### Air

*Air* analyses reported respiratory illness versus exposure to nitrogen dioxide in 103 children. The parameters `alpha`, `beta` and `sigma2` are known in advance, and the data are grouped into three categories.

See [WinBUGS examples volume 2](https://legacy.voteview.com/pdf/WINBUGSmanual_2.pdf) (pdf) for details.

#### data
```{r air_data, highlight = FALSE}
y <- c(21, 20, 15)
n <- c(48, 34, 21)
Z <- c(10, 30, 50)
alpha <- 4.48
beta <- 0.76
sigma2 <- 81.14
sigma <- sqrt(sigma2)
tau <- 1 / sigma2
J <- 3
```

#### greta code
```{r air_greta}
theta <- normal(0, 32, dim = 2)
mu <- alpha + beta * Z
X <- normal(mu, sigma)
p <- ilogit(theta[1] + theta[2] * X)
distribution(y) <- binomial(n, p)
```

#### BUGS/JAGS code
<div class="bugs">
```
for(j in 1 : J) {
   y[j] ~ dbin(p[j], n[j])
   logit(p[j]) <- theta[1] + theta[2] * X[j]
   X[j] ~ dnorm(mu[j], tau)
   mu[j] <- alpha + beta * Z[j]
}
theta[1] ~ dnorm(0.0, 0.001)
theta[2] ~ dnorm(0.0, 0.001)
```
</div>

#### Stan code
<div class="stan">
```{r air_stan, echo = FALSE}
cat(readLines('https://raw.githubusercontent.com/stan-dev/example-models/master/bugs_examples/vol2/air/air.stan'), sep = '\n')
```
</div>
