Beta regression modeling with optional prior distributions for the
coefficients, intercept, and auxiliary parameter phi
(if applicable).
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(), prior_intercept = normal(), prior_z = normal(),
prior_intercept_z = normal(), prior_phi = exponential(),
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(),
prior_intercept = normal(), prior_z = normal(),
prior_intercept_z = normal(), prior_phi = exponential(),
prior_PD = FALSE, algorithm = c("sampling", "optimizing", "meanfield",
"fullrank"), adapt_delta = NULL, QR = FALSE)
Arguments
formula, data, subset |
Same as betareg ,
but we strongly advise against omitting the data
argument. Unless data is specified (and is a data frame) many
post-estimation functions (including update , loo ,
kfold ) are not guaranteed to work properly. |
na.action |
Same as betareg , but
rarely specified. |
link |
Character specification of the link function used in the model
for mu (specified through x ). Currently, "logit", "probit",
"cloglog", "cauchit", "log", and "loglog" are supported. |
link.phi |
If applicable, character specification of the link function
used in the model for phi (specified through z ). Currently,
"identity", "log" (default), and "sqrt" are supported. Since the "sqrt"
link function is known to be unstable, it is advisable to specify a
different link function (or to model phi as a scalar parameter
instead of via a linear predictor by excluding z from the
formula and excluding link.phi ). |
model, offset, weights |
Same as betareg . |
x, y |
In stan_betareg , logical scalars indicating whether to
return the design matrix and response vector. In stan_betareg.fit ,
a design matrix and response vector. |
... |
Further arguments passed to the function in the rstan
package (sampling , vb , or
optimizing ), corresponding to the estimation method
named by algorithm . For example, if algorithm is
"sampling" it is possibly to specify iter , chains ,
cores , refresh , etc. |
prior |
The prior distribution for the regression coefficients.
prior should be a call to one of the various functions provided by
rstanarm for specifying priors. The subset of these functions that
can be used for the prior on the coefficients can be grouped into several
"families":
Family | Functions |
Student t family | normal , student_t , cauchy |
Hierarchical shrinkage family | hs , hs_plus |
Laplace family | laplace , lasso |
Product normal family | product_normal |
See the priors help page for details on the families and
how to specify the arguments for all of the functions in the table above.
To omit a prior ---i.e., to use a flat (improper) uniform prior---
prior can be set to NULL , although this is rarely a good
idea.
Note: Unless QR=TRUE , if prior is from the Student t
family or Laplace family, and if the autoscale argument to the
function used to specify the prior (e.g. normal ) is left at
its default and recommended value of TRUE , then the default or
user-specified prior scale(s) may be adjusted internally based on the
scales of the predictors. See the priors help page and the
Prior Distributions vignette for details on the rescaling and the
prior_summary function for a summary of the priors used for a
particular model. |
prior_intercept |
The prior distribution for the intercept.
prior_intercept can be a call to normal , student_t or
cauchy . See the priors help page for details on
these functions. To omit a prior on the intercept ---i.e., to use a flat
(improper) uniform prior--- prior_intercept can be set to
NULL .
Note: If using a dense representation of the design matrix
---i.e., if the sparse argument is left at its default value of
FALSE --- then the prior distribution for the intercept is set so it
applies to the value when all predictors are centered. If you prefer
to specify a prior on the intercept without the predictors being
auto-centered, then you have to omit the intercept from the
formula and include a column of ones as a predictor,
in which case some element of prior specifies the prior on it,
rather than prior_intercept . Regardless of how
prior_intercept is specified, the reported estimates of the
intercept always correspond to a parameterization without centered
predictors (i.e., same as in glm ). |
prior_z |
Prior distribution for the coefficients in the model for
phi (if applicable). Same options as for prior . |
prior_intercept_z |
Prior distribution for the intercept in the model
for phi (if applicable). Same options as for prior_intercept . |
prior_phi |
The prior distribution for phi if it is not
modeled as a function of predictors. If z variables are specified
then prior_phi is ignored and prior_intercept_z and
prior_z are used to specify the priors on the intercept and
coefficients in the model for phi . When applicable, prior_phi
can be a call to exponential to use an exponential distribution, or
one of normal , student_t or cauchy to use half-normal,
half-t, or half-Cauchy prior. See priors for details on these
functions. To omit a prior ---i.e., to use a flat (improper) uniform
prior--- set prior_phi to NULL . |
prior_PD |
A logical scalar (defaulting to FALSE ) indicating
whether to draw from the prior predictive distribution instead of
conditioning on the outcome. |
algorithm |
A string (possibly abbreviated) indicating the
estimation approach to use. Can be "sampling" for MCMC (the
default), "optimizing" for optimization, "meanfield" for
variational inference with independent normal distributions, or
"fullrank" for variational inference with a multivariate normal
distribution. See rstanarm-package for more details on the
estimation algorithms. NOTE: not all fitting functions support all four
algorithms. |
adapt_delta |
Only relevant if algorithm="sampling" . See
the adapt_delta help page for details. |
QR |
A logical scalar defaulting to FALSE , but if TRUE
applies a scaled qr decomposition to the design matrix. The
transformation does not change the likelihood of the data but is
recommended for computational reasons when there are multiple predictors.
See the QR-argument documentation page for details on how
rstanarm does the transformation and important information about how
to interpret the prior distributions of the model parameters when using
QR=TRUE . |
z |
For stan_betareg.fit , a regressor matrix for phi .
Defaults to an intercept only. |
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.
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.
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
Examples
### 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
)
#> Initial log joint probability = -324.957
#> Optimization terminated normally:
#> Convergence detected: relative gradient magnitude is below tolerance
print(fit, digits = 2)
#> stan_betareg
#> family: beta [logit, link.phi=log]
#> formula: y ~ x | z
#> observations: 200
#> ------
#> Median MAD_SD
#> (Intercept) 0.87 0.11
#> x 0.23 0.05
#> (phi)_(Intercept) 1.24 0.21
#> (phi)_z 0.43 0.09
#> mean_PPD 0.79 0.01
#>
#> ------
#> For info on the priors used see help('prior_summary.stanreg').