rstanarm-package: 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.

Arguments

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.

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.

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. http://stat.columbia.edu/~gelman/book/

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

Stan Development Team. (2017). Stan Modeling Language Users Guide and Reference Manual. http://mc-stan.org/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: http://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. (online).

Gabry, J., Simpson, D., Vehtari, A., Betancourt, M., and Gelman, A. (2018). Visualization in Bayesian workflow. Journal of the Royal Statistical Society Series A, accepted for publication. arXiv preprint: http://arxiv.org/abs/1709.01449.

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