Bayesian inference for GLMs with group-specific coefficients that have unknown covariance matrices with flexible priors.

```
stan_glmer(
formula,
data = NULL,
family = gaussian,
subset,
weights,
na.action = getOption("na.action", "na.omit"),
offset,
contrasts = NULL,
...,
prior = default_prior_coef(family),
prior_intercept = default_prior_intercept(family),
prior_aux = exponential(autoscale = TRUE),
prior_covariance = decov(),
prior_PD = FALSE,
algorithm = c("sampling", "meanfield", "fullrank"),
adapt_delta = NULL,
QR = FALSE,
sparse = FALSE
)
```stan_lmer(
formula,
data = NULL,
subset,
weights,
na.action = getOption("na.action", "na.omit"),
offset,
contrasts = NULL,
...,
prior = default_prior_coef(family),
prior_intercept = default_prior_intercept(family),
prior_aux = exponential(autoscale = TRUE),
prior_covariance = decov(),
prior_PD = FALSE,
algorithm = c("sampling", "meanfield", "fullrank"),
adapt_delta = NULL,
QR = FALSE
)

stan_glmer.nb(
formula,
data = NULL,
subset,
weights,
na.action = getOption("na.action", "na.omit"),
offset,
contrasts = NULL,
link = "log",
...,
prior = default_prior_coef(family),
prior_intercept = default_prior_intercept(family),
prior_aux = exponential(autoscale = TRUE),
prior_covariance = decov(),
prior_PD = FALSE,
algorithm = c("sampling", "meanfield", "fullrank"),
adapt_delta = NULL,
QR = FALSE
)

formula, data

Same as for `glmer`

. *We
strongly advise against omitting the data argument*. Unless

`data`

is specified (and is a data frame) many post-estimation
functions (including `update`

, `loo`

, `kfold`

) are not
guaranteed to work properly.family

Same as for `glmer`

except it is also
possible to use `family=mgcv::betar`

to estimate a Beta regression
with `stan_glmer`

.

subset, weights, offset

Same as `glm`

.

na.action, contrasts

Same as `glm`

, but rarely
specified.

...

prior

The prior distribution for the (non-hierarchical) regression coefficients.

The default priors are described in the vignette
*Prior
Distributions for rstanarm Models*.
If not using the default, `prior`

should be a call to one of the
various functions provided by rstanarm for specifying priors. The
subset of these functions that can be used for the prior on the
coefficients can be grouped into several "families":

Family |
Functions |

Student t family |
`normal` , `student_t` , `cauchy` |

Hierarchical shrinkage family |
`hs` , `hs_plus` |

Laplace family |
`laplace` , `lasso` |

Product normal family |
`product_normal` |

See the priors help page for details on the families and
how to specify the arguments for all of the functions in the table above.
To omit a prior ---i.e., to use a flat (improper) uniform prior---
`prior`

can be set to `NULL`

, although this is rarely a good
idea.

**Note:** Unless `QR=TRUE`

, if `prior`

is from the Student t
family or Laplace family, and if the `autoscale`

argument to the
function used to specify the prior (e.g. `normal`

) is left at
its default and recommended value of `TRUE`

, then the default or
user-specified prior scale(s) may be adjusted internally based on the
scales of the predictors. See the priors help page and the
*Prior Distributions* vignette for details on the rescaling and the
`prior_summary`

function for a summary of the priors used for a
particular model.

prior_intercept

The prior distribution for the intercept (after centering all predictors, see note below).

The default prior is described in the vignette
*Prior
Distributions for rstanarm Models*.
If not using the default, `prior_intercept`

can be a call to
`normal`

, `student_t`

or `cauchy`

. See the
priors help page for details on these functions. To omit a
prior on the intercept ---i.e., to use a flat (improper) uniform prior---
`prior_intercept`

can be set to `NULL`

.

**Note:** If using a dense representation of the design matrix
---i.e., if the `sparse`

argument is left at its default value of
`FALSE`

--- then the prior distribution for the intercept is set so it
applies to the value *when all predictors are centered* (you don't
need to manually center them). This is explained further in
[Prior Distributions for rstanarm Models](https://mc-stan.org/rstanarm/articles/priors.html)
If you prefer to specify a prior on the intercept without the predictors
being auto-centered, then you have to omit the intercept from the
`formula`

and include a column of ones as a predictor,
in which case some element of `prior`

specifies the prior on it,
rather than `prior_intercept`

. Regardless of how
`prior_intercept`

is specified, the reported *estimates* of the
intercept always correspond to a parameterization without centered
predictors (i.e., same as in `glm`

).

prior_aux

The prior distribution for the "auxiliary" parameter (if
applicable). The "auxiliary" parameter refers to a different parameter
depending on the `family`

. For Gaussian models `prior_aux`

controls `"sigma"`

, the error
standard deviation. For negative binomial models `prior_aux`

controls
`"reciprocal_dispersion"`

, which is similar to the
`"size"`

parameter of `rnbinom`

:
smaller values of `"reciprocal_dispersion"`

correspond to
greater dispersion. For gamma models `prior_aux`

sets the prior on
to the `"shape"`

parameter (see e.g.,
`rgamma`

), and for inverse-Gaussian models it is the
so-called `"lambda"`

parameter (which is essentially the reciprocal of
a scale parameter). Binomial and Poisson models do not have auxiliary
parameters.

The default prior is described in the vignette
*Prior
Distributions for rstanarm Models*.
If not using the default, `prior_aux`

can be a call to
`exponential`

to use an exponential distribution, or `normal`

,
`student_t`

or `cauchy`

, which results in a half-normal, half-t,
or half-Cauchy prior. See `priors`

for details on these
functions. To omit a prior ---i.e., to use a flat (improper) uniform
prior--- set `prior_aux`

to `NULL`

.

prior_covariance

Cannot be `NULL`

; see `decov`

for
more information about the default arguments.

prior_PD

A logical scalar (defaulting to `FALSE`

) indicating
whether to draw from the prior predictive distribution instead of
conditioning on the outcome.

algorithm

A string (possibly abbreviated) indicating the
estimation approach to use. Can be `"sampling"`

for MCMC (the
default), `"optimizing"`

for optimization, `"meanfield"`

for
variational inference with independent normal distributions, or
`"fullrank"`

for variational inference with a multivariate normal
distribution. See `rstanarm-package`

for more details on the
estimation algorithms. NOTE: not all fitting functions support all four
algorithms.

adapt_delta

Only relevant if `algorithm="sampling"`

. See
the adapt_delta help page for details.

QR

A logical scalar defaulting to `FALSE`

, but if `TRUE`

applies a scaled `qr`

decomposition to the design matrix. The
transformation does not change the likelihood of the data but is
recommended for computational reasons when there are multiple predictors.
See the QR-argument documentation page for details on how
rstanarm does the transformation and important information about how
to interpret the prior distributions of the model parameters when using
`QR=TRUE`

.

sparse

A logical scalar (defaulting to `FALSE`

) indicating
whether to use a sparse representation of the design (X) matrix.
If `TRUE`

, the the design matrix is not centered (since that would
destroy the sparsity) and likewise it is not possible to specify both
`QR = TRUE`

and `sparse = TRUE`

. Depending on how many zeros
there are in the design matrix, setting `sparse = TRUE`

may make
the code run faster and can consume much less RAM.

link

For `stan_glmer.nb`

only, the link function to use. See
`neg_binomial_2`

.

A stanreg object is returned
for `stan_glmer, stan_lmer, stan_glmer.nb`

.

A list with classes `stanreg`

, `glm`

, `lm`

,
and `lmerMod`

. The conventions for the parameter names are the
same as in the lme4 package with the addition that the standard
deviation of the errors is called `sigma`

and the variance-covariance
matrix of the group-specific deviations from the common parameters is
called `Sigma`

, even if this variance-covariance matrix only has
one row and one column (in which case it is just the group-level variance).

The `stan_glmer`

function is similar in syntax to
`glmer`

but rather than performing (restricted) maximum
likelihood estimation of generalized linear models, Bayesian estimation is
performed via MCMC. The Bayesian model adds priors on the
regression coefficients (in the same way as `stan_glm`

) and
priors on the terms of a decomposition of the covariance matrices of the
group-specific parameters. See `priors`

for more information
about the priors.

The `stan_lmer`

function is equivalent to `stan_glmer`

with
`family = gaussian(link = "identity")`

.

The `stan_glmer.nb`

function, which takes the extra argument
`link`

, is a wrapper for `stan_glmer`

with ```
family =
neg_binomial_2(link)
```

.

Gelman, A. and Hill, J. (2007). *Data Analysis Using
Regression and Multilevel/Hierarchical Models.* Cambridge University Press,
Cambridge, UK. (Ch. 11-15)

Muth, C., Oravecz, Z., and Gabry, J. (2018)
User-friendly Bayesian regression modeling: A tutorial with rstanarm and shinystan.
*The Quantitative Methods for Psychology*. 14(2), 99--119.
https://www.tqmp.org/RegularArticles/vol14-2/p099/p099.pdf

`stanreg-methods`

and
`glmer`

.

The vignette for `stan_glmer`

and the *Hierarchical
Partial Pooling* vignette. https://mc-stan.org/rstanarm/articles/

```
# NOT RUN {
if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
# see help(example_model) for details on the model below
if (!exists("example_model")) example(example_model)
print(example_model, digits = 1)
}
# }
```

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