This vignette explains how to model continuous outcomes on the open
unit interval using the stan_betareg function in the
rstanarm package.
Steps 3 and 4 are covered in more depth by the vignette entitled “How to Use the rstanarm Package”. This vignette focuses on Step 1 when the likelihood is the product of beta distributions.
Beta regression uses the beta distribution as the likelihood for the data, $$ f(y_i | a, b) = \frac{y_i^{(a-1)}(1-y_i)^{(b-1)}}{B(a,b)} $$ where B(⋅) is the beta function. The shape parameters for the distribution are a and b and enter into the model according to the following transformations, $$ a = \mu\cdot\phi \\ b = (1-\mu)\cdot\phi $$
Let g1(⋅) be some link function. Then, in the specification of the shape parameters above, μ = g1−1(Xβ), where X is a N × K dimensional matrix of predictors, and β is a K dimensional vector of parameters associated with each predictor.
In the simplest case (with only one set of regressors), ϕ is a scalar parameter. Alternatively, it is possible to model ϕ using a second set of regressors Z. In this context let g2(⋅) be some link function that is not necessarily identical to g1(⋅). Then ϕ = g2−1(Zγ), where γ is a J dimensional vector of parameters associated with the N × J dimensional matrix of predictors Z.
After substituting the shape parameter values in, the likelihood used in beta regression takes the following form, $$ f(y_i | \mu, \phi) = \frac{y_i^{(\mu\phi-1)}(1-y_i)^{((1-\mu)\phi-1)}}{B(\mu\phi,(1-\mu)\phi)} $$
A full Bayesian analysis requires specifying prior distributions
f(β) and
f(ϕ) for the vector
of regression coefficients and ϕ. When using
stan_betareg, these distributions can be set using the
prior_intercept, prior, and
prior_phi arguments. The stan_betareg function
supports a variety of prior distributions, which are explained in the
rstanarm documentation
(help(priors, package = 'rstanarm')).
When modeling ϕ with a
linear predictor a full Bayesian analysis requires specifying the prior
distributions f(β) and f(γ). In
stan_betareg the prior distributions on γ can be set using the
prior_intercept_z and prior_z arguments.
As an example, suppose we have K predictors and believe — prior to
seeing the data — that β1, …, βK
and ϕ are as likely to be
positive as they are to be negative, but are highly unlikely to be far
from zero. These beliefs can be represented by normal distributions with
mean zero and a small scale (standard deviation). To give ϕ and each of the βs this prior (with a scale of 1,
say), in the call to stan_betareg we would include the
arguments prior_intercept = normal(0,1),
prior = normal(0,1), and
prior_phi = normal(0,1).
If, on the other hand, we have less a priori confidence that the parameters will be close to zero then we could use a larger scale for the normal distribution and/or a distribution with heavier tails than the normal like the Student t distribution. Step 1 in the “How to Use the rstanarm Package” vignette discusses one such example.
After fitting the model we can use the prior_summary
function to print information about the prior distributions used when
fitting the model.
When using only a single set of regressors, the posterior distribution of β and ϕ is proportional to the product of the likelihood contributions, the K priors on the βk parameters, and ϕ, $$ f(\boldsymbol{\beta},\phi|\mathbf{y},\mathbf{X}) \propto \prod_{i=1}^N f(y_i | a, b) \times \prod_{k=1}^K f(\beta_k) \times f(\phi) $$
When using two sets of regressors, the posterior distribution of β and γ is proportional to the product of the likelihood contribution, the K priors on the βk parameters, and the J priors on the γj parameters,
$$ f(\boldsymbol{\beta},\boldsymbol{\gamma}|\mathbf{y},\mathbf{X}) \propto \prod_{i=1}^N f(y_i | a, b) \times \prod_{k=1}^K f(\beta_k) \times \prod_{j=1}^J f(\gamma_j) $$
In this example the outcome variable y is simulated in a way that warrants the use of beta regression. It is worth mentioning that the data generation process is quite convoluted, which is apparent in the identification of the likelihood above.
The data simulated below uses the logistic link function on the first set of regressors and the log link function on the second set of regressors.
SEED <- 1234
set.seed(SEED)
eta <- c(1, -0.2)
gamma <- c(1.8, 0.4)
N <- 200
x <- rnorm(N, 2, 2)
z <- rnorm(N, 0, 2)
mu <- binomial(link = logit)$linkinv(eta[1] + eta[2]*x)
phi <- binomial(link = log)$linkinv(gamma[1] + gamma[2]*z)
y <- rbeta(N, mu * phi, (1 - mu) * phi)
dat <- data.frame(cbind(y, x, z))
hist(dat$y, col = "darkgrey", border = F, main = "Distribution of Outcome Variable", xlab = "y", breaks = 20, freq = F)The model can be fit by calling stan_betareg, using the
appropriate link functions.
library(rstanarm)
fit1 <- stan_betareg(y ~ x | z, data = dat, link = "logit", link.phi = "log",
cores = 2, seed = 12345)
fit2 <- stan_betareg(y ~ -1 + x , data = dat, link = "logit", link.phi = "log",
cores = 2, seed = 12345)
round(coef(fit1), 2)
round(coef(fit2), 2)For clarity we can use prior_summary to print the
information about the prior distributions used to fit the models. The
priors used in fit1 are provided below.
The usual posterior analyses are available in rstanarm. The plots below illustrate simulated values of the outcome variable. The incorrect model noticeably fails to capture the top of the distribution consistently in comparison to the true model.
library(ggplot2)
library(bayesplot)
bayesplot_grid(
pp_check(fit1), pp_check(fit2),
xlim = c(0,1),
ylim = c(0,4),
titles = c("True Model: y ~ x | z", "False Model: y ~ x - 1"),
grid_args = list(ncol = 2)
)We can also compare models by evaluating the expected log pointwise
predictive density (elpd), which can be calculated using
the loo method, which provides an interface for
rstanarm models to the functionality in the
loo package.
The difference in elpd is negative indicating that the
expected predictive accuracy for the first model is higher.
In some applied contexts it may be necessary to work with an outcome
variable that is a proportion. If the proportion is bound on the open
unit interval then beta regression can be considered a reasonable
estimation method. The betareg package provides a dataset
on the proportion of crude oil converted to gasoline after distillation
and fractionation. This variable is defined as yield. Below
stan_betareg is used to model yield as a function of
temperature, pressure, and the batch of conditions.
library(rstanarm)
data("GasolineYield", package = "betareg")
gas_fit1 <- stan_betareg(yield ~ temp + batch, data = GasolineYield, link = "logit",
seed = 12345)
gas_fit2 <- stan_betareg(yield ~ temp + batch | pressure,
data = GasolineYield, link = "logit",
seed = 12345)
round(coef(gas_fit1), 2)
round(coef(gas_fit2), 2)The plots below illustrate simulated values of gasoline yield. While the first model accounts for variation in batch conditions its predictions looks somewhat uniform rather than resembling the peaked and right-skewed behavior of the true data. The second model does a somewhat better job at capturing the shape of the distribution, however its location is off as it is centered around 0.50 rather than 0.20.
library(ggplot2)
bayesplot_grid(
pp_check(gas_fit1), pp_check(gas_fit2),
xlim = c(0,1),
ylim = c(0,5),
titles = c("gas_fit1", "gas_fit2"),
grid_args = list(ncol = 2)
)Evaluating the expected log predictive distribution using
loo reveals that the second of the two models is
preferred.
Ferrari, SLP and Cribari-Neto, F (2004) “Beta Regression for Modeling Rates and Proportions”. Journal of Applied Statistics. Vol. 31, No. 07, p799-815.