Statistical & Clinical Analysis
Newcombe Hybrid Wilson Score Confidence Interval for Two-Proportion Risk Difference
Construct boundary-respecting 95% confidence limits for independent risk difference RD = p_1 - p_2 using Robert Newcombe’s (1998) Method 10 square-and-add Wilson score intervals.
Why FDA/EMA biostatisticians prefer Newcombe’s Method 10 over naive Wald RD ± z*SQRT(p1*q1/n1 + p2*q2/n2)!
Robert G. Newcombe (1998, Statistics in Medicine — benchmarking 11 confidence-interval methods for the difference between two independent proportions p_1 - p_2) showed that naive Wald intervals frequently overshoot [-1, +1] and suffer severe under-coverage near boundary proportions! Newcombe’s Method 10 ("square-and-add" MOVER) first computes single-arm Wilson score limits [l_1, u_1] for Arm 1 and [l_2, u_2] for Arm 2, then combines asymmetric arm half-widths in quadrature:
Lower RD = (p_1 - p_2) - SQRT((p_1 - l_1)^2 + (u_2 - p_2)^2) and Upper RD = (p_1 - p_2) + SQRT((u_1 - p_1)^2 + (p_2 - l_2)^2)—guaranteeing limits strictly inside [-1, +1] even when x_1 = 0 or x_2 = 0! For 18/50 (36.0%) vs 6/50 (12.0%, raw RD = +0.2400), Newcombe’s 95% CI is [+0.0727, +0.3926] (and [-0.1342, +0.1342] when both arms are 6/50)!
| Two-Arm Binomial Comparison (z = 1.96) | Point Risk Difference p_1 - p_2 | Newcombe Method 10 Lower 95% | Newcombe Method 10 Upper 95% |
|---|---|---|---|
| 18/50 (36%) vs 6/50 (12%) | +0.2400 (+24.0% ARD) | +0.0727 | +0.3926 (Excludes 0 -> Significant) |
| 6/50 (12%) vs 6/50 (12%) — Equal | 0.0000 | -0.1342 | +0.1342 (Symmetric Null) |
| Invalid Count x < 0, x > n, or n <= 0 | Error Guard | Error Guard | "Invalid Two-Arm Counts" |
Newcombe Method 10 Lower 95% Risk-Difference Confidence Limit
Evaluates (p_1 - p_2) - SQRT((p_1 - l_1)^2 + (u_2 - p_2)^2) using single-arm Wilson limits B7 (l_1) and B10 (u_2).
=IF(OR(B2 < 0, B3 <= 0, B2 > B3, B4 < 0, B5 <= 0, B4 > B5, B6 <= 0), "Invalid Two-Arm Counts", ROUND((B2 / B3 - B4 / B5) - SQRT(((B2 / B3 - B7) ^ 2) + ((B10 - B4 / B5) ^ 2)), 4))Newcombe Method 10 Upper 95% Risk-Difference Confidence Limit
Evaluates (p_1 - p_2) + SQRT((u_1 - p_1)^2 + (p_2 - l_2)^2) using Wilson limits B8 (u_1) and B9 (l_2).
=IF(OR(B2 < 0, B3 <= 0, B2 > B3, B4 < 0, B5 <= 0, B4 > B5, B6 <= 0), "Invalid Two-Arm Counts", ROUND((B2 / B3 - B4 / B5) + SQRT(((B8 - B2 / B3) ^ 2) + ((B4 / B5 - B9) ^ 2)), 4))Try the example
Use the original values or change them to test your understanding.