brmsformula: Set up a model formula for use in 'brms'
In brms: Bayesian Regression Models using 'Stan'
brmsformula R Documentation
Set up a model formula for use in brms
Description
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.
Usage
brmsformula(
formula,
...,
flist = NULL,
family = NULL,
autocor = NULL,
nl = NULL,
loop = NULL,
center = NULL,
cmc = NULL,
sparse = NULL,
decomp = NULL,
unused = NULL
)
Arguments
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. Alternatively,
it is possible to fix parameters to certain values by passing
numbers or character strings in which case arguments have to be named
to provide the parameter names. See 'Details' for more information.
flist
Optional list of formulas, which are treated in the
same way as formulas passed via the ... argument.
family
Same argument as in brm.
If family is specified in brmsformula, it will
overwrite the value specified in other functions.
autocor
An optional formula which contains
autocorrelation terms as described in autocor-terms
or alternatively a cor_brms object (deprecated).
If autocor is specified in brmsformula, it will
overwrite the value specified in other functions.
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.
loop
Logical; Only used in non-linear models.
Indicates if the computation of the non-linear formula should be
done inside (TRUE) or outside (FALSE) a loop
over observations. Defaults to TRUE.
center
Logical; Indicates if the population-level design
matrix should be centered, which usually increases sampling efficiency.
See the 'Details' section for more information.
Defaults to TRUE for distributional parameters
and to FALSE for non-linear parameters.
cmc
Logical; Indicates whether automatic cell-mean coding
should be enabled when removing the intercept by adding 0
to the right-hand of model formulas. Defaults to TRUE to
mirror the behavior of standard R formula parsing.
sparse
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.
decomp
Optional name of the decomposition used for the
population-level design matrix. Defaults to NULL that is
no decomposition. Other options currently available are
"QR" for the QR decomposition that helps in fitting models
with highly correlated predictors.
unused
An optional formula which contains variables
that are unused in the model but should still be stored in the
model's data frame. This can be useful, for example,
if those variables are required for post-processing the model.
Details
General formula structure
The formula argument accepts formulas of the following syntax:
response | aterms ~ pterms + (gterms | group)
The pterms part contains effects that are assumed to be the same
across observations. We call them 'population-level' or 'overall' effects,
or (adopting frequentist vocabulary) 'fixed' effects. The optional
gterms part may contain effects that are assumed to vary across
grouping variables specified in group. We call them 'group-level' or
'varying' 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 we can also run models without any group-level
effects.) Instead of | you may use || in grouping terms to
prevent correlations from being modeled. Equivalently, the cor
argument of the gr function can be used for this purpose,
for example, (1 + x || g) is equivalent to
(1 + x | gr(g, cor = FALSE)).
It is also possible to model different group-level terms of the same
grouping factor as correlated (even across different formulas, 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|i|g) and (1+z|i|g)
somewhere in the formulas passed to brmsformula, correlations
between the corresponding group-level effects will be estimated. In the
above example, i is not a variable in the data but just a symbol to
indicate correlations between multiple group-level terms. Equivalently, the
id argument of the gr function can be used as well,
for example, (1 + x | gr(g, id = "i")).
If levels of the grouping factor belong to different sub-populations,
it may be reasonable to assume a different covariance matrix for each
of the sub-populations. For instance, the variation within the
treatment group and within the control group in a randomized control
trial might differ. Suppose that y is the outcome, and
x is the factor indicating the treatment and control group.
Then, we could estimate different hyper-parameters of the varying
effects (in this case a varying intercept) for treatment and control
group via y ~ x + (1 | gr(subject, by = x)).
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. Moreover,
if a covariate x varies across the levels of the grouping-factors
g1 and g2, we can save the respective covariate values
in the variables x1 and x2 and then model the varying
effect as (1 + mmc(x1, x2) | mm(g1, g2)).
Special predictor terms
Flexible non-linear smooth 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
relationships, for instance temporal or spatial autocorrelation.
However, they are computationally demanding and are thus not recommended
for very large datasets or approximations need to be used.
The pterms and gterms parts may contain four non-standard
effect types namely monotonic, measurement error, missing value, and
category specific effects, which can be specified using terms of the
form mo(predictor), me(predictor, sd_predictor),
mi(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 three 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 assumed 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 researchers are well aware of this problem,
measurement error in predictors is ignored in most
regression analyses, possibly because only few packages allow
for modeling 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 formulas.
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.
The me function is soft deprecated in favor of the more flexible
and consistent mi function (see below).
When a variable contains missing values, the corresponding rows will
be excluded from the data by default (row-wise exclusion). However,
quite often we want to keep these rows and instead estimate the missing values.
There are two approaches for this: (a) Impute missing values before
the model fitting for instance via multiple imputation (see
brm_multiple for a way to handle multiple imputed datasets).
(b) Impute missing values on the fly during model fitting. The latter
approach is explained in the following. Using a variable with missing
values as predictors requires two things, First, we need to specify that
the predictor contains missings that should to be imputed.
If, say, y is the primary response, x is a
predictor with missings and z is a predictor without missings,
we go for y ~ mi(x) + z. Second, we need to model x
as an additional response with corresponding predictors and the
addition term mi(). In our example, we could write
x | mi() ~ z. Measurement error may be included via
the sdy argument, say, x | mi(sdy = se) ~ z.
See mi for examples with real data.
Autocorrelation terms
Autocorrelation terms can be directly specified inside the pterms
part as well. Details can be found in autocor-terms.
Additional response information
Another special of the brms formula syntax is the optional
aterms part, which may contain multiple terms of the form
fun(<variable>) separated by + each providing special
information on the response variable. fun can be replaced with
either se, weights, subset, cens, trunc,
trials, cat, dec, rate, vreal, or
vint. Their meanings are explained below
(see also addition-terms).
For families gaussian, student and skew_normal, it is
possible to specify standard errors of the observations, 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 formulas 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 parameter 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.
For multivariate models, subset may be used in the aterms
part, to use different subsets of the data in different univariate
models. For instance, if sub is a logical variable and
y is the response of one of the univariate models, we may
write y | subset(sub) ~ predictors so that y is
predicted only for those observations for which sub evaluates
to TRUE.
For log-linear models such as poisson models, rate may be used
in the aterms part to specify the denominator of a response that
is expressed as a rate. The numerator is given by the actual response
variable and has a distribution according to the family as usual. Using
rate(denom) is equivalent to adding offset(log(denom)) to
the linear predictor of the main parameter but the former is arguably
more convenient and explicit.
With the exception 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 equivalently -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 exception 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.
For all continuous families, missing values in the responses can be imputed
within Stan by using the addition term mi. This is mostly
useful in combination with mi predictor terms as explained
above under 'Special predictor terms'.
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).
Please note that the cbind() syntax will not work
in brms in the expected way because this syntax is reserved
for other purposes.
For all ordinal families, aterms may contain a term
thres(number) to specify the number thresholds (e.g,
thres(6)), which should be equal to the total number of response
categories - 1. If not given, the number of thresholds is calculated from
the data. If different threshold vectors should be used for different
subsets of the data, the gr argument can be used to provide the
grouping variable (e.g, thres(6, gr = item), if item is the
grouping variable). In this case, the number of thresholds can also be a
variable in the data with different values per group.
A deprecated quasi alias of thres() is cat() with which the
total number of response categories (i.e., number of thresholds + 1) can be
specified.
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'.
All families support the index addition term to uniquely identify
each observation of the corresponding response variable. Currently,
index is primarily useful in combination with the subset
addition and mi terms.
For custom families, it is possible to pass an arbitrary number of real and
integer vectors via the addition terms vreal and vint,
respectively. An example is provided in
vignette('brms_customfamilies'). To pass multiple vectors of the
same data type, provide them separated by commas inside a single
vreal or vint statement.
Multiple addition terms of different types may be specified at the same
time using the + operator. For example, the formula
formula = yi | se(sei) + cens(censored) ~ 1 implies a censored
meta-analytic model.
The addition argument disp (short for dispersion)
has been removed in version 2.0. You may instead use the
distributional regression approach by specifying
sigma ~ 1 + offset(log(xdisp)) or
shape ~ 1 + offset(log(xdisp)), where xdisp is
the variable being previously passed to disp.
Parameterization of the population-level intercept
By default, 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. You can turn off this special handling of the
intercept by setting argument center to FALSE. 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 specify 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).
By default, non-linear covariates are treated as real vectors in Stan.
However, if the data of the covariates is of type 'integer' in R (which
can be enforced by the 'as.integer' function), the Stan type will be
changed to an integer array. That way, covariates can also be used
for indexing purposes in Stan.
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").
Parameter mu exists for every family and can be used as an
alternative to specifying terms in formula. If both mu and
formula are given, the right-hand side of formula is ignored.
Accordingly, specifying terms on the right-hand side of both formula
and mu at the same time is deprecated. In future versions,
formula might be updated by mu.
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,
skew_normal, 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 and frechet families); 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 component 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). By default, distributional parameters
are modeled on the log scale if they can be positive only or on the logit
scale if the can only be within the unit interval.
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.
In categorical models, distributional parameters do not have
fixed names. Instead, they are named after the response categories
(excluding the first one, which serves as the reference category),
with the prefix 'mu'. If, for instance, categories are named
cat1, cat2, and cat3, the distributional parameters
will be named mucat2 and mucat3.
Some distributional parameters currently supported by brmsformula
have to be positive (a negative standard deviation or precision parameter
does not make any sense) or are bounded between 0 and 1 (for zero-inflated /
hurdle probabilities, quantiles, or the initial 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 such 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, if the parameter actually has a restricted range.
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 so-called 'softmax'
transformation is applied on the linear predictor terms to form a
probability vector.
For more information on mixture models, see
the documentation of mixture.
Formula syntax for multivariate models
Multivariate models may be specified using mvbind notation
or with help of the mvbf function.
Suppose that y1 and y2 are response variables
and x is a predictor. Then mvbind(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.
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,
mvbind(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:
mvbind(y1, y2) ~ x + (1+x|2|g). Of course, you could also use
any value other than 2 as ID.
It is also possible to specify different formulas for different responses.
If, for instance, y1 should be predicted by x and y2
should be predicted by z, we could write mvbf(y1 ~ x, y2 ~ z).
Alternatively, multiple brmsformula objects can be added to
specify a joint multivariate model (see 'Examples').
Value
An object of class brmsformula, which
is essentially a list containing all model
formulas as well as some additional information.
See Also
mvbrmsformula, brmsformula-helpers
Examples
# 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)
# alternative but equivalent way to specify the above model
bf(y ~ a1 - a2^x, a1 ~ 1 + (1 | gr(g, id = 2)),
a2 ~ x + (x | gr(g, id = 2)), nl = TRUE)
# define a multivariate model
bf(mvbind(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()
# specify a multivariate model using the '+' operator
bf(y1 ~ x + (1|g)) +
gaussian() + cor_ar(~1|g) +
bf(y2 ~ z) + poisson()
# specify correlated residuals of a gaussian and a poisson model
form1 <- bf(y1 ~ 1 + x + (1|c|obs), sigma = 1) + gaussian()
form2 <- bf(y2 ~ 1 + x + (1|c|obs)) + poisson()
# model missing values in predictors
bf(bmi ~ age * mi(chl)) +
bf(chl | mi() ~ age) +
set_rescor(FALSE)
# model sigma as a function of the mean
bf(y ~ eta, nl = TRUE) +
lf(eta ~ 1 + x) +
nlf(sigma ~ tau * sqrt(eta)) +
lf(tau ~ 1)
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| brmsformula | R Documentation |
Set up a model formula for use in brms
Description
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.
Usage
brmsformula(
formula,
...,
flist = NULL,
family = NULL,
autocor = NULL,
nl = NULL,
loop = NULL,
center = NULL,
cmc = NULL,
sparse = NULL,
decomp = NULL,
unused = NULL
)
Arguments
formula |
An object of class |
... |
Additional |
flist |
Optional list of formulas, which are treated in the
same way as formulas passed via the |
family |
Same argument as in |
autocor |
An optional |
nl |
Logical; Indicates whether |
loop |
Logical; Only used in non-linear models.
Indicates if the computation of the non-linear formula should be
done inside ( |
center |
Logical; Indicates if the population-level design
matrix should be centered, which usually increases sampling efficiency.
See the 'Details' section for more information.
Defaults to |
cmc |
Logical; Indicates whether automatic cell-mean coding
should be enabled when removing the intercept by adding |
sparse |
Logical; indicates whether the population-level design matrices
should be treated as sparse (defaults to |
decomp |
Optional name of the decomposition used for the
population-level design matrix. Defaults to |
unused |
An optional |
Details
General formula structure
The formula argument accepts formulas of the following syntax:
response | aterms ~ pterms + (gterms | group)
The pterms part contains effects that are assumed to be the same
across observations. We call them 'population-level' or 'overall' effects,
or (adopting frequentist vocabulary) 'fixed' effects. The optional
gterms part may contain effects that are assumed to vary across
grouping variables specified in group. We call them 'group-level' or
'varying' 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 we can also run models without any group-level
effects.) Instead of | you may use || in grouping terms to
prevent correlations from being modeled. Equivalently, the cor
argument of the gr function can be used for this purpose,
for example, (1 + x || g) is equivalent to
(1 + x | gr(g, cor = FALSE)).
It is also possible to model different group-level terms of the same
grouping factor as correlated (even across different formulas, 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|i|g) and (1+z|i|g)
somewhere in the formulas passed to brmsformula, correlations
between the corresponding group-level effects will be estimated. In the
above example, i is not a variable in the data but just a symbol to
indicate correlations between multiple group-level terms. Equivalently, the
id argument of the gr function can be used as well,
for example, (1 + x | gr(g, id = "i")).
If levels of the grouping factor belong to different sub-populations,
it may be reasonable to assume a different covariance matrix for each
of the sub-populations. For instance, the variation within the
treatment group and within the control group in a randomized control
trial might differ. Suppose that y is the outcome, and
x is the factor indicating the treatment and control group.
Then, we could estimate different hyper-parameters of the varying
effects (in this case a varying intercept) for treatment and control
group via y ~ x + (1 | gr(subject, by = x)).
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. Moreover,
if a covariate x varies across the levels of the grouping-factors
g1 and g2, we can save the respective covariate values
in the variables x1 and x2 and then model the varying
effect as (1 + mmc(x1, x2) | mm(g1, g2)).
Special predictor terms
Flexible non-linear smooth 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
relationships, for instance temporal or spatial autocorrelation.
However, they are computationally demanding and are thus not recommended
for very large datasets or approximations need to be used.
The pterms and gterms parts may contain four non-standard
effect types namely monotonic, measurement error, missing value, and
category specific effects, which can be specified using terms of the
form mo(predictor), me(predictor, sd_predictor),
mi(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 three 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 assumed 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 researchers are well aware of this problem,
measurement error in predictors is ignored in most
regression analyses, possibly because only few packages allow
for modeling 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 formulas.
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.
The me function is soft deprecated in favor of the more flexible
and consistent mi function (see below).
When a variable contains missing values, the corresponding rows will
be excluded from the data by default (row-wise exclusion). However,
quite often we want to keep these rows and instead estimate the missing values.
There are two approaches for this: (a) Impute missing values before
the model fitting for instance via multiple imputation (see
brm_multiple for a way to handle multiple imputed datasets).
(b) Impute missing values on the fly during model fitting. The latter
approach is explained in the following. Using a variable with missing
values as predictors requires two things, First, we need to specify that
the predictor contains missings that should to be imputed.
If, say, y is the primary response, x is a
predictor with missings and z is a predictor without missings,
we go for y ~ mi(x) + z. Second, we need to model x
as an additional response with corresponding predictors and the
addition term mi(). In our example, we could write
x | mi() ~ z. Measurement error may be included via
the sdy argument, say, x | mi(sdy = se) ~ z.
See mi for examples with real data.
Autocorrelation terms
Autocorrelation terms can be directly specified inside the pterms
part as well. Details can be found in autocor-terms.
Additional response information
Another special of the brms formula syntax is the optional
aterms part, which may contain multiple terms of the form
fun(<variable>) separated by + each providing special
information on the response variable. fun can be replaced with
either se, weights, subset, cens, trunc,
trials, cat, dec, rate, vreal, or
vint. Their meanings are explained below
(see also addition-terms).
For families gaussian, student and skew_normal, it is
possible to specify standard errors of the observations, 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 formulas 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 parameter 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.
For multivariate models, subset may be used in the aterms
part, to use different subsets of the data in different univariate
models. For instance, if sub is a logical variable and
y is the response of one of the univariate models, we may
write y | subset(sub) ~ predictors so that y is
predicted only for those observations for which sub evaluates
to TRUE.
For log-linear models such as poisson models, rate may be used
in the aterms part to specify the denominator of a response that
is expressed as a rate. The numerator is given by the actual response
variable and has a distribution according to the family as usual. Using
rate(denom) is equivalent to adding offset(log(denom)) to
the linear predictor of the main parameter but the former is arguably
more convenient and explicit.
With the exception 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 equivalently -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 exception 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.
For all continuous families, missing values in the responses can be imputed
within Stan by using the addition term mi. This is mostly
useful in combination with mi predictor terms as explained
above under 'Special predictor terms'.
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).
Please note that the cbind() syntax will not work
in brms in the expected way because this syntax is reserved
for other purposes.
For all ordinal families, aterms may contain a term
thres(number) to specify the number thresholds (e.g,
thres(6)), which should be equal to the total number of response
categories - 1. If not given, the number of thresholds is calculated from
the data. If different threshold vectors should be used for different
subsets of the data, the gr argument can be used to provide the
grouping variable (e.g, thres(6, gr = item), if item is the
grouping variable). In this case, the number of thresholds can also be a
variable in the data with different values per group.
A deprecated quasi alias of thres() is cat() with which the
total number of response categories (i.e., number of thresholds + 1) can be
specified.
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'.
All families support the index addition term to uniquely identify
each observation of the corresponding response variable. Currently,
index is primarily useful in combination with the subset
addition and mi terms.
For custom families, it is possible to pass an arbitrary number of real and
integer vectors via the addition terms vreal and vint,
respectively. An example is provided in
vignette('brms_customfamilies'). To pass multiple vectors of the
same data type, provide them separated by commas inside a single
vreal or vint statement.
Multiple addition terms of different types may be specified at the same
time using the + operator. For example, the formula
formula = yi | se(sei) + cens(censored) ~ 1 implies a censored
meta-analytic model.
The addition argument disp (short for dispersion)
has been removed in version 2.0. You may instead use the
distributional regression approach by specifying
sigma ~ 1 + offset(log(xdisp)) or
shape ~ 1 + offset(log(xdisp)), where xdisp is
the variable being previously passed to disp.
Parameterization of the population-level intercept
By default, 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. You can turn off this special handling of the
intercept by setting argument center to FALSE. 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 specify 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).
By default, non-linear covariates are treated as real vectors in Stan. However, if the data of the covariates is of type 'integer' in R (which can be enforced by the 'as.integer' function), the Stan type will be changed to an integer array. That way, covariates can also be used for indexing purposes in Stan.
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").
Parameter mu exists for every family and can be used as an
alternative to specifying terms in formula. If both mu and
formula are given, the right-hand side of formula is ignored.
Accordingly, specifying terms on the right-hand side of both formula
and mu at the same time is deprecated. In future versions,
formula might be updated by mu.
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,
skew_normal, 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 and frechet families); 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 component 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). By default, distributional parameters
are modeled on the log scale if they can be positive only or on the logit
scale if the can only be within the unit interval.
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.
In categorical models, distributional parameters do not have
fixed names. Instead, they are named after the response categories
(excluding the first one, which serves as the reference category),
with the prefix 'mu'. If, for instance, categories are named
cat1, cat2, and cat3, the distributional parameters
will be named mucat2 and mucat3.
Some distributional parameters currently supported by brmsformula
have to be positive (a negative standard deviation or precision parameter
does not make any sense) or are bounded between 0 and 1 (for zero-inflated /
hurdle probabilities, quantiles, or the initial 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 such 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, if the parameter actually has a restricted range.
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 so-called 'softmax'
transformation is applied on the linear predictor terms to form a
probability vector.
For more information on mixture models, see
the documentation of mixture.
Formula syntax for multivariate models
Multivariate models may be specified using mvbind notation
or with help of the mvbf function.
Suppose that y1 and y2 are response variables
and x is a predictor. Then mvbind(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.
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,
mvbind(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:
mvbind(y1, y2) ~ x + (1+x|2|g). Of course, you could also use
any value other than 2 as ID.
It is also possible to specify different formulas for different responses.
If, for instance, y1 should be predicted by x and y2
should be predicted by z, we could write mvbf(y1 ~ x, y2 ~ z).
Alternatively, multiple brmsformula objects can be added to
specify a joint multivariate model (see 'Examples').
Value
An object of class brmsformula, which
is essentially a list containing all model
formulas as well as some additional information.
See Also
mvbrmsformula, brmsformula-helpers
Examples
# 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)
# alternative but equivalent way to specify the above model
bf(y ~ a1 - a2^x, a1 ~ 1 + (1 | gr(g, id = 2)),
a2 ~ x + (x | gr(g, id = 2)), nl = TRUE)
# define a multivariate model
bf(mvbind(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()
# specify a multivariate model using the '+' operator
bf(y1 ~ x + (1|g)) +
gaussian() + cor_ar(~1|g) +
bf(y2 ~ z) + poisson()
# specify correlated residuals of a gaussian and a poisson model
form1 <- bf(y1 ~ 1 + x + (1|c|obs), sigma = 1) + gaussian()
form2 <- bf(y2 ~ 1 + x + (1|c|obs)) + poisson()
# model missing values in predictors
bf(bmi ~ age * mi(chl)) +
bf(chl | mi() ~ age) +
set_rescor(FALSE)
# model sigma as a function of the mean
bf(y ~ eta, nl = TRUE) +
lf(eta ~ 1 + x) +
nlf(sigma ~ tau * sqrt(eta)) +
lf(tau ~ 1)
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