Applied Regression Modeling via RStan

Description

Stan Development Team

The rstanarm package is an appendage to the rstan package that enables many of the most common applied regression models to be estimated using Markov Chain Monte Carlo, variational approximations to the posterior distribution, or optimization. The rstanarm package allows these models to be specified using the customary R modeling syntax (e.g., like that of glm with a formula and a data.frame).

The sections below provide an overview of the modeling functions and estimation algorithms used by rstanarm.

Details

The set of models supported by rstanarm is large (and will continue to grow), but also limited enough so that it is possible to integrate them tightly with the pp_check function for graphical posterior predictive checks with bayesplot and the posterior_predict function to easily estimate the effect of specific manipulations of predictor variables or to predict the outcome in a training set.

The objects returned by the rstanarm modeling functions are called stanreg objects. In addition to all of the typical methods defined for fitted model objects, stanreg objects can be passed to the loo function in the loo package for model comparison or to the launch_shinystan function in the shinystan package in order to visualize the posterior distribution using the ShinyStan graphical user interface. See the rstanarm vignettes for more details about the entire process.

Prior distributions

See priors help page and the vignette Prior Distributions for rstanarm Models for an overview of the various choices the user can make for prior distributions. The package vignettes for the modeling functions also provide examples of using many of the available priors as well as more detailed descriptions of some of the novel priors used by rstanarm.

Modeling functions

The model estimating functions are described in greater detail in their individual help pages and vignettes. Here we provide a very brief overview:

stan_lm, stan_aov, stan_biglm

Similar to lm or aov but with novel regularizing priors on the model parameters that are driven by prior beliefs about R^2, the proportion of variance in the outcome attributable to the predictors in a linear model.

stan_glm, stan_glm.nb

Similar to glm but with various possible prior distributions for the coefficients and, if applicable, a prior distribution for any auxiliary parameter in a Generalized Linear Model (GLM) that is characterized by a family object (e.g. the shape parameter in Gamma models). It is also possible to estimate a negative binomial model in a similar way to the glm.nb function in the MASS package.

stan_glmer, stan_glmer.nb, stan_lmer

Similar to the glmer, glmer.nb and lmer functions in the lme4 package in that GLMs are augmented to have group-specific terms that deviate from the common coefficients according to a mean-zero multivariate normal distribution with a highly-structured but unknown covariance matrix (for which rstanarm introduces an innovative prior distribution). MCMC provides more appropriate estimates of uncertainty for models that consist of a mix of common and group-specific parameters.

stan_nlmer

Similar to nlmer in the lme4 package for nonlinear "mixed-effects" models, but the group-specific coefficients have flexible priors on their unknown covariance matrices.

stan_gamm4

Similar to gamm4 in the gamm4 package, which augments a GLM (possibly with group-specific terms) with nonlinear smooth functions of the predictors to form a Generalized Additive Mixed Model (GAMM). Rather than calling glmer like gamm4 does, stan_gamm4 essentially calls stan_glmer, which avoids the optimization issues that often crop up with GAMMs and provides better estimates for the uncertainty of the parameter estimates.

stan_polr

Similar to polr in the MASS package in that it models an ordinal response, but the Bayesian model also implies a prior distribution on the unknown cutpoints. Can also be used to model binary outcomes, possibly while estimating an unknown exponent governing the probability of success.

stan_betareg

Similar to betareg in that it models an outcome that is a rate (proportion) but, rather than performing maximum likelihood estimation, full Bayesian estimation is performed by default, with customizable prior distributions for all parameters.

stan_clogit

Similar to clogit in that it models an binary outcome where the number of successes and failures is fixed within each stratum by the research design. There are some minor syntactical differences relative to clogit that allow stan_clogit to accept group-specific terms as in stan_glmer.

stan_mvmer

A multivariate form of stan_glmer, whereby the user can specify one or more submodels each consisting of a GLM with group-specific terms. If more than one submodel is specified (i.e. there is more than one outcome variable) then a dependence is induced by assuming that the group-specific terms for each grouping factor are correlated across submodels.

stan_jm

Estimates shared parameter joint models for longitudinal and time-to-event (i.e. survival) data. The joint model can be univariate (i.e. one longitudinal outcome) or multivariate (i.e. more than one longitudinal outcome). A variety of parameterisations are available for linking the longitudinal and event processes (i.e. a variety of association structures).

Estimation algorithms

The modeling functions in the rstanarm package take an algorithm argument that can be one of the following:

Sampling (algorithm="sampling")

Uses Markov Chain Monte Carlo (MCMC) — in particular, Hamiltonian Monte Carlo (HMC) with a tuned but diagonal mass matrix — to draw from the posterior distribution of the parameters. See sampling (rstan) for more details. This is the slowest but most reliable of the available estimation algorithms and it is the default and recommended algorithm for statistical inference.

Mean-field (algorithm="meanfield")

Uses mean-field variational inference to draw from an approximation to the posterior distribution. In particular, this algorithm finds the set of independent normal distributions in the unconstrained space that — when transformed into the constrained space — most closely approximate the posterior distribution. Then it draws repeatedly from these independent normal distributions and transforms them into the constrained space. The entire process is much faster than HMC and yields independent draws but is not recommended for final statistical inference. It can be useful to narrow the set of candidate models in large problems, particularly when specifying QR=TRUE in stan_glm, stan_glmer, and stan_gamm4, but is only an approximation to the posterior distribution.

Full-rank (algorithm="fullrank")

Uses full-rank variational inference to draw from an approximation to the posterior distribution by finding the multivariate normal distribution in the unconstrained space that — when transformed into the constrained space — most closely approximates the posterior distribution. Then it draws repeatedly from this multivariate normal distribution and transforms the draws into the constrained space. This process is slower than meanfield variational inference but is faster than HMC. Although still an approximation to the posterior distribution and thus not recommended for final statistical inference, the approximation is more realistic than that of mean-field variational inference because the parameters are not assumed to be independent in the unconstrained space. Nevertheless, fullrank variational inference is a more difficult optimization problem and the algorithm is more prone to non-convergence or convergence to a local optimum.

Optimizing (algorithm="optimizing")

Finds the posterior mode using a C++ implementation of the LBGFS algorithm. See optimizing for more details. If there is no prior information, then this is equivalent to maximum likelihood, in which case there is no great reason to use the functions in the rstanarm package over the emulated functions in other packages. However, if priors are specified, then the estimates are penalized maximum likelihood estimates, which may have some redeeming value. Currently, optimization is only supported for stan_glm.

References

Bates, D., Maechler, M., Bolker, B., and Walker, S. (2015). Fitting linear mixed-Effects models using lme4. Journal of Statistical Software. 67(1), 1–48.

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013). Bayesian Data Analysis. Chapman & Hall/CRC Press, London, third edition. https://stat.columbia.edu/~gelman/book/

Gelman, A. and Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press, Cambridge, UK. https://stat.columbia.edu/~gelman/arm/

Stan Development Team. Stan Modeling Language Users Guide and Reference Manual. https://mc-stan.org/users/documentation/.

Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. doi:10.1007/s11222-016-9696-4. arXiv preprint: https://arxiv.org/abs/1507.04544

Yao, Y., Vehtari, A., Simpson, D., and Gelman, A. (2018) Using stacking to average Bayesian predictive distributions. Bayesian Analysis, advance publication, doi:10.1214/17-BA1091.

Gabry, J. , Simpson, D. , Vehtari, A. , Betancourt, M. and Gelman, A. (2019), Visualization in Bayesian workflow. J. R. Stat. Soc. A, 182: 389-402. doi:10.1111/rssa.12378, arXiv preprint, code on GitHub)

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

See Also

• https://mc-stan.org/ for more information on the Stan C++ package used by rstanarm for model fitting.

• https://github.com/stan-dev/rstanarm/issues/ to submit a bug report or feature request.

• https://discourse.mc-stan.org to ask a question about rstanarm on the Stan-users forum.

adapt_delta: Target average acceptance probability

Description

Details about the adapt_delta argument to rstanarm's modeling functions.

Details

For the No-U-Turn Sampler (NUTS), the variant of Hamiltonian Monte Carlo used used by rstanarm, adapt_delta is the target average proposal acceptance probability during Stan's adaptation period. adapt_delta is ignored by rstanarm if the algorithm argument is not set to "sampling".

The default value of adapt_delta is 0.95, except when the prior for the regression coefficients is R2, hs, or hs_plus, in which case the default is 0.99.

These defaults are higher (more conservative) than the default of adapt_delta=0.8 used in the rstan package, which may result in slower sampling speeds but will be more robust to posterior distributions with high curvature.

In general you should not need to change adapt_delta unless you see a warning message about divergent transitions, in which case you can increase adapt_delta from the default to a value closer to 1 (e.g. from 0.95 to 0.99, or from 0.99 to 0.999, etc). The step size used by the numerical integrator is a function of adapt_delta in that increasing adapt_delta will result in a smaller step size and fewer divergences. Increasing adapt_delta will typically result in a slower sampler, but it will always lead to a more robust sampler.

References

Stan Development Team. Stan Modeling Language Users Guide and Reference Manual. https://mc-stan.org/users/documentation/.

Brief Guide to Stan's Warnings: https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup

Extract the posterior sample

Description

For models fit using MCMC (algorithm="sampling"), the posterior sample —the post-warmup draws from the posterior distribution— can be extracted from a fitted model object as a matrix, data frame, or array. The as.matrix and as.data.frame methods merge all chains together, whereas the as.array method keeps the chains separate. For models fit using optimization ("optimizing") or variational inference ("meanfield" or "fullrank"), there is no posterior sample but rather a matrix (or data frame) of 1000 draws from either the asymptotic multivariate Gaussian sampling distribution of the parameters or the variational approximation to the posterior distribution.

Usage

## S3 method for class 'stanreg'
as.matrix(x, ..., pars = NULL, regex_pars = NULL)

## S3 method for class 'stanreg'
as.array(x, ..., pars = NULL, regex_pars = NULL)

## S3 method for class 'stanreg'
as.data.frame(x, ..., pars = NULL, regex_pars = NULL)

Arguments

Value

A matrix, data.frame, or array, the dimensions of which depend on pars and regex_pars, as well as the model and estimation algorithm (see the Description section above).

See Also

stanreg-draws-formats, stanreg-methods

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {

if (!exists("example_model")) example(example_model)
# Extract posterior sample after MCMC
draws <- as.matrix(example_model)
print(dim(draws))

# For example, we can see that the median of the draws for the intercept 
# is the same as the point estimate rstanarm uses
print(median(draws[, "(Intercept)"]))
print(example_model$coefficients[["(Intercept)"]])

# The as.array method keeps the chains separate
draws_array <- as.array(example_model)
print(dim(draws_array)) # iterations x chains x parameters

# Extract draws from asymptotic Gaussian sampling distribution 
# after optimization
fit <- stan_glm(mpg ~ wt, data = mtcars, algorithm = "optimizing")
draws <- as.data.frame(fit)
print(colnames(draws))
print(nrow(draws)) # 1000 draws are taken

# Extract draws from variational approximation to the posterior distribution
fit2 <- update(fit, algorithm = "meanfield")
draws <- as.data.frame(fit2, pars = "wt")
print(colnames(draws))
print(nrow(draws)) # 1000 draws are taken

}

Estimation algorithms available for rstanarm models

Description

Estimation algorithms available for rstanarm models

Estimation algorithms

The modeling functions in the rstanarm package take an algorithm argument that can be one of the following:

Sampling (algorithm="sampling")

Uses Markov Chain Monte Carlo (MCMC) — in particular, Hamiltonian Monte Carlo (HMC) with a tuned but diagonal mass matrix — to draw from the posterior distribution of the parameters. See sampling (rstan) for more details. This is the slowest but most reliable of the available estimation algorithms and it is the default and recommended algorithm for statistical inference.

Mean-field (algorithm="meanfield")

Uses mean-field variational inference to draw from an approximation to the posterior distribution. In particular, this algorithm finds the set of independent normal distributions in the unconstrained space that — when transformed into the constrained space — most closely approximate the posterior distribution. Then it draws repeatedly from these independent normal distributions and transforms them into the constrained space. The entire process is much faster than HMC and yields independent draws but is not recommended for final statistical inference. It can be useful to narrow the set of candidate models in large problems, particularly when specifying QR=TRUE in stan_glm, stan_glmer, and stan_gamm4, but is only an approximation to the posterior distribution.

Full-rank (algorithm="fullrank")

Uses full-rank variational inference to draw from an approximation to the posterior distribution by finding the multivariate normal distribution in the unconstrained space that — when transformed into the constrained space — most closely approximates the posterior distribution. Then it draws repeatedly from this multivariate normal distribution and transforms the draws into the constrained space. This process is slower than meanfield variational inference but is faster than HMC. Although still an approximation to the posterior distribution and thus not recommended for final statistical inference, the approximation is more realistic than that of mean-field variational inference because the parameters are not assumed to be independent in the unconstrained space. Nevertheless, fullrank variational inference is a more difficult optimization problem and the algorithm is more prone to non-convergence or convergence to a local optimum.

Optimizing (algorithm="optimizing")

Finds the posterior mode using a C++ implementation of the LBGFS algorithm. See optimizing for more details. If there is no prior information, then this is equivalent to maximum likelihood, in which case there is no great reason to use the functions in the rstanarm package over the emulated functions in other packages. However, if priors are specified, then the estimates are penalized maximum likelihood estimates, which may have some redeeming value. Currently, optimization is only supported for stan_glm.

See Also

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

Modeling functions available in rstanarm

Description

Modeling functions available in rstanarm

Modeling functions

The model estimating functions are described in greater detail in their individual help pages and vignettes. Here we provide a very brief overview:

stan_lm, stan_aov, stan_biglm

Similar to lm or aov but with novel regularizing priors on the model parameters that are driven by prior beliefs about R^2, the proportion of variance in the outcome attributable to the predictors in a linear model.

stan_glm, stan_glm.nb

Similar to glm but with various possible prior distributions for the coefficients and, if applicable, a prior distribution for any auxiliary parameter in a Generalized Linear Model (GLM) that is characterized by a family object (e.g. the shape parameter in Gamma models). It is also possible to estimate a negative binomial model in a similar way to the glm.nb function in the MASS package.

stan_glmer, stan_glmer.nb, stan_lmer

Similar to the glmer, glmer.nb and lmer functions in the lme4 package in that GLMs are augmented to have group-specific terms that deviate from the common coefficients according to a mean-zero multivariate normal distribution with a highly-structured but unknown covariance matrix (for which rstanarm introduces an innovative prior distribution). MCMC provides more appropriate estimates of uncertainty for models that consist of a mix of common and group-specific parameters.

stan_nlmer

Similar to nlmer in the lme4 package for nonlinear "mixed-effects" models, but the group-specific coefficients have flexible priors on their unknown covariance matrices.

stan_gamm4

Similar to gamm4 in the gamm4 package, which augments a GLM (possibly with group-specific terms) with nonlinear smooth functions of the predictors to form a Generalized Additive Mixed Model (GAMM). Rather than calling glmer like gamm4 does, stan_gamm4 essentially calls stan_glmer, which avoids the optimization issues that often crop up with GAMMs and provides better estimates for the uncertainty of the parameter estimates.

stan_polr

Similar to polr in the MASS package in that it models an ordinal response, but the Bayesian model also implies a prior distribution on the unknown cutpoints. Can also be used to model binary outcomes, possibly while estimating an unknown exponent governing the probability of success.

stan_betareg

Similar to betareg in that it models an outcome that is a rate (proportion) but, rather than performing maximum likelihood estimation, full Bayesian estimation is performed by default, with customizable prior distributions for all parameters.

stan_clogit

Similar to clogit in that it models an binary outcome where the number of successes and failures is fixed within each stratum by the research design. There are some minor syntactical differences relative to clogit that allow stan_clogit to accept group-specific terms as in stan_glmer.

stan_mvmer

A multivariate form of stan_glmer, whereby the user can specify one or more submodels each consisting of a GLM with group-specific terms. If more than one submodel is specified (i.e. there is more than one outcome variable) then a dependence is induced by assuming that the group-specific terms for each grouping factor are correlated across submodels.

stan_jm

Estimates shared parameter joint models for longitudinal and time-to-event (i.e. survival) data. The joint model can be univariate (i.e. one longitudinal outcome) or multivariate (i.e. more than one longitudinal outcome). A variety of parameterisations are available for linking the longitudinal and event processes (i.e. a variety of association structures).

See Also

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

Compute a Bayesian version of R-squared or LOO-adjusted R-squared for regression models.

Description

Compute a Bayesian version of R-squared or LOO-adjusted R-squared for regression models.

Usage

## S3 method for class 'stanreg'
bayes_R2(object, ..., re.form = NULL)

## S3 method for class 'stanreg'
loo_R2(object, ...)

Arguments

Value

A vector of R-squared values with length equal to the posterior sample size (the posterior distribution of R-squared).

References

Andrew Gelman, Ben Goodrich, Jonah Gabry, and Aki Vehtari (2018). R-squared for Bayesian regression models. The American Statistician, to appear. doi:10.1080/00031305.2018.1549100 (Preprint, Notebook)

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
fit <- stan_glm(
  mpg ~ wt + cyl, 
  data = mtcars, 
  QR = TRUE, 
  chains = 2, 
  refresh = 0
)
rsq <- bayes_R2(fit)
print(median(rsq))
hist(rsq)

loo_rsq <- loo_R2(fit)
print(median(loo_rsq))

# multilevel binomial model
if (!exists("example_model")) example(example_model)
print(example_model)
median(bayes_R2(example_model))
median(bayes_R2(example_model, re.form = NA)) # exclude group-level
}

Example joint longitudinal and time-to-event model

Description

A model for use in the rstanarm examples related to stan_jm.

Format

Calling example("example_jm") will run the model in the Examples section, below, and the resulting stanmvreg object will then be available in the global environment. The chains and iter arguments are specified to make this example be small in size. In practice, we recommend that they be left unspecified in order to use the default values or increased if there are convergence problems. The cores argument is optional and on a multicore system, the user may well want to set that equal to the number of chains being executed.

Examples

  # set.seed(123)
  if (.Platform$OS.type != "windows" || .Platform$r_arch !="i386")
  example_jm <- 
     stan_jm(formulaLong = logBili ~ year + (1 | id), 
             dataLong = pbcLong[1:101,],
             formulaEvent = survival::Surv(futimeYears, death) ~ sex + trt, 
             dataEvent = pbcSurv[1:15,],
             time_var = "year",
             # this next line is only to keep the example small in size!
             chains = 1, seed = 12345, iter = 100, refresh = 0)

Example model

Description

A model for use in rstanarm examples.

Format

Calling example("example_model") will run the model in the Examples section, below, and the resulting stanreg object will then be available in the global environment. The chains and iter arguments are specified to make this example be small in size. In practice, we recommend that they be left unspecified in order to use the default values (4 and 2000 respectively) or increased if there are convergence problems. The cores argument is optional and on a multicore system, the user may well want to set that equal to the number of chains being executed.

See Also

cbpp for a description of the data.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
example_model <- 
  stan_glmer(cbind(incidence, size - incidence) ~ size + period + (1|herd),
             data = lme4::cbpp, family = binomial, QR = TRUE,
             # this next line is only to keep the example small in size!
             chains = 2, cores = 1, seed = 12345, iter = 1000, refresh = 0)
example_model
}

family method for stanmvreg objects

Description

family method for stanmvreg objects

Usage

## S3 method for class 'stanmvreg'
family(object, m = NULL, ...)

Arguments

family method for stanreg objects

Description

family method for stanreg objects

Usage

## S3 method for class 'stanreg'
family(object, ...)

Arguments

formula method for stanreg objects

Description

formula method for stanreg objects

Usage

## S3 method for class 'stanreg'
formula(x, ..., m = NULL)

Arguments

Extract X, Y or Z from a stanreg object

Description

Extract X, Y or Z from a stanreg object

Usage

get_y(object, ...)

get_x(object, ...)

get_z(object, ...)

Arguments

Value

For get_x and get_z, a matrix. For get_y, either a vector or a matrix, depending on how the response variable was specified.

K-fold cross-validation

Description

The kfold method performs exact K-fold cross-validation. First the data are randomly partitioned into K subsets of equal size (or as close to equal as possible), or the user can specify the folds argument to determine the partitioning. Then the model is refit K times, each time leaving out one of the K subsets. If K is equal to the total number of observations in the data then K-fold cross-validation is equivalent to exact leave-one-out cross-validation (to which loo is an efficient approximation).

Usage

## S3 method for class 'stanreg'
kfold(
  x,
  K = 10,
  ...,
  folds = NULL,
  save_fits = FALSE,
  cores = getOption("mc.cores", 1)
)

Arguments

Value

An object with classes 'kfold' and 'loo' that has a similar structure as the objects returned by the loo and waic methods and is compatible with the loo_compare function for comparing models.

References

Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. doi:10.1007/s11222-016-9696-4. arXiv preprint: https://arxiv.org/abs/1507.04544

Yao, Y., Vehtari, A., Simpson, D., and Gelman, A. (2018) Using stacking to average Bayesian predictive distributions. Bayesian Analysis, advance publication, doi:10.1214/17-BA1091.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {

fit1 <- stan_glm(mpg ~ wt, data = mtcars, refresh = 0)
fit2 <- stan_glm(mpg ~ wt + cyl, data = mtcars, refresh = 0)
fit3 <- stan_glm(mpg ~ disp * as.factor(cyl), data = mtcars, refresh = 0)

# 10-fold cross-validation
# (if possible also specify the 'cores' argument to use multiple cores)
(kfold1 <- kfold(fit1, K = 10))
kfold2 <- kfold(fit2, K = 10)
kfold3 <- kfold(fit3, K = 10) 
loo_compare(kfold1, kfold2, kfold3)

# stratifying by a grouping variable
# (note: might get some divergences warnings with this model but 
# this is just intended as a quick example of how to code this)
fit4 <- stan_lmer(mpg ~ disp + (1|cyl), data = mtcars, refresh = 0)
table(mtcars$cyl)
folds_cyl <- loo::kfold_split_stratified(K = 3, x = mtcars$cyl)
table(cyl = mtcars$cyl, fold = folds_cyl)
kfold4 <- kfold(fit4, folds = folds_cyl, cores = 2)
print(kfold4)

}
# Example code demonstrating the different ways to specify the number 
# of cores and how the cores are used
# 
# options(mc.cores = NULL)
# 
# # spread the K models over N_CORES cores (method 1)
# kfold(fit, K, cores = N_CORES)
# 
# # spread the K models over N_CORES cores (method 2)
# options(mc.cores = N_CORES)
# kfold(fit, K)
#  
# # fit K models sequentially using N_CORES cores for the Markov chains each time
# options(mc.cores = N_CORES)
# kfold(fit, K, cores = 1)

Using the ShinyStan GUI with rstanarm models

Description

The ShinyStan interface provides visual and numerical summaries of model parameters and convergence diagnostics.

Usage

## S3 method for class 'stanreg'
launch_shinystan(
  object,
  ppd = TRUE,
  seed = 1234,
  model_name = NULL,
  note = NULL,
  rstudio = getOption("shinystan.rstudio"),
  ...
)

Arguments

Details

The launch_shinystan function will accept a stanreg object as input. Currently, almost any model fit using one of rstanarm's model-fitting functions can be used with ShinyStan. The only exception is that ShinyStan does not currently support rstanarm models fit using algorithm='optimizing'. See the shinystan package documentation for more information.

Faster launch times

For some rstanarm models ShinyStan may take a very long time to launch. If this is the case with one of your models you may be able to speed up launch_shinystan in one of several ways:

Prevent ShinyStan from preparing graphical posterior predictive checks:

When used with a stanreg object (rstanarm model object) ShinyStan will draw from the posterior predictive distribution and prepare graphical posterior predictive checks before launching. That way when you go to the PPcheck page the plots are immediately available. This can be time consuming for models fit to very large datasets and you can prevent this behavior by creating a shinystan object before calling launch_shinystan. To do this use as.shinystan with optional argument ppd set to FALSE (see the Examples section below). When you then launch ShinyStan and go to the PPcheck page the plots will no longer be automatically generated and you will be presented with the standard interface requiring you to first specify the appropriate y and yrep, which can be done for many but not all rstanarm models.

Use a shinystan object:

Even if you don't want to prevent ShinyStan from preparing graphical posterior predictive checks, first creating a shinystan object using as.shinystan can reduce future launch times. That is, launch_shinystan(sso) will be faster than launch_shinystan(fit), where sso is a shinystan object and fit is a stanreg object. It still may take some time for as.shinystan to create sso initially, but each time you subsequently call launch_shinystan(sso) it will reuse sso instead of internally creating a shinystan object every time. See the Examples section below.

References

Gabry, J. , Simpson, D. , Vehtari, A. , Betancourt, M. and Gelman, A. (2019), Visualization in Bayesian workflow. J. R. Stat. Soc. A, 182: 389-402. doi:10.1111/rssa.12378, arXiv preprint, code on GitHub)

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") {
## Not run: 
if (!exists("example_model")) example(example_model) 

# Launch the ShinyStan app without saving the resulting shinystan object
if (interactive()) launch_shinystan(example_model)

# Launch the ShinyStan app (saving resulting shinystan object as sso)
if (interactive()) sso <- launch_shinystan(example_model)

# First create shinystan object then call launch_shinystan
sso <- shinystan::as.shinystan(example_model)
if (interactive()) launch_shinystan(sso)

# Prevent ShinyStan from preparing graphical posterior predictive checks that
# can be time consuming. example_model is small enough that it won't matter
# much here but in general this can help speed up launch_shinystan
sso <- shinystan::as.shinystan(example_model, ppd = FALSE)
if (interactive()) launch_shinystan(sso)

## End(Not run)
}

Pointwise log-likelihood matrix

Description

For models fit using MCMC only, the log_lik method returns the S by N pointwise log-likelihood matrix, where S is the size of the posterior sample and N is the number of data points, or in the case of the stanmvreg method (when called on stan_jm model objects) an S by Npat matrix where Npat is the number of individuals.

Usage

## S3 method for class 'stanreg'
log_lik(object, newdata = NULL, offset = NULL, ...)

## S3 method for class 'stanmvreg'
log_lik(object, m = 1, newdata = NULL, ...)

## S3 method for class 'stanjm'
log_lik(object, newdataLong = NULL, newdataEvent = NULL, ...)

Arguments

Value

For the stanreg and stanmvreg methods an S by N matrix, where S is the size of the posterior sample and N is the number of data points. For the stanjm method an S by Npat matrix where Npat is the number of individuals.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {

 roaches$roach100 <- roaches$roach1 / 100
 fit <- stan_glm(
    y ~ roach100 + treatment + senior,
    offset = log(exposure2),
    data = roaches,
    family = poisson(link = "log"),
    prior = normal(0, 2.5),
    prior_intercept = normal(0, 10),
    iter = 500, # just to speed up example,
    refresh = 0
 )
 ll <- log_lik(fit)
 dim(ll)
 all.equal(ncol(ll), nobs(fit))

 # using newdata argument
 nd <- roaches[1:2, ]
 nd$treatment[1:2] <- c(0, 1)
 ll2 <- log_lik(fit, newdata = nd, offset = c(0, 0))
 head(ll2)
 dim(ll2)
 all.equal(ncol(ll2), nrow(nd))

}

Logit and inverse logit

Description

Logit and inverse logit

Usage

logit(x)

invlogit(x)

Arguments

Value

A numeric vector the same length as x.

Compute weighted expectations using LOO

Description

These functions are wrappers around the E_loo function (loo package) that provide compatibility for rstanarm models.

Usage

## S3 method for class 'stanreg'
loo_predict(
  object,
  type = c("mean", "var", "quantile"),
  probs = 0.5,
  ...,
  psis_object = NULL
)

## S3 method for class 'stanreg'
loo_linpred(
  object,
  type = c("mean", "var", "quantile"),
  probs = 0.5,
  transform = FALSE,
  ...,
  psis_object = NULL
)

## S3 method for class 'stanreg'
loo_predictive_interval(object, prob = 0.9, ..., psis_object = NULL)

Arguments

Value

A list with elements value and pareto_k.

For loo_predict and loo_linpred the value component is a vector with one element per observation.

For loo_predictive_interval the value component is a matrix with one row per observation and two columns (like predictive_interval). loo_predictive_interval(..., prob = p) is equivalent to loo_predict(..., type = "quantile", probs = c(a, 1-a)) with a = (1 - p)/2, except it transposes the result and adds informative column names.

See E_loo and pareto-k-diagnostic for details on the pareto_k diagnostic.

References

Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. doi:10.1007/s11222-016-9696-4. arXiv preprint: https://arxiv.org/abs/1507.04544

Yao, Y., Vehtari, A., Simpson, D., and Gelman, A. (2018) Using stacking to average Bayesian predictive distributions. Bayesian Analysis, advance publication, doi:10.1214/17-BA1091.

Gabry, J. , Simpson, D. , Vehtari, A. , Betancourt, M. and Gelman, A. (2019), Visualization in Bayesian workflow. J. R. Stat. Soc. A, 182: 389-402. doi:10.1111/rssa.12378, arXiv preprint, code on GitHub)

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
## Not run: 
if (!exists("example_model")) example(example_model)

# optionally, log-weights can be pre-computed and reused
psis_result <- loo::psis(log_ratios = -log_lik(example_model))

loo_probs <- loo_linpred(example_model, type = "mean", transform = TRUE, psis_object = psis_result)
str(loo_probs)

loo_pred_var <- loo_predict(example_model, type = "var", psis_object = psis_result)
str(loo_pred_var)

loo_pred_ints <- loo_predictive_interval(example_model, prob = 0.8, psis_object = psis_result)
str(loo_pred_ints)

## End(Not run)
}

Information criteria and cross-validation

Description

For models fit using MCMC, compute approximate leave-one-out cross-validation (LOO, LOOIC) or, less preferably, the Widely Applicable Information Criterion (WAIC) using the loo package. (For K-fold cross-validation see kfold.stanreg.) Functions for model comparison, and model weighting/averaging are also provided.

Note: these functions are not guaranteed to work properly unless the data argument was specified when the model was fit. Also, as of loo version 2.0.0 the default number of cores is now only 1, but we recommend using as many (or close to as many) cores as possible by setting the cores argument or using options(mc.cores = VALUE) to set it for an entire session.

Usage

## S3 method for class 'stanreg'
loo(
  x,
  ...,
  cores = getOption("mc.cores", 1),
  save_psis = FALSE,
  k_threshold = NULL
)

## S3 method for class 'stanreg'
waic(x, ...)

## S3 method for class 'stanreg'
loo_compare(x, ..., criterion = c("loo", "kfold", "waic"), detail = FALSE)

## S3 method for class 'stanreg_list'
loo_compare(x, ..., criterion = c("loo", "kfold", "waic"), detail = FALSE)

## S3 method for class 'stanreg_list'
loo_model_weights(x, ..., cores = getOption("mc.cores", 1), k_threshold = NULL)

compare_models(..., loos = list(), detail = FALSE)

Arguments

Value

The structure of the objects returned by loo and waic methods are documented in detail in the Value section in loo and waic (from the loo package).

loo_compare returns a matrix with class 'compare.loo'. See the Comparing models section below for more details.

Approximate LOO CV

The loo method for stanreg objects provides an interface to the loo package for approximate leave-one-out cross-validation (LOO). The LOO Information Criterion (LOOIC) has the same purpose as the Akaike Information Criterion (AIC) that is used by frequentists. Both are intended to estimate the expected log predictive density (ELPD) for a new dataset. However, the AIC ignores priors and assumes that the posterior distribution is multivariate normal, whereas the functions from the loo package do not make this distributional assumption and integrate over uncertainty in the parameters. This only assumes that any one observation can be omitted without having a major effect on the posterior distribution, which can be judged using the diagnostic plot provided by the plot.loo method and the warnings provided by the print.loo method (see the How to Use the rstanarm Package vignette for an example of this process).

How to proceed when loo gives warnings (k_threshold)

The k_threshold argument to the loo method for rstanarm models is provided as a possible remedy when the diagnostics reveal problems stemming from the posterior's sensitivity to particular observations. Warnings about Pareto k estimates indicate observations for which the approximation to LOO is problematic (this is described in detail in Vehtari, Gelman, and Gabry (2017) and the loo package documentation). The k_threshold argument can be used to set the k value above which an observation is flagged. If k_threshold is not NULL and there are J observations with k estimates above k_threshold then when loo is called it will refit the original model J times, each time leaving out one of the J problematic observations. The pointwise contributions of these observations to the total ELPD are then computed directly and substituted for the previous estimates from these J observations that are stored in the object created by loo. Another option to consider is K-fold cross-validation, which is documented on a separate page (see kfold).

Note: in the warning messages issued by loo about large Pareto k estimates we recommend setting k_threshold to at least 0.7. There is a theoretical reason, explained in Vehtari, Gelman, and Gabry (2017), for setting the threshold to the stricter value of 0.5, but in practice they find that errors in the LOO approximation start to increase non-negligibly when k > 0.7.

Comparing models

"loo" (or "waic" or "kfold") objects can be passed to the loo_compare function in the loo package to perform model comparison. rstanarm also provides a loo_compare.stanreg method that can be used if the "loo" (or "waic" or "kfold") object has been added to the fitted model object (see the Examples section below for how to do this). This second method allows rstanarm to perform some extra checks that can't be done by the loo package itself (e.g., verifying that all models to be compared were fit using the same outcome variable).

loo_compare will return a matrix with one row per model and columns containing the ELPD difference and the standard error of the difference. In the first row of the matrix will be the model with the largest ELPD (smallest LOOIC) and will contain zeros (there is no difference between this model and itself). For each of the remaining models the ELPD difference and SE are reported relative to the model with the best ELPD (the first row). See the Details section at the loo_compare page in the loo package for more information.

Model weights

The loo_model_weights method can be used to compute model weights for a "stanreg_list" object, which is a list of fitted model objects made with stanreg_list. The end of the Examples section has a demonstration. For details see the loo_model_weights documentation in the loo package.

References

Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. doi:10.1007/s11222-016-9696-4. arXiv preprint: https://arxiv.org/abs/1507.04544

Yao, Y., Vehtari, A., Simpson, D., and Gelman, A. (2018) Using stacking to average Bayesian predictive distributions. Bayesian Analysis, advance publication, doi:10.1214/17-BA1091.

Gabry, J. , Simpson, D. , Vehtari, A. , Betancourt, M. and Gelman, A. (2019), Visualization in Bayesian workflow. J. R. Stat. Soc. A, 182: 389-402. doi:10.1111/rssa.12378, arXiv preprint, code on GitHub)

See Also

• The new loo package vignettes and various rstanarm vignettes for more examples using loo and related functions with rstanarm models.

• pareto-k-diagnostic in the loo package for more on Pareto k diagnostics.

• log_lik.stanreg to directly access the pointwise log-likelihood matrix.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {

fit1 <- stan_glm(mpg ~ wt, data = mtcars, refresh = 0)
fit2 <- stan_glm(mpg ~ wt + cyl, data = mtcars, refresh = 0)

# (for bigger models use as many cores as possible)
loo1 <- loo(fit1, cores = 1)
print(loo1)
loo2 <- loo(fit2, cores = 1)
print(loo2)

# when comparing models the loo objects can be passed to loo_compare
# as individual arguments or as a list of loo objects
loo_compare(loo1, loo2)
loo_compare(list(loo1, loo2))

# if the fitted model objects contain a loo object in the component "loo"
# then the model objects can be passed directly or as a stanreg_list
fit1$loo <- loo1
fit2$loo <- loo2
loo_compare(fit1, fit2)

# if the fitted model objects contain a loo object _and_ a waic or kfold
# object, then the criterion argument determines which of them the comparison
# is based on 
fit1$waic <- waic(fit1)
fit2$waic <- waic(fit2)
loo_compare(fit1, fit2, criterion = "waic")

# the models can also be combined into a stanreg_list object, and more 
# informative model names can be provided to use when printing
model_list <- stanreg_list(fit1, fit2, model_names = c("Fewer predictors", "More predictors"))
loo_compare(model_list)

fit3 <- stan_glm(mpg ~ disp * as.factor(cyl), data = mtcars, refresh = 0)
loo3 <- loo(fit3, cores = 2, k_threshold = 0.7)
loo_compare(loo1, loo2, loo3)

# setting detail=TRUE will also print model formulas if used with
# loo_compare.stanreg or loo_compare.stanreg_list
fit3$loo <- loo3
model_list <- stanreg_list(fit1, fit2, fit3)
loo_compare(model_list, detail=TRUE)

# Computing model weights
#
# if the objects in model_list already have 'loo' components then those
# will be used. otherwise loo will be computed for each model internally
# (in which case the 'cores' argument may also be used and is passed to loo())
loo_model_weights(model_list)  # defaults to method="stacking"
loo_model_weights(model_list,  method = "pseudobma")
loo_model_weights(model_list,  method = "pseudobma", BB = FALSE)

# you can also pass precomputed loo objects directly to loo_model_weights
loo_list <- list(A = loo1, B = loo2, C = loo3) # names optional (affects printing)
loo_model_weights(loo_list)

}

model.frame method for stanmvreg objects

Description

model.frame method for stanmvreg objects

Usage

## S3 method for class 'stanmvreg'
model.frame(formula, fixed.only = FALSE, m = NULL, ...)

Arguments

model.frame method for stanreg objects

Description

model.frame method for stanreg objects

Usage

## S3 method for class 'stanreg'
model.frame(formula, fixed.only = FALSE, ...)

Arguments

model.matrix method for stanreg objects

Description

model.matrix method for stanreg objects

Usage

## S3 method for class 'stanreg'
model.matrix(object, ...)

Arguments

Family function for negative binomial GLMs

Description

Specifies the information required to fit a Negative Binomial GLM in a similar way to negative.binomial. However, here the overdispersion parameter theta is not specified by the user and always estimated (really the reciprocal of the dispersion parameter is estimated). A call to this function can be passed to the family argument of stan_glm or stan_glmer to estimate a Negative Binomial model. Alternatively, the stan_glm.nb and stan_glmer.nb wrapper functions may be used, which call neg_binomial_2 internally.

Usage

neg_binomial_2(link = "log")

Arguments

Value

An object of class family very similar to that of poisson but with a different family name.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386")
stan_glm(Days ~ Sex/(Age + Eth*Lrn), data = MASS::quine, seed = 123,
         family = neg_binomial_2, QR = TRUE, algorithm = "optimizing") 
                
# or, equivalently, call stan_glm.nb() without specifying the family

Methods for stanreg objects

Description

The methods documented on this page are actually some of the least important methods defined for stanreg objects. The most important methods are documented separately, each with its own page. Links to those pages are provided in the See Also section, below.

Usage

## S3 method for class 'stanmvreg'
nobs(object, ...)

## S3 method for class 'stanreg'
coef(object, ...)

## S3 method for class 'stanreg'
confint(object, parm, level = 0.95, ...)

## S3 method for class 'stanreg'
fitted(object, ...)

## S3 method for class 'stanreg'
nobs(object, ...)

## S3 method for class 'stanreg'
residuals(object, ...)

## S3 method for class 'stanreg'
se(object, ...)

## S3 method for class 'stanreg'
update(object, formula., ..., evaluate = TRUE)

## S3 method for class 'stanreg'
vcov(object, correlation = FALSE, ...)

## S3 method for class 'stanreg'
fixef(object, ...)

## S3 method for class 'stanreg'
ngrps(object, ...)

## S3 method for class 'stanreg'
nsamples(object, ...)

## S3 method for class 'stanreg'
ranef(object, ...)

## S3 method for class 'stanreg'
sigma(object, ...)

## S3 method for class 'stanreg'
VarCorr(x, sigma = 1, ...)

Arguments

Details

The methods documented on this page are similar to the methods defined for objects of class 'lm', 'glm', 'glmer', etc. However there are a few key differences:

residuals

Residuals are always of type "response" (not "deviance" residuals or any other type). However, in the case of stan_polr with more than two response categories, the residuals are the difference between the latent utility and its linear predictor.

coef

Medians are used for point estimates. See the Point estimates section in print.stanreg for more details.

se

The se function returns standard errors based on mad. See the Uncertainty estimates section in print.stanreg for more details.

confint

For models fit using optimization, confidence intervals are returned via a call to confint.default. If algorithm is "sampling", "meanfield", or "fullrank", the confint will throw an error because the posterior_interval function should be used to compute Bayesian uncertainty intervals.

nsamples

The number of draws from the posterior distribution obtained

See Also

• The print, summary, and prior_summary methods for stanreg objects for information on the fitted model.

• launch_shinystan to use the ShinyStan GUI to explore a fitted rstanarm model.

• The plot method to plot estimates and diagnostics.

• The pp_check method for graphical posterior predictive checking.

• The posterior_predict and predictive_error methods for predictions and predictive errors.

• The posterior_interval and predictive_interval methods for uncertainty intervals for model parameters and predictions.

• The loo, kfold, and log_lik methods for leave-one-out or K-fold cross-validation, model comparison, and computing the log-likelihood of (possibly new) data.

• The as.matrix, as.data.frame, and as.array methods to access posterior draws.

Pairs method for stanreg objects

Description

Interface to bayesplot's mcmc_pairs function for use with rstanarm models. Be careful not to specify too many parameters to include or the plot will be both hard to read and slow to render.

Usage

## S3 method for class 'stanreg'
pairs(
  x,
  pars = NULL,
  regex_pars = NULL,
  condition = pairs_condition(nuts = "accept_stat__"),
  ...
)

Arguments

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {

if (!exists("example_model")) example(example_model)

bayesplot::color_scheme_set("purple")

# see 'condition' argument above for details on the plots below and 
# above the diagonal. default is to split by accept_stat__.
pairs(example_model, pars = c("(Intercept)", "log-posterior"))

# for demonstration purposes, intentionally fit a model that
# will (almost certainly) have some divergences
fit <- stan_glm(
  mpg ~ ., data = mtcars,
  iter = 1000,
  # this combo of prior and adapt_delta should lead to some divergences
  prior = hs(),
  adapt_delta = 0.9,
  refresh = 0
)

pairs(fit, pars = c("wt", "sigma", "log-posterior"))

# requires hexbin package
# pairs(
# fit, 
#   pars = c("wt", "sigma", "log-posterior"), 
#   transformations = list(sigma = "log"), # show log(sigma) instead of sigma
#  off_diag_fun = "hex" # use hexagonal heatmaps instead of scatterplots
# )

bayesplot::color_scheme_set("brightblue")
pairs(
  fit, 
  pars = c("(Intercept)", "wt", "sigma", "log-posterior"), 
  transformations = list(sigma = "log"), 
  off_diag_args = list(size = 3/4, alpha = 1/3), # size and transparency of scatterplot points
  np_style = pairs_style_np(div_color = "black", div_shape = 2) # color and shape of the divergences
)

# Using the condition argument to show divergences above the diagonal 
pairs(
  fit, 
  pars = c("(Intercept)", "wt", "log-posterior"), 
  condition = pairs_condition(nuts = "divergent__")
)


}

Plot the estimated subject-specific or marginal longitudinal trajectory

Description

This generic plot method for predict.stanjm objects will plot the estimated subject-specific or marginal longitudinal trajectory using the data frame returned by a call to posterior_traj. To ensure that enough data points are available to plot the longitudinal trajectory, it is assumed that the call to posterior_traj would have used the default interpolate = TRUE, and perhaps also extrapolate = TRUE (the latter being optional, depending on whether or not the user wants to see extrapolation of the longitudinal trajectory beyond the last observation time).

Usage

## S3 method for class 'predict.stanjm'
plot(
  x,
  ids = NULL,
  limits = c("ci", "pi", "none"),
  xlab = NULL,
  ylab = NULL,
  vline = FALSE,
  plot_observed = FALSE,
  facet_scales = "free_x",
  ci_geom_args = NULL,
  grp_overlay = FALSE,
  ...
)

Arguments

Value

A ggplot object, also of class plot.predict.stanjm. This object can be further customised using the ggplot2 package. It can also be passed to the function plot_stack_jm.

See Also

posterior_traj, plot_stack_jm, posterior_survfit, plot.survfit.stanjm

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {

  # Run example model if not already loaded
  if (!exists("example_jm")) example(example_jm)
  
  # For a subset of individuals in the estimation dataset we will
  # obtain subject-specific predictions for the longitudinal submodel 
  # at evenly spaced times between 0 and their event or censoring time.
  pt1 <- posterior_traj(example_jm, ids = c(7,13,15), interpolate = TRUE)
  plot(pt1)                  # credible interval for mean response
  plot(pt1, limits = "pi")   # prediction interval for raw response
  plot(pt1, limits = "none") # no uncertainty interval
  
  # We can also extrapolate the longitudinal trajectories.
  pt2 <- posterior_traj(example_jm, ids = c(7,13,15), interpolate = TRUE,
                        extrapolate = TRUE)
  plot(pt2)
  plot(pt2, vline = TRUE)    # add line indicating event or censoring time
  plot(pt2, vline = TRUE, plot_observed = TRUE)  # overlay observed longitudinal data
 
  # We can change or add attributes to the plot
  plot1 <- plot(pt2, ids = c(7,13,15), xlab = "Follow up time",
                     vline = TRUE, plot_observed = TRUE, 
                     facet_scales = "fixed", color = "blue", linetype = 2,
                     ci_geom_args = list(fill = "red"))
  plot1
       
  # Since the returned plot is also a ggplot object, we can
  # modify some of its attributes after it has been returned
  plot1 + 
    ggplot2::theme(strip.background = ggplot2::element_blank()) +
    ggplot2::labs(title = "Some plotted longitudinal trajectories")

}

Plot method for stanreg objects

Description

The plot method for stanreg-objects provides a convenient interface to the MCMC module in the bayesplot package for plotting MCMC draws and diagnostics. It is also straightforward to use the functions from the bayesplot package directly rather than via the plot method. Examples of both methods of plotting are given below.

Usage

## S3 method for class 'stanreg'
plot(x, plotfun = "intervals", pars = NULL, regex_pars = NULL, ...)

Arguments

Value

Either a ggplot object that can be further customized using the ggplot2 package, or an object created from multiple ggplot objects (e.g. a gtable object created by arrangeGrob).

References

Gabry, J. , Simpson, D. , Vehtari, A. , Betancourt, M. and Gelman, A. (2019), Visualization in Bayesian workflow. J. R. Stat. Soc. A, 182: 389-402. doi:10.1111/rssa.12378, arXiv preprint, code on GitHub)

See Also

• The vignettes in the bayesplot package for many examples.

• MCMC-overview (bayesplot) for links to the documentation for all the available plotting functions.

• color_scheme_set (bayesplot) to change the color scheme used for plotting.

• pp_check for graphical posterior predictive checks.

• plot_nonlinear for models with nonlinear smooth functions fit using stan_gamm4.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {

# Use rstanarm example model
if (!exists("example_model")) example(example_model)
fit <- example_model

#####################################
### Intervals and point estimates ###
#####################################
plot(fit) # same as plot(fit, "intervals"), plot(fit, "mcmc_intervals")

p <- plot(fit, pars = "size", regex_pars = "period",
          prob = 0.5, prob_outer = 0.9)
p + ggplot2::ggtitle("Posterior medians \n with 50% and 90% intervals")

# Shaded areas under densities
bayesplot::color_scheme_set("brightblue")
plot(fit, "areas", regex_pars = "period",
     prob = 0.5, prob_outer = 0.9)

# Make the same plot by extracting posterior draws and calling
# bayesplot::mcmc_areas directly
x <- as.array(fit, regex_pars = "period")
bayesplot::mcmc_areas(x, prob = 0.5, prob_outer = 0.9)

# Ridgelines version of the areas plot
bayesplot::mcmc_areas_ridges(x, regex_pars = "period", prob = 0.9)


##################################
### Histograms & density plots ###
##################################
plot_title <- ggplot2::ggtitle("Posterior Distributions")
plot(fit, "hist", regex_pars = "period") + plot_title
plot(fit, "dens_overlay", pars = "(Intercept)",
     regex_pars = "period") + plot_title

####################
### Scatterplots ###
####################
bayesplot::color_scheme_set("teal")
plot(fit, "scatter", pars = paste0("period", 2:3))
plot(fit, "scatter", pars = c("(Intercept)", "size"),
     size = 3, alpha = 0.5) +
     ggplot2::stat_ellipse(level = 0.9)


####################################################
### Rhat, effective sample size, autocorrelation ###
####################################################
bayesplot::color_scheme_set("red")

# rhat
plot(fit, "rhat")
plot(fit, "rhat_hist")

# ratio of effective sample size to total posterior sample size
plot(fit, "neff")
plot(fit, "neff_hist")

# autocorrelation by chain
plot(fit, "acf", pars = "(Intercept)", regex_pars = "period")
plot(fit, "acf_bar", pars = "(Intercept)", regex_pars = "period")


##################
### Traceplots ###
##################
# NOTE: rstanarm doesn't store the warmup draws (to save space because they
# are not so essential for diagnosing the particular models implemented in
# rstanarm) so the iterations in the traceplot are post-warmup iterations

bayesplot::color_scheme_set("pink")
(trace <- plot(fit, "trace", pars = "(Intercept)"))

# change traceplot colors to ggplot defaults or custom values
trace + ggplot2::scale_color_discrete()
trace + ggplot2::scale_color_manual(values = c("maroon", "skyblue2"))

# changing facet layout 
plot(fit, "trace", pars = c("(Intercept)", "period2"),
     facet_args = list(nrow = 2))
# same plot by calling bayesplot::mcmc_trace directly
x <- as.array(fit, pars = c("(Intercept)", "period2"))
bayesplot::mcmc_trace(x, facet_args = list(nrow = 2))


############
### More ###
############

# regex_pars examples
plot(fit, regex_pars = "herd:1\\]")
plot(fit, regex_pars = "herd:[279]")
plot(fit, regex_pars = "herd:[279]|period2")
plot(fit, regex_pars = c("herd:[279]", "period2"))


# For graphical posterior predictive checks see
# help("pp_check.stanreg")
}

Plot the estimated subject-specific or marginal survival function

Description

This generic plot method for survfit.stanjm objects will plot the estimated subject-specific or marginal survival function using the data frame returned by a call to posterior_survfit. The call to posterior_survfit should ideally have included an "extrapolation" of the survival function, obtained by setting the extrapolate argument to TRUE.

The plot_stack_jm function takes arguments containing the plots of the estimated subject-specific longitudinal trajectory (or trajectories if a multivariate joint model was estimated) and the plot of the estimated subject-specific survival function and combines them into a single figure. This is most easily understood by running the Examples below.

Usage

## S3 method for class 'survfit.stanjm'
plot(
  x,
  ids = NULL,
  limits = c("ci", "none"),
  xlab = NULL,
  ylab = NULL,
  facet_scales = "free",
  ci_geom_args = NULL,
  ...
)

plot_stack_jm(yplot, survplot)

Arguments

Value

The plot method returns a ggplot object, also of class plot.survfit.stanjm. This object can be further customised using the ggplot2 package. It can also be passed to the function plot_stack_jm.

plot_stack_jm returns an object of class bayesplot_grid that includes plots of the estimated subject-specific longitudinal trajectories stacked on top of the associated subject-specific survival curve.

See Also

posterior_survfit, plot_stack_jm, posterior_traj, plot.predict.stanjm

plot.predict.stanjm, plot.survfit.stanjm, posterior_predict, posterior_survfit

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {

  # Run example model if not already loaded
  if (!exists("example_jm")) example(example_jm)

  # Obtain subject-specific conditional survival probabilities
  # for all individuals in the estimation dataset.
  ps1 <- posterior_survfit(example_jm, extrapolate = TRUE)

  # We then plot the conditional survival probabilities for
  # a subset of individuals
  plot(ps1, ids = c(7,13,15))
  # We can change or add attributes to the plot
  plot(ps1, ids = c(7,13,15), limits = "none")
  plot(ps1, ids = c(7,13,15), xlab = "Follow up time")
  plot(ps1, ids = c(7,13,15), ci_geom_args = list(fill = "red"),
       color = "blue", linetype = 2)
  plot(ps1, ids = c(7,13,15), facet_scales = "fixed")

  # Since the returned plot is also a ggplot object, we can
  # modify some of its attributes after it has been returned
  plot1 <- plot(ps1, ids = c(7,13,15))
  plot1 +
    ggplot2::theme(strip.background = ggplot2::element_blank()) +
    ggplot2::coord_cartesian(xlim = c(0, 15)) +
    ggplot2::labs(title = "Some plotted survival functions")

  # We can also combine the plot(s) of the estimated
  # subject-specific survival functions, with plot(s)
  # of the estimated longitudinal trajectories for the
  # same individuals
  ps1 <- posterior_survfit(example_jm, ids = c(7,13,15))
  pt1 <- posterior_traj(example_jm, , ids = c(7,13,15))
  plot_surv <- plot(ps1)
  plot_traj <- plot(pt1, vline = TRUE, plot_observed = TRUE)
  plot_stack_jm(plot_traj, plot_surv)

  # Lastly, let us plot the standardised survival function
  # based on all individuals in our estimation dataset
  ps2 <- posterior_survfit(example_jm, standardise = TRUE, times = 0,
                          control = list(epoints = 20))
  plot(ps2)

}
if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {

  if (!exists("example_jm")) example(example_jm)
  ps1 <- posterior_survfit(example_jm, ids = c(7,13,15))
  pt1 <- posterior_traj(example_jm, ids = c(7,13,15), extrapolate = TRUE)
  plot_surv <- plot(ps1)
  plot_traj <- plot(pt1, vline = TRUE, plot_observed = TRUE)
  plot_stack_jm(plot_traj, plot_surv)

}

Posterior uncertainty intervals

Description

For models fit using MCMC (algorithm="sampling") or one of the variational approximations ("meanfield" or "fullrank"), the posterior_interval function computes Bayesian posterior uncertainty intervals. These intervals are often referred to as credible intervals, but we use the term uncertainty intervals to highlight the fact that wider intervals correspond to greater uncertainty.

Usage

## S3 method for class 'stanreg'
posterior_interval(
  object,
  prob = 0.9,
  type = "central",
  pars = NULL,
  regex_pars = NULL,
  ...
)

Arguments

Details

Interpretation

Unlike for a frenquentist confidence interval, it is valid to say that, conditional on the data and model, we believe that with probability p the value of a parameter is in its 100p% posterior interval. This intuitive interpretation of Bayesian intervals is often erroneously applied to frequentist confidence intervals. See Morey et al. (2015) for more details on this issue and the advantages of using Bayesian posterior uncertainty intervals (also known as credible intervals).

Default 90% intervals

We default to reporting 90% intervals rather than 95% intervals for several reasons:

• Computational stability: 90% intervals are more stable than 95% intervals (for which each end relies on only 2.5% of the posterior draws).

• Relation to Type-S errors (Gelman and Carlin, 2014): 95% of the mass in a 90% central interval is above the lower value (and 95% is below the upper value). For a parameter \theta, it is therefore easy to see if the posterior probability that \theta > 0 (or \theta < 0) is larger or smaller than 95%.

Of course, if 95% intervals are desired they can be computed by specifying prob=0.95.

Types of intervals

Currently posterior_interval only computes central intervals because other types of intervals are rarely useful for the models that rstanarm can estimate. Additional possibilities may be provided in future releases as more models become available.

Value

A matrix with two columns and as many rows as model parameters (or the subset of parameters specified by pars and/or regex_pars). For a given value of prob, p, the columns correspond to the lower and upper 100p% interval limits and have the names 100\alpha/2% and 100(1 - \alpha/2)%, where \alpha = 1-p. For example, if prob=0.9 is specified (a 90% interval), then the column names will be "5%" and "95%", respectively.

References

Gelman, A. and Carlin, J. (2014). Beyond power calculations: assessing Type S (sign) and Type M (magnitude) errors. Perspectives on Psychological Science. 9(6), 641–51.

Morey, R. D., Hoekstra, R., Rouder, J., Lee, M. D., and Wagenmakers, E. (2016). The fallacy of placing confidence in confidence intervals. Psychonomic Bulletin & Review. 23(1), 103–123.

See Also

confint.stanreg, which, for models fit using optimization, can be used to compute traditional confidence intervals.

predictive_interval for predictive intervals.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
if (!exists("example_model")) example(example_model)
posterior_interval(example_model)
posterior_interval(example_model, regex_pars = "herd")
posterior_interval(example_model, pars = "period2", prob = 0.5)
}

Posterior distribution of the (possibly transformed) linear predictor

Description

Extract the posterior draws of the linear predictor, possibly transformed by the inverse-link function. This function is occasionally useful, but it should be used sparingly: inference and model checking should generally be carried out using the posterior predictive distribution (i.e., using posterior_predict).

Usage

## S3 method for class 'stanreg'
posterior_linpred(
  object,
  transform = FALSE,
  newdata = NULL,
  draws = NULL,
  re.form = NULL,
  offset = NULL,
  XZ = FALSE,
  ...
)

## S3 method for class 'stanreg'
posterior_epred(
  object,
  newdata = NULL,
  draws = NULL,
  re.form = NULL,
  offset = NULL,
  XZ = FALSE,
  ...
)

Arguments

Details

The posterior_linpred function returns the posterior distribution of the linear predictor, while the posterior_epred function returns the posterior distribution of the conditional expectation. In the special case of a Gaussian likelihood with an identity link function, these two concepts are the same. The posterior_epred function is a less noisy way to obtain expectations over the output of posterior_predict.

Value

The default is to return a draws by nrow(newdata) matrix of simulations from the posterior distribution of the (possibly transformed) linear predictor. The exception is if the argument XZ is set to TRUE (see the XZ argument description above).

Note

For models estimated with stan_clogit, the number of successes per stratum is ostensibly fixed by the research design. Thus, when calling posterior_linpred with new data and transform = TRUE, the data.frame passed to the newdata argument must contain an outcome variable and a stratifying factor, both with the same name as in the original data.frame. Then, the probabilities will condition on this outcome in the new data.

See Also

posterior_predict to draw from the posterior predictive distribution of the outcome, which is typically preferable.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
if (!exists("example_model")) example(example_model)
print(family(example_model))

# linear predictor on log-odds scale
linpred <- posterior_linpred(example_model)
colMeans(linpred)

# probabilities
# same as posterior_linpred(example_model, transform = TRUE)
probs <- posterior_epred(example_model) 
colMeans(probs)

# not conditioning on any group-level parameters
probs2 <- posterior_epred(example_model, re.form = NA)
apply(probs2, 2, median)
}

Draw from posterior predictive distribution

Description

The posterior predictive distribution is the distribution of the outcome implied by the model after using the observed data to update our beliefs about the unknown parameters in the model. Simulating data from the posterior predictive distribution using the observed predictors is useful for checking the fit of the model. Drawing from the posterior predictive distribution at interesting values of the predictors also lets us visualize how a manipulation of a predictor affects (a function of) the outcome(s). With new observations of predictor variables we can use the posterior predictive distribution to generate predicted outcomes.

Usage

## S3 method for class 'stanreg'
posterior_predict(
  object,
  newdata = NULL,
  draws = NULL,
  re.form = NULL,
  fun = NULL,
  seed = NULL,
  offset = NULL,
  ...
)

## S3 method for class 'stanmvreg'
posterior_predict(
  object,
  m = 1,
  newdata = NULL,
  draws = NULL,
  re.form = NULL,
  fun = NULL,
  seed = NULL,
  offset = NULL,
  ...
)

Arguments

Value

A draws by nrow(newdata) matrix of simulations from the posterior predictive distribution. Each row of the matrix is a vector of predictions generated using a single draw of the model parameters from the posterior distribution.

Note

For binomial models with a number of trials greater than one (i.e., not Bernoulli models), if newdata is specified then it must include all variables needed for computing the number of binomial trials to use for the predictions. For example if the left-hand side of the model formula is cbind(successes, failures) then both successes and failures must be in newdata. The particular values of successes and failures in newdata do not matter so long as their sum is the desired number of trials. If the left-hand side of the model formula were cbind(successes, trials - successes) then both trials and successes would need to be in newdata, probably with successes set to 0 and trials specifying the number of trials. See the Examples section below and the How to Use the rstanarm Package for examples.

For models estimated with stan_clogit, the number of successes per stratum is ostensibly fixed by the research design. Thus, when doing posterior prediction with new data, the data.frame passed to the newdata argument must contain an outcome variable and a stratifying factor, both with the same name as in the original data.frame. Then, the posterior predictions will condition on this outcome in the new data.

See Also

pp_check for graphical posterior predictive checks. Examples of posterior predictive checking can also be found in the rstanarm vignettes and demos.

predictive_error and predictive_interval.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
if (!exists("example_model")) example(example_model)
yrep <- posterior_predict(example_model)
table(yrep)


# Using newdata
counts <- c(18,17,15,20,10,20,25,13,12)
outcome <- gl(3,1,9)
treatment <- gl(3,3)
dat <- data.frame(counts, treatment, outcome)
fit3 <- stan_glm(
  counts ~ outcome + treatment, 
  data = dat,
  family = poisson(link="log"),
  prior = normal(0, 1, autoscale = FALSE), 
  prior_intercept = normal(0, 5, autoscale = FALSE),
  refresh = 0
)
nd <- data.frame(treatment = factor(rep(1,3)), outcome = factor(1:3))
ytilde <- posterior_predict(fit3, nd, draws = 500)
print(dim(ytilde))  # 500 by 3 matrix (draws by nrow(nd))

ytilde <- data.frame(
  count = c(ytilde),
  outcome = rep(nd$outcome, each = 500)
)
ggplot2::ggplot(ytilde, ggplot2::aes(x=outcome, y=count)) +
  ggplot2::geom_boxplot() +
  ggplot2::ylab("predicted count")


# Using newdata with a binomial model.
# example_model is binomial so we need to set
# the number of trials to use for prediction.
# This could be a different number for each
# row of newdata or the same for all rows.
# Here we'll use the same value for all.
nd <- lme4::cbpp
print(formula(example_model))  # cbind(incidence, size - incidence) ~ ...
nd$size <- max(nd$size) + 1L   # number of trials
nd$incidence <- 0  # set to 0 so size - incidence = number of trials
ytilde <- posterior_predict(example_model, newdata = nd)


# Using fun argument to transform predictions
mtcars2 <- mtcars
mtcars2$log_mpg <- log(mtcars2$mpg)
fit <- stan_glm(log_mpg ~ wt, data = mtcars2, refresh = 0)
ytilde <- posterior_predict(fit, fun = exp)

}

Estimate subject-specific or standardised survival probabilities

Description

This function allows us to generate estimated survival probabilities based on draws from the posterior predictive distribution. By default the survival probabilities are conditional on an individual's group-specific coefficients (i.e. their individual-level random effects). If prediction data is provided via the newdataLong and newdataEvent arguments, then the default behaviour is to sample new group-specific coefficients for the individuals in the new data using a Monte Carlo scheme that conditions on their longitudinal outcome data provided in newdataLong (sometimes referred to as "dynamic predictions", see Rizopoulos (2011)). This default behaviour can be stopped by specifying dynamic = FALSE, in which case the predicted survival probabilities will be marginalised over the distribution of the group-specific coefficients. This has the benefit that the user does not need to provide longitudinal outcome measurements for the new individuals, however, it does mean that the predictions will incorporate all the uncertainty associated with between-individual variation, since the predictions aren't conditional on any observed data for the individual. In addition, by default, the predicted subject-specific survival probabilities are conditional on observed values of the fixed effect covariates (ie, the predictions will be obtained using either the design matrices used in the original stan_jm model call, or using the covariate values provided in the newdataLong and newdataEvent arguments). However, if you wish to average over the observed distribution of the fixed effect covariates then this is possible – such predictions are sometimes referred to as standardised survival probabilties – see the standardise argument below.

Usage

posterior_survfit(
  object,
  newdataLong = NULL,
  newdataEvent = NULL,
  extrapolate = TRUE,
  control = list(),
  condition = NULL,
  last_time = NULL,
  prob = 0.95,
  ids,
  times = NULL,
  standardise = FALSE,
  dynamic = TRUE,
  scale = 1.5,
  draws = NULL,
  seed = NULL,
  ...
)

Arguments

Value

A data frame of class survfit.stanjm. The data frame includes columns for each of the following: (i) the median of the posterior predictions of the estimated survival probabilities (survpred); (ii) each of the lower and upper limits of the corresponding uncertainty interval for the estimated survival probabilities (ci_lb and ci_ub); (iii) a subject identifier (id_var), unless standardised survival probabilities were estimated; (iv) the time that the estimated survival probability is calculated for (time_var). The returned object also includes a number of additional attributes.

Note

Note that if any variables were transformed (e.g. rescaled) in the data used to fit the model, then these variables must also be transformed in newdataLong and newdataEvent. This only applies if variables were transformed before passing the data to one of the modeling functions and not if transformations were specified inside the model formula.

References

Rizopoulos, D. (2011). Dynamic predictions and prospective accuracy in joint models for longitudinal and time-to-event data. Biometrics 67, 819.

See Also

plot.survfit.stanjm for plotting the estimated survival probabilities, ps_check for for graphical checks of the estimated survival function, and posterior_traj for estimating the marginal or subject-specific longitudinal trajectories, and plot_stack_jm for combining plots of the estimated subject-specific longitudinal trajectory and survival function.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {

  # Run example model if not already loaded
  if (!exists("example_jm")) example(example_jm)

  # Obtain subject-specific survival probabilities for a few
  # selected individuals in the estimation dataset who were
  # known to survive up until their censoring time. By default
  # the posterior_survfit function will estimate the conditional
  # survival probabilities, that is, conditional on having survived
  # until the event or censoring time, and then by default will
  # extrapolate the survival predictions forward from there.
  ps1 <- posterior_survfit(example_jm, ids = c(7,13,15))
  # We can plot the estimated survival probabilities using the
  # associated plot function
  plot(ps1)

  # If we wanted to estimate the survival probabilities for the
  # same three individuals as the previous example, but this time
  # we won't condition on them having survived up until their
  # censoring time. Instead, we will estimate their probability
  # of having survived between 0 and 5 years given their covariates
  # and their estimated random effects.
  # The easiest way to achieve the time scale we want (ie, 0 to 5 years)
  # is to specify that we want the survival time estimated at time 0
  # and then extrapolated forward 5 years. We also specify that we
  # do not want to condition on their last known survival time.
  ps2 <- posterior_survfit(example_jm, ids = c(7,13,15), times = 0,
    extrapolate = TRUE, condition = FALSE, control = list(edist = 5))

  # Instead we may want to estimate subject-specific survival probabilities
  # for a set of new individuals. To demonstrate this, we will simply take
  # the first two individuals in the estimation dataset, but pass their data
  # via the newdata arguments so that posterior_survfit will assume we are
  # predicting survival for new individuals and draw new random effects
  # under a Monte Carlo scheme (see Rizopoulos (2011)).
  ndL <- pbcLong[pbcLong$id %in% c(1,2),]
  ndE <- pbcSurv[pbcSurv$id %in% c(1,2),]
  ps3 <- posterior_survfit(example_jm,
    newdataLong = ndL, newdataEvent = ndE,
    last_time = "futimeYears", seed = 12345)
  head(ps3)
  # We can then compare the estimated random effects for these
  # individuals based on the fitted model and the Monte Carlo scheme
  ranef(example_jm)$Long1$id[1:2,,drop=FALSE] # from fitted model
  colMeans(attr(ps3, "b_new"))                # from Monte Carlo scheme

  # Lastly, if we wanted to obtain "standardised" survival probabilities,
  # (by averaging over the observed distribution of the fixed effect
  # covariates, as well as averaging over the estimated random effects
  # for individuals in our estimation sample or new data) then we can
  # specify 'standardise = TRUE'. We can then plot the resulting
  # standardised survival curve.
  ps4 <- posterior_survfit(example_jm, standardise = TRUE,
                           times = 0, extrapolate = TRUE)
  plot(ps4)

}

Estimate the subject-specific or marginal longitudinal trajectory

Description

This function allows us to generate an estimated longitudinal trajectory (either subject-specific, or by marginalising over the distribution of the group-specific parameters) based on draws from the posterior predictive distribution.

Usage

posterior_traj(
  object,
  m = 1,
  newdata = NULL,
  newdataLong = NULL,
  newdataEvent = NULL,
  interpolate = TRUE,
  extrapolate = FALSE,
  control = list(),
  last_time = NULL,
  prob = 0.95,
  ids,
  dynamic = TRUE,
  scale = 1.5,
  draws = NULL,
  seed = NULL,
  return_matrix = FALSE,
  ...
)

Arguments

Details

The posterior_traj function acts as a wrapper to the posterior_predict function, but allows predictions to be easily generated at time points that are interpolated and/or extrapolated between time zero (baseline) and the last known survival time for the individual, thereby providing predictions that correspond to a smooth estimate of the longitudinal trajectory (useful for the plotting via the associated plot.predict.stanjm method). In addition it returns a data frame by default, whereas the posterior_predict function returns a matrix; see the Value section below for details. Also, posterior_traj allows predictions to only be generated for a subset of individuals, via the ids argument.

Value

When return_matrix = FALSE, a data frame of class predict.stanjm. The data frame includes a column for the median of the posterior predictions of the mean longitudinal response (yfit), a column for each of the lower and upper limits of the uncertainty interval corresponding to the posterior predictions of the mean longitudinal response (ci_lb and ci_ub), and a column for each of the lower and upper limits of the prediction interval corresponding to the posterior predictions of the (raw) longitudinal response. The data frame also includes columns for the subject ID variable, and each of the predictor variables. The returned object also includes a number of attributes.

When return_matrix = TRUE, the returned object is the same as that described for posterior_predict.

See Also

plot.predict.stanjm, posterior_predict, posterior_survfit

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {

  # Run example model if not already loaded
  if (!exists("example_jm")) example(example_jm)
  
  # Obtain subject-specific predictions for all individuals 
  # in the estimation dataset
  pt1 <- posterior_traj(example_jm, interpolate = FALSE, extrapolate = FALSE)
  head(pt1)
  
  # Obtain subject-specific predictions only for a few selected individuals
  pt2 <- posterior_traj(example_jm, ids = c(1,3,8))
  
  # If we wanted to obtain subject-specific predictions in order to plot the 
  # longitudinal trajectories, then we might want to ensure a full trajectory 
  # is obtained by interpolating and extrapolating time. We can then use the 
  # generic plot function to plot the subject-specific predicted trajectories
  # for the first three individuals. Interpolation and extrapolation is 
  # carried out by default.
  pt3 <- posterior_traj(example_jm)
  head(pt3) # predictions at additional time points compared with pt1 
  plot(pt3, ids = 1:3)
  
  # If we wanted to extrapolate further in time, but decrease the number of 
  # discrete time points at which we obtain predictions for each individual, 
  # then we could specify a named list in the 'control' argument
  pt4 <- posterior_traj(example_jm, control = list(ipoints = 10, epoints = 10, eprop = 0.5))
  
  # If we have prediction data for a new individual, and we want to
  # estimate the longitudinal trajectory for that individual conditional
  # on this new data (perhaps extrapolating forward from our last
  # longitudinal measurement) then we can do that. It requires drawing
  # new individual-specific parameters, based on the full likelihood,
  # so we must supply new data for both the longitudinal and event 
  # submodels. These are sometimes known as dynamic predictions.
  ndL <- pbcLong[pbcLong$id == 8, , drop = FALSE]
  ndE <- pbcSurv[pbcSurv$id == 8, , drop = FALSE]
  ndL$id <- "new_subject" # new id can't match one used in training data
  ndE$id <- "new_subject"
  pt5 <- posterior_traj(example_jm, 
                        newdataLong = ndL,
                        newdataEvent = ndE)
                        
  # By default it is assumed that the last known survival time for 
  # the individual is the time of their last biomarker measurement,
  # but if we know they survived to some later time then we can
  # condition on that information using the last_time argument
  pt6 <- posterior_traj(example_jm, 
                        newdataLong = ndL,
                        newdataEvent = ndE, 
                        last_time = "futimeYears")
  
  # Alternatively we may want to estimate the marginal longitudinal
  # trajectory for a given set of covariates. To do this, we can pass
  # the desired covariate values in a new data frame (however the only
  # covariate in our fitted model was the time variable, year). To make sure  
  # that we marginalise over the random effects, we need to specify an ID value
  # which does not correspond to any of the individuals who were used in the
  # model estimation and specify the argument dynamic=FALSE.
  # The marginal prediction is obtained by generating subject-specific 
  # predictions using a series of random draws from the random 
  # effects distribution, and then integrating (ie, averaging) over these. 
  # Our marginal prediction will therefore capture the between-individual 
  # variation associated with the random effects.
  
  nd <- data.frame(id = rep("new1", 11), year = (0:10 / 2))
  pt7 <- posterior_traj(example_jm, newdataLong = nd, dynamic = FALSE)
  head(pt7)  # note the greater width of the uncertainty interval compared 
             # with the subject-specific predictions in pt1, pt2, etc
  
  # Alternatively, we could have estimated the "marginal" trajectory by 
  # ignoring the random effects (ie, assuming the random effects were set 
  # to zero). This will generate a predicted longitudinal trajectory only
  # based on the fixed effect component of the model. In essence, for a 
  # linear mixed effects model (ie, a model that uses an identity link 
  # function), we should obtain a similar point estimate ("yfit") to the
  # estimates obtained in pt5 (since the mean of the estimated random effects
  # distribution will be approximately 0). However, it is important to note that
  # the uncertainty interval will be much more narrow, since it completely
  # ignores the between-individual variability captured by the random effects.
  # Further, if the model uses a non-identity link function, then the point
  # estimate ("yfit") obtained only using the fixed effect component of the
  # model will actually provide a biased estimate of the marginal prediction.
  # Nonetheless, to demonstrate how we can obtain the predictions only using 
  # the fixed effect component of the model, we simply specify 're.form = NA'. 
  # (We will use the same covariate values as used in the prediction for 
  # example for pt5).
  
  pt8 <- posterior_traj(example_jm, newdataLong = nd, dynamic = FALSE, 
                        re.form = NA)
  head(pt8)  # note the much narrower ci, compared with pt5

}

Juxtapose prior and posterior

Description

Plot medians and central intervals comparing parameter draws from the prior and posterior distributions. If the plotted priors look different than the priors you think you specified it is likely either because of internal rescaling or the use of the QR argument (see the documentation for the prior_summary method for details on these special cases).

Usage

posterior_vs_prior(object, ...)

## S3 method for class 'stanreg'
posterior_vs_prior(
  object,
  pars = NULL,
  regex_pars = NULL,
  prob = 0.9,
  color_by = c("parameter", "vs", "none"),
  group_by_parameter = FALSE,
  facet_args = list(),
  ...
)

Arguments

Value

A ggplot object that can be further customized using the ggplot2 package.

References

Gabry, J. , Simpson, D. , Vehtari, A. , Betancourt, M. and Gelman, A. (2019), Visualization in Bayesian workflow. J. R. Stat. Soc. A, 182: 389-402. doi:10.1111/rssa.12378, arXiv preprint, code on GitHub)

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
## Not run: 
if (!exists("example_model")) example(example_model)
# display non-varying (i.e. not group-level) coefficients
posterior_vs_prior(example_model, pars = "beta")

# show group-level (varying) parameters and group by parameter
posterior_vs_prior(example_model, pars = "varying",
                   group_by_parameter = TRUE, color_by = "vs")

# group by parameter and allow axis scales to vary across facets
posterior_vs_prior(example_model, regex_pars = "period",
                   group_by_parameter = TRUE, color_by = "none",
                   facet_args = list(scales = "free"))

# assign to object and customize with functions from ggplot2
(gg <- posterior_vs_prior(example_model, pars = c("beta", "varying"), prob = 0.8))

gg + 
 ggplot2::geom_hline(yintercept = 0, size = 0.3, linetype = 3) + 
 ggplot2::coord_flip() + 
 ggplot2::ggtitle("Comparing the prior and posterior")
 
# compare very wide and very narrow priors using roaches example
# (see help(roaches, "rstanarm") for info on the dataset)
roaches$roach100 <- roaches$roach1 / 100
wide_prior <- normal(0, 10)
narrow_prior <- normal(0, 0.1)
fit_pois_wide_prior <- stan_glm(y ~ treatment + roach100 + senior, 
                                offset = log(exposure2), 
                                family = "poisson", data = roaches, 
                                prior = wide_prior)
posterior_vs_prior(fit_pois_wide_prior, pars = "beta", prob = 0.5, 
                   group_by_parameter = TRUE, color_by = "vs", 
                   facet_args = list(scales = "free"))
                   
fit_pois_narrow_prior <- update(fit_pois_wide_prior, prior = narrow_prior)
posterior_vs_prior(fit_pois_narrow_prior, pars = "beta", prob = 0.5, 
                   group_by_parameter = TRUE, color_by = "vs", 
                   facet_args = list(scales = "free"))
                   

# look at cutpoints for ordinal model
fit_polr <- stan_polr(tobgp ~ agegp, data = esoph, method = "probit",
                      prior = R2(0.2, "mean"), init_r = 0.1)
(gg_polr <- posterior_vs_prior(fit_polr, regex_pars = "\\|", color_by = "vs",
                               group_by_parameter = TRUE))
# flip the x and y axes
gg_polr + ggplot2::coord_flip()

## End(Not run)
}

Graphical posterior predictive checks

Description

Interface to the PPC (posterior predictive checking) module in the bayesplot package, providing various plots comparing the observed outcome variable y to simulated datasets y^{rep} from the posterior predictive distribution. The pp_check method for stanreg-objects prepares the arguments required for the specified bayesplot PPC plotting function and then calls that function. It is also straightforward to use the functions from the bayesplot package directly rather than via the pp_check method. Examples of both are given below.

Usage

## S3 method for class 'stanreg'
pp_check(object, plotfun = "dens_overlay", nreps = NULL, seed = NULL, ...)

Arguments

Value

pp_check returns a ggplot object that can be further customized using the ggplot2 package.

Note

For binomial data, plots of y and y^{rep} show the proportion of 'successes' rather than the raw count. Also for binomial models see ppc_error_binned for binned residual plots.

References

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013). Bayesian Data Analysis. Chapman & Hall/CRC Press, London, third edition. (Ch. 6)

Gabry, J. , Simpson, D. , Vehtari, A. , Betancourt, M. and Gelman, A. (2019), Visualization in Bayesian workflow. J. R. Stat. Soc. A, 182: 389-402. doi:10.1111/rssa.12378, arXiv preprint, code on GitHub)

See Also

• The vignettes in the bayesplot package for many examples. Examples of posterior predictive checks can also be found in the rstanarm vignettes and demos.

• PPC-overview (bayesplot) for links to the documentation for all the available plotting functions.

• posterior_predict for drawing from the posterior predictive distribution.

• color_scheme_set to change the color scheme of the plots.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
fit <- stan_glmer(
  mpg ~ wt + am + (1|cyl),
  data = mtcars,
  iter = 400, # iter and chains small just to keep example quick
  chains = 2,
  refresh = 0
)

# Compare distribution of y to distributions of multiple yrep datasets
pp_check(fit)
pp_check(fit, plotfun = "boxplot", nreps = 10, notch = FALSE)
pp_check(fit, plotfun = "hist", nreps = 3)


# Same plot (up to RNG noise) using bayesplot package directly
bayesplot::ppc_hist(y = mtcars$mpg, yrep = posterior_predict(fit, draws = 3))

# Check histograms of test statistics by level of grouping variable 'cyl'
pp_check(fit, plotfun = "stat_grouped", stat = "median", group = "cyl")

# Defining a custom test statistic
q25 <- function(y) quantile(y, probs = 0.25)
pp_check(fit, plotfun = "stat_grouped", stat = "q25", group = "cyl")

# Scatterplot of two test statistics
pp_check(fit, plotfun = "stat_2d", stat = c("mean", "sd"))

# Scatterplot of y vs. average yrep
pp_check(fit, plotfun = "scatter_avg") # y vs. average yrep
# Same plot (up to RNG noise) using bayesplot package directly
bayesplot::ppc_scatter_avg(y = mtcars$mpg, yrep = posterior_predict(fit))

# Scatterplots of y vs. several individual yrep datasets
pp_check(fit, plotfun = "scatter", nreps = 3)

# Same plot (up to RNG noise) using bayesplot package directly
bayesplot::ppc_scatter(y = mtcars$mpg, yrep = posterior_predict(fit, draws = 3))

# yrep intervals with y points overlaid
# by default 1:length(y) used on x-axis but can also specify an x variable
pp_check(fit, plotfun = "intervals")
pp_check(fit, plotfun = "intervals", x = "wt") + ggplot2::xlab("wt")

# Same plot (up to RNG noise) using bayesplot package directly
bayesplot::ppc_intervals(y = mtcars$mpg, yrep = posterior_predict(fit),
                         x = mtcars$wt) + ggplot2::xlab("wt")

# predictive errors
pp_check(fit, plotfun = "error_hist", nreps = 6)
pp_check(fit, plotfun = "error_scatter_avg_vs_x", x = "wt") +
  ggplot2::xlab("wt")

# Example of a PPC for ordinal models (stan_polr)
fit2 <- stan_polr(tobgp ~ agegp, data = esoph, method = "probit",
                  prior = R2(0.2, "mean"), init_r = 0.1,
                  refresh = 0)
pp_check(fit2, plotfun = "bars", nreps = 500, prob = 0.5)
pp_check(fit2, plotfun = "bars_grouped", group = esoph$agegp,
         nreps = 500, prob = 0.5)

}

Model validation via simulation

Description

The pp_validate function is based on the methods described in Cook, Gelman, and Rubin (2006) for validating software developed to fit particular Bayesian models. Here we take the perspective that models themselves are software and thus it is useful to apply this validation approach to individual models.

Usage

pp_validate(object, nreps = 20, seed = 12345, ...)

Arguments

Details

We repeat nreps times the process of simulating parameters and data from the model and refitting the model to this simulated data. For each of the nreps replications we do the following:

1. Refit the model but without conditioning on the data (setting prior_PD=TRUE), obtaining draws \theta^{true} from the prior distribution of the model parameters.

2. Given \theta^{true}, simulate data y^\ast from the prior predictive distribution (calling posterior_predict on the fitted model object obtained in step 1).

3. Fit the model to the simulated outcome y^\ast, obtaining parameters \theta^{post}.

For any individual parameter, the quantile of the "true" parameter value with respect to its posterior distribution should be uniformly distributed. The validation procedure entails looking for deviations from uniformity by computing statistics for a test that the quantiles are uniformly distributed. The absolute values of the computed test statistics are plotted for batches of parameters (e.g., non-varying coefficients are grouped into a batch called "beta", parameters that vary by group level are in batches named for the grouping variable, etc.). See Cook, Gelman, and Rubin (2006) for more details on the validation procedure.

Value

A ggplot object that can be further customized using the ggplot2 package.

Note

In order to make it through nreps replications without running into numerical difficulties you may have to restrict the range for randomly generating initial values for parameters when you fit the original model. With any of rstanarm's modeling functions this can be done by specifying the optional argument init_r as some number less than the default of 2.

References

Cook, S., Gelman, A., and Rubin, D. (2006). Validation of software for Bayesian models using posterior quantiles. Journal of Computational and Graphical Statistics. 15(3), 675–692.

See Also

pp_check for graphical posterior predictive checks and posterior_predict to draw from the posterior predictive distribution.

color_scheme_set to change the color scheme of the plot.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
## Not run: 
if (!exists("example_model")) example(example_model)
try(pp_validate(example_model)) # fails with default seed / priors

## End(Not run)
}

Predict method for stanreg objects

Description

This method is primarily intended to be used only for models fit using optimization. For models fit using MCMC or one of the variational approximations, see posterior_predict.

Usage

## S3 method for class 'stanreg'
predict(
  object,
  ...,
  newdata = NULL,
  type = c("link", "response"),
  se.fit = FALSE
)

Arguments

Value

A vector if se.fit is FALSE and a list if se.fit is TRUE.

See Also

posterior_predict

In-sample or out-of-sample predictive errors

Description

This is a convenience function for computing y - y^{rep} (in-sample, for observed y) or y - \tilde{y} (out-of-sample, for new or held-out y). The method for stanreg objects calls posterior_predict internally, whereas the method for matrices accepts the matrix returned by posterior_predict as input and can be used to avoid multiple calls to posterior_predict.

Usage

## S3 method for class 'stanreg'
predictive_error(
  object,
  newdata = NULL,
  draws = NULL,
  re.form = NULL,
  seed = NULL,
  offset = NULL,
  ...
)

## S3 method for class 'matrix'
predictive_error(object, y, ...)

## S3 method for class 'ppd'
predictive_error(object, y, ...)

Arguments

Value

A draws by nrow(newdata) matrix. If newdata is not specified then it will be draws by nobs(object).

Note

The Note section in posterior_predict about newdata for binomial models also applies for predictive_error, with one important difference. For posterior_predict if the left-hand side of the model formula is cbind(successes, failures) then the particular values of successes and failures in newdata don't matter, only that they add to the desired number of trials. This is not the case for predictive_error. For predictive_error the particular value of successes matters because it is used as y when computing the error.

See Also

posterior_predict to draw from the posterior predictive distribution without computing predictive errors.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
if (!exists("example_model")) example(example_model)
err1 <- predictive_error(example_model, draws = 50)
hist(err1)

# Using newdata with a binomial model
formula(example_model)
nd <- data.frame(
 size = c(10, 20), 
 incidence = c(5, 10), 
 period = factor(c(1,2)), 
 herd = c(1, 15)
)
err2 <- predictive_error(example_model, newdata = nd, draws = 10, seed = 1234)

# stanreg vs matrix methods
fit <- stan_glm(mpg ~ wt, data = mtcars, iter = 300)
preds <- posterior_predict(fit, seed = 123)
all.equal(
  predictive_error(fit, seed = 123),
  predictive_error(preds, y = fit$y)
)
}

Predictive intervals

Description

For models fit using MCMC (algorithm="sampling") or one of the variational approximations ("meanfield" or "fullrank"), the predictive_interval function computes Bayesian predictive intervals. The method for stanreg objects calls posterior_predict internally, whereas the method for matrices accepts the matrix returned by posterior_predict as input and can be used to avoid multiple calls to posterior_predict.

Usage

## S3 method for class 'stanreg'
predictive_interval(
  object,
  prob = 0.9,
  newdata = NULL,
  draws = NULL,
  re.form = NULL,
  fun = NULL,
  seed = NULL,
  offset = NULL,
  ...
)

## S3 method for class 'matrix'
predictive_interval(object, prob = 0.9, ...)

## S3 method for class 'ppd'
predictive_interval(object, prob = 0.9, ...)

Arguments

Value

A matrix with two columns and as many rows as are in newdata. If newdata is not provided then the matrix will have as many rows as the data used to fit the model. For a given value of prob, p, the columns correspond to the lower and upper 100p% central interval limits and have the names 100\alpha/2% and 100(1 - \alpha/2)%, where \alpha = 1-p. For example, if prob=0.9 is specified (a 90% interval), then the column names will be "5%" and "95%", respectively.

See Also

predictive_error, posterior_predict, posterior_interval

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
fit <- stan_glm(mpg ~ wt, data = mtcars, iter = 300)
predictive_interval(fit)
predictive_interval(fit, newdata = data.frame(wt = range(mtcars$wt)), 
                    prob = 0.5)

# stanreg vs matrix methods
preds <- posterior_predict(fit, seed = 123)
all.equal(
  predictive_interval(fit, seed = 123),
  predictive_interval(preds)
)
}

Print method for stanreg objects

Description

The print method for stanreg objects displays a compact summary of the fitted model. See the Details section below for descriptions of the different components of the printed output. For additional summary statistics and diagnostics use the summary method.

Usage

## S3 method for class 'stanreg'
print(x, digits = 1, detail = TRUE, ...)

## S3 method for class 'stanmvreg'
print(x, digits = 3, ...)

Arguments

Details

Point estimates

Regardless of the estimation algorithm, point estimates are medians computed from simulations. For models fit using MCMC ("sampling") the posterior sample is used. For optimization ("optimizing"), the simulations are generated from the asymptotic Gaussian sampling distribution of the parameters. For the "meanfield" and "fullrank" variational approximations, draws from the variational approximation to the posterior are used. In all cases, the point estimates reported are the same as the values returned by coef.

Uncertainty estimates (MAD_SD)

The standard deviations reported (labeled MAD_SD in the print output) are computed from the same set of draws described above and are proportional to the median absolute deviation (mad) from the median. Compared to the raw posterior standard deviation, the MAD_SD will be more robust for long-tailed distributions. These are the same as the values returned by se.

Additional output

• For GLMs with group-specific terms (see stan_glmer) the printed output also shows point estimates of the standard deviations of the group effects (and correlations if there are both intercept and slopes that vary by group).

• For analysis of variance models (see stan_aov) models, an ANOVA-like table is also displayed.

• For joint longitudinal and time-to-event (see stan_jm) models the estimates are presented separately for each of the distinct submodels.

Value

Returns x, invisibly.

See Also

summary.stanreg, stanreg-methods

Generic print method for survfit.stanjm objects

Description

Generic print method for survfit.stanjm objects

Usage

## S3 method for class 'survfit.stanjm'
print(x, digits = 4, ...)

Arguments

Summarize the priors used for an rstanarm model

Description

The prior_summary method provides a summary of the prior distributions used for the parameters in a given model. In some cases the user-specified prior does not correspond exactly to the prior used internally by rstanarm (see the sections below). Especially in these cases, but also in general, it can be much more useful to visualize the priors. Visualizing the priors can be done using the posterior_vs_prior function, or alternatively by fitting the model with the prior_PD argument set to TRUE (to draw from the prior predictive distribution instead of conditioning on the outcome) and then plotting the parameters.

Usage

## S3 method for class 'stanreg'
prior_summary(object, digits = 2, ...)

Arguments

Value

A list of class "prior_summary.stanreg", which has its own print method.

Intercept (after predictors centered)

For rstanarm modeling functions that accept a prior_intercept argument, the specified prior for the intercept term applies to the intercept after rstanarm internally centers the predictors so they each have mean zero. The estimate of the intercept returned to the user correspond to the intercept with the predictors as specified by the user (unmodified by rstanarm), but when specifying the prior the intercept can be thought of as the expected outcome when the predictors are set to their means. The only exception to this is for models fit with the sparse argument set to TRUE (which is only possible with a subset of the modeling functions and never the default).

Adjusted scales

For some models you may see "adjusted scale" in the printed output and adjusted scales included in the object returned by prior_summary. These adjusted scale values are the prior scales actually used by rstanarm and are computed by adjusting the prior scales specified by the user to account for the scales of the predictors (as described in the documentation for the autoscale argument). To disable internal prior scale adjustments set the autoscale argument to FALSE when setting a prior using one of the distributions that accepts an autoscale argument. For example, normal(0, 5, autoscale=FALSE) instead of just normal(0, 5).

Coefficients in Q-space

For the models fit with an rstanarm modeling function that supports the QR argument (see e.g, stan_glm), if QR is set to TRUE then the prior distributions for the regression coefficients specified using the prior argument are not relative to the original predictor variables X but rather to the variables in the matrix Q obtained from the QR decomposition of X.

In particular, if prior = normal(location,scale), then this prior on the coefficients in Q-space can be easily translated into a joint multivariate normal (MVN) prior on the coefficients on the original predictors in X. Letting \theta denote the coefficients on Q and \beta the coefficients on X then if \theta \sim N(\mu, \sigma) the corresponding prior on \beta is \beta \sim MVN(R\mu, R'R\sigma^2), where \mu and \sigma are vectors of the appropriate length. Technically, rstanarm uses a scaled QR decomposition to ensure that the columns of the predictor matrix used to fit the model all have unit scale, when the autoscale argument to the function passed to the prior argument is TRUE (the default), in which case the matrices actually used are Q^\ast = Q \sqrt{n-1} and R^\ast = \frac{1}{\sqrt{n-1}} R. If autoscale = FALSE we instead scale such that the lower-right element of R^\ast is 1, which is useful if you want to specify a prior on the coefficient of the last predictor in its original units (see the documentation for the QR argument).

If you are interested in the prior on \beta implied by the prior on \theta, we strongly recommend visualizing it as described above in the Description section, which is simpler than working it out analytically.

See Also

The priors help page and the Prior Distributions vignette.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
if (!exists("example_model")) example(example_model) 
prior_summary(example_model)

priors <- prior_summary(example_model)
names(priors)
priors$prior$scale
priors$prior$adjusted_scale

# for a glm with adjusted scales (see Details, above), compare 
# the default (rstanarm adjusting the scales) to setting 
# autoscale=FALSE for prior on coefficients
fit <- stan_glm(mpg ~ wt + am, data = mtcars, 
                prior = normal(0, c(2.5, 4)), 
                prior_intercept = normal(0, 5), 
                iter = 10, chains = 1) # only for demonstration 
prior_summary(fit)

fit2 <- update(fit, prior = normal(0, c(2.5, 4), autoscale=FALSE), 
               prior_intercept = normal(0, 5, autoscale=FALSE))
prior_summary(fit2)
}

Prior distributions and options

Description

The functions described on this page are used to specify the prior-related arguments of the various modeling functions in the rstanarm package (to view the priors used for an existing model see prior_summary).

The default priors used in the various rstanarm modeling functions are intended to be weakly informative in that they provide moderate regularization and help stabilize computation. For many applications the defaults will perform well, but prudent use of more informative priors is encouraged. Uniform prior distributions are possible (e.g. by setting stan_glm's prior argument to NULL) but, unless the data is very strong, they are not recommended and are not non-informative, giving the same probability mass to implausible values as plausible ones.

More information on priors is available in the vignette Prior Distributions for rstanarm Models as well as the vignettes for the various modeling functions. For details on the priors used for multilevel models in particular see the vignette Estimating Generalized (Non-)Linear Models with Group-Specific Terms with rstanarm and also the Covariance matrices section lower down on this page.

Usage

normal(location = 0, scale = NULL, autoscale = FALSE)

student_t(df = 1, location = 0, scale = NULL, autoscale = FALSE)

cauchy(location = 0, scale = NULL, autoscale = FALSE)

hs(df = 1, global_df = 1, global_scale = 0.01, slab_df = 4, slab_scale = 2.5)

hs_plus(
  df1 = 1,
  df2 = 1,
  global_df = 1,
  global_scale = 0.01,
  slab_df = 4,
  slab_scale = 2.5
)

laplace(location = 0, scale = NULL, autoscale = FALSE)

lasso(df = 1, location = 0, scale = NULL, autoscale = FALSE)

product_normal(df = 2, location = 0, scale = 1)

exponential(rate = 1, autoscale = FALSE)

decov(regularization = 1, concentration = 1, shape = 1, scale = 1)

lkj(regularization = 1, scale = 10, df = 1, autoscale = TRUE)

dirichlet(concentration = 1)

R2(location = NULL, what = c("mode", "mean", "median", "log"))

default_prior_intercept(family)

default_prior_coef(family)

Arguments

Details

The details depend on the family of the prior being used:

Student t family

Family members:

• normal(location, scale)

• student_t(df, location, scale)

• cauchy(location, scale)

Each of these functions also takes an argument autoscale.

For the prior distribution for the intercept, location, scale, and df should be scalars. For the prior for the other coefficients they can either be vectors of length equal to the number of coefficients (not including the intercept), or they can be scalars, in which case they will be recycled to the appropriate length. As the degrees of freedom approaches infinity, the Student t distribution approaches the normal distribution and if the degrees of freedom are one, then the Student t distribution is the Cauchy distribution.

If scale is not specified it will default to 2.5, unless the probit link function is used, in which case these defaults are scaled by a factor of dnorm(0)/dlogis(0), which is roughly 1.6.

If the autoscale argument is TRUE, then the scales will be further adjusted as described above in the documentation of the autoscale argument in the Arguments section.

Hierarchical shrinkage family

Family members:

• hs(df, global_df, global_scale, slab_df, slab_scale)

• hs_plus(df1, df2, global_df, global_scale, slab_df, slab_scale)

The hierarchical shrinkage priors are normal with a mean of zero and a standard deviation that is also a random variable. The traditional hierarchical shrinkage prior utilizes a standard deviation that is distributed half Cauchy with a median of zero and a scale parameter that is also half Cauchy. This is called the "horseshoe prior". The hierarchical shrinkage (hs) prior in the rstanarm package instead utilizes a regularized horseshoe prior, as described by Piironen and Vehtari (2017), which recommends setting the global_scale argument equal to the ratio of the expected number of non-zero coefficients to the expected number of zero coefficients, divided by the square root of the number of observations.

The hierarhical shrinkpage plus (hs_plus) prior is similar except that the standard deviation that is distributed as the product of two independent half Cauchy parameters that are each scaled in a similar way to the hs prior.

The hierarchical shrinkage priors have very tall modes and very fat tails. Consequently, they tend to produce posterior distributions that are very concentrated near zero, unless the predictor has a strong influence on the outcome, in which case the prior has little influence. Hierarchical shrinkage priors often require you to increase the adapt_delta tuning parameter in order to diminish the number of divergent transitions. For more details on tuning parameters and divergent transitions see the Troubleshooting section of the How to Use the rstanarm Package vignette.

Laplace family

Family members:

• laplace(location, scale)

• lasso(df, location, scale)

Each of these functions also takes an argument autoscale.

The Laplace distribution is also known as the double-exponential distribution. It is a symmetric distribution with a sharp peak at its mean / median / mode and fairly long tails. This distribution can be motivated as a scale mixture of normal distributions and the remarks above about the normal distribution apply here as well.

The lasso approach to supervised learning can be expressed as finding the posterior mode when the likelihood is Gaussian and the priors on the coefficients have independent Laplace distributions. It is commonplace in supervised learning to choose the tuning parameter by cross-validation, whereas a more Bayesian approach would be to place a prior on “it”, or rather its reciprocal in our case (i.e. smaller values correspond to more shrinkage toward the prior location vector). We use a chi-square prior with degrees of freedom equal to that specified in the call to lasso or, by default, 1. The expectation of a chi-square random variable is equal to this degrees of freedom and the mode is equal to the degrees of freedom minus 2, if this difference is positive.

It is also common in supervised learning to standardize the predictors before training the model. We do not recommend doing so. Instead, it is better to specify autoscale = TRUE, which will adjust the scales of the priors according to the dispersion in the variables. See the documentation of the autoscale argument above and also the prior_summary page for more information.

Product-normal family

Family members:

• product_normal(df, location, scale)

The product-normal distribution is the product of at least two independent normal variates each with mean zero, shifted by the location parameter. It can be shown that the density of a product-normal variate is symmetric and infinite at location, so this prior resembles a “spike-and-slab” prior for sufficiently large values of the scale parameter. For better or for worse, this prior may be appropriate when it is strongly believed (by someone) that a regression coefficient “is” equal to the location, parameter even though no true Bayesian would specify such a prior.

Each element of df must be an integer of at least 2 because these “degrees of freedom” are interpreted as the number of normal variates being multiplied and then shifted by location to yield the regression coefficient. Higher degrees of freedom produce a sharper spike at location.

Each element of scale must be a non-negative real number that is interpreted as the standard deviation of the normal variates being multiplied and then shifted by location to yield the regression coefficient. In other words, the elements of scale may differ, but the k-th standard deviation is presumed to hold for all the normal deviates that are multiplied together and shifted by the k-th element of location to yield the k-th regression coefficient. The elements of scale are not the prior standard deviations of the regression coefficients. The prior variance of the regression coefficients is equal to the scale raised to the power of 2 times the corresponding element of df. Thus, larger values of scale put more prior volume on values of the regression coefficient that are far from zero.

Dirichlet family

Family members:

• dirichlet(concentration)

The Dirichlet distribution is a multivariate generalization of the beta distribution. It is perhaps the easiest prior distribution to specify because the concentration parameters can be interpreted as prior counts (although they need not be integers) of a multinomial random variable.

The Dirichlet distribution is used in stan_polr for an implicit prior on the cutpoints in an ordinal regression model. More specifically, the Dirichlet prior pertains to the prior probability of observing each category of the ordinal outcome when the predictors are at their sample means. Given these prior probabilities, it is straightforward to add them to form cumulative probabilities and then use an inverse CDF transformation of the cumulative probabilities to define the cutpoints.

If a scalar is passed to the concentration argument of the dirichlet function, then it is replicated to the appropriate length and the Dirichlet distribution is symmetric. If concentration is a vector and all elements are 1, then the Dirichlet distribution is jointly uniform. If all concentration parameters are equal but greater than 1 then the prior mode is that the categories are equiprobable, and the larger the value of the identical concentration parameters, the more sharply peaked the distribution is at the mode. The elements in concentration can also be given different values to represent that not all outcome categories are a priori equiprobable.

Covariance matrices

Family members:

• decov(regularization, concentration, shape, scale)

• lkj(regularization, scale, df)

(Also see vignette for stan_glmer, Estimating Generalized (Non-)Linear Models with Group-Specific Terms with rstanarm)

Covariance matrices are decomposed into correlation matrices and variances. The variances are in turn decomposed into the product of a simplex vector and the trace of the matrix. Finally, the trace is the product of the order of the matrix and the square of a scale parameter. This prior on a covariance matrix is represented by the decov function.

The prior for a correlation matrix is called LKJ whose density is proportional to the determinant of the correlation matrix raised to the power of a positive regularization parameter minus one. If regularization = 1 (the default), then this prior is jointly uniform over all correlation matrices of that size. If regularization > 1, then the identity matrix is the mode and in the unlikely case that regularization < 1, the identity matrix is the trough.

The trace of a covariance matrix is equal to the sum of the variances. We set the trace equal to the product of the order of the covariance matrix and the square of a positive scale parameter. The particular variances are set equal to the product of a simplex vector — which is non-negative and sums to 1 — and the scalar trace. In other words, each element of the simplex vector represents the proportion of the trace attributable to the corresponding variable.

A symmetric Dirichlet prior is used for the simplex vector, which has a single (positive) concentration parameter, which defaults to 1 and implies that the prior is jointly uniform over the space of simplex vectors of that size. If concentration > 1, then the prior mode corresponds to all variables having the same (proportion of total) variance, which can be used to ensure the the posterior variances are not zero. As the concentration parameter approaches infinity, this mode becomes more pronounced. In the unlikely case that concentration < 1, the variances are more polarized.

If all the variables were multiplied by a number, the trace of their covariance matrix would increase by that number squared. Thus, it is reasonable to use a scale-invariant prior distribution for the positive scale parameter, and in this case we utilize a Gamma distribution, whose shape and scale are both 1 by default, implying a unit-exponential distribution. Set the shape hyperparameter to some value greater than 1 to ensure that the posterior trace is not zero.

If regularization, concentration, shape and / or scale are positive scalars, then they are recycled to the appropriate length. Otherwise, each can be a positive vector of the appropriate length, but the appropriate length depends on the number of covariance matrices in the model and their sizes. A one-by-one covariance matrix is just a variance and thus does not have regularization or concentration parameters, but does have shape and scale parameters for the prior standard deviation of that variable.

Note that for stan_mvmer and stan_jm models an additional prior distribution is provided through the lkj function. This prior is in fact currently used as the default for those modelling functions (although decov is still available as an option if the user wishes to specify it through the prior_covariance argument). The lkj prior uses the same decomposition of the covariance matrices into correlation matrices and variances, however, the variances are not further decomposed into a simplex vector and the trace; instead the standard deviations (square root of the variances) for each of the group specific parameters are given a half Student t distribution with the scale and df parameters specified through the scale and df arguments to the lkj function. The scale parameter default is 10 which is then autoscaled, whilst the df parameter default is 1 (therefore equivalent to a half Cauchy prior distribution for the standard deviation of each group specific parameter). This prior generally leads to similar results as the decov prior, but it is also likely to be **less** diffuse compared with the decov prior; therefore it sometimes seems to lead to faster estimation times, hence why it has been chosen as the default prior for stan_mvmer and stan_jm where estimation times can be long.

R2 family

Family members:

• R2(location, what)

The stan_lm, stan_aov, and stan_polr functions allow the user to utilize a function called R2 to convey prior information about all the parameters. This prior hinges on prior beliefs about the location of R^2, the proportion of variance in the outcome attributable to the predictors, which has a Beta prior with first shape hyperparameter equal to half the number of predictors and second shape hyperparameter free. By specifying what to be the prior mode (the default), mean, median, or expected log of R^2, the second shape parameter for this Beta distribution is determined internally. If what = 'log', location should be a negative scalar; otherwise it should be a scalar on the (0,1) interval.

For example, if R^2 = 0.5, then the mode, mean, and median of the Beta distribution are all the same and thus the second shape parameter is also equal to half the number of predictors. The second shape parameter of the Beta distribution is actually the same as the shape parameter in the LKJ prior for a correlation matrix described in the previous subsection. Thus, the smaller is R^2, the larger is the shape parameter, the smaller are the prior correlations among the outcome and predictor variables, and the more concentrated near zero is the prior density for the regression coefficients. Hence, the prior on the coefficients is regularizing and should yield a posterior distribution with good out-of-sample predictions if the prior location of R^2 is specified in a reasonable fashion.

Value

A named list to be used internally by the rstanarm model fitting functions.

References

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013). Bayesian Data Analysis. Chapman & Hall/CRC Press, London, third edition. https://stat.columbia.edu/~gelman/book/

Gelman, A., Jakulin, A., Pittau, M. G., and Su, Y. (2008). A weakly informative default prior distribution for logistic and other regression models. Annals of Applied Statistics. 2(4), 1360–1383.

Piironen, J., and Vehtari, A. (2017). Sparsity information and regularization in the horseshoe and other shrinkage priors. https://arxiv.org/abs/1707.01694

Stan Development Team. Stan Modeling Language Users Guide and Reference Manual. https://mc-stan.org/users/documentation/.

See Also

The various vignettes for the rstanarm package also discuss and demonstrate the use of some of the supported prior distributions.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
fmla <- mpg ~ wt + qsec + drat + am

# Draw from prior predictive distribution (by setting prior_PD = TRUE)
prior_pred_fit <- stan_glm(fmla, data = mtcars, prior_PD = TRUE,
                           chains = 1, seed = 12345, iter = 250, # for speed only
                           prior = student_t(df = 4, 0, 2.5), 
                           prior_intercept = cauchy(0,10), 
                           prior_aux = exponential(1/2))
plot(prior_pred_fit, "hist")


# Can assign priors to names
N05 <- normal(0, 5)
fit <- stan_glm(fmla, data = mtcars, prior = N05, prior_intercept = N05)


# Visually compare normal, student_t, cauchy, laplace, and product_normal
compare_priors <- function(scale = 1, df_t = 2, xlim = c(-10, 10)) {
  dt_loc_scale <- function(x, df, location, scale) { 
    1/scale * dt((x - location)/scale, df)  
  }
  dlaplace <- function(x, location, scale) {
    0.5 / scale * exp(-abs(x - location) / scale)
  }
  dproduct_normal <- function(x, scale) {
    besselK(abs(x) / scale ^ 2, nu = 0) / (scale ^ 2 * pi)
  }
  stat_dist <- function(dist, ...) {
    ggplot2::stat_function(ggplot2::aes_(color = dist), ...)
  }
  ggplot2::ggplot(data.frame(x = xlim), ggplot2::aes(x)) + 
    stat_dist("normal", size = .75, fun = dnorm, 
              args = list(mean = 0, sd = scale)) +
    stat_dist("student_t", size = .75, fun = dt_loc_scale, 
              args = list(df = df_t, location = 0, scale = scale)) +
    stat_dist("cauchy", size = .75, linetype = 2, fun = dcauchy, 
              args = list(location = 0, scale = scale)) + 
    stat_dist("laplace", size = .75, linetype = 2, fun = dlaplace,
              args = list(location = 0, scale = scale)) +
    stat_dist("product_normal", size = .75, linetype = 2, fun = dproduct_normal,
              args = list(scale = 1))            
}
# Cauchy has fattest tails, followed by student_t, laplace, and normal
compare_priors()

# The student_t with df = 1 is the same as the cauchy
compare_priors(df_t = 1) 

# Even a scale of 5 is somewhat large. It gives plausibility to rather 
# extreme values
compare_priors(scale = 5, xlim = c(-20,20)) 

# If you use a prior like normal(0, 1000) to be "non-informative" you are 
# actually saying that a coefficient value of e.g. -500 is quite plausible
compare_priors(scale = 1000, xlim = c(-1000,1000))
}

Graphical checks of the estimated survival function

Description

This function plots the estimated marginal survival function based on draws from the posterior predictive distribution of the fitted joint model, and then overlays the Kaplan-Meier curve based on the observed data.

Usage

ps_check(
  object,
  check = "survival",
  limits = c("ci", "none"),
  draws = NULL,
  seed = NULL,
  xlab = NULL,
  ylab = NULL,
  ci_geom_args = NULL,
  ...
)

Arguments

Value

A ggplot object that can be further customized using the ggplot2 package.

See Also

posterior_survfit for the estimated marginal or subject-specific survival function based on draws of the model parameters from the posterior distribution, posterior_predict for drawing from the posterior predictive distribution for the longitudinal submodel, and pp_check for graphical checks of the longitudinal submodel.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {

if (!exists("example_jm")) example(example_jm)
# Compare estimated survival function to Kaplan-Meier curve
ps <- ps_check(example_jm)
ps + 
 ggplot2::scale_color_manual(values = c("red", "black")) + # change colors
 ggplot2::scale_size_manual(values = c(0.5, 3)) + # change line sizes 
 ggplot2::scale_fill_manual(values = c(NA, NA)) # remove fill

}

The QR argument

Description

Details about the QR argument to rstanarm's modeling functions.

Details

The QR argument is a logical scalar defaulting to FALSE, but if TRUE applies a scaled qr decomposition to the design matrix, X = Q^\ast R^\ast. If autoscale = TRUE (the default) in the call to the function passed to the prior argument, then Q^\ast = Q \sqrt{n-1} and R^\ast = \frac{1}{\sqrt{n-1}} R. When autoscale = FALSE, R is scaled such that the lower-right element of R^\ast is 1.

The coefficients relative to Q^\ast are obtained and then premultiplied by the inverse of R^{\ast} to obtain coefficients relative to the original predictors, X. Thus, when autoscale = FALSE, the coefficient on the last column of X is the same as the coefficient on the last column of Q^\ast.

These transformations do not change the likelihood of the data but are recommended for computational reasons when there are multiple predictors. Importantly, while the columns of X are almost generally correlated, the columns of Q^\ast are uncorrelated by design, which often makes sampling from the posterior easier. However, because when QR is TRUE the prior argument applies to the coefficients relative to Q^\ast (and those are not very interpretable), setting QR=TRUE is only recommended if you do not have an informative prior for the regression coefficients or if the only informative prior is on the last regression coefficient (in which case you should set autoscale = FALSE when specifying such priors).

For more details see the Stan case study The QR Decomposition For Regression Models at https://mc-stan.org/users/documentation/case-studies/qr_regression.html.

References

Stan Development Team. Stan Modeling Language Users Guide and Reference Manual. https://mc-stan.org/users/documentation/.

Objects exported from other packages

Description

These objects are imported from other packages. Follow the links below to see their documentation.

survival

Surv

Datasets for rstanarm examples

Description

Small datasets for use in rstanarm examples and vignettes.

Format

bball1970

Data on hits and at-bats from the 1970 Major League Baseball season for 18 players.

Source: Efron and Morris (1975).

18 obs. of 5 variables

• Player Player's last name

• Hits Number of hits in the first 45 at-bats of the season

• AB Number of at-bats (45 for all players)

• RemainingAB Number of remaining at-bats (different for most players)

• RemainingHits Number of remaining hits

bball2006

Hits and at-bats for the entire 2006 American League season of Major League Baseball.

Source: Carpenter (2009)

302 obs. of 2 variables

• y Number of hits

• K Number of at-bats

kidiq

Data from a survey of adult American women and their children (a subsample from the National Longitudinal Survey of Youth).

Source: Gelman and Hill (2007)

434 obs. of 4 variables

• kid_score Child's IQ score

• mom_hs Indicator for whether the mother has a high school degree

• mom_iq Mother's IQ score

• mom_age Mother's age

mortality

Surgical mortality rates in 12 hospitals performing cardiac surgery in babies.

Source: Spiegelhalter et al. (1996).

12 obs. of 2 variables

• y Number of deaths

• K Number of surgeries

pbcLong,pbcSurv

Longitudinal biomarker and time-to-event survival data for 40 patients with primary biliary cirrhosis who participated in a randomised placebo controlled trial of D-penicillamine conducted at the Mayo Clinic between 1974 and 1984.

Source: Therneau and Grambsch (2000)

304 obs. of 8 variables (pbcLong) and 40 obs. of 7 variables (pbcSurv)

• age in years

• albumin serum albumin (g/dl)

• logBili logarithm of serum bilirubin

• death indicator of death at endpoint

• futimeYears time (in years) between baseline and the earliest of death, transplantion or censoring

• id numeric ID unique to each individual

• platelet platelet count

• sex gender (m = male, f = female)

• status status at endpoint (0 = censored, 1 = transplant, 2 = dead)

• trt binary treatment code (0 = placebo, 1 = D-penicillamine)

• year time (in years) of the longitudinal measurements, taken as time since baseline)

radon

Data on radon levels in houses in the state of Minnesota.

Source: Gelman and Hill (2007)

919 obs. of 4 variables

• log_radon Radon measurement from the house (log scale)

• log_uranium Uranium level in the county (log scale)

• floor Indicator for radon measurement made on the first floor of the house (0 = basement, 1 = first floor)

• county County name (factor)

roaches

Data on the efficacy of a pest management system at reducing the number of roaches in urban apartments.

Source: Gelman and Hill (2007)

262 obs. of 6 variables

• y Number of roaches caught

• roach1 Pretreatment number of roaches

• treatment Treatment indicator

• senior Indicator for only elderly residents in building

• exposure2 Number of days for which the roach traps were used

tumors

Tarone (1982) provides a data set of tumor incidence in historical control groups of rats; specifically endometrial stromal polyps in female lab rats of type F344.

Source: Gelman and Hill (2007)

71 obs. of 2 variables

• y Number of rats with tumors

• K Number of rats

wells

A survey of 3200 residents in a small area of Bangladesh suffering from arsenic contamination of groundwater. Respondents with elevated arsenic levels in their wells had been encouraged to switch their water source to a safe public or private well in the nearby area and the survey was conducted several years later to learn which of the affected residents had switched wells.

Souce: Gelman and Hill (2007)

3020 obs. of 5 variables

• switch Indicator for well-switching

• arsenic Arsenic level in respondent's well

• dist Distance (meters) from the respondent's house to the nearest well with safe drinking water.

• assoc Indicator for member(s) of household participate in community organizations

• educ Years of education (head of household)

References

Carpenter, B. (2009) Bayesian estimators for the beta-binomial model of batting ability. https://web.archive.org/web/20220618114439/https://lingpipe-blog.com/2009/09/23/

Efron, B. and Morris, C. (1975) Data analysis using Stein's estimator and its generalizations. Journal of the American Statistical Association 70(350), 311–319.

Gelman, A. and Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press, Cambridge, UK. https://stat.columbia.edu/~gelman/arm/

Spiegelhalter, D., Thomas, A., Best, N., & Gilks, W. (1996) BUGS 0.5 Examples. MRC Biostatistics Unit, Institute of Public health, Cambridge, UK.

Tarone, R. E. (1982) The use of historical control information in testing for a trend in proportions. Biometrics 38(1):215–220.

Therneau, T. and Grambsch, P. (2000) Modeling Survival Data: Extending the Cox Model. Springer-Verlag, New York, US.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
# Using 'kidiq' dataset 
fit <- stan_lm(kid_score ~ mom_hs * mom_iq, data = kidiq, 
               prior = R2(location = 0.30, what = "mean"),
               # the next line is only to make the example go fast enough
               chains = 1, iter = 500, seed = 12345)
pp_check(fit, nreps = 20)

bayesplot::color_scheme_set("brightblue")
pp_check(fit, plotfun = "stat_grouped", stat = "median", 
         group = factor(kidiq$mom_hs, labels = c("No HS", "HS")))

}

Deprecated functions

Description

These functions are deprecated and will be removed in a future release. The Arguments section below provides details on how the functionality obtained via each of the arguments has been replaced.

Usage

prior_options(
  prior_scale_for_dispersion = 5,
  min_prior_scale = 1e-12,
  scaled = TRUE
)

Arguments

Extract standard errors

Description

Generic function for extracting standard errors from fitted models.

Usage

se(object, ...)

Arguments

Value

Standard errors of model parameters.

See Also

se.stanreg

Bayesian regularized linear models via Stan

Description

Bayesian inference for linear modeling with regularizing priors on the model parameters that are driven by prior beliefs about R^2, the proportion of variance in the outcome attributable to the predictors. See priors for an explanation of this critical point. stan_glm with family="gaussian" also estimates a linear model with normally-distributed errors and allows for various other priors on the coefficients.

Usage

stan_aov(
  formula,
  data,
  projections = FALSE,
  contrasts = NULL,
  ...,
  prior = R2(stop("'location' must be specified")),
  prior_PD = FALSE,
  algorithm = c("sampling", "meanfield", "fullrank"),
  adapt_delta = NULL
)

stan_lm(
  formula,
  data,
  subset,
  weights,
  na.action,
  model = TRUE,
  x = FALSE,
  y = FALSE,
  singular.ok = TRUE,
  contrasts = NULL,
  offset,
  ...,
  prior = R2(stop("'location' must be specified")),
  prior_intercept = NULL,
  prior_PD = FALSE,
  algorithm = c("sampling", "meanfield", "fullrank"),
  adapt_delta = NULL
)

stan_lm.wfit(
  x,
  y,
  w,
  offset = NULL,
  singular.ok = TRUE,
  ...,
  prior = R2(stop("'location' must be specified")),
  prior_intercept = NULL,
  prior_PD = FALSE,
  algorithm = c("sampling", "meanfield", "fullrank"),
  adapt_delta = NULL
)

stan_lm.fit(
  x,
  y,
  offset = NULL,
  singular.ok = TRUE,
  ...,
  prior = R2(stop("'location' must be specified")),
  prior_intercept = NULL,
  prior_PD = FALSE,
  algorithm = c("sampling", "meanfield", "fullrank"),
  adapt_delta = NULL
)

Arguments

Details

The stan_lm function is similar in syntax to the lm function but rather than choosing the parameters to minimize the sum of squared residuals, samples from the posterior distribution are drawn using MCMC (if algorithm is "sampling"). The stan_lm function has a formula-based interface and would usually be called by users but the stan_lm.fit and stan_lm.wfit functions might be called by other functions that parse the data themselves and are analogous to lm.fit and lm.wfit respectively.

In addition to estimating sigma — the standard deviation of the normally-distributed errors — this model estimates a positive parameter called log-fit_ratio. If it is positive, the marginal posterior variance of the outcome will exceed the sample variance of the outcome by a multiplicative factor equal to the square of fit_ratio. Conversely if log-fit_ratio is negative, then the model underfits. Given the regularizing nature of the priors, a slight underfit is good.

Finally, the posterior predictive distribution is generated with the predictors fixed at their sample means. This quantity is useful for checking convergence because it is reasonably normally distributed and a function of all the parameters in the model.

The stan_aov function is similar to aov, but does a Bayesian analysis of variance that is basically equivalent to stan_lm with dummy variables. stan_aov has a somewhat customized print method that prints an ANOVA-like table in addition to the output printed for stan_lm models.

Value

A stanreg object is returned for stan_lm, stan_aov.

A stanfit object (or a slightly modified stanfit object) is returned if stan_lm.fit or stan_lm.wfit is called directly.

References

Lewandowski, D., Kurowicka D., and Joe, H. (2009). Generating random correlation matrices based on vines and extended onion method. Journal of Multivariate Analysis. 100(9), 1989–2001.

See Also

The vignettes for stan_lm and stan_aov, which have more thorough descriptions and examples. https://mc-stan.org/rstanarm/articles/

Also see stan_glm, which — if family = gaussian(link="identity") — also estimates a linear model with normally-distributed errors but specifies different priors.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {

op <- options(contrasts = c("contr.helmert", "contr.poly"))
fit_aov <- stan_aov(yield ~ block + N*P*K, data = npk,
         prior = R2(0.5), seed = 12345)
options(op)
print(fit_aov)

}
if (.Platform$OS.type != "windows" || .Platform$r_arch !="i386") {
(fit <- stan_lm(mpg ~ wt + qsec + am, data = mtcars, prior = R2(0.75), 
                # the next line is only to make the example go fast enough
                chains = 1, iter = 300, seed = 12345, refresh = 0))
plot(fit, "hist", pars = c("wt", "am", "qsec", "sigma"), 
     transformations = list(sigma = "log"))
}

Bayesian beta regression models via Stan

Description

Beta regression modeling with optional prior distributions for the coefficients, intercept, and auxiliary parameter phi (if applicable).

Usage

stan_betareg(
  formula,
  data,
  subset,
  na.action,
  weights,
  offset,
  link = c("logit", "probit", "cloglog", "cauchit", "log", "loglog"),
  link.phi = NULL,
  model = TRUE,
  y = TRUE,
  x = FALSE,
  ...,
  prior = normal(autoscale = TRUE),
  prior_intercept = normal(autoscale = TRUE),
  prior_z = normal(autoscale = TRUE),
  prior_intercept_z = normal(autoscale = TRUE),
  prior_phi = exponential(autoscale = TRUE),
  prior_PD = FALSE,
  algorithm = c("sampling", "optimizing", "meanfield", "fullrank"),
  adapt_delta = NULL,
  QR = FALSE
)

stan_betareg.fit(
  x,
  y,
  z = NULL,
  weights = rep(1, NROW(x)),
  offset = rep(0, NROW(x)),
  link = c("logit", "probit", "cloglog", "cauchit", "log", "loglog"),
  link.phi = NULL,
  ...,
  prior = normal(autoscale = TRUE),
  prior_intercept = normal(autoscale = TRUE),
  prior_z = normal(autoscale = TRUE),
  prior_intercept_z = normal(autoscale = TRUE),
  prior_phi = exponential(autoscale = TRUE),
  prior_PD = FALSE,
  algorithm = c("sampling", "optimizing", "meanfield", "fullrank"),
  adapt_delta = NULL,
  QR = FALSE
)

Arguments

Details

The stan_betareg function is similar in syntax to betareg but rather than performing maximum likelihood estimation, full Bayesian estimation is performed (if algorithm is "sampling") via MCMC. The Bayesian model adds priors (independent by default) on the coefficients of the beta regression model. The stan_betareg function calls the workhorse stan_betareg.fit function, but it is also possible to call the latter directly.

Value

A stanreg object is returned for stan_betareg.

A stanfit object (or a slightly modified stanfit object) is returned if stan_betareg.fit is called directly.

References

Ferrari, SLP and Cribari-Neto, F (2004). Beta regression for modeling rates and proportions. Journal of Applied Statistics. 31(7), 799–815.

See Also

stanreg-methods and betareg.

The vignette for stan_betareg. https://mc-stan.org/rstanarm/articles/

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
### Simulated data
N <- 200
x <- rnorm(N, 2, 1)
z <- rnorm(N, 2, 1)
mu <- binomial(link = "logit")$linkinv(1 + 0.2*x)
phi <- exp(1.5 + 0.4*z)
y <- rbeta(N, mu * phi, (1 - mu) * phi)
hist(y, col = "dark grey", border = FALSE, xlim = c(0,1))
fake_dat <- data.frame(y, x, z)

fit <- stan_betareg(
  y ~ x | z, data = fake_dat, 
  link = "logit", 
  link.phi = "log", 
  algorithm = "optimizing" # just for speed of example
 ) 
print(fit, digits = 2)
}

Bayesian regularized linear but big models via Stan

Description

This is the same model as with stan_lm but it utilizes the output from biglm in the biglm package in order to proceed when the data is too large to fit in memory.

Usage

stan_biglm(
  biglm,
  xbar,
  ybar,
  s_y,
  ...,
  prior = R2(stop("'location' must be specified")),
  prior_intercept = NULL,
  prior_PD = FALSE,
  algorithm = c("sampling", "meanfield", "fullrank"),
  adapt_delta = NULL
)

stan_biglm.fit(
  b,
  R,
  SSR,
  N,
  xbar,
  ybar,
  s_y,
  has_intercept = TRUE,
  ...,
  prior = R2(stop("'location' must be specified")),
  prior_intercept = NULL,
  prior_PD = FALSE,
  algorithm = c("sampling", "meanfield", "fullrank", "optimizing"),
  adapt_delta = NULL,
  importance_resampling = TRUE,
  keep_every = 1
)

Arguments

Details

The stan_biglm function is intended to be used in the same circumstances as the biglm function in the biglm package but with an informative prior on the R^2 of the regression. Like biglm, the memory required to estimate the model depends largely on the number of predictors rather than the number of observations. However, stan_biglm and stan_biglm.fit have additional required arguments that are not necessary in biglm, namely xbar, ybar, and s_y. If any observations have any missing values on any of the predictors or the outcome, such observations do not contribute to these statistics.

Value

The output of both stan_biglm and stan_biglm.fit is an object of stanfit-class rather than stanreg-objects, which is more limited and less convenient but necessitated by the fact that stan_biglm does not bring the full design matrix into memory. Without the full design matrix,some of the elements of a stanreg-objects object cannot be calculated, such as residuals. Thus, the functions in the rstanarm package that input stanreg-objects, such as posterior_predict cannot be used.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
# create inputs
ols <- lm(mpg ~ wt + qsec + am, data = mtcars, # all row are complete so ...
          na.action = na.exclude)              # not necessary in this case
b <- coef(ols)[-1]
R <- qr.R(ols$qr)[-1,-1]
SSR <- crossprod(ols$residuals)[1]
not_NA <- !is.na(fitted(ols))
N <- sum(not_NA)
xbar <- colMeans(mtcars[not_NA,c("wt", "qsec", "am")])
y <- mtcars$mpg[not_NA]
ybar <- mean(y)
s_y <- sd(y)
post <- stan_biglm.fit(b, R, SSR, N, xbar, ybar, s_y, prior = R2(.75),
                       # the next line is only to make the example go fast
                       chains = 1, iter = 500, seed = 12345)
cbind(lm = b, stan_lm = rstan::get_posterior_mean(post)[13:15,]) # shrunk
}

Conditional logistic (clogit) regression models via Stan

Description

A model for case-control studies with optional prior distributions for the coefficients, intercept, and auxiliary parameters.

Usage

stan_clogit(
  formula,
  data,
  subset,
  na.action = NULL,
  contrasts = NULL,
  ...,
  strata,
  prior = normal(autoscale = TRUE),
  prior_covariance = decov(),
  prior_PD = FALSE,
  algorithm = c("sampling", "optimizing", "meanfield", "fullrank"),
  adapt_delta = NULL,
  QR = FALSE,
  sparse = FALSE
)

Arguments

Details

The stan_clogit function is mostly similar in syntax to clogit but rather than performing maximum likelihood estimation of generalized linear models, full Bayesian estimation is performed (if algorithm is "sampling") via MCMC. The Bayesian model adds priors (independent by default) on the coefficients of the GLM.

The data.frame passed to the data argument must be sorted by the variable passed to the strata argument.

The formula may have group-specific terms like in stan_glmer but should not allow the intercept to vary by the stratifying variable, since there is no information in the data with which to estimate such deviations in the intercept.

Value

A stanreg object is returned for stan_clogit.

See Also

stanreg-methods and clogit.

The vignette for Bernoulli and binomial models, which has more details on using stan_clogit. https://mc-stan.org/rstanarm/articles/

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
dat <- infert[order(infert$stratum), ] # order by strata
post <- stan_clogit(case ~ spontaneous + induced + (1 | education), 
                    strata = stratum,
                    data = dat,
                    subset = parity <= 2,
                    QR = TRUE,
                    chains = 2, iter = 500) # for speed only

nd <- dat[dat$parity > 2, c("case", "spontaneous", "induced", "education", "stratum")]
# next line would fail without case and stratum variables                                 
pr <- posterior_epred(post, newdata = nd) # get predicted probabilities

# not a random variable b/c probabilities add to 1 within strata
all.equal(rep(sum(nd$case), nrow(pr)), rowSums(pr)) 
}

Bayesian generalized linear additive models with optional group-specific terms via Stan

Description

Bayesian inference for GAMMs with flexible priors.

Usage

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
)

Arguments

Details

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.

Value

A stanreg object is returned for stan_gamm4.

plot_nonlinear returns a ggplot object.

References

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

See Also

stanreg-methods and gamm4.

The vignette for stan_glmer, which also discusses stan_gamm4. https://mc-stan.org/rstanarm/articles/

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
# from example(gamm4, package = "gamm4"), prefixing gamm4() call with stan_

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)

}

Bayesian generalized linear models via Stan

Description

Generalized linear modeling with optional prior distributions for the coefficients, intercept, and auxiliary parameters.

Usage

stan_glm(
  formula,
  family = gaussian(),
  data,
  weights,
  subset,
  na.action = NULL,
  offset = NULL,
  model = TRUE,
  x = FALSE,
  y = TRUE,
  contrasts = NULL,
  ...,
  prior = default_prior_coef(family),
  prior_intercept = default_prior_intercept(family),
  prior_aux = exponential(autoscale = TRUE),
  prior_PD = FALSE,
  algorithm = c("sampling", "optimizing", "meanfield", "fullrank"),
  mean_PPD = algorithm != "optimizing" && !prior_PD,
  adapt_delta = NULL,
  QR = FALSE,
  sparse = FALSE
)

stan_glm.nb(
  formula,
  data,
  weights,
  subset,
  na.action = NULL,
  offset = NULL,
  model = TRUE,
  x = FALSE,
  y = TRUE,
  contrasts = NULL,
  link = "log",
  ...,
  prior = default_prior_coef(family),
  prior_intercept = default_prior_intercept(family),
  prior_aux = exponential(autoscale = TRUE),
  prior_PD = FALSE,
  algorithm = c("sampling", "optimizing", "meanfield", "fullrank"),
  mean_PPD = algorithm != "optimizing",
  adapt_delta = NULL,
  QR = FALSE
)

stan_glm.fit(
  x,
  y,
  weights = rep(1, NROW(y)),
  offset = rep(0, NROW(y)),
  family = gaussian(),
  ...,
  prior = default_prior_coef(family),
  prior_intercept = default_prior_intercept(family),
  prior_aux = exponential(autoscale = TRUE),
  prior_smooth = exponential(autoscale = FALSE),
  prior_ops = NULL,
  group = list(),
  prior_PD = FALSE,
  algorithm = c("sampling", "optimizing", "meanfield", "fullrank"),
  mean_PPD = algorithm != "optimizing" && !prior_PD,
  adapt_delta = NULL,
  QR = FALSE,
  sparse = FALSE,
  importance_resampling = algorithm != "sampling",
  keep_every = algorithm != "sampling"
)

Arguments

Details

The stan_glm function is similar in syntax to glm but rather than performing maximum likelihood estimation of generalized linear models, full Bayesian estimation is performed (if algorithm is "sampling") via MCMC. The Bayesian model adds priors (independent by default) on the coefficients of the GLM. The stan_glm function calls the workhorse stan_glm.fit function, but it is also possible to call the latter directly.

The stan_glm.nb function, which takes the extra argument link, is a wrapper for stan_glm with family = neg_binomial_2(link).

Value

A stanreg object is returned for stan_glm, stan_glm.nb.

A stanfit object (or a slightly modified stanfit object) is returned if stan_glm.fit is called directly.

References

Gelman, A. and Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press, Cambridge, UK. (Ch. 3-6)

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

See Also

stanreg-methods and glm.

The various vignettes for stan_glm at https://mc-stan.org/rstanarm/articles/.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch != "i386") {
### Linear regression
mtcars$mpg10 <- mtcars$mpg / 10
fit <- stan_glm(
  mpg10 ~ wt + cyl + am,            
  data = mtcars, 
  QR = TRUE,
  # for speed of example only (default is "sampling")
  algorithm = "fullrank",
  refresh = 0 
 ) 
                
plot(fit, prob = 0.5)
plot(fit, prob = 0.5, pars = "beta")
plot(fit, "hist", pars = "sigma")

### Logistic regression
head(wells)
wells$dist100 <- wells$dist / 100
fit2 <- stan_glm(
  switch ~ dist100 + arsenic, 
  data = wells, 
  family = binomial(link = "logit"), 
  prior_intercept = normal(0, 10),
  QR = TRUE,
  refresh = 0,
  # for speed of example only
  chains = 2, iter = 200 
)
print(fit2)
prior_summary(fit2)

# ?bayesplot::mcmc_areas
plot(fit2, plotfun = "areas", prob = 0.9,
     pars = c("(Intercept)", "arsenic"))

# ?bayesplot::ppc_error_binned
pp_check(fit2, plotfun = "error_binned") 


### Poisson regression (example from help("glm")) 
count_data <- data.frame(
 counts = c(18,17,15,20,10,20,25,13,12),
 outcome = gl(3,1,9),
 treatment = gl(3,3)
)
fit3 <- stan_glm(
  counts ~ outcome + treatment, 
  data = count_data, 
  family = poisson(link="log"),
  prior = normal(0, 2),
  refresh = 0,
  # for speed of example only
  chains = 2, iter = 250 
) 
print(fit3)

bayesplot::color_scheme_set("viridis")
plot(fit3)
plot(fit3, regex_pars = c("outcome", "treatment"))
plot(fit3, plotfun = "combo", regex_pars = "treatment") # ?bayesplot::mcmc_combo
posterior_vs_prior(fit3, regex_pars = c("outcome", "treatment"))

### Gamma regression (example from help("glm"))
clotting <- data.frame(log_u = log(c(5,10,15,20,30,40,60,80,100)),
                       lot1 = c(118,58,42,35,27,25,21,19,18),
                       lot2 = c(69,35,26,21,18,16,13,12,12))
fit4 <- stan_glm(
  lot1 ~ log_u, 
  data = clotting, 
  family = Gamma(link="log"),
  iter = 500, # for speed of example only
  refresh = 0
 ) 
print(fit4, digits = 2)

fit5 <- update(fit4, formula = lot2 ~ log_u)

# ?bayesplot::ppc_dens_overlay
bayesplot::bayesplot_grid(
  pp_check(fit4, seed = 123), 
  pp_check(fit5, seed = 123),
  titles = c("lot1", "lot2")
) 


### Negative binomial regression
fit6 <- stan_glm.nb(
  Days ~ Sex/(Age + Eth*Lrn), 
  data = MASS::quine, 
  link = "log", 
  prior_aux = exponential(1.5, autoscale=TRUE),
  chains = 2, iter = 200, # for speed of example only
  refresh = 0
) 

prior_summary(fit6)
bayesplot::color_scheme_set("brightblue")
plot(fit6)
pp_check(fit6, plotfun = "hist", nreps = 5) # ?bayesplot::ppc_hist

# 80% interval of estimated reciprocal_dispersion parameter
posterior_interval(fit6, pars = "reciprocal_dispersion", prob = 0.8)
plot(fit6, "areas", pars = "reciprocal_dispersion", prob = 0.8)

}

Bayesian generalized linear models with group-specific terms via Stan

Description

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

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).

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).

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

See Also

stanreg-methods and glmer.

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

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)
}

Bayesian joint longitudinal and time-to-event models via Stan

Description

Fits a shared parameter joint model for longitudinal and time-to-event (e.g. survival) data under a Bayesian framework using Stan.

Usage

stan_jm(
  formulaLong,
  dataLong,
  formulaEvent,
  dataEvent,
  time_var,
  id_var,
  family = gaussian,
  assoc = "etavalue",
  lag_assoc = 0,
  grp_assoc,
  scale_assoc = NULL,
  epsilon = 1e-05,
  basehaz = c("bs", "weibull", "piecewise"),
  basehaz_ops,
  qnodes = 15,
  init = "prefit",
  weights,
  priorLong = normal(autoscale = TRUE),
  priorLong_intercept = normal(autoscale = TRUE),
  priorLong_aux = cauchy(0, 5, autoscale = TRUE),
  priorEvent = normal(autoscale = TRUE),
  priorEvent_intercept = normal(autoscale = TRUE),
  priorEvent_aux = cauchy(autoscale = TRUE),
  priorEvent_assoc = normal(autoscale = TRUE),
  prior_covariance = lkj(autoscale = TRUE),
  prior_PD = FALSE,
  algorithm = c("sampling", "meanfield", "fullrank"),
  adapt_delta = NULL,
  max_treedepth = 10L,
  QR = FALSE,
  sparse = FALSE,
  ...
)

Arguments

Details

The stan_jm function can be used to fit a joint model (also known as a shared parameter model) for longitudinal and time-to-event data under a Bayesian framework. The underlying estimation is carried out using the Bayesian C++ package Stan (https://mc-stan.org/).

The joint model may be univariate (with only one longitudinal submodel) or multivariate (with more than one longitudinal submodel). For the longitudinal submodel a (possibly multivariate) generalised linear mixed model is assumed with any of the family choices allowed by glmer. If a multivariate joint model is specified (by providing a list of formulas in the formulaLong argument), then the multivariate longitudinal submodel consists of a multivariate generalized linear model (GLM) with group-specific terms that are assumed to be correlated across the different GLM submodels. That is, within a grouping factor (for example, patient ID) the group-specific terms are assumed to be correlated across the different GLM submodels. It is possible to specify a different outcome type (for example a different family and/or link function) for each of the GLM submodels, by providing a list of family objects in the family argument. Multi-level clustered data are allowed, and that additional clustering can occur at a level higher than the individual-level (e.g. patients clustered within clinics), or at a level lower than the individual-level (e.g. tumor lesions clustered within patients). If the clustering occurs at a level lower than the individual, then the user needs to indicate how the lower level clusters should be handled when forming the association structure between the longitudinal and event submodels (see the grp_assoc argument described above).

For the event submodel a parametric proportional hazards model is assumed. The baseline hazard can be estimated using either a cubic B-splines approximation (basehaz = "bs", the default), a Weibull distribution (basehaz = "weibull"), or a piecewise constant baseline hazard (basehaz = "piecewise"). If the B-spline or piecewise constant baseline hazards are used, then the degrees of freedom or the internal knot locations can be (optionally) specified. If the degrees of freedom are specified (through the df argument) then the knot locations are automatically generated based on the distribution of the observed event times (not including censoring times). Otherwise internal knot locations can be specified directly through the knots argument. If neither df or knots is specified, then the default is to set df equal to 6. It is not possible to specify both df and knots.

Time-varying covariates are allowed in both the longitudinal and event submodels. These should be specified in the data in the same way as they normally would when fitting a separate longitudinal model using lmer or a separate time-to-event model using coxph. These time-varying covariates should be exogenous in nature, otherwise they would perhaps be better specified as an additional outcome (i.e. by including them as an additional longitudinal outcome in the joint model).

Bayesian estimation of the joint model is performed via MCMC. The Bayesian model includes independent priors on the regression coefficients for both the longitudinal and event submodels, including the association parameter(s) (in much the same way as the regression parameters in 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 distributions that are available.

Gauss-Kronrod quadrature is used to numerically evaluate the integral over the cumulative hazard in the likelihood function for the event submodel. The accuracy of the numerical approximation can be controlled using the number of quadrature nodes, specified through the qnodes argument. Using a higher number of quadrature nodes will result in a more accurate approximation.

Association structures

The association structure for the joint model can be based on any of the following parameterisations:

• current value of the linear predictor in the longitudinal submodel ("etavalue")

• first derivative (slope) of the linear predictor in the longitudinal submodel ("etaslope")

• the area under the curve of the linear predictor in the longitudinal submodel ("etaauc")

• current expected value of the longitudinal submodel ("muvalue")

• the area under the curve of the expected value from the longitudinal submodel ("muauc")

• shared individual-level random effects ("shared_b")

• shared individual-level random effects which also incorporate the corresponding fixed effect as well as any corresponding random effects for clustering levels higher than the individual) ("shared_coef")

• interactions between association terms and observed data/covariates ("etavalue_data", "etaslope_data", "muvalue_data", "muslope_data"). These are described further below.

• interactions between association terms corresponding to different longitudinal outcomes in a multivariate joint model ("etavalue_etavalue(#)", "etavalue_muvalue(#)", "muvalue_etavalue(#)", "muvalue_muvalue(#)"). These are described further below.

• no association structure (equivalent to fitting separate longitudinal and event models) ("null" or NULL)

More than one association structure can be specified, however, not all possible combinations are allowed. Note that for the lagged association structures baseline values (time = 0) are used for the instances where the time lag results in a time prior to baseline. When using the "etaauc" or "muauc" association structures, the area under the curve is evaluated using Gauss-Kronrod quadrature with 15 quadrature nodes. By default, "shared_b" and "shared_coef" contribute all random effects to the association structure; however, a subset of the random effects can be chosen by specifying their indices between parentheses as a suffix, for example, "shared_b(1)" or "shared_b(1:3)" or "shared_b(1,2,4)", and so on.

In addition, several association terms ("etavalue", "etaslope", "muvalue", "muslope") can be interacted with observed data/covariates. To do this, use the association term's main handle plus a suffix of "_data" then followed by the model matrix formula in parentheses. For example if we had a variable in our dataset for gender named sex then we might want to obtain different estimates for the association between the current slope of the marker and the risk of the event for each gender. To do this we would specify assoc = c("etaslope", "etaslope_data(~ sex)").

It is also possible, when fitting a multivariate joint model, to include interaction terms between the association terms themselves (this only applies for interacting "etavalue" or "muvalue"). For example, if we had a joint model with two longitudinal markers, we could specify assoc = list(c("etavalue", "etavalue_etavalue(2)"), "etavalue"). The first element of list says we want to use the value of the linear predictor for the first marker, as well as it's interaction with the value of the linear predictor for the second marker. The second element of the list says we want to also include the expected value of the second marker (i.e. as a "main effect"). Therefore, the linear predictor for the event submodel would include the "main effects" for each marker as well as their interaction.

There are additional examples in the Examples section below.

Value

A stanjm object is returned.

See Also

stanreg-objects, stanmvreg-methods, print.stanmvreg, summary.stanmvreg, posterior_traj, posterior_survfit, posterior_predict, posterior_interval, pp_check, ps_check, stan_mvmer.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch !="i386") {


#####
# Univariate joint model, with association structure based on the 
# current value of the linear predictor
f1 <- stan_jm(formulaLong = logBili ~ year + (1 | id), 
              dataLong = pbcLong,
              formulaEvent = Surv(futimeYears, death) ~ sex + trt, 
              dataEvent = pbcSurv,
              time_var = "year",
              # this next line is only to keep the example small in size!
              chains = 1, cores = 1, seed = 12345, iter = 1000)
print(f1) 
summary(f1) 
        
#####
# Univariate joint model, with association structure based on the 
# current value and slope of the linear predictor
f2 <- stan_jm(formulaLong = logBili ~ year + (year | id), 
              dataLong = pbcLong,
              formulaEvent = Surv(futimeYears, death) ~ sex + trt, 
              dataEvent = pbcSurv,
              assoc = c("etavalue", "etaslope"),
              time_var = "year",
              chains = 1, cores = 1, seed = 12345, iter = 1000)
print(f2)  

#####
# Univariate joint model, with association structure based on the 
# lagged value of the linear predictor, where the lag is 2 time 
# units (i.e. 2 years in this example)
f3 <- stan_jm(formulaLong = logBili ~ year + (1 | id), 
              dataLong = pbcLong,
              formulaEvent = Surv(futimeYears, death) ~ sex + trt, 
              dataEvent = pbcSurv,
              time_var = "year",
              assoc = "etavalue", lag_assoc = 2,
              chains = 1, cores = 1, seed = 12345, iter = 1000)
print(f3) 

#####
# Univariate joint model, where the association structure includes 
# interactions with observed data. Here we specify that we want to use 
# an association structure based on the current value of the linear 
# predictor from the longitudinal submodel (i.e. "etavalue"), but we 
# also want to interact this with the treatment covariate (trt) from
# pbcLong data frame, so that we can estimate a different association 
# parameter (i.e. estimated effect of log serum bilirubin on the log 
# hazard of death) for each treatment group
f4 <- stan_jm(formulaLong = logBili ~ year + (1 | id), 
              dataLong = pbcLong,
              formulaEvent = Surv(futimeYears, death) ~ sex + trt, 
              dataEvent = pbcSurv,
              time_var = "year",
              assoc = c("etavalue", "etavalue_data(~ trt)"),
              chains = 1, cores = 1, seed = 12345, iter = 1000)
print(f4)

######
# Multivariate joint model, with association structure based 
# on the current value and slope of the linear predictor in the 
# first longitudinal submodel and the area under the marker 
# trajectory for the second longitudinal submodel
mv1 <- stan_jm(
        formulaLong = list(
          logBili ~ year + (1 | id), 
          albumin ~ sex + year + (year | id)),
        dataLong = pbcLong,
        formulaEvent = Surv(futimeYears, death) ~ sex + trt, 
        dataEvent = pbcSurv,
        assoc = list(c("etavalue", "etaslope"), "etaauc"), 
        time_var = "year",
        chains = 1, cores = 1, seed = 12345, iter = 100)
print(mv1)

#####
# Multivariate joint model, where the association structure is formed by 
# including the expected value of each longitudinal marker (logBili and 
# albumin) in the linear predictor of the event submodel, as well as their 
# interaction effect (i.e. the interaction between the two "etavalue" terms). 
# Note that whether such an association structure based on a marker by 
# marker interaction term makes sense will depend on the context of your 
# application -- here we just show it for demostration purposes).
mv2 <- stan_jm(
        formulaLong = list(
          logBili ~ year + (1 | id), 
          albumin ~ sex + year + (year | id)),
        dataLong = pbcLong,
        formulaEvent = Surv(futimeYears, death) ~ sex + trt, 
        dataEvent = pbcSurv,
        assoc = list(c("etavalue", "etavalue_etavalue(2)"), "etavalue"),
        time_var = "year", 
        chains = 1, cores = 1, seed = 12345, iter = 100)
        
#####
# Multivariate joint model, with one bernoulli marker and one
# Gaussian marker. We will artificially create the bernoulli
# marker by dichotomising log serum bilirubin
pbcLong$ybern <- as.integer(pbcLong$logBili >= mean(pbcLong$logBili))
mv3 <- stan_jm(
        formulaLong = list(
          ybern ~ year + (1 | id), 
          albumin ~ sex + year + (year | id)),
        dataLong = pbcLong,
        formulaEvent = Surv(futimeYears, death) ~ sex + trt, 
        dataEvent = pbcSurv,
        family = list(binomial, gaussian),
        time_var = "year", 
        chains = 1, cores = 1, seed = 12345, iter = 1000)

}

Bayesian multivariate generalized linear models with correlated group-specific terms via Stan

Description

Bayesian inference for multivariate GLMs with group-specific coefficients that are assumed to be correlated across the GLM submodels.

Usage

stan_mvmer(
  formula,
  data,
  family = gaussian,
  weights,
  prior = normal(autoscale = TRUE),
  prior_intercept = normal(autoscale = TRUE),
  prior_aux = cauchy(0, 5, autoscale = TRUE),
  prior_covariance = lkj(autoscale = TRUE),
  prior_PD = FALSE,
  algorithm = c("sampling", "meanfield", "fullrank"),
  adapt_delta = NULL,
  max_treedepth = 10L,
  init = "random",
  QR = FALSE,
  sparse = FALSE,
  ...
)

Arguments

Details

The stan_mvmer function can be used to fit a multivariate generalized linear model (GLM) with group-specific terms. The model consists of distinct GLM submodels, each which contains group-specific terms; within a grouping factor (for example, patient ID) the grouping-specific terms are assumed to be correlated across the different GLM submodels. It is possible to specify a different outcome type (for example a different family and/or link function) for each of the GLM submodels.

Bayesian estimation of the model is performed via MCMC, in the same way as for stan_glmer. Also, similar to stan_glmer, an unstructured covariance matrix is used for the group-specific terms within a given grouping factor, with priors on the terms of a decomposition of the covariance matrix.See priors for more information about the priors distributions that are available for the covariance matrices, the regression coefficients and the intercept and auxiliary parameters.

Value

A stanmvreg object is returned.

See Also

stan_glmer, stan_jm, stanreg-objects, stanmvreg-methods, print.stanmvreg, summary.stanmvreg, posterior_predict, posterior_interval.

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch !="i386") {

#####
# A multivariate GLM with two submodels. For the grouping factor 'id', the 
# group-specific intercept from the first submodel (logBili) is assumed to
# be correlated with the group-specific intercept and linear slope in the 
# second submodel (albumin)
f1 <- stan_mvmer(
        formula = list(
          logBili ~ year + (1 | id), 
          albumin ~ sex + year + (year | id)),
        data = pbcLong, 
        # this next line is only to keep the example small in size!
        chains = 1, cores = 1, seed = 12345, iter = 1000)
summary(f1) 

#####
# A multivariate GLM with one bernoulli outcome and one
# gaussian outcome. We will artificially create the bernoulli
# outcome by dichotomising log serum bilirubin
pbcLong$ybern <- as.integer(pbcLong$logBili >= mean(pbcLong$logBili))
f2 <- stan_mvmer(
        formula = list(
          ybern ~ year + (1 | id), 
          albumin ~ sex + year + (year | id)),
        data = pbcLong,
        family = list(binomial, gaussian),
        chains = 1, cores = 1, seed = 12345, iter = 1000)

}

Bayesian nonlinear models with group-specific terms via Stan

Description

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

Usage

stan_nlmer(
  formula,
  data = NULL,
  subset,
  weights,
  na.action,
  offset,
  contrasts = NULL,
  ...,
  prior = normal(autoscale = TRUE),
  prior_aux = exponential(autoscale = TRUE),
  prior_covariance = decov(),
  prior_PD = FALSE,
  algorithm = c("sampling", "meanfield", "fullrank"),
  adapt_delta = NULL,
  QR = FALSE,
  sparse = FALSE
)

Arguments

Details

The stan_nlmer function is similar in syntax to nlmer but rather than performing (approximate) maximum marginal likelihood estimation, Bayesian estimation is by default performed via MCMC. The Bayesian model adds independent priors on the "coefficients" — which are really intercepts — in the same way as stan_nlmer 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 supported transformation functions are limited to the named "self-starting" functions in the stats library: SSasymp, SSasympOff, SSasympOrig, SSbiexp, SSfol, SSfpl, SSgompertz, SSlogis, SSmicmen, and SSweibull.

Value

A stanreg object is returned for stan_nlmer.

See Also

stanreg-methods and nlmer.

The vignette for stan_glmer, which also discusses stan_nlmer models. https://mc-stan.org/rstanarm/articles/

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch !="i386") {

data("Orange", package = "datasets")
Orange$circumference <- Orange$circumference / 100
Orange$age <- Orange$age / 100
fit <- stan_nlmer(
  circumference ~ SSlogis(age, Asym, xmid, scal) ~ Asym|Tree, 
  data = Orange, 
  # for speed only
  chains = 1, 
  iter = 1000
 ) 
print(fit)
posterior_interval(fit)
plot(fit, regex_pars = "b\\[")

}

Bayesian ordinal regression models via Stan

Description

Bayesian inference for ordinal (or binary) regression models under a proportional odds assumption.

Usage

stan_polr(
  formula,
  data,
  weights,
  ...,
  subset,
  na.action = getOption("na.action", "na.omit"),
  contrasts = NULL,
  model = TRUE,
  method = c("logistic", "probit", "loglog", "cloglog", "cauchit"),
  prior = R2(stop("'location' must be specified")),
  prior_counts = dirichlet(1),
  shape = NULL,
  rate = NULL,
  prior_PD = FALSE,
  algorithm = c("sampling", "meanfield", "fullrank"),
  adapt_delta = NULL,
  do_residuals = NULL
)

stan_polr.fit(
  x,
  y,
  wt = NULL,
  offset = NULL,
  method = c("logistic", "probit", "loglog", "cloglog", "cauchit"),
  ...,
  prior = R2(stop("'location' must be specified")),
  prior_counts = dirichlet(1),
  shape = NULL,
  rate = NULL,
  prior_PD = FALSE,
  algorithm = c("sampling", "meanfield", "fullrank"),
  adapt_delta = NULL,
  do_residuals = algorithm == "sampling"
)

Arguments

Details

The stan_polr function is similar in syntax to polr but rather than performing maximum likelihood estimation of a proportional odds model, Bayesian estimation is performed (if algorithm = "sampling") via MCMC. The stan_polr function calls the workhorse stan_polr.fit function, but it is possible to call the latter directly.

As for stan_lm, it is necessary to specify the prior location of R^2. In this case, the R^2 pertains to the proportion of variance in the latent variable (which is discretized by the cutpoints) attributable to the predictors in the model.

Prior beliefs about the cutpoints are governed by prior beliefs about the outcome when the predictors are at their sample means. Both of these are explained in the help page on priors and in the rstanarm vignettes.

Unlike polr, stan_polr also allows the "ordinal" outcome to contain only two levels, in which case the likelihood is the same by default as for stan_glm with family = binomial but the prior on the coefficients is different. However, stan_polr allows the user to specify the shape and rate hyperparameters, in which case the probability of success is defined as the logistic CDF of the linear predictor, raised to the power of alpha where alpha has a gamma prior with the specified shape and rate. This likelihood is called “scobit” by Nagler (1994) because if alpha is not equal to 1, then the relationship between the linear predictor and the probability of success is skewed. If shape or rate is NULL, then alpha is assumed to be fixed to 1.

Otherwise, it is usually advisible to set shape and rate to the same number so that the expected value of alpha is 1 while leaving open the possibility that alpha may depart from 1 a little bit. It is often necessary to have a lot of data in order to estimate alpha with much precision and always necessary to inspect the Pareto shape parameters calculated by loo to see if the results are particularly sensitive to individual observations.

Users should think carefully about how the outcome is coded when using a scobit-type model. When alpha is not 1, the asymmetry implies that the probability of success is most sensitive to the predictors when the probability of success is less than 0.63. Reversing the coding of the successes and failures allows the predictors to have the greatest impact when the probability of failure is less than 0.63. Also, the gamma prior on alpha is positively skewed, but you can reverse the coding of the successes and failures to circumvent this property.

Value

A stanreg object is returned for stan_polr.

A stanfit object (or a slightly modified stanfit object) is returned if stan_polr.fit is called directly.

References

Nagler, J., (1994). Scobit: An Alternative Estimator to Logit and Probit. American Journal of Political Science. 230 – 255.

See Also

stanreg-methods and polr.

The vignette for stan_polr. https://mc-stan.org/rstanarm/articles/

Examples

if (.Platform$OS.type != "windows" || .Platform$r_arch !="i386") {
 fit <- stan_polr(tobgp ~ agegp, data = esoph, method = "probit",
          prior = R2(0.2, "mean"), init_r = 0.1, seed = 12345,
          algorithm = "fullrank") # for speed only
 print(fit)
 plot(fit)
}

Methods for stanmvreg objects

Description

S3 methods for stanmvreg objects. There are also several methods (listed in See Also, below) with their own individual help pages. The main difference between these methods and the stanreg methods is that the methods described here generally include an additional argument m which allows the user to specify which submodel they wish to return the result for. If the argument m is set to NULL then the result will generally be a named list with each element of the list containing the result for one of the submodels.

Usage

## S3 method for class 'stanmvreg'
coef(object, m = NULL, ...)

## S3 method for class 'stanmvreg'
fitted(object, m = NULL, ...)

## S3 method for class 'stanmvreg'
residuals(object, m = NULL, ...)

## S3 method for class 'stanmvreg'
se(object, m = NULL, ...)

## S3 method for class 'stanmvreg'
formula(x, fixed.only = FALSE, random.only = FALSE, m = NULL, ...)

## S3 method for class 'stanmvreg'
update(object, formula., ..., evaluate = TRUE)

## S3 method for class 'stanjm'
update(object, formulaLong., formulaEvent., ..., evaluate = TRUE)

## S3 method for class 'stanmvreg'
fixef(object, m = NULL, remove_stub = TRUE, ...)

## S3 method for class 'stanmvreg'
ngrps(object, ...)

## S3 method for class 'stanmvreg'
ranef(object, m = NULL, ...)

## S3 method for class 'stanmvreg'
sigma(object, m = NULL, ...)

Arguments

Details

Most of these methods are similar to the methods defined for objects of class 'lm', 'glm', 'glmer', etc. However there are a few exceptions:

coef

Medians are used for point estimates. See the Point estimates section in print.stanmvreg for more details. coef returns a list equal to the length of the number of submodels. The first elements of the list are the coefficients from each of the fitted longitudinal submodels and are the same layout as those returned by coef method of the lme4 package, that is, the sum of the random and fixed effects coefficients for each explanatory variable for each level of each grouping factor. The final element of the returned list is a vector of fixed effect coefficients from the event submodel.

se

The se function returns standard errors based on mad. See the Uncertainty estimates section in print.stanmvreg for more details.

confint

Not supplied, since the posterior_interval function should be used instead to compute Bayesian uncertainty intervals.

residuals

Residuals are always of type "response" (not "deviance" residuals or any other type).

See Also

• The print, summary, and prior_summary methods for stanmvreg objects for information on the fitted model.

• The plot method to plot estimates and diagnostics.

• The pp_check method for graphical posterior predictive checking of the longitudinal or glmer submodels.

• The ps_check method for graphical posterior predictive checking of the event submodel.

• The posterior_traj for predictions for the longitudinal submodel (for models estimated using stan_jm), as well as it's associated plot method.

• The posterior_survfit for predictions for the event submodel, including so-called "dynamic" predictions (for models estimated using stan_jm), as well as it's associated plot method.

• The posterior_predict for predictions for the glmer submodel (for models estimated using stan_mvmer).

• The posterior_interval for uncertainty intervals for model parameters.

• The loo, and log_lik methods for leave-one-out model comparison, and computing the log-likelihood of (possibly new) data.

• The as.matrix, as.data.frame, and as.array methods to access posterior draws.

Other S3 methods for stanmvreg objects, which have separate documentation, including print.stanmvreg, and summary.stanmvreg.

Also posterior_interval for an alternative to confint, and posterior_predict, posterior_traj and posterior_survfit for predictions based on the fitted joint model.

Create lists of fitted model objects, combine them, or append new models to existing lists of models.

Description

Create lists of fitted model objects, combine them, or append new models to existing lists of models.

Usage

stanreg_list(..., model_names = NULL)

stanmvreg_list(..., model_names = NULL)

stanjm_list(..., model_names = NULL)

## S3 method for class 'stanreg_list'
print(x, ...)

Arguments

Value

A list of class "stanreg_list", "stanmvreg_list", or "stanjm_list", containing the fitted model objects and some metadata stored as attributes.

See Also

loo_model_weights for usage of stanreg_list.

Create a draws object from a stanreg object

Description

Convert a stanreg object to a format supported by the posterior package.

Usage

## S3 method for class 'stanreg'
as_draws(x, ...)

## S3 method for class 'stanreg'
as_draws_matrix(x, ...)

## S3 method for class 'stanreg'
as_draws_array(x, ...)

## S3 method for class 'stanreg'
as_draws_df(x, ...)

## S3 method for class 'stanreg'
as_draws_list(x, ...)

## S3 method for class 'stanreg'
as_draws_rvars(x, ...)

Arguments

Details

To subset iterations, chains, or draws, use subset_draws after making the draws object. To subset variables use ... to pass the pars and/or regex_pars arguments to as.matrix.stanreg or as.array.stanreg (these are called internally by as_draws.stanreg), or use subset_draws after making the draws object.

Value

A draws object from the posterior package. See the posterior package documentation and vignettes for details on working with these objects.

Examples

fit <- stan_glm(mpg ~ wt + as.factor(cyl), data = mtcars)
as_draws_matrix(fit) # matrix format combines all chains 
as_draws_df(fit, regex_pars = "cyl")
posterior::summarize_draws(as_draws_array(fit))

Fitted model objects

Description

The rstanarm model-fitting functions return an object of class 'stanreg', which is a list containing at a minimum the components listed below. Each stanreg object will also have additional classes (e.g. 'aov', 'betareg', 'glm', 'polr', etc.) and several additional components depending on the model and estimation algorithm.

Some additional details apply to models estimated using the stan_mvmer or stan_jm modelling functions. The stan_mvmer modelling function returns an object of class 'stanmvreg', which inherits the 'stanreg' class, but has a number of additional elements described in the subsection below. The stan_jm modelling function returns an object of class 'stanjm', which inherits both the 'stanmvreg' and 'stanreg' classes, but has a number of additional elements described in the subsection below. Both the 'stanjm' and 'stanmvreg' classes have several of their own methods for situations in which the default 'stanreg' methods are not suitable; see the See Also section below.

Elements for stanreg objects

coefficients

Point estimates, as described in print.stanreg.

ses

Standard errors based on mad, as described in print.stanreg.

residuals

Residuals of type 'response'.

fitted.values

Fitted mean values. For GLMs the linear predictors are transformed by the inverse link function.

linear.predictors

Linear fit on the link scale. For linear models this is the same as fitted.values.

covmat

Variance-covariance matrix for the coefficients based on draws from the posterior distribution, the variational approximation, or the asymptotic sampling distribution, depending on the estimation algorithm.

model,x,y

If requested, the the model frame, model matrix and response variable used, respectively.

family

The family object used.

call

The matched call.

formula

The model formula.

data,offset,weights

The data, offset, and weights arguments.

algorithm

The estimation method used.

prior.info

A list with information about the prior distributions used.

stanfit,stan_summary

The object of stanfit-class returned by RStan and a matrix of various summary statistics from the stanfit object.

rstan_version

The version of the rstan package that was used to fit the model.