betareg.Rmd

---
title: "Modeling Rates/Proportions using Beta Regression with rstanarm"
author: "Imad Ali, Jonah Gabry and Ben Goodrich"
date: "`r Sys.Date()`"
output: 
  html_vignette: 
    toc: yes
---
<!--
%\VignetteEngine{knitr::rmarkdown}
%\VignetteIndexEntry{stan_betareg: Models for Rate/Proportion Data}
-->
```{r, child="children/SETTINGS-knitr.txt"}
```
```{r, child="children/SETTINGS-gg.txt"}
```

# Introduction

This vignette explains how to model continuous outcomes on the open unit
interval using the `stan_betareg` function in the __rstanarm__ package.

```{r, child="children/four_steps.txt"}
```

Steps 3 and 4 are covered in more depth by the vignette entitled ["How to Use
the __rstanarm__ Package"](rstanarm.html). This vignette focuses on Step 1 when
the likelihood is the product of beta distributions.

# Likelihood

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 $B(\cdot)$ 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 $g_1(\cdot)$ be some link function. Then, in the specification of the shape
parameters above, $\mu = g_1^{-1}(\mathbf{X}\boldsymbol{\beta})$, where $\boldsymbol{X}$
is a $N\times K$ dimensional matrix of predictors, and $\boldsymbol{\beta}$ is a $K$
dimensional vector of parameters associated with each predictor.

In the simplest case (with only one set of regressors), $\phi$ is a scalar
parameter. Alternatively, it is possible to model $\phi$ using a second set of
regressors $\mathbf{Z}$. In this context let $g_2(\cdot)$ be some link function that is not
necessarily identical to $g_1(\cdot)$. Then $\phi = g_2^{-1}(\mathbf{Z}\boldsymbol{\gamma})$,
where $\boldsymbol{\gamma}$ is a $J$ dimensional vector of parameters associated with
the $N\times J$ dimensional matrix of predictors $\mathbf{Z}$.

After substituting the shape parameter values in, the likelihood used in beta regression takes the
following form,
$$
f(y_i | \mu, \phi) = \frac{y_i^{(\mu\phi-1)}(1-y_i)^{((1-\mu)\phi-1)}}{B(\mu\phi,(1-\mu)\phi)}
$$

# Priors

A full Bayesian analysis requires specifying prior distributions
$f(\boldsymbol{\beta})$ and $f(\phi)$ for the vector of
regression coefficients and $\phi$. 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 $\phi$ with a linear predictor a full Bayesian analysis requires
specifying the prior distributions $f(\boldsymbol{\beta})$ and $f(\boldsymbol{\gamma})$.
In `stan_betareg` the prior distributions on $\boldsymbol{\gamma}$ 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 $\beta_1, \dots, \beta_K$ and $\phi$ 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 $\phi$ and each of the $\beta$s 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.

# Posterior

When using only a *single set of regressors*, the posterior distribution of
$\boldsymbol{\beta}$ and $\phi$ is proportional to the product of the likelihood
contributions, the $K$ priors on the $\beta_k$ parameters, and $\phi$,
$$
f(\boldsymbol{\beta},\phi|\mathbf{y},\mathbf{X}) \propto 
\prod_{i=1}^N f(y_i | a, b) \times 
\prod_{k=1}^K f(\beta_k) \times
f(\phi)
$$

When using *two sets of regressors*, the posterior distribution of $\boldsymbol{\beta}$ and
$\boldsymbol{\gamma}$ is proportional to the product of the likelihood contribution, the $K$
priors on the $\beta_k$ parameters, and the $J$ priors on the $\gamma_j$ parameters,

$$
f(\boldsymbol{\beta},\boldsymbol{\gamma}|\mathbf{y},\mathbf{X}) \propto 
\prod_{i=1}^N f(y_i | a, b) \times 
\prod_{k=1}^K f(\beta_k) \times
\prod_{j=1}^J f(\gamma_j)
$$

# An Example Using Simulated Data

In this example the outcome variable $\mathbf{y}$ 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.

```{r simulated-data, fig.height=5}
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.
```{r simulated-fit, results = "hide"}
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)
```

``` {r simulated-fit-print, echo=FALSE}
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. 

``` {r print-priors}
prior_summary(fit1)
```

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.
```{r simulated-analysis, fig.height=5}
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.

``` {r simulated-loo}
loo1 <- loo(fit1)
loo2 <- loo(fit2)
loo_compare(loo1, loo2)
```

The difference in `elpd` is negative indicating that the expected predictive
accuracy for the first model is higher.

# An Example Using Gasoline Data
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.

```{r, gas-fit, results="hide"}
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)
```

``` {r, gas-print, echo=FALSE}
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.

```{r gas-analysis, fig.height=5}
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)
)
```
``` {r, gas-loo}
gas_loo1 <- loo(gas_fit1)
gas_loo2 <- loo(gas_fit2)
loo_compare(gas_loo1, gas_loo2)
```

Evaluating the expected log predictive distribution using `loo` 
reveals that the second of the two models is preferred.

# References

Ferrari, SLP and Cribari-Neto, F (2004) "Beta Regression for Modeling Rates
and Proportions". _Journal of Applied Statistics._ Vol. 31, No. 07, p799-815.