BUGS models

The BUGS project provide a number of example models written in the BUGS modelling language. These models will run in WinBUGS and OpenBUGS, and likely also in JAGS. The Stan wiki provides Stan implementations of these models.

The following sections provide greta implementations of some of these example models, alongside the BUGS code from WinBUGS examples volume 2 (pdf) and Stan code and an R version of the data from the Stan example models wiki.

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 (pdf) for details.

data

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

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

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)

Stan code

data {
  real alpha; 
  real beta; 
  real<lower=0> sigma2; 
  int<lower=0> J; 
  int y[J]; 
  vector[J] Z;
  int n[J]; 
} 

transformed data {
  real<lower=0> sigma; 
  sigma <- sqrt(sigma2); 
} 

parameters {
   real theta1; 
   real theta2; 
   vector[J] X; 
} 

model {
  real p[J];
  theta1 ~ normal(0, 32);   // 32^2 = 1024 
  theta2 ~ normal(0, 32); 
  X ~ normal(alpha + beta * Z, sigma);
  y ~ binomial_logit(n, theta1 + theta2 * X);
}

Beetles

Beetles considers dose-response data from an experiment applying carbon disulphide to 8 beetles. The original example compares three different link functions; the logit, probit and complementary log-log. Here, only the code for the logit link is shown. You can implement the other two link functions in greta by changing ilogit to iprobit or icloglog.

See WinBUGS examples volume 2 (pdf) for details.

data

x <- c(1.6907, 1.7242, 1.7552, 1.7842, 1.8113, 1.8369, 1.8610, 1.8839)
n <- c(59, 60, 62, 56, 63, 59, 62, 60)
r <- c(6, 13, 18, 28, 52, 53, 61, 60)
N <- 8

greta code

alpha_star = normal(0, 32)
beta = normal(0, 32)
p <- ilogit(alpha_star + beta * (x - mean(x)))
distribution(r) = binomial(n, p)

alpha <- alpha_star - beta * mean(x)
rhat <- p * n

BUGS/JAGS code

for( i in 1 : N ) {
  r[i] ~ dbin(p[i],n[i])
  logit(p[i]) <- alpha.star + beta * (x[i] - mean(x[]))
  rhat[i] <- n[i] * p[i]
  culmative.r[i] <- culmative(r[i], r[i])
}
alpha <- alpha.star - beta * mean(x[])
beta ~ dnorm(0.0,0.001)
alpha.star ~ dnorm(0.0,0.001)

Stan code

data {
    int<lower=0> N;
    int<lower=0> n[N];
    int<lower=0> r[N];
    vector[N] x;
}

transformed data {
    vector[N] centered_x;
    real mean_x;
    mean_x <- mean(x);
    centered_x <- x - mean_x;
}

parameters {
    real alpha_star;
    real beta;
}

transformed parameters {
    vector[N] m;
    m <- alpha_star + beta * centered_x;
}

model {
  alpha_star ~ normal(0.0, 1.0E4);  
  beta ~ normal(0.0, 1.0E4);
  r ~ binomial_logit(n, m);
}

generated quantities {
  real alpha; 
  real p[N];
  real llike[N];
  real rhat[N];
  for (i in 1:N)  {
    p[i] <- inv_logit(m[i]);
    llike[i]  <- r[i]*log(p[i]) + (n[i]-r[i])*log(1-p[i]);  
    rhat[i] <- p[i]*n[i];  // fitted values
  }
  alpha <- alpha_star - beta*mean_x;              
}