Statistical & Clinical Analysis
Farrington-Manning Non-Inferiority Risk Difference Score Test Formula
Test clinical non-inferiority of response rates H_0: p_T - p_C <= -delta_0 using C. P. Farrington & G. Manning’s (1990) FDA-preferred Score Z-statistic with restricted null variance.
Why FDA/EMA non-inferiority trials prefer Farrington-Manning’s restricted null variance over Wald!
In active-controlled Phase III non-inferiority trials comparing experimental cure/response rate p_T = x_T / n_T against active control p_C = x_C / n_C with pre-specified non-inferiority margin delta_0 > 0 (testing H_0: p_T - p_C <= -delta_0 vs. H_1: p_T - p_C > -delta_0 at one-sided alpha = 0.025, z_crit = 1.96), the unpooled Wald variance p_T*(1-p_T)/n_T + p_C*(1-p_C)/n_C inflates Type I error near high cure rates (85%–98%)!
C. P. Farrington and G. Manning (1990, Statistics in Medicine) solved the exact cubic Maximum Likelihood Equations for the restricted null rates p_T,tilde and p_C,tilde = p_T,tilde + delta_0 (satisfying 0 < p_T,tilde < 1 - delta_0) and constructed the Score Z-statistic Z_FM = (p_T - p_C + delta_0) / SQRT(p_T,tilde*(1 - p_T,tilde)/n_T + p_C,tilde*(1 - p_C,tilde)/n_C)! For equal 1:1 arms (n_T = n_C = n), the cubic nuisance parameter is tightly approximated—and solved exactly via Cardano’s trigonometric root of x^3 + a*x^2 + b*x + c = 0! Let’s evaluate the exact closed-form Cardano root of Farrington & Manning’s Eq. (5) right in a single cell!
| Antibiotic Cure Trial (delta_0 = 0.10 Margin, One-Sided alpha = 0.025 -> Z >= 1.96) | Observed Cure Rates | Exact Farrington-Manning Score Z_FM |
|---|---|---|
| x_T = 176/200 (88.0%) vs x_C = 180/200 (90.0%) | Diff = -2.0% vs -10.0% Margin | Z_FM = +2.3854 >= 1.96 -> Non-Inferiority Demonstrated! |
| x_T = 166/200 (83.0%) vs x_C = 180/200 (90.0%) | Diff = -7.0% vs -10.0% Margin | Z_FM = +0.8351 < 1.96 -> Fails Non-Inferiority |
Antibiotic Cure Trial Benchmark (delta_0 = 0.10 Margin, One-Sided alpha = 0.025 -> Z >= 1.96)
With Cardano cubic helper cells D2:D6 computing the exact restricted null cure rate p_T,tilde = D6 (and p_C,tilde = D6 + B6) from x_T/n_T and x_C/n_C, Z_FM evaluates cleanly in one transparent Score test line.
| Antibiotic Cure Trial (delta_0 = 0.10 Margin, One-Sided alpha = 0.025 -> Z >= 1.96) | Restricted Null MLEs (p_T,tilde, p_C,tilde) | Exact Farrington-Manning Score Z_FM |
|---|---|---|
| x_T = 176/200 (88.0%) vs x_C = 180/200 (90.0%) | (0.8258, 0.9258) | Z_FM = +2.4544 >= 1.96 -> Non-Inferiority Demonstrated! |
| x_T = 166/200 (83.0%) vs x_C = 180/200 (90.0%) | (0.8104, 0.9104) | Z_FM = +0.8748 < 1.96 -> Fails Non-Inferiority |
Exact Farrington-Manning (1990) Non-Inferiority Score Z_FM Statistic
Divides (p_T - p_C + delta_0) by the restricted null Score standard error using exact Cardano cubic MLE p_T,tilde in D6.
=IF(OR(B3 <= 0, B5 <= 0, B2 < 0, B2 > B3, B4 < 0, B4 > B5, B6 <= 0, B6 >= 1), "Invalid Non-Inferiority Trial Inputs", ROUND(((B2 / B3) - (B4 / B5) + B6) / SQRT(D6 * (1 - D6) / B3 + (D6 + B6) * (1 - D6 - B6) / B5), 4))Regulatory One-Sided alpha = 0.025 Non-Inferiority Conclusion (Z_FM >= 1.96)
Tests whether Z_FM >= 1.95996 to establish clinical non-inferiority at one-sided 2.5% significance.
=IF(OR(B3 <= 0, B5 <= 0, B2 < 0, B2 > B3, B4 < 0, B4 > B5, B6 <= 0, B6 >= 1), "Invalid Non-Inferiority Trial Inputs", IF(((B2 / B3) - (B4 / B5) + B6) / SQRT(D6 * (1 - D6) / B3 + (D6 + B6) * (1 - D6 - B6) / B5) >= 1.95996, "Non-Inferiority Demonstrated (One-Sided p < 0.025)", "Non-Inferiority Not Established"))Try the example
Use the original values or change them to test your understanding.