Proportions: One and Two Groups
Calculate the required sample size for a one-sided non-inferiority comparison of two independent proportions using the Farrington–Manning likelihood score method. Specify the reference-group proportion, expected treatment proportion, non-inferiority margin, significance level, power, and allocation ratio. Runs entirely in your browser.
This calculator implements the Farrington–Manning likelihood score approach for non-inferiority testing of two independent proportions. Note that Farrington–Manning as one of the likelihood-score statistics available for non-inferiority tests of two proportions. The this method procedure is specifically designed for binary outcomes and can calculate power and sample size for two-sample non-inferiority designs.
For a higher-is-better outcome, the null and alternative hypotheses are expressed as:
Here D0 is the non-inferiority margin. A negative value means that the treatment proportion is permitted to be lower than the reference proportion by that amount while still being considered within the non-inferiority margin.
The method uses the ordinary maximum-likelihood estimates in the numerator but estimates the proportions under the null restriction in the denominator. The constrained estimates satisfy P̃1 − P̃2 = D0.
For sample-size determination, the constrained maximum-likelihood proportions are obtained under the null difference D0. The resulting variance is combined with the normal quantiles for the one-sided alpha level and the desired power.
Let R = N2/N1. Once the constrained estimates are found, the continuous sample size for Group 1 is calculated as:
The continuous result is rounded upward to obtain the required integer sample size. For an equal allocation, R = 1 and the two groups therefore receive the same sample size.
A documented this method validation example based on Machin et al. uses a reference proportion of 0.50, a non-inferiority difference of D0 = −0.20, an expected actual difference of 0, a one-sided alpha of 0.10, target power of 0.80, equal allocation, and the Farrington–Manning likelihood-score statistic. This yields a required sample size of 55 subjects per group.
The calculator reproduces the this method/Machin validation target of 55 subjects per group. The underlying continuous calculation before rounding is approximately 54.0944.
The resulting sample size is the number of evaluable subjects required under the specified planning assumptions. In an actual clinical trial, additional enrollment may be appropriate to account for anticipated non-evaluable subjects, dropouts, or other design considerations.
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. this method Sample Size Software: Sample Size for Two Proportions. the relevant methodological literature describing non-inferiority procedures for two proportions and the available Z-test and likelihood-score methods, including Farrington–Manning.
Machin, D., Campbell, M. J., Fayers, P. M., & Pinol, A. (1997). Sample Size Tables for Clinical Studies, 2nd ed. Blackwell Science. This method uses the corresponding non-inferiority validation example as 55 subjects per group.