Bayesian generalized linear models with group-specific terms via Stan

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

Usage

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
)

Arguments

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.

...

For stan_glmer, further arguments passed to sampling (e.g. iter, chains, cores, etc.) or to vb (if algorithm is "meanfield" or "fullrank"). For stan_lmer and stan_glmer.nb, ... should also contain all relevant arguments to pass to stan_glmer (except family).

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.

Value

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).

Details

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).

References

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

Examples

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)
}
#> stan_glmer
#>  family:       binomial [logit]
#>  formula:      cbind(incidence, size - incidence) ~ size + period + (1 | herd)
#>  observations: 56
#> ------
#>             Median MAD_SD
#> (Intercept) -1.5    0.6  
#> size         0.0    0.0  
#> period2     -1.0    0.3  
#> period3     -1.1    0.4  
#> period4     -1.6    0.5  
#> 
#> Error terms:
#>  Groups Name        Std.Dev.
#>  herd   (Intercept) 0.76    
#> Num. levels: herd 15 
#> 
#> ------
#> * For help interpreting the printed output see ?print.stanreg
#> * For info on the priors used see ?prior_summary.stanreg