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, ..., flist = NULL, family = NULL, autocor = NULL,
nl = NULL, nonlinear = NULL)
```

formula

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 in 'Details'.

...

Additional `formula`

objects to specify
predictors of non-linear and distributional parameters.
Formulas can either be named directly or contain
names on their left-hand side.
The following are distributional parameters of specific families
(all other parameters are treated as non-linear parameters):
`sigma`

(residual standard deviation or scale of
the `gaussian`

, `student`

, `lognormal`

`exgaussian`

, and `asym_laplace`

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);
`kappa`

(precision parameter of the `von_mises`

family);
`beta`

(mean parameter of the exponential componenent
of the `exgaussian`

family);
`quantile`

(quantile parameter of the `asym_laplace`

family);
`zi`

(zero-inflation probability);
`hu`

(hurdle probability);
`zoi`

(zero-one-inflation probability);
`coi`

(conditional one-inflation probability);
`disc`

(discrimination) for ordinal models;
`bs`

, `ndt`

, and `bias`

(boundary separation,
non-decision time, and initial bias of the `wiener`

diffusion model).
All distributional parameters are modeled
on the log or logit scale to ensure correct definition
intervals after transformation.
See 'Details' for more explanation.

flist

Optional list of formulas, which are treated in the
same way as formulas passed via the `...`

argument.

family

autocor

nl

Logical; Indicates whether `formula`

should be
treated as specifying a non-linear model. By default, `formula`

is treated as an ordinary linear model formula.

nonlinear

(Deprecated) An optional list of formulas, specifying
linear models for non-linear parameters. If `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.
As of brms 1.4.0, we recommend specifying non-linear
parameters directly within `formula`

.

An object of class `brmsformula`

, which
is essentially a `list`

containing all model
formulas as well as some additional information.

**General formula structure**

The `formula`

argument accepts formulae of the following syntax:

`response | aterms ~ 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_overview")`

and
`vignette("brms_multilevel")`

.

**Group-level terms**

Multiple grouping factors each with multiple group-level effects
are possible (of course can also run models without any
group-level effects).
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 `|<ID>|`

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.

You can specify multi-membership terms
using the `mm`

function. For instance,
a multi-membership term with two members could be
`(1|mm(g1, g2))`

, where `g1`

and `g2`

specify
the first and second member, respectively.

**Special predictor terms**

Smoothing terms can modeled using the `s`

and `t2`

functions 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`

.

Gaussian process terms can be fitted using the `gp`

function in the `pterms`

part of the model formula. Similar to
smooth terms, Gaussian processes can be used to model complex non-linear
relationsships, for instance temporal or spatial autocorrelation.
However, they are computationally demanding and are thus not recommended
for very large datasets.

The `pterms`

and `gterms`

parts may contain three non-standard
effect types namely monotonic, measurement error, and category specific effects,
which can be specified using terms of the form `mo(predictor)`

,
`me(predictor, sd_predictor)`

, and `cs(<predictors>)`

,
respectively. Category specific effects can only be estimated in
ordinal models and are explained in more detail in the package's
main vignette (type `vignette("brms_overview")`

).
The other two effect types are explained in the following.

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. For more details and examples
see `vignette("brms_monotonic")`

.

Quite often, predictors are measured and as such naturally contain
measurement error. Although most reseachers are well aware of this problem,
measurement error in predictors is ignored in most
regression analyses, possibly because only few packages allow
for modelling it. Notably, measurement error can be handled in
structural equation models, but many more general regression models
(such as those featured by brms) cannot be transferred
to the SEM framework. In brms, effects of noise-free predictors
can be modeled using the `me`

(for 'measurement error') function.
If, say, `y`

is the response variable and
`x`

is a measured predictor with known measurement error
`sdx`

, we can simply include it on the right-hand side of the
model formula via `y ~ me(x, sdx)`

.
This can easily be extended to more general formulae.
If `x2`

is another measured predictor with corresponding error
`sdx2`

and `z`

is a predictor without error
(e.g., an experimental setting), we can model all main effects
and interactions of the three predictors in the well known manner:
`y ~ me(x, sdx) * me(x2, sdx2) * z`

. In future version of brms,
a vignette will be added to explain more details about these
so called 'error-in-variables' models and provide real world examples.

**Additional response information**

Another speciality of the brms formula syntax is the optional
`aterms`

part, 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`

, `trunc`

, or `dec`

.
Their meanings are explained below
(see also `addition-terms`

).

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.
By default, the standard errors replace the paramter `sigma`

.
To model `sigma`

in addition to the known standard errors,
set argument `sigma`

in function `se`

to `TRUE`

,
for instance, `yi | se(sei, sigma = TRUE) ~ 1`

.

For all families, weighted regression may be performed using
`weights`

in the `aterms`

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.

(DEPRECATED) 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`

,
`exgaussian`

, `asym_laplace`

, `Gamma`

,
`weibull`

, and `negbinomial`

.
For the former three 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.
Instead of using addition argument `disp`

, you may
equivalently use the distributional regression approach
by specifying `sigma ~ 1 + offset(log(xdisp))`

or
`shape ~ 1 + offset(log(xdisp))`

, where `xdisp`

is
the variable being passed to `disp`

.

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 observations, say `10`

,
we may also write `success | trials(10)`

.

For all ordinal families, `aterms`

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.

In Wiener diffusion models (family `wiener`

) the addition term
`dec`

is mandatory to specify the (vector of) binary decisions
corresponding to the reaction times. Non-zero values will be treated
as a response on the upper boundary of the diffusion process and zeros
will be treated as a response on the lower boundary. Alternatively,
the variable passed to `dec`

might also be a character vector
consisting of `'lower'`

and `'upper'`

.

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.

**Formula syntax for multivariate and categorical models**

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 distributional 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`

.
As a result, priors on the intercept also have to be specified separately.
Furthermore, to increase sampling efficiency, the population-level
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
`<X_means, b>`

, where `<,>`

is the scalar product.
The bias is corrected after fitting the model, but be aware
that you are effectively defining a prior on the intercept
of the centered design matrix not on the real intercept.
For more details on setting priors on population-level intercepts,
see `set_prior`

.

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 population-level effect
and thus priors defined on `b`

will also apply to it.
Note that this parameterization may be less efficient
than the default parameterization discussed above.

**Formula syntax for non-linear models**

In brms, it is possible to specify non-linear models
of arbitrary complexity.
The non-linear model can 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).
To tell `brms`

that this is a non-linear model,
we set argument `nl`

to `TRUE`

.
Now we have to specfiy a model for each of the non-linear parameters.
Let's say we just want to estimate those three parameters
with no further covariates or random effects. Then we can pass
`alpha + beta + lambda ~ 1`

or equivalently
(and more flexible) `alpha ~ 1, beta ~ 1, lambda ~ 1`

to the `...`

argument.
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 `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`

.
For some examples of non-linear models, see `vignette("brms_nonlinear")`

.

**Formula syntax for predicting distributional parameters**

It is also possible to predict 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)`

.
For some examples of distributional models, see `vignette("brms_distreg")`

.

Alternatively, one may fix distributional parameters to certain values.
However, this is mainly useful when models become too
complicated and otherwise have convergence issues.
We thus suggest to be generally careful when making use of this option.
The `quantile`

parameter of the `asym_laplace`

distribution
is a good example where it is useful. By fixing `quantile`

,
one can perform quantile regression for the specified quantile.
For instance, `quantile = 0.25`

allows predicting the 25%-quantile.
Furthermore, the `bias`

parameter in drift-diffusion models,
is assumed to be `0.5`

(i.e. no bias) in many applications.
To achieve this, simply write `bias = 0.5`

.
Other possible applications are the Cauchy
distribution as a special case of the Student-t distribution with
`nu = 1`

, or the geometric distribution as a special case of
the negative binomial distribution with `shape = 1`

.
Furthermore, the parameter `disc`

('discrimination') in ordinal
models is fixed to `1`

by default and not estimated,
but may be modeled as any other distributional parameter if desired
(see examples). For reasons of identification, `'disc'`

can only be positive, which is achieved by applying the log-link.

All distributional 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, quantiles, or the intial bias parameter of
drift-diffusion models).
However, linear predictors can be positive or negative, and thus the log link
(for positive parameters) or logit link (for probability parameters) are used
by default to ensure that distributional parameters are within their valid intervals.
This implies that, by default, effects for distributional 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.
Alternatively, it is possible to use the identity link to predict parameters
on their original scale, directly. However, this is much more likely to lead
to problems in the model fitting.

See also `brmsfamily`

for an overview of
valid link functions.

**Formula syntax for mixture models**

The specification of mixture models closely resembles that
of non-mixture models. If not specified otherwise (see below),
all mean parameters of the mixture components are predicted
using the right-hand side of `formula`

. All types of predictor
terms allowed in non-mixture models are allowed in mixture models
as well.

distributional parameters of mixture distributions have the same
name as those of the corresponding ordinary distributions, but with
a number at the end to indicate the mixture component. For instance, if
you use family `mixture(gaussian, gaussian)`

, the distributional
parameters are `sigma1`

and `sigma2`

.
distributional parameters of the same class can be fixed to the same value.
For the above example, we could write `sigma2 = "sigma1"`

to make
sure that both components have the same residual standard deviation,
which is in turn estimated from the data.

In addition, there are two types of special distributional parameters.
The first are named `mu<ID>`

, that allow for modeling different
predictors for the mean parameters of different mixture components.
For instance, if you want to predict the mean of the first component
using predictor `x`

and the mean of the second component using
predictor `z`

, you can write `mu1 ~ x`

as well as `mu2 ~ z`

.
The second are named `theta<ID>`

, which constitute the mixing
proportions. If the mixing proportions are fixed to certain values,
they are internally normalized to form a probability vector.
If one seeks to predict the mixing proportions, all but
one of the them has to be predicted, while the remaining one is used
as the reference category to identify the model. The `softmax`

function is applied on the linear predictor terms to form a
probability vector.

For more information on mixture models, see
the documentation of `mixture`

.

```
# NOT RUN {
# 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 simple non-linear model
bf(y ~ a1 - a2^x, a1 + a2 ~ 1, nl = TRUE)
# predict a1 and a2 differently
bf(y ~ a1 - a2^x, a1 ~ 1, a2 ~ x + (x|g), nl = TRUE)
# correlated group-level effects across parameters
bf(y ~ a1 - a2^x, a1 ~ 1 + (1|2|g), a2 ~ x + (x|2|g), nl = TRUE)
# 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 ~ mo(x) + more_predictors)
# for ordinal models only
# specify a predictor as category specific
bf(y ~ cs(x) + more_predictors)
# add a category specific group-level intercept
bf(y ~ cs(x) + (cs(1)|g))
# specify parameter 'disc'
bf(y ~ person + item, disc ~ item)
# specify variables containing measurement error
bf(y ~ me(x, sdx))
# specify predictors on all parameters of the wiener diffusion model
# the main formula models the drift rate 'delta'
bf(rt | dec(decision) ~ x, bs ~ x, ndt ~ x, bias ~ x)
# fix the bias parameter to 0.5
bf(rt | dec(decision) ~ x, bias = 0.5)
# specify different predictors for different mixture components
mix <- mixture(gaussian, gaussian)
bf(y ~ 1, mu1 ~ x, mu2 ~ z, family = mix)
# fix both residual standard deviations to the same value
bf(y ~ x, sigma2 = "sigma1", family = mix)
# use the '+' operator to specify models
bf(y ~ 1) +
nlf(sigma ~ a * exp(b * x), a ~ x) +
lf(b ~ z + (1|g), dpar = "sigma") +
gaussian()
# }
```

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