Non-Inferiority and Equivalence
A Farrington–Manning likelihood-score test for two independent proportions. Enter the observed successes and sample sizes, specify the prespecified non-inferiority margin, and obtain the score statistic, one-sided p-value, and confidence bound directly in your browser.
The calculator uses the Farrington–Manning likelihood-score test for a non-zero difference between two independent binomial proportions. Note that Farrington and Manning's likelihood-score test as one of its available non-inferiority test statistics for two proportions. The null hypothesis for the higher-is-better case is H0: p1 − p2 ≤ δ0, with rejection in the upper tail.
The observed difference is compared with the prespecified non-inferiority difference. The numerator is the observed difference minus the null difference, while the denominator uses constrained maximum-likelihood estimates under the restriction p̃1 − p̃2 = δ0.
Here p̂1 = x1/n1 and p̂2 = x2/n2. The constrained estimates p̃1 and p̃2 are the maximum-likelihood estimates subject to p̃1 − p̃2 = δ0. The one-sided p-value is 1 − Φ(zFM).
Non-inferiority is concluded when the one-sided p-value is below α. Equivalently, the lower confidence bound for p1 − p2 is above the non-inferiority margin. The confidence bound reported here uses z1−α; for the default α = 0.025, it is a one-sided 97.5% lower bound.
The constrained likelihood is solved directly for the restricted Group 2 proportion using a monotone score equation and bisection. The resulting restricted estimates are then used in the Farrington–Manning variance. This avoids relying on an external statistical library and remains stable near ordinary interior parameter values.
Farrington, C. P., & Manning, G. (1990). Test statistics and sample size formulae for comparative binomial trials with null hypothesis of non-zero risk difference or non-unity relative risk. Statistics in Medicine, 9(12), 1447–1454. DOI: 10.1002/sim.4780091208.
the software, LLC. this method Sample Size Software: Non-Inferiority Tests for the Difference Between Two Proportions, Chapter 210. the relevant methodological literature lists the Farrington–Manning likelihood-score statistic and gives the corresponding constrained-estimate formulation.
Independent implementation check: the CRAN PropTestR documentation gives the example x1=82, n1=100, x2=85, n2=100, δ0=−0.10, alternative="greater", with z=1.3198 and p=0.09345. This calculator reproduces those values.