Each simulated trial reports a log risk ratio yi with standard error SEi. A meta-analysis pools them into one estimate, weighting each study by the inverse of its variance so precise (large, low-SE) trials count more:
Fixed-effect: w_i = 1/SE_i²
pooled = Σ(w_i·y_i) / Σw_i
Var(pooled) = 1/Σw_i
Heterogeneity: Q = Σ w_i·(y_i − pooled_FE)², df = k−1
I² = max(0, (Q−df)/Q) × 100%
τ²_DL = max(0, (Q−df) / (Σw_i − Σw_i²/Σw_i))
Random-effects: w_i* = 1/(SE_i² + τ²_DL)
pooled_RE = Σ(w_i*·y_i) / Σw_i*
- Fixed-effect assumes every trial estimates the same true effect and differences are pure sampling noise — it gives narrow trials almost all the weight.
- Random-effects (DerSimonian–Laird) assumes each trial estimates its own true effect drawn from a distribution with variance τ², adding that extra uncertainty to every weight — the more heterogeneous the trials, the more the model flattens the weighting toward equal shares and widens the pooled CI.
- I² is the percentage of total variation across studies caused by real heterogeneity rather than chance — under 25% is usually called low, over 75% high.
- The funnel plot (bottom-left panel) plots each study's effect against its standard error; in the absence of bias, points scatter symmetrically inside the pseudo-95% triangle around the pooled estimate — asymmetry is a classic red flag for publication bias or small-study effects.
- The τ² slider controls how much the trials' true underlying effects are made to disagree before sampling noise is added, letting you watch I², the gauge and the funnel spread all diverge in real time.
This is the exact inverse-variance / DerSimonian–Laird machinery behind every published Cochrane-style forest plot pooling clinical trial results.