meta-analysis

# Meta-Analysis Models

## Fixed Effects Meta-Analysis

### Stan
```stan
data {
  int<lower=0> K;           // Number of studies
  vector[K] y;              // Effect estimates
  vector<lower=0>[K] se;    // Standard errors
}
parameters {
  real theta;               // Common effect
}
model {
  theta ~ normal(0, 10);
  y ~ normal(theta, se);
}
```

### JAGS
```
model {
  for (i in 1:K) {
    y[i] ~ dnorm(theta, prec[i])
    prec[i] <- pow(se[i], -2)
  }
  theta ~ dnorm(0, 0.0001)
}
```

## Random Effects Meta-Analysis

### Stan (Non-centered, recommended)
```stan
data {
  int<lower=0> K;
  vector[K] y;
  vector<lower=0>[K] se;
}
parameters {
  real mu;                  // Overall mean
  real<lower=0> tau;        // Between-study SD
  vector[K] eta;            // Study effects (standardized)
}
transformed parameters {
  vector[K] theta = mu + tau * eta;
}
model {
  // Priors
  mu ~ normal(0, 10);
  tau ~ cauchy(0, 0.5);     // Half-Cauchy
  eta ~ std_normal();

  // Likelihood
  y ~ normal(theta, se);
}
generated quantities {
  real theta_new = normal_rng(mu, tau);  // Predictive
  real I2 = square(tau) / (square(tau) + mean(square(se)));
}
```

### JAGS
```
model {
  for (i in 1:K) {
    y[i] ~ dnorm(theta[i], prec[i])
    prec[i] <- pow(se[i], -2)
    theta[i] ~ dnorm(mu, tau.theta)
  }

  mu ~ dnorm(0, 0.0001)
  tau.theta <- pow(sigma.theta, -2)
  sigma.theta ~ dunif(0, 10)

  # Heterogeneity
  tau2 <- pow(sigma.theta, 2)
}
```

## Binary Outcomes

### Stan (Log-Odds)
```stan
data {
  int<lower=0> K;
  array[K] int<lower=0> r1;   // Events in treatment
  array[K] int<lower=0> n1;   // Total in treatment
  array[K] int<lower=0> r2;   // Events in control
  array[K] int<lower=0> n2;   // Total in control
}
parameters {
  real d;                     // Overall log-OR
  real<lower=0> tau;
  vector[K] delta;            // Study-specific log-OR
  vector[K] mu;               // Baseline log-odds
}
model {
  d ~ normal(0, 10);
  tau ~ cauchy(0, 0.5);
  delta ~ normal(d, t