Meta-Analysis Forest Plot: Pooling Clinical Trials

Trial Cohort

Number of studies 8

True effect (log RR) μ -0.30

Between-study heterogeneity τ² 0.05

Pooling Model

Pooled Result

Pooled log RR

0.00

Pooled RR

1.00

95% CI (log RR)

[0.00, 0.00]

I² heterogeneity

Q (df, p)

0.0

Study estimate — box size ∝ weight Pooled diamond — width = 95% CI Line of no effect (log RR = 0)

How it works

Each simulated trial reports a log risk ratio y_i with standard error SE_i. 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 τ² slider controls how much the trials' true underlying effects are made to disagree before sampling noise is added, letting you watch I² and the two pooling models diverge in real time.

This is the exact inverse-variance / DerSimonian–Laird machinery behind every published Cochrane-style forest plot pooling clinical trial results.

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