Stan code

Model description

The Gaussian random-effects model is the standard Bayesian meta-analysis model. Study-level true effects $\theta_i$ are drawn from a normal distribution with mean $\mu$ and standard deviation $\tau$ . Observed effects $y_i$ are noisy realisations of $\theta_i$ .

Mathematical specification

Likelihood:

$y_i \mid \theta_i \sim \mathcal{N}(\theta_i,\, s_i^2)$

Random effects:

$\theta_i \sim \mathcal{N}(\mu,\, \tau^2) \quad \Longleftrightarrow \quad \theta_i = \mu + \tau z_i, \quad z_i \sim \mathcal{N}(0, 1)$

Priors:

$\mu \sim \mathcal{N}(0,\, 1), \qquad \tau \sim \text{Half-Cauchy}(0,\, 0.5)$

Derived quantities:

$u_i = \tau z_i \quad (\text{study-level deviation}), \qquad b\_\text{Intercept} = \mu$

data {
  int<lower=1> N;
  int<lower=1> K;
  vector[N] y;
  vector<lower=0>[N] se;
  array[N] int<lower=1> study;
}

parameters {
  real mu;
  real<lower=0> tau;
  vector[K] z;
}

transformed parameters {
  vector[K] u = tau * z;
}

model {
  target += normal_lpdf(mu  | 0, 1);
  target += cauchy_lpdf(tau | 0, 0.5);
  target += std_normal_lpdf(z);

  target += normal_lpdf(y | mu + u[study], se);
}

generated quantities {
  real b_Intercept = mu;
}

How bayesma calls this model

This is the default for model_type = "random_effect" with re_dist = "normal". The Stan data list includes the study index array, mapping each row in the data to a study. For two-stage meta-analysis, each study contributes one row; $N = K$ . For one-stage models with multiple arms per study, $N > K$ .

The default prior on $\tau$ is Half-Cauchy(0, 0.5). This places substantial prior mass below $\tau = 1$ while allowing larger values. It can be changed with tau_prior:

bayesma(data, model_type = "random_effect", tau_prior = half_normal(0, 0.25))

Non-centred parameterisation

The z parameterisation separates $\tau$ from the shape of the random effects. When $\tau$ is near zero, the centred parameterisation (using u directly) creates a funnel-shaped posterior where the $u_i$ must all approach zero together as $\tau \to 0$ . The NCP avoids this by sampling $z$ on a fixed $\mathcal{N}(0,1)$ scale.

Identifiability and small- $k$ behaviour

When $k < 5$ , $\tau$ is weakly identified. The posterior for $\tau$ reflects the prior more heavily. In this case:

Report the sensitivity of conclusions to the $\tau$ prior.
Consider using a more informative $\tau$ prior from an external source (e.g., Turner et al. 2015).
The posterior mean of $\mu$ is still a valid estimator, but its credible interval may be misleading.

Known sampling difficulties

The funnel-shaped posterior near $\tau = 0$ can cause divergent transitions in the centred parameterisation. The NCP (used by default) resolves this in most cases. If divergences persist:

Increase adapt_delta to 0.99.
Place a more informative prior on $\tau$ (e.g., half_normal(0, 0.5) instead of half_cauchy(0, 0.5)).

Stan code — Gaussian random-effects model