Stan code

Model description

The common-effect meta-analysis model assumes all studies estimate the same true effect $\mu$ . Observed differences between study estimates arise solely from sampling error. There is no between-study heterogeneity parameter.

Mathematical specification

Likelihood:

$y_i \mid \mu \sim \mathcal{N}(\mu,\, s_i^2), \quad i = 1, \ldots, k$

Prior:

$\mu \sim \mathcal{N}(0,\, 1)$

Derived quantity:

$b\_\text{Intercept} = \mu$

The posterior is analytically tractable (conjugate Gaussian) but bayesma fits it via MCMC for consistency with the rest of the pipeline.

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

parameters {
  real mu;
}

model {
  target += normal_lpdf(mu | 0, 1);
  target += normal_lpdf(y  | mu, se);
}

generated quantities {
  real b_Intercept = mu;
}

How bayesma calls this model

bayesma() selects this model when model_type = "common_effect". The data list passed to Stan contains:

list(
  N  = nrow(data),
  y  = data$yi,       # pre-computed effect estimates
  se = data$sei       # pre-computed standard errors
)

Internally bayesma_stan_data() computes $y_i$ and $s_i$ from the raw outcome columns (events, means, SDs, sample sizes) before passing them to Stan.

Parameterisation notes

The prior $\mu \sim \mathcal{N}(0, 1)$ is weakly informative on the log-OR or SMD scale. For studies on other scales (e.g., raw mean differences in physical units), the prior should be widened via mu_prior = normal(0, sigma) where $\sigma$ reflects the plausible range of effects.
The common-effect model has a single parameter. With $k \geq 5$ , the likelihood dominates the prior and the posterior is largely data-driven.
No random-effects structure; no $\tau$ parameter; no non-centred parameterisation needed.

Identifiability

The model is fully identified with a single observation ( $k = 1$ ). As $k$ increases, the posterior for $\mu$ concentrates around the precision-weighted average of the $y_i$ .

Known sampling difficulties

None. The posterior is unimodal and near-Gaussian. MCMC mixes rapidly with the default settings.