brmsformula(formula, ..., nonlinear = NULL)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'.formula objects 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 gaussian and student families);
shape (shape parameter of the Gamma,
weibull, negbinomial and related
zero-inflated / hurdle families); nu
(degrees of freedom parameter of the student family);
phi (precision parameter of the beta
and zero_inflated_beta families);
zi (zero-inflation probability);
hu (hurdle probability).
All auxiliary parameters are modeled
on the log or logit scale to ensure correct definition
intervals after transformation.NULL (the default)
formula is treated as an ordinary formula.
If not NULL, formula is treated as a non-linear model
and nonlinear should 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'.brmsformula, which inherits
from class formula but contains additional attributes.
formula argument accepts formulae of the following syntax:
response | addition ~ Pterms + (Gterms | group)
The 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 group. We
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 vignette("brms")).
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+x|2|g) and
(1+z|2|g) somewhere in the formulae passed to brmsformula,
correlations between the corresponding group-level effects
will be estimated.
Smoothing terms can modeled using the s
and t2 functions of the mgcv package
in the 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 gam
and gamm.
The Pterms and Gterms parts may contain two non-standard
effect types namely monotonic and category specific effects,
which can be specified using terms of the form monotonic()
and cse() respectively. The latter can only be applied in
ordinal models and is explained in more detail in the package's vignette
(type 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 + each
providing special information on the response variable. fun can be
replaced with either se, weights, disp, trials,
cat, cens, or trunc. Their meanings are explained below.
For families gaussian and student, it is
possible to specify standard errors of the observation, thus allowing
to perform meta-analysis. Suppose that the variable yi contains
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)
or yi | se(sei) ~ 1 + mod1 + mod2 + (1 + mod1 + mod2|study),
where mod1 and 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
and that yi is the response variable.
Then, formula 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
families gaussian, student,
lognormal, Gamma, weibull, and negbinomial.
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 shape
is multiplied by the values given in disp. As shape
can be understood as a precision parameter (inverse of the variance),
higher values will lead to higher weights in this case.
For families binomial and zero_inflated_binomial,
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
to success | trials(n) in brms syntax. If the number of trials
is constant across all observation (say 10),
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, cat(7)).
If not given, the number of categories is calculated from the data.
With the expection of categorical and ordinal families,
left, right, and interval censoring can be modeled through
y | cens(censored) ~ predictors. The censoring variable
(named censored in this example) should contain the values
'left', 'none', 'right', and 'interval'
(or equivalenty -1, 0, 1, and 2) to indicate that
the corresponding observation is left censored, not censored, right censored,
or interval censored. For interval censored data, a second variable
(let's call it y2) has to be passed to cens. In this case,
the formula has the structure y | cens(censored, y2) ~ predictors.
While the lower bounds are given in y,
the upper bounds are given in y2 for interval
censored data. Intervals are assumed to be open on the left and closed
on the right: (y, y2].
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. Mutiple addition terms may be specified at the same time using
the + operator, for instance formula = yi | se(sei) + cens(censored) ~ 1
for a censored meta-analytic model.
For families gaussian and student,
multivariate models may be specified using cbind notation.
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.
Suppose that y1 and y2 are response variables
and x is a predictor.
Then 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
on y1 and another for the effect on y2.
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 be 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
(named zi and hu respectively)
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 b.
also have to be specified separately
(see set_prior for more details).
Furthermore, to increase sampling efficiency, the fixed effects
design matrix 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 intercept.
Instead of 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
Using the 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 formula
argument. Suppose, that we want to predict the response y
through the predictor x, where x is linked to y
through y = alpha - beta * lambda^x, with parameters
alpha, beta, and 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 nonlinear
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 z and
observations nested within the grouping factor g, we may write for
instance 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 z and g only for the
prediction of beta, but we might also use them for the other
non-linear parameters (provided that the resulting model is still
scientifically reasonable).
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 set_prior.
Formula syntax for predicting auxiliary parameters
It is also possible to predict auxiliary parameters of the response
distribution such as the residual standard deviation sigma
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 brmsformula
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 /
hurdle proabilities).
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.
# 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)
# add a category specific group-level intercept
bf(y ~ cse(x) + (cse(1)|g))
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