---
title: "Estimating Generalized Linear Models for Binary and Binomial Data with rstanarm"
author: "Jonah Gabry and Ben Goodrich"
date: "`r Sys.Date()`"
output: 
  html_vignette: 
    toc: yes
params:
  EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true")
---
<!--
%\VignetteEngine{knitr::rmarkdown}
%\VignetteIndexEntry{stan_glm: GLMs for Binary and Binomial Data}
-->
```{r, child="children/SETTINGS-knitr.txt"}
```
```{r, child="children/SETTINGS-gg.txt"}
```
```{r, child="children/SETTINGS-rstan.txt"}
```
```{r, child="children/SETTINGS-loo.txt"}
```

# Introduction

This vignette explains how to estimate generalized linear models (GLMs) for 
binary (Bernoulli) and Binomial response variables using the `stan_glm`
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 conditionally independent binomial distributions (possibly with
only one trial per observation).

# Likelihood

For a binomial GLM the likelihood for one observation $y$ can be written as a
conditionally binomial PMF
$$\binom{n}{y} \pi^{y} (1 - \pi)^{n - y},$$
where $n$ is the known number of trials, $\pi = g^{-1}(\eta)$ is the probability
of success and $\eta = \alpha + \mathbf{x}^\top \boldsymbol{\beta}$ is a linear
predictor. For a sample of size $N$, the likelihood of the entire sample is the
product of $N$ individual likelihood contributions.

Because $\pi$ is a probability, for a binomial model the _link_ function $g$
maps between the unit interval (the support of $\pi$) and the set of all real
numbers $\mathbb{R}$. When applied to a linear predictor $\eta$ with values in
$\mathbb{R}$, the inverse link function $g^{-1}(\eta)$ therefore returns a valid
probability between 0 and 1.

The two most common link functions used for binomial GLMs are the
[logit](https://en.wikipedia.org/wiki/Logit) and 
[probit](https://en.wikipedia.org/wiki/Probit)
functions. With the logit (or log-odds) link function $g(x) =
\ln{\left(\frac{x}{1-x}\right)}$, the likelihood for a single observation
becomes

$$\binom{n}{y}\left(\text{logit}^{-1}(\eta)\right)^y 
\left(1 - \text{logit}^{-1}(\eta)\right)^{n-y} = 
\binom{n}{y} \left(\frac{e^{\eta}}{1 + e^{\eta}}\right)^{y}
\left(\frac{1}{1 + e^{\eta}}\right)^{n - y}$$

and the probit link function $g(x) = \Phi^{-1}(x)$ yields the likelihood

$$\binom{n}{y} \left(\Phi(\eta)\right)^{y}
\left(1 - \Phi(\eta)\right)^{n - y},$$

where $\Phi$ is the CDF of the standard normal distribution. The differences 
between the logit and probit functions are minor and -- if, as __rstanarm__ does
by default, the probit is scaled so its slope at the origin matches the logit's 
-- the two link functions should yield similar results. With `stan_glm`, 
binomial models with a logit link function can typically be fit slightly faster 
than the identical model with a probit link because of how the two models are 
implemented in Stan. Unless the user has a specific reason to prefer the probit
link, we recommend the logit simply because it will be slightly faster and more
numerically stable.

In theory, there are infinitely many possible link functions, although in
practice only a few are typically used. Other common choices are the `cauchit`
and `cloglog` functions, which can also be used with `stan_glm` (every link 
function compatible with`glm` will work with `stan_glm`).

# Priors

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

# Posterior

With independent prior distributions, the joint posterior distribution for
$\alpha$ and $\boldsymbol{\beta}$ is proportional to the product of the priors 
and the $N$ likelihood contributions:

$$f\left(\alpha,\boldsymbol{\beta} | \mathbf{y},\mathbf{X}\right) \propto
  f\left(\alpha\right) \times \prod_{k=1}^K f\left(\beta_k\right) \times
  \prod_{i=1}^N {
  g^{-1}\left(\eta_i\right)^{y_i} 
  \left(1 - g^{-1}\left(\eta_i\right)\right)^{n_i-y_i}}.$$
  
This is posterior distribution that `stan_glm` will draw from when using MCMC.
  
# Logistic Regression Example

When the logit link function is used the model is often referred to as a 
logistic regression model (the inverse logit function is the CDF of the standard
logistic distribution). As an example, here we will show how to carry out a 
few parts of the analysis from Chapter 5.4 of 
[Gelman and Hill (2007)](http://www.stat.columbia.edu/~gelman/arm/) 
using `stan_glm`.

Gelman and Hill describe 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. The goal of the analysis presented by 
Gelman and Hill is to learn about the factors associated with switching wells.

To start, we'll use `dist` (the distance from the respondent's house to the 
nearest well with safe drinking water) as the only predictor of `switch` (1 if
switched, 0 if not). Then we'll expand the model by adding the arsenic level
of the water in the resident's own well as a predictor and compare this larger
model to the original.

After loading the `wells` data, we first rescale the `dist` variable (measured
in meters) so that it is measured in units of 100 meters. If we leave `dist` in
its original units then the corresponding regression coefficient will represent
the effect of the marginal meter, which is too small to have a useful
interpretation.

```{r binom-arsenic-data}
library(rstanarm)
data(wells)
wells$dist100 <- wells$dist / 100
```

Before estimating any models we can visualize the distribution of 
`dist100` in the data:

```{r, binom-arsenic-plot-dist100, fig.height=3}
ggplot(wells, aes(x = dist100, y = ..density.., fill = switch == 1)) +
  geom_histogram() + 
  scale_fill_manual(values = c("gray30", "skyblue"))
```

In the plot above the blue bars correspond to the 
`r sum(rstanarm::wells$switch == 1)` residents who said they switched wells and darker bars show the distribution of `dist100` for the 
`r sum(rstanarm::wells$switch == 0)` residents who didn't switch. As we would expect, for the
residents who switched wells, the distribution of `dist100` is more concentrated
at smaller distances.

A Bayesian version of Gelman and Hill's initial logistic regression model
can be estimated using the `stan_glm` function. Here we'll use a Student t 
prior with 7 degrees of freedom and a scale of 2.5, which, as discussed above,
is a reasonable default prior when coefficients should be close to zero but 
have some chance of being large. 

```{r, binom-arsenic-mcmc, results="hide"}
t_prior <- student_t(df = 7, location = 0, scale = 2.5)
fit1 <- stan_glm(switch ~ dist100, data = wells, 
                 family = binomial(link = "logit"), 
                 prior = t_prior, prior_intercept = t_prior,  
                 chains = CHAINS, cores = CORES, seed = SEED, iter = ITER)
```
```{r, binom-arsenic-print, echo=FALSE}
(coef_fit1 <- round(coef(fit1), 3))
```
The `formula`, `data` and `family` arguments to `stan_glm` are specified in
exactly the same way as for `glm`. We've also added the optional additional 
arguments `chains` (how many chains we want to execute), `cores` (how many 
cores we want the computer to utilize) and `seed` 
(for reproducibility). You can read about other possible arguments in the
`stan_glm` documentation (`help(stan_glm, package = 'rstanarm')`).

To get a sense for the uncertainty in our estimates we can use the 
`posterior_interval` function to get Bayesian uncertainty intervals. The
uncertainty intervals are computed by finding the relevant quantiles of the
draws from the posterior distribution. For example, to compute 50% intervals we
use:

```{r, binom-arsenic-ci}
round(posterior_interval(fit1, prob = 0.5), 2)
```

For more on `posterior_interval` and interpreting the parameter estimates from a Bayesian
model see Step 2 in the ["How to Use the __rstanarm__ Package"](rstanarm.html) vignette.


Using the coefficient estimates we can plot the predicted probability of 
`switch = 1` (as a function of `dist100`) together with the observed outcomes:

```{r, binom-arsenic-plot-model}
# Predicted probability as a function of x
pr_switch <- function(x, ests) plogis(ests[1] + ests[2] * x)
# A function to slightly jitter the binary data
jitt <- function(...) {
  geom_point(aes_string(...), position = position_jitter(height = 0.05, width = 0.1), 
             size = 2, shape = 21, stroke = 0.2)
}
ggplot(wells, aes(x = dist100, y = switch, color = switch)) + 
  scale_y_continuous(breaks = c(0, 0.5, 1)) +
  jitt(x="dist100") + 
  stat_function(fun = pr_switch, args = list(ests = coef(fit1)), 
                size = 2, color = "gray35")
```

The plot shows that under this model the predicted probability of switching is 
a decent bit above 50% for residents living very close to wells with safe 
drinking water. As expected, larger values of `dist100` are associated with 
lower predicted probabilities of switching. At the extreme 
($\approx 300$ meters), the probability is about 25%.

Next, we incorporate an additional predictor into the model: the arsenic level
of water in the respondent's well. According to Gelman and Hill, 
"At the levels present in the Bangladesh drinking water, the health risks 
from arsenic are roughly proportional to exposure, and so we would expect 
switching to be more likely from wells with high arsenic levels" (pg. 90). 
We only need to change the formula, so we can use the `update` function:

```{r binom-arsenic-mcmc2, results="hide"}
fit2 <- update(fit1, formula = switch ~ dist100 + arsenic) 
```
```{r}
(coef_fit2 <- round(coef(fit2), 3))
```

As expected the coefficient on `arsenic` is positive. The plot
below shows distance on the x-axis and arsenic level on the y-axis with the 
predicted probability of well-switching mapped to the color of the background 
tiles (the lighter the color the higher the probability). The observed value of
`switch` is indicated by the color of the points.

```{r,echo=FALSE}
theme_update(legend.position = "right")
```
```{r, binom-arsenic-plot-model2}
pr_switch2 <- function(x, y, ests) plogis(ests[1] + ests[2] * x + ests[3] * y)
grid <- expand.grid(dist100 = seq(0, 4, length.out = 100), 
                    arsenic = seq(0, 10, length.out = 100))
grid$prob <- with(grid, pr_switch2(dist100, arsenic, coef(fit2)))
ggplot(grid, aes(x = dist100, y = arsenic)) + 
  geom_tile(aes(fill = prob)) + 
  geom_point(data = wells, aes(color = factor(switch)), size = 2, alpha = 0.85) + 
  scale_fill_gradient() +
  scale_color_manual("switch", values = c("white", "black"), labels = c("No", "Yes"))
```

We can see that the black points (`switch=1`) are predominantly clustered in the
upper-left region of the plot where the predicted probability of switching is
highest.

Another way we can visualize the data and model is to follow Gelman and Hill and
create separate plots for varying the arsenic level and distance. Here we'll
plot curves representing the predicted probability of switching for the minimum,
maximum and quartile values of both variables.

```{r,echo=FALSE}
theme_update(legend.position = "none")
```
```{r, binom-arsenic-plot-model2-alt}
if (require(gridExtra)) {
  # Quantiles
  q_ars <- quantile(wells$dist100, seq(0, 1, 0.25))
  q_dist <- quantile(wells$arsenic, seq(0, 1, 0.25))  
  base <- ggplot(wells) + xlim(c(0, NA)) +
    scale_y_continuous(breaks = c(0, 0.5, 1))
  vary_arsenic <- base + jitt(x="arsenic", y="switch", color="switch")
  vary_dist <- base + jitt(x="dist100", y="switch", color="switch")
  for (i in 1:5) {
    vary_dist <- 
      vary_dist + stat_function(fun = pr_switch2, color = "gray35", 
                                args = list(ests = coef(fit2), y = q_dist[i]))
    vary_arsenic <-
      vary_arsenic + stat_function(fun = pr_switch2, color = "gray35", 
                                   args = list(ests = coef(fit2), x = q_ars[i]))
  }
  grid.arrange(vary_dist, vary_arsenic, ncol = 2)
}
```

We can compare our two models (with and without `arsenic`) using an 
approximation to Leave-One-Out (LOO) cross-validation, which is a method
for estimating out of sample predictive performance and is implemented 
by the `loo` function in the __loo__ package:

```{r, binom-arsenic-loo}
(loo1 <- loo(fit1))
(loo2 <- loo(fit2))
compare_models(loo1, loo2)
```

These results favor `fit2` over `fit1`, as the estimated
difference in `elpd` (the expected log pointwise predictive density for a new
dataset) is so much larger than its standard error. 
LOO penalizes models for adding additional predictors (this helps
counter overfitting), but in this case `fit2` represents enough of an
improvement over `fit1` that the penalty for including `arsenic` is negligible
(as it should be if `arsenic` is an important predictor).

The [vignette](lm.html) for the `stan_lm` function also has an example of using the `loo`
function where the results are quite a bit different from what we see here and
some important additional considerations are discussed.

# Binomial Models

Although the example in this vignette focused on a binary response variable, we
can use nearly identical code if we have the sum of multiple binary variables.
For example, image a hypothetical dataset similar to the well-switching data but
spanning multiple villages. Each observation (each row) of this `data.frame`
corresponds to an entire village: `switch[i]` is the number of 'yes' responses
to the well-switching question for village `i`, `dist100[i]` is the average
distance to the closest well with clean water for village `i`, etc. We also now
have a variable `n` where `n[i]` is the number of respondents from village `i`.

For this data we can estimate a similar model to the one we used in the binary
case by changing the formula to

  `cbind(switch, n - switch) ~ dist100 + arsenic`

The left-hand side is now a 2-column matrix where the first column is the 
number of 'yes' responses and the second column is the number of 'no' responses
(or more generally, the number of successes and number of failures). 
The same model can also be specified using the proportion of 'yes' responses
and the total number of responses in each village. This corresponds to the formula

  `prop_switch ~ dist100 + arsenic`

where `prop_switch = switch / n` is the proportion of 'yes' responses. The total
number of responses is provided using the `weights` argument. In this case we 
would add `weights = n` to the call to `stan_glm`. 

An example of a similar model can also be found in __Step 1__ of the 
["How to Use the __rstanarm__ Package"](rstanarm.html) vignette.

# Going Further

In the hypothetical scenario above, if we also have access to the observations
for each individual in all of the villages (not just the aggregate data), then a
natural extension would be to consider a multilevel model that takes advantage
of the inherent multilevel structure of the data (individuals nested within
villages). The [vignette](glmer.html) for the `stan_glmer` function discusses these models.

# References

Gelman, A. and Hill, J. (2007). _Data Analysis Using Regression and
Multilevel/Hierarchical Models._ Cambridge University Press, Cambridge, UK.
