Statistical Model

For study (i = 1,\ldots,k), the default model is a normal-normal Bayesian random-effects model. The likelihood is marginalised over the latent study effects for more stable small-(k) sampling:

[ y_i \sim \mathrm{Normal}\left(\mu, \sqrt{s_i^{2 + \tau}2}\right), \qquad \mu \sim \mathrm{Normal}(0, \sigma_\mu), \qquad \tau \sim \mathrm{HalfNormal}(\sigma_\tau). ]

Here (y_i) is the observed model-scale effect and (s_i) is its known standard error. The pooled mean is (\mu), and (\tau) is between-study heterogeneity.

Posterior study-level (\theta_i) draws are exposed as derived quantities using the Normal conditional distribution implied by (y_i), (\mu), (\tau), and (s_i).

The model also samples future_true_effect, representing the latent true effect for a comparable future study. It excludes sampling error, so it is not a prediction for the next observed study estimate.

Small-sample stance

When study counts are small, posterior intervals, prior sensitivity, and uncertainty scenario sensitivity are more informative than a single headline pooled estimate.

prior_diagnostics.csv includes a fixed-effect precision approximation for the Normal prior on mu and prior-predictive checks for the tau prior. Treat warnings as prompts to inspect prior sensitivity, not as formal pass/fail tests.