Bayesian inference for GAMMs with flexible priors.

```
stan_gamm4(
formula,
random = NULL,
family = gaussian(),
data,
weights = NULL,
subset = NULL,
na.action,
knots = NULL,
drop.unused.levels = TRUE,
...,
prior = default_prior_coef(family),
prior_intercept = default_prior_intercept(family),
prior_smooth = exponential(autoscale = FALSE),
prior_aux = exponential(autoscale = TRUE),
prior_covariance = decov(),
prior_PD = FALSE,
algorithm = c("sampling", "meanfield", "fullrank"),
adapt_delta = NULL,
QR = FALSE,
sparse = FALSE
)
```plot_nonlinear(
x,
smooths,
...,
prob = 0.9,
facet_args = list(),
alpha = 1,
size = 0.75
)

formula, random, family, data, knots, drop.unused.levels

Same as for
`gamm4`

. *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.subset, weights, na.action

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_smooth

The prior distribution for the hyperparameters in GAMs,
with lower values yielding less flexible smooth functions.
`prior_smooth`

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_smooth`

to `NULL`

. The number of hyperparameters depends
on the model specification but a scalar prior will be recylced as necessary
to the appropriate length.

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.

x

An object produced by `stan_gamm4`

.

smooths

An optional character vector specifying a subset of the smooth
functions specified in the call to `stan_gamm4`

. The default is
include all smooth terms.

prob

For univarite smooths, a scalar between 0 and 1 governing the width of the uncertainty interval.

facet_args

An optional named list of arguments passed to
`facet_wrap`

(other than the `facets`

argument).

alpha, size

For univariate smooths, passed to
`geom_ribbon`

. For bivariate smooths, `size/2`

is
passed to `geom_contour`

.

A stanreg object is returned
for `stan_gamm4`

.

`plot_nonlinear`

returns a ggplot object.

The `stan_gamm4`

function is similar in syntax to
`gamm4`

in the gamm4 package. But rather than performing
(restricted) maximum likelihood estimation with the lme4 package,
the `stan_gamm4`

function utilizes MCMC to perform Bayesian
estimation. The Bayesian model adds priors on the common regression
coefficients (in the same way as `stan_glm`

), priors on the
standard deviations of the smooth terms, and a prior on the decomposition
of the covariance matrices of any group-specific parameters (as in
`stan_glmer`

). Estimating these models via MCMC avoids
the optimization issues that often crop up with GAMMs and provides better
estimates for the uncertainty in the parameter estimates.

See `gamm4`

for more information about the model
specicification and `priors`

for more information about the
priors on the main coefficients. The `formula`

should include at least
one smooth term, which can be specified in any way that is supported by the
`jagam`

function in the mgcv package. The
`prior_smooth`

argument should be used to specify a prior on the unknown
standard deviations that govern how smooth the smooth function is. The
`prior_covariance`

argument can be used to specify the prior on the
components of the covariance matrix for any (optional) group-specific terms.
The `gamm4`

function in the gamm4 package uses
group-specific terms to implement the departure from linearity in the smooth
terms, but that is not the case for `stan_gamm4`

where the group-specific
terms are exactly the same as in `stan_glmer`

.

The `plot_nonlinear`

function creates a ggplot object with one facet for
each smooth function specified in the call to `stan_gamm4`

in the case
where all smooths are univariate. A subset of the smooth functions can be
specified using the `smooths`

argument, which is necessary to plot a
bivariate smooth or to exclude the bivariate smooth and plot the univariate
ones. In the bivariate case, a plot is produced using
`geom_contour`

. In the univariate case, the resulting
plot is conceptually similar to `plot.gam`

except the
outer lines here demark the edges of posterior uncertainty intervals
(credible intervals) rather than confidence intervals and the inner line
is the posterior median of the function rather than the function implied
by a point estimate. To change the colors used in the plot see
`color_scheme_set`

.

Crainiceanu, C., Ruppert D., and Wand, M. (2005). Bayesian analysis for
penalized spline regression using WinBUGS. *Journal of Statistical
Software*. **14**(14), 1--22.
https://www.jstatsoft.org/article/view/v014i14

`stanreg-methods`

and
`gamm4`

.

The vignette for `stan_glmer`

, which also discusses
`stan_gamm4`

. https://mc-stan.org/rstanarm/articles/

```
# NOT RUN {
if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
# from example(gamm4, package = "gamm4"), prefixing gamm4() call with stan_
# }
# NOT RUN {
dat <- mgcv::gamSim(1, n = 400, scale = 2) ## simulate 4 term additive truth
## Now add 20 level random effect `fac'...
dat$fac <- fac <- as.factor(sample(1:20, 400, replace = TRUE))
dat$y <- dat$y + model.matrix(~ fac - 1) %*% rnorm(20) * .5
br <- stan_gamm4(y ~ s(x0) + x1 + s(x2), data = dat, random = ~ (1 | fac),
chains = 1, iter = 500) # for example speed
print(br)
plot_nonlinear(br)
plot_nonlinear(br, smooths = "s(x0)", alpha = 2/3)
# }
# NOT RUN {
}
# }
```

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