Set up a model formula for use in the brms package
Set up a model formula for use in the brms package allowing to define (potentially non-linear) additive multilevel models for all parameters of the assumed response distribution.
brmsformula(formula, ..., nonlinear = NULL)
- An object of class
formula(or one that can be coerced to that class): a symbolic description of the model to be fitted. The details of model specification are given under 'Details'.
formulaobjects to specify predictors of special model parts and auxiliary parameters. Formulas can either be named directly or contain names on their left-hand side. Currently, the following names are accepted:
sigma(residual standard deviation of the
shape(shape parameter of the
negbinomialand related zero-inflated / hurdle families);
nu(degrees of freedom parameter of the
phi(precision parameter of the
hu(hurdle probability). All auxiliary parameters are modeled on the log or logit scale to ensure correct definition intervals after transformation.
- An optional list of formuluas, specifying
linear models for non-linear parameters. If
formulais treated as an ordinary formula. If not
formulais treated as a non-linear model and
nonlinearshould contain a formula for each non-linear parameter, which has the parameter on the left hand side and its linear predictor on the right hand side. Alternatively, it can be a single formula with all non-linear parameters on the left hand side (separated by a
+) and a common linear predictor on the right hand side. More information is given under 'Details'.
formula argument accepts formulae of the following syntax:
response | addition ~ Pterms + (Gterms | group)
Pterms part contains effects that are assumed to be the
same across obervations. We call them 'population-level' effects
or (adopting frequentist vocabulary) 'fixed' effects. The optional
Gterms part may contain effects that are assumed to vary
accross grouping variables specified in
call them 'group-level' effects or (adopting frequentist
vocabulary) 'random' effects, although the latter name is misleading
in a Bayesian context (for more details type
Multiple grouping factors each with multiple group-level effects
are possible. Instead of
| you may use
|| in grouping terms
to prevent correlations from being modeled.
Alternatively, it is possible to model different group-level terms of
the same grouping factor as correlated (even across different formulae,
e.g. in non-linear models) by using
| instead of
All group-level terms sharing the same ID will be modeled as correlated.
If, for instance, one specifies the terms
(1+z|2|g) somewhere in the formulae passed to
correlations between the corresponding group-level effects
will be estimated.
Smoothing terms can modeled using the
t2 functions of the mgcv package
Pterms part of the model formula.
This allows to fit generalized additive mixed models (GAMMs) with brms.
The implementation is similar to that used in the gamm4 package.
For more details on this model class see
Pterms part may contain two non-standard types
of population-level effects namely monotonic and category specific effects,
which can be specified using terms of the form
cse( respectively. The latter can only be applied in
ordinal models and is explained in more detail in the package's vignette
vignette("brms")). The former effect type is explained here.
A monotonic predictor must either be integer valued or an ordered factor,
which is the first difference to an ordinary continuous predictor.
More importantly, predictor categories (or integers) are not assumend to be
equidistant with respect to their effect on the response variable.
Instead, the distance between adjacent predictor categories (or integers)
is estimated from the data and may vary across categories.
This is realized by parameterizing as follows:
One parameter takes care of the direction and size of the effect similar
to an ordinary regression parameter, while an additional parameter vector
estimates the normalized distances between consecutive predictor categories.
A main application of monotonic effects are ordinal predictors that
can this way be modeled without (falsely) treating them as continuous
or as unordered categorical predictors.
The third exception is the optional
addition term, which may contain
multiple terms of the form
fun(variable) seperated by
providing special information on the response variable.
fun can be
replaced with either
trunc. Their meanings are explained below.
student, it is
possible to specify standard errors of the observation, thus allowing
to perform meta-analysis. Suppose that the variable
the effect sizes from the studies and
sei the corresponding
standard errors. Then, fixed and random effects meta-analyses can
be conducted using the formulae
yi | se(sei) ~ 1 and
yi | se(sei) ~ 1 + (1|study), respectively, where
study is a variable uniquely identifying every study.
If desired, meta-regression can be performed via
yi | se(sei) ~ 1 + mod1 + mod2 + (1|study)
yi | se(sei) ~ 1 + mod1 + mod2 + (1 + mod1 + mod2|study), where
mod2 represent moderator variables.
For all families, weighted regression may be performed using
weights in the addition part. Internally, this is
implemented by multiplying the log-posterior values of each
observation by their corresponding weights.
Suppose that variable
wei contains the weights
yi is the response variable.
yi | weights(wei) ~ predictors
implements a weighted regression.
The addition argument
disp (short for dispersion) serves a
similar purpose than
weight. However, it has a different
implementation and is less general as it is only usable for the
For the former four families, the residual standard deviation
sigma is multiplied by the values given in
disp, so that higher values lead to lower weights.
Contrariwise, for the latter three families, the parameter
is multiplied by the values given in
can be understood as a precision parameter (inverse of the variance),
higher values will lead to higher weights in this case.
addition should contain a variable indicating the number of trials
underlying each observation. In
lme4 syntax, we may write for instance
cbind(success, n - success), which is equivalent
success | trials(n) in
brms syntax. If the number of trials
is constant across all observation (say
we may also write
success | trials(10).
For all ordinal families,
addition may contain a term
cat(number) to specify the number categories (e.g,
If not given, the number of categories is calculated from the data.
With the expection of
categorical and ordinal families,
left and right censoring can be modeled through
yi | cens(censored) ~ predictors.
The censoring variable (named
censored in this example) should
contain the values
(or equivalenty -1, 0, and 1) to indicate that the corresponding observation is
left censored, not censored, or right censored.
With the expection of
categorical and ordinal families, the response
distribution can be truncated using the
trunc function in the addition part.
If the response variable is truncated between, say, 0 and 100, we can specify this via
yi | trunc(lb = 0, ub = 100) ~ predictors.
Instead of numbers, variables in the data set can also be passed allowing
for varying truncation points across observations.
Defining only one of the two arguments in
trunc leads to one-sided truncation.
addition terms may be specified at the same time using
+ operator, for instance
formula = yi | se(sei) + cens(censored) ~ 1
for a censored meta-analytic model.
multivariate models may be specified using
In brms 1.0.0, the multvariate 'trait' syntax was removed
from the package as it repeatedly confused users, required much
special case coding, and was hard to maintain. Below the new
syntax is described.
y2 are response variables
x is a predictor.
cbind(y1,y2) ~ x specifies a multivariate model,
The effects of all terms specified at the RHS of the formula
are assumed to vary across response variables (this was not the
case by default in brms < 1.0.0). For instance, two parameters will
be estimated for
x, one for the effect
y1 and another for the effect on
This is also true for group-level effects. When writing, for instance,
cbind(y1,y2) ~ x + (1+x|g), group-level effects will be
estimated separately for each response. To model these effects
as correlated across responses, use the ID syntax (see above).
For the present example, this would look as follows:
cbind(y1,y2) ~ x + (1+x|2|g). Of course, you could also use
any value other than
2 as ID. It is not yet possible
to model terms as only affecting certain responses (and not others),
but this will comebe implemented in the future.
Categorical models use the same syntax as multivariate
models. As in most other implementations of categorical models,
values of one category (the first in brms) are fixed
to identify the model. Thus, all terms on the RHS of
the formula correspond to
K - 1 effects
K = number of categories), one for each non-fixed category.
Group-level effects may be specified as correlated across
categories using the ID syntax.
As of brms 1.0.0, zero-inflated and hurdle models are specfied
in the same way as as their non-inflated counterparts.
However, they have additional auxiliary parameters
modeling the zero-inflation / hurdle probability depending on which
model you choose. These parameters can also be affected by predictors
in the same way the response variable itself. See the end of the
Details section for information on how to accomplish that.
Parameterization of the population-level intercept
The population-level intercept (if incorporated) is estimated separately
and not as part of population-level parameter vector
also have to be specified separately
set_prior for more details).
Furthermore, to increase sampling efficiency, the fixed effects
X is centered around its column means
X_means if the intercept is incorporated.
This leads to a temporary bias in the intercept equal to
<,>,> is the scalar product.
The bias is corrected after fitting the model, but be aware
that you are effectively defining a prior on the temporary
intercept of the centered design matrix not on the real intercept.
This behavior can be avoided by using the reserved
(and internally generated) variable
y ~ x, you may write
y ~ 0 + intercept + x. This way, priors can be
defined on the real intercept, directly. In addition,
the intercept is just treated as an ordinary fixed effect
and thus priors defined on
b will also apply to it.
Note that this parameterization may be a bit less efficient
than the default parameterization discussed above.
Formula syntax for non-linear models
nonlinear argument, it is possible to specify
non-linear models in brms. Contrary to what the name might suggest,
nonlinear should not contain the non-linear model itself
but rather information on the non-linear parameters.
The non-linear model will just be specified within the
argument. Suppose, that we want to predict the response
through the predictor
x is linked to
y = alpha - beta * lambda^x, with parameters
lambda. This is certainly a
non-linear model being defined via
formula = y ~ alpha - beta * lambda^x (addition arguments
can be added in the same way as for ordinary formulas).
Now we have to tell brms the names of the non-linear parameters
and specfiy a (linear mixed) model for each of them using the
argument. Let's say we just want to estimate those three parameters
with no further covariates or random effects. Then we can write
nonlinear = alpha + beta + lambda ~ 1 or equivalently
(and more flexible)
nonlinear = list(alpha ~ 1, beta ~ 1, lambda ~ 1).
This can, of course, be extended. If we have another predictor
observations nested within the grouping factor
g, we may write for
nonlinear = list(alpha ~ 1, beta ~ 1 + z + (1|g), lambda ~ 1).
The formula syntax described above applies here as well.
In this example, we are using
g only for the
beta, but we might also use them for the other
non-linear parameters (provided that the resulting model is still
Non-linear models may not be uniquely identified and / or show bad convergence.
For this reason it is mandatory to specify priors on the non-linear parameters.
For instructions on how to do that, see
Formula syntax for predicting auxiliary parameters
It is also possible to predict auxiliary parameters of the response
distribution such as the residual standard deviation
in gaussian models or the hurdle probability
hu in hurdle models.
The syntax closely resembles that of a non-linear
parameter, for instance
sigma ~ x + s(z) + (1+x|g).
All auxiliary parameters currently supported by
have to positive (a negative standard deviation or precision parameter
doesn't make any sense) or are bounded between 0 and 1 (for zero-inflated /
However, linear predictors can be positive or negative, and thus
the log link (for positive parameters) or logit link (for probability parameters)
are used to ensure that auxiliary parameters are within their valid intervals.
This implies that effects for auxiliary parameters are estimated on the
log / logit scale and one has to apply the inverse link function to get
to the effects on the original scale.
An object of class
brmsformula, which inherits from class
formulabut contains additional attributes.
# multilevel model with smoothing terms brmsformula(y ~ x1*x2 + s(z) + (1+x1|1) + (1|g2)) # additionally predict 'sigma' brmsformula(y ~ x1*x2 + s(z) + (1+x1|1) + (1|g2), sigma ~ x1 + (1|g2)) # use the shorter alias 'bf' (formula1 <- brmsformula(y ~ x + (x|g))) (formula2 <- bf(y ~ x + (x|g))) # will be TRUE identical(formula1, formula2) # incorporate censoring bf(y | cens(censor_variable) ~ predictors) # define a non-linear model bf(y ~ a1 - a2^x, nonlinear = list(a1 ~ 1, a2 ~ x + (x|g))) # correlated group-level effects across parameters bf(y ~ a1 - a2^x, nonlinear = list(a1 ~ 1 + (1|2|g), a2 ~ x + (x|2|g))) # define a multivariate model bf(cbind(y1, y2) ~ x * z + (1|g)) # define a zero-inflated model # also predicting the zero-inflation part bf(y ~ x * z + (1+x|ID1|g), zi ~ x + (1|ID1|g)) # specify a predictor as monotonic bf(y ~ mono(x) + more_predictors) # specify a predictor as category specific # for ordinal models only bf(y ~ cse(x) + more_predictors)