Statistical & Clinical Analysis
Agresti-Caffo "+1 Success, +1 Failure per Arm" Risk Difference 95% CI Formula
Replace degenerate small-sample Wald 2-proportion confidence intervals with Alan Agresti & Brian Caffo’s (2000) adjusted interval: add 1 pseudo-success and 1 pseudo-failure to EACH arm (p_tilde = (x + 1) / (n + 2)) before computing Wald bounds!
Why Wald’s 2-proportion CI collapses completely at zero events—and how Agresti & Caffo’s "+4 total" adjustment fixes coverage!
Just as Agresti & Coull (1998) rescued single-proportion confidence intervals by adding 2 successes and 2 failures, Alan Agresti and Brian Caffo (2000, The American Statistician) discovered a remarkably simple cure for the notorious under-coverage of two-independent-sample Risk Difference intervals (p_1 - p_2) in small clinical trials, pilot studies, and A/B tests—especially when one arm observes 0 events (x_2 = 0/15, where naive Wald variance p_2*(1-p_2)/n_2 collapses to 0)!
Agresti & Caffo proved that adding exactly ONE pseudo-success and ONE pseudo-failure to EACH independent group—forming p_1,tilde = (x_1 + 1) / (n_1 + 2) and p_2,tilde = (x_2 + 1) / (n_2 + 2) (4 pseudo-observations total across the 2x2 table)—achieves near-nominal 95% coverage across virtually all sample sizes and true rates! The 95% Agresti-Caffo interval is (p_1,tilde - p_2,tilde) ± 1.95996 * SQRT(p_1,tilde*(1 - p_1,tilde)/(n_1 + 2) + p_2,tilde*(1 - p_2,tilde)/(n_2 + 2))!
| Pilot Trial Data (Arm 1: 6/20 = 30.0% vs Arm 2: 0/20 = 0.0%) | Adjusted Difference p_1,tilde - p_2,tilde | 95% Agresti-Caffo Bounds [Lower, Upper] |
|---|---|---|
| 6/20 vs 0/20 (Zero-Event Control Arm!) | 7/22 - 1/22 = +0.2727 (+27.27%) | [+0.0609, +0.4846] (Excludes 0 Cleanly!) |
| 3/20 vs 1/20 (Sparse Events) | 4/22 - 2/22 = +0.0909 (+9.09%) | [-0.1095, +0.2913] (Includes 0 — Non-Significant) |
Agresti-Caffo (2000) Lower 95% Confidence Bound on Risk Difference (p_1 - p_2)
Computes p_1,tilde = (B2 + 1)/(B3 + 2) and p_2,tilde = (B4 + 1)/(B5 + 2), then evaluates the lower 95% confidence limit.
=IF(OR(B3 <= 0, B5 <= 0, B2 < 0, B2 > B3, B4 < 0, B4 > B5), "Invalid 2-Sample Counts", ROUND(((B2 + 1) / (B3 + 2) - (B4 + 1) / (B5 + 2)) - 1.959964 * SQRT(((B2 + 1) / (B3 + 2)) * (1 - (B2 + 1) / (B3 + 2)) / (B3 + 2) + ((B4 + 1) / (B5 + 2)) * (1 - (B4 + 1) / (B5 + 2)) / (B5 + 2)), 4))Agresti-Caffo (2000) Upper 95% Confidence Bound on Risk Difference (p_1 - p_2)
Evaluates the upper 95% Agresti-Caffo confidence limit.
=IF(OR(B3 <= 0, B5 <= 0, B2 < 0, B2 > B3, B4 < 0, B4 > B5), "Invalid 2-Sample Counts", ROUND(((B2 + 1) / (B3 + 2) - (B4 + 1) / (B5 + 2)) + 1.959964 * SQRT(((B2 + 1) / (B3 + 2)) * (1 - (B2 + 1) / (B3 + 2)) / (B3 + 2) + ((B4 + 1) / (B5 + 2)) * (1 - (B4 + 1) / (B5 + 2)) / (B5 + 2)), 4))Try the example
Use the original values or change them to test your understanding.