      720      968
#> Trt1          -0.29      0.18    -0.63     0.05 1.01      548      828
#> zBase:Trt1     0.02      0.17    -0.30     0.36 1.01      720     1258
#> 
#> 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).

# plot the MCMC chains as well as the posterior distributions
plot(fit1)


# predict responses based on the fitted model
head(predict(fit1))
#>      Estimate Est.Error Q2.5 Q97.5
#> [1,]  3.48525  1.995063    0     8
#> [2,]  3.54450  2.034712    0     8
#> [3,]  2.80625  1.867105    0     7
#> [4,]  3.25575  1.939789    0     8
#> [5,] 13.84325  4.151565    6    23
#> [6,]  5.46925  2.607826    1    11

# plot conditional effects for each predictor
plot(conditional_effects(fit1), ask = FALSE)




# investigate model fit
loo(fit1)
#> Warning: Found 8 observations with a pareto_k > 0.7 in model 'fit1'. We recommend to set 'moment_match = TRUE' in order to perform moment matching for problematic observations. 
#> 
#> Computed from 4000 by 236 log-likelihood matrix.
#> 
#>          Estimate   SE
#> elpd_loo   -670.7 36.3
#> p_loo        93.3 14.0
#> looic      1341.3 72.7
#> ------
#> MCSE of elpd_loo is NA.
#> MCSE and ESS estimates assume MCMC draws (r_eff in [0.3, 2.6]).
#> 
#> Pareto k diagnostic values:
#>                          Count Pct.    Min. ESS
#> (-Inf, 0.7]   (good)     228   96.6%   160     
#>    (0.7, 1]   (bad)        6    2.5%   <NA>    
#>    (1, Inf)   (very bad)   2    0.8%   <NA>    
#> See help('pareto-k-diagnostic') for details.
pp_check(fit1)
#> Using 10 posterior draws for ppc type 'dens_overlay' by default.



# 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)
#> Compiling Stan program...
#> Start sampling
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 0.000286 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 2.86 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1: 
#> Chain 1: 
#> Chain 1: Iteration:    1 / 2000 [  0%]  (Warmup)
#> Chain 1: Iteration:  200 / 2000 [ 10%]  (Warmup)
#> Chain 1: Iteration:  400 / 2000 [ 20%]  (Warmup)
#> Chain 1: Iteration:  600 / 2000 [ 30%]  (Warmup)
#> Chain 1: Iteration:  800 / 2000 [ 40%]  (Warmup)
#> Chain 1: Iteration: 1000 / 2000 [ 50%]  (Warmup)
#> Chain 1: Iteration: 1001 / 2000 [ 50%]  (Sampling)
#> Chain 1: Iteration: 1200 / 2000 [ 60%]  (Sampling)
#> Chain 1: Iteration: 1400 / 2000 [ 70%]  (Sampling)
#> Chain 1: Iteration: 1600 / 2000 [ 80%]  (Sampling)
#> Chain 1: Iteration: 1800 / 2000 [ 90%]  (Sampling)
#> Chain 1: Iteration: 2000 / 2000 [100%]  (Sampling)
#> Chain 1: 
#> Chain 1:  Elapsed Time: 3.03 seconds (Warm-up)
#> Chain 1:                2.925 seconds (Sampling)
#> Chain 1:                5.955 seconds (Total)
#> Chain 1: 
#> 
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summary(fit2)
#>  Family: sratio 
#>   Links: mu = logit 
#> Formula: rating ~ period + carry + cs(treat) 
#>    Data: inhaler (Number of observations: 572) 
#>   Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
#>          total post-warmup draws = 2000
#> 
#> Regression Coefficients:
#>              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept[1]     0.55      0.09     0.38     0.73 1.00     2298     1517
#> Intercept[2]     2.36      0.29     1.83     2.98 1.00     1947     1213
#> Intercept[3]     0.48      0.58    -0.67     1.64 1.00     1892     1328
#> period           0.22      0.17    -0.11     0.55 1.00     2497     1272
#> carry           -0.22      0.16    -0.53     0.12 1.00     1675     1684
#> treat[1]        -0.78      0.24    -1.28    -0.34 1.00     1912     1650
#> treat[2]        -1.03      0.59    -2.29     0.07 1.00     1592     1500
#> treat[3]         1.15      1.15    -1.14     3.44 1.00     1594     1100
#> 
#> Further Distributional Parameters:
#>      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> disc     1.00      0.00     1.00     1.00   NA       NA       NA
#> 
#> 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).
plot(fit2, ask = FALSE)


WAIC(fit2)
#> Warning: 
#> 4 (0.7%) p_waic estimates greater than 0.4. We recommend trying loo instead.
#> 
#> Computed from 2000 by 572 log-likelihood matrix.
#> 
#>           Estimate   SE
#> elpd_waic   -459.6 17.5
#> p_waic         8.3  1.2
#> waic         919.2 35.1
#> 
#> 4 (0.7%) p_waic estimates greater than 0.4. We recommend trying loo instead. 


# 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())
#> Compiling Stan program...
#> Start sampling
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1: Rejecting initial value:
#> Chain 1:   Log probability evaluates to log(0), i.e. negative infinity.
#> Chain 1:   Stan can't start sampling from this initial value.
#> Chain 1: 
#> Chain 1: Gradient evaluation took 2.3e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.23 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 2:   Log probability evaluates to log(0), i.e. negative infinity.
#> Chain 2:   Stan can't start sampling from this initial value.
#> Chain 2: 
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#> Chain 2: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3: Rejecting initial value:
#> Chain 3:   Log probability evaluates to log(0), i.e. negative infinity.
#> Chain 3:   Stan can't start sampling from this initial value.
#> Chain 3: 
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#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
#> Chain 4: Rejecting initial value:
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#> Chain 4:   Stan can't start sampling from this initial value.
#> Chain 4: 
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#> Chain 4:                25.905 seconds (Total)
#> Chain 4: 
#> Warning: There were 1000 transitions after warmup that exceeded the maximum treedepth. Increase max_treedepth above 10. See
#> https://mc-stan.org/misc/warnings.html#maximum-treedepth-exceeded
#> Warning: Examine the pairs() plot to diagnose sampling problems
#> Warning: The largest R-hat is 1.59, indicating chains have not mixed.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#r-hat
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
summary(fit3)
#> Warning: Inference for the model posterior has not converged (some Rhats are > 1.05). Be careful when analysing the results! We recommend running more iterations or setting stronger priors.
#>  Family: lognormal 
#>   Links: mu = identity 
#> Formula: time | cens(censored) ~ age * sex + disease + (1 | patient) 
#>    Data: kidney (Number of observations: 76) 
#>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
#>          total post-warmup draws = 4000
#> 
#> Multilevel Hyperparameters:
#> ~patient (Number of levels: 38) 
#>               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sd(Intercept)     0.45      0.22     0.04     0.90 1.54      345     1387
#> 
#> Regression Coefficients:
#>               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept        -9.31     20.67   -45.08     4.36 1.58        7       12
#> age              -0.09      0.19    -0.41     0.06 1.55        7       20
#> sexfemale         2.07      1.35     0.49     4.84 1.36        9      113
#> diseaseGN         0.06      0.96    -1.38     1.53 1.54        7       11
#> diseaseAN        -0.71      0.52    -1.47     0.36 1.21       14     1434
#> diseasePKD        0.27      0.80    -0.67     1.83 1.31       10      167
#> age:sexfemale     0.48      0.88    -0.07     2.00 1.59        7       11
#> 
#> Further Distributional Parameters:
#>       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma     1.35      0.35     0.93     1.93 1.59        7       17
#> 
#> 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).
plot(fit3, ask = FALSE)


plot(conditional_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"))
#> Compiling Stan program...
#> Start sampling
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 2.5e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.25 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
#> Chain 2: 
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#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.21 seconds.
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#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3: 
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#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
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#> Chain 4: 
summary(fit4)
#>  Family: binomial 
#>   Links: mu = probit 
#> Formula: success | trials(ntrials) ~ x 
#>    Data: data4 (Number of observations: 100) 
#>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
#>          total post-warmup draws = 4000
#> 
#> Regression Coefficients:
#>           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept    -0.22      0.06    -0.33    -0.11 1.00     3413     2925
#> x            -0.01      0.05    -0.12     0.09 1.00     3676     2881
#> 
#> 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).


# Non-linear Gaussian model
fit5 <- brm(
  bf(cum ~ ult * (1 - exp(-(dev/theta)^omega)),
     ult ~ 1 + (1|AY), omega ~ 1, theta ~ 1,
     nl = TRUE),
  data = loss, family = gaussian(),
  prior = c(
    prior(normal(5000, 1000), nlpar = "ult"),
    prior(normal(1, 2), nlpar = "omega"),
    prior(normal(45, 10), nlpar = "theta")
  ),
  control = list(adapt_delta = 0.9)
)
#> Compiling Stan program...
#> Start sampling
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1: Rejecting initial value:
#> Chain 1:   Error evaluating the log probability at the initial value.
#> Chain 1: Exception: normal_lpdf: Location parameter[1] is nan, but must be finite! (in 'anon_model', line 68, column 4 to column 41)
#> Chain 1: Rejecting initial value:
#> Chain 1:   Error evaluating the log probability at the initial value.
#> Chain 1: Exception: normal_lpdf: Location parameter[1] is nan, but must be finite! (in 'anon_model', line 68, column 4 to column 41)
#> Chain 1: 
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#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.35 seconds.
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#> Chain 1: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
#> Chain 2: Rejecting initial value:
#> Chain 2:   Error evaluating the log probability at the initial value.
#> Chain 2: Exception: normal_lpdf: Location parameter[1] is nan, but must be finite! (in 'anon_model', line 68, column 4 to column 41)
#> Chain 2: Rejecting initial value:
#> Chain 2:   Error evaluating the log probability at the initial value.
#> Chain 2: Exception: normal_lpdf: Location parameter[1] is nan, but must be finite! (in 'anon_model', line 68, column 4 to column 41)
#> Chain 2: Rejecting initial value:
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#> Chain 2: Exception: normal_lpdf: Location parameter[1] is nan, but must be finite! (in 'anon_model', line 68, column 4 to column 41)
#> Chain 2: Rejecting initial value:
#> Chain 2:   Error evaluating the log probability at the initial value.
#> Chain 2: Exception: normal_lpdf: Location parameter[1] is nan, but must be finite! (in 'anon_model', line 68, column 4 to column 41)
#> Chain 2: Rejecting initial value:
#> Chain 2:   Error evaluating the log probability at the initial value.
#> Chain 2: Exception: normal_lpdf: Location parameter[1] is nan, but must be finite! (in 'anon_model', line 68, column 4 to column 41)
#> Chain 2: 
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#> Chain 2: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3: Rejecting initial value:
#> Chain 3:   Error evaluating the log probability at the initial value.
#> Chain 3: Exception: normal_lpdf: Location parameter[1] is nan, but must be finite! (in 'anon_model', line 68, column 4 to column 41)
#> Chain 3: 
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#> Chain 3: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
#> Chain 4: Rejecting initial value:
#> Chain 4:   Error evaluating the log probability at the initial value.
#> Chain 4: Exception: normal_lpdf: Location parameter[1] is nan, but must be finite! (in 'anon_model', line 68, column 4 to column 41)
#> Chain 4: Rejecting initial value:
#> Chain 4:   Error evaluating the log probability at the initial value.
#> Chain 4: Exception: normal_lpdf: Location parameter[1] is nan, but must be finite! (in 'anon_model', line 68, column 4 to column 41)
#> Chain 4: Rejecting initial value:
#> Chain 4:   Error evaluating the log probability at the initial value.
#> Chain 4: Exception: normal_lpdf: Location parameter[1] is nan, but must be finite! (in 'anon_model', line 68, column 4 to column 41)
#> Chain 4: 
#> Chain 4: Gradient evaluation took 6.9e-05 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.69 seconds.
#> Chain 4: Adjust your expectations accordingly!
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#> Chain 4: 
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#> Chain 4:                3.662 seconds (Total)
#> Chain 4: 
summary(fit5)
#>  Family: gaussian 
#>   Links: mu = identity 
#> Formula: cum ~ ult * (1 - exp(-(dev/theta)^omega)) 
#>          ult ~ 1 + (1 | AY)
#>          omega ~ 1
#>          theta ~ 1
#>    Data: loss (Number of observations: 55) 
#>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
#>          total post-warmup draws = 4000
#> 
#> Multilevel Hyperparameters:
#> ~AY (Number of levels: 10) 
#>                   Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sd(ult_Intercept)   751.87    237.81   427.80  1332.92 1.00     1286     1743
#> 
#> Regression Coefficients:
#>                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> ult_Intercept    5295.83    299.78  4725.94  5916.82 1.01      934     1567
#> omega_Intercept     1.34      0.05     1.24     1.44 1.00     2455     2372
#> theta_Intercept    46.18      2.16    42.40    50.81 1.00     2191     2052
#> 
#> Further Distributional Parameters:
#>       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma   139.77     15.20   114.20   172.91 1.00     3063     2671
#> 
#> 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).
conditional_effects(fit5)



# 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)
#> Compiling Stan program...
#> Start sampling
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 2.4e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.24 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
#> Chain 2: 
#> Chain 2: Gradient evaluation took 1.6e-05 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.16 seconds.
#> Chain 2: Adjust your expectations accordingly!
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#> Chain 2: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3: 
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#> Chain 3: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
#> Chain 4: 
#> Chain 4: Gradient evaluation took 1.6e-05 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.16 seconds.
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#> Chain 4: 
summary(fit6)
#>  Family: gaussian 
#>   Links: mu = identity; sigma = log 
#> Formula: y ~ x 
#>          sigma ~ 0 + x
#>    Data: data_het (Number of observations: 100) 
#>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
#>          total post-warmup draws = 4000
#> 
#> Regression Coefficients:
#>           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept    -0.08      0.14    -0.35     0.19 1.00     5924     3099
#> xb            1.20      0.34     0.53     1.84 1.00     2878     2354
#> sigma_xa     -0.04      0.10    -0.22     0.17 1.00     3358     2775
#> sigma_xb      0.75      0.10     0.56     0.97 1.00     3747     2097
#> 
#> 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).
plot(fit6)

conditional_effects(fit6)


# extract estimated residual SDs of both groups
sigmas <- exp(as.data.frame(fit6, variable = "^b_sigma_", regex = TRUE))
ggplot(stack(sigmas), aes(values)) +
  geom_density(aes(fill = ind))
#> Error in ggplot(stack(sigmas), aes(values)): could not find function "ggplot"


# Quantile regression predicting the 25%-quantile
fit7 <- brm(bf(y ~ x, quantile = 0.25), data = data_het,
            family = asym_laplace())
#> Compiling Stan program...
#> Start sampling
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 4e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.4 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
#> Chain 2: 
#> Chain 2: Gradient evaluation took 3.4e-05 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.34 seconds.
#> Chain 2: Adjust your expectations accordingly!
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#> Chain 2: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3: 
#> Chain 3: Gradient evaluation took 3.5e-05 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.35 seconds.
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#> Chain 3: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
#> Chain 4: 
#> Chain 4: Gradient evaluation took 4e-05 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.4 seconds.
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#> Chain 4: 
summary(fit7)
#>  Family: asym_laplace 
#>   Links: mu = identity 
#> Formula: y ~ x 
#>          quantile = 0.25
#>    Data: data_het (Number of observations: 100) 
#>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
#>          total post-warmup draws = 4000
#> 
#> Regression Coefficients:
#>           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept    -0.77      0.15    -1.07    -0.47 1.00     3754     3031
#> xb            0.72      0.30     0.12     1.29 1.00     2291     2289
#> 
#> Further Distributional Parameters:
#>          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma        0.50      0.05     0.41     0.61 1.00     2795     2543
#> quantile     0.25      0.00     0.25     0.25   NA       NA       NA
#> 
#> 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).
conditional_effects(fit7)


# use the future package for more flexible parallelization
library(future)
plan(multisession, workers = 4)
fit7a <- update(fit7, future = TRUE)
#> Start sampling
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 6.8e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.68 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
#> Chain 2: 
#> Chain 2: Gradient evaluation took 5.7e-05 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.57 seconds.
#> Chain 2: Adjust your expectations accordingly!
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#> Chain 2: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3: 
#> Chain 3: Gradient evaluation took 5.2e-05 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.52 seconds.
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#> Chain 3: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
#> Chain 4: 
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#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.9 seconds.
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#> Chain 4: 

# we may also use mirai allowing the use of remote machines as
# well, including the use of computer clusters managed by queuing
# systems
mirai::daemons(4) # distributed computing with url and remote argument
plan(future.mirai::mirai_cluster)
fit7b <- update(fit7, future = TRUE)
#> Start sampling
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 7.1e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.71 seconds.
#> Chain 1: Adjust your expectations accordingly!
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mirai::daemons(0) # shuts down the 4 workers

# fit a model manually via rstan
scode <- stancode(count ~ Trt, data = epilepsy)
sdata <- standata(count ~ Trt, data = epilepsy)
stanfit <- rstan::stan(model_code = scode, data = sdata)
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# feed the Stan model back into brms
fit8 <- brm(count ~ Trt, data = epilepsy, empty = TRUE)
fit8$fit <- stanfit
fit8 <- rename_pars(fit8)
summary(fit8)
#>  Family: gaussian 
#>   Links: mu = identity 
#> Formula: count ~ Trt 
#>    Data: epilepsy (Number of observations: 236) 
#>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
#>          total post-warmup draws = 4000
#> 
#> Regression Coefficients:
#>           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept     8.43      1.20     6.09    10.74 1.00     4109     2787
#> Trt1         -0.61      1.65    -3.87     2.61 1.00     4594     2902
#> 
#> Further Distributional Parameters:
#>       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma    12.37      0.56    11.33    13.53 1.00     4671     3366
#> 
#> 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).
# }