Statistical & Clinical Analysis
Peto One-Step Pooled Odds Ratio Formula for Rare-Event Clinical Trials
Pool rare-event and zero-cell randomized trials without arbitrary 0.5 continuity corrections using Richard Peto’s O - E score and hypergeometric variance V: OR_Peto = EXP(Σ(O - E) / ΣV).
Why Cochrane recommends Peto’s O - E method when event rates are < 1% and standard log-OR divides by zero!
When an adverse event or mortality endpoint occurs in 0, 1, or 2 patients per arm, Woolf’s log((a*d)/(b*c)) crashes on zero cells and +0.5 continuity corrections bias pooled estimates toward 1.0! Yusuf, Peto, Lewis, Collins, and Sleight (1985, Progress in Cardiovascular Diseases) introduced the exact hypergeometric score statistic for each 2×2 trial (a events in n_1 treated, c events in n_0 control, total events d = a + c, total N = n_1 + n_0): observed-minus-expected O_i - E_i = a - n_1 * d / N and hypergeometric variance V_i = n_1 * n_0 * d * (N - d) / (N^2 * (N - 1))!
Summing across k trials gives Peto’s one-step pooled Odds Ratio OR_Peto = EXP(Σ(O_i - E_i) / ΣV_i) and pooled Z-statistic Z = Σ(O_i - E_i) / SQRT(ΣV_i)! Across 3 balanced trials (2/120 vs 8/120; 1/150 vs 6/150; 3/200 vs 11/200), OR_Peto = 0.2821 with Z = -3.467 (and exact OR_Peto = 1.0000 when arm counts match)!
| Rare-Event RCT Evidence Set | Pooled Σ(O - E) & ΣV | Peto Pooled Odds Ratio OR_Peto | Pooled Z-Score |
|---|---|---|---|
| 3 RCTs (6/470 Treated vs 25/470 Control) | Σ(O - E) = -9.50 / ΣV = 7.503 | 0.2821 (71.8% Odds Reduction) | Z = -3.467 (p < 0.001) |
| Null Identical Event Counts Across Arms | Σ(O - E) = 0.00 | 1.0000 (Exact Null) | Z = 0.000 |
| Zero Total Events Across All Studies (ΣV = 0) | Undefined 0 / 0 | "No Informative Events Across Trials" | Zero-Information Guard |
Peto One-Step Pooled Odds Ratio OR_Peto = EXP(Σ(O - E) / ΣV)
Exponentiates the ratio of total O - E score SUM(F2:F4) to total hypergeometric variance SUM(G2:G4).
=IF(SUM(G2:G4) <= 0, "No Informative Events Across Trials", ROUND(EXP(SUM(F2:F4) / SUM(G2:G4)), 4))Peto Pooled Two-Sided Score Test Statistic Z = Σ(O - E) / SQRT(ΣV)
Computes the standard normal test statistic against the null hypothesis OR = 1.
=IF(SUM(G2:G4) <= 0, "No Informative Events Across Trials", ROUND(SUM(F2:F4) / SQRT(SUM(G2:G4)), 3))Try the example
Use the original values or change them to test your understanding.