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Compare 2 Means: 2-Sample Non-Inferiority or Superiority

This calculator is useful for the types of tests known as non-inferiority and superiority tests. Whether the null hypothesis represents 'non-inferiority' or 'superiority' depends on the context and whether the non-inferiority/superiority margin, δ\delta, is positive or negative. In this setting, we wish to test whether the mean in group 'A', μA\mu_A, is non-inferior/superior to the mean in group 'B', μB\mu_B. We collect a sample from both groups, and thus will conduct a two-sample test. The idea is that statistically significant differences between the means may not be of interest unless the difference is greater than a threshold, δ\delta. This is particularly popular in clinical studies, where the margin is chosen based on clinical judgement and subject-domain knowledge. The hypotheses to test are H0:μA−μB≤δH_0: \mu_A - \mu_B \le \delta
H1:μA−μB>δH_1: \mu_A - \mu_B > \delta
, where δ\delta is the superiority or non-inferiority margin and the ratio between the sample sizes of the two groups is κ=nA/nB\kappa = n_A/n_B

Formulas

nA=κnB   and   nB=(1+1κ) n_A=\kappa n_B \;\text{ and }\; n_B=\left(1+\frac{1}{\kappa}\right)
(σz1−α+z1−βμA−μB−δ)21−β=Φ(z−z1−α)+Φ(−z−z1−α),z=μA−μB−δσ1nA+1nB\left(\sigma\frac{z_{1-\alpha}+z_{1-\beta}}{\mu_A-\mu_B-\delta}\right)^2 1-\beta = \Phi\left(z-z_{1-\alpha}\right)+\Phi\left(-z-z_{1-\alpha}\right) \quad ,\quad z=\frac{\mu_A-\mu_B-\delta}{\sigma\sqrt{\frac{1}{n_A}+\frac{1}{n_B}}}

R Code

1muA=5
2muB=5
3delta=5
4kappa=1
5sd=10
6alpha=0.05
7beta=0.20
8(nB=(1+1/kappa)*(sd*(qnorm(1-alpha)+qnorm(1-beta))/(muA-muB-delta))^2)
9ceiling(nB) # 50
10z=(muA-muB-delta)/(sd*sqrt((1+1/kappa)/nB))
11(Power=pnorm(z-qnorm(1-alpha))+pnorm(-z-qnorm(1-alpha)))

References

Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series. page 61.