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Calculate Sample Size Needed to Compare 2 Means: 2-Sample Equivalence

This calculator is useful when we wish to test whether the means of two groups are equivalent, without concern of which group's mean is larger. Suppose we collect a sample from a group 'A' and a group 'B'; that is we collect two samples, and will conduct a two-sample test. For example, we may wish to test whether a new product is equivalent to an existing, industry standard product. Here, the 'burden of proof', so to speak, falls on the new product; that is, equivalence is actually represented by the alternative, rather than the null hypothesis.
H0:∣μA−μB∣≥δH_0:|\mu_A-\mu_B|\ge\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=n1n2\kappa=\frac{n_1}{n_2}

Formulas

This calculator uses the following formulas to compute sample size and power, respectively: nA=κnB   and   nB=(1+1κ)(σz1−α+z1−β/2∣μA−μB∣−δ)2 n_A=\kappa n_B \;\text{ and }\; n_B=\left(1+\frac{1}{\kappa}\right) \left(\sigma\frac{z_{1-\alpha}+z_{1-\beta/2}}{|\mu_A-\mu_B|-\delta}\right)^2
1−β=2[Φ(z−z1−α)+Φ(−z−z1−α)]−1,z=∣μA−μB∣−δσ1nA+1nB1-\beta= 2\left[\Phi\left(z-z_{1-\alpha}\right)+\Phi\left(-z-z_{1-\alpha}\right)\right]-1 \quad ,\quad z=\frac{|\mu_A-\mu_B|-\delta}{\sigma\sqrt{\frac{1}{n_A}+\frac{1}{n_B}}}
where κ=nA/nB\kappa=n_A/n_B is the matching ratio σ\sigma is standard deviation Φ\Phi is the standard Normal distribution function Φ−1\Phi^{-1} is the standard Normal quantile function α\alpha is Type I error β\beta is Type II error, meaning 1−β1-\beta is power δ\delta is the testing margin

R Code

1muA=5
2muB=4
3delta=5
4kappa=1
5sd=10
6alpha=0.05
7beta=0.20
8(nB=(1+1/kappa)*(sd*(qnorm(1-alpha)+qnorm(1-beta/2))/(abs(muA-muB)-delta))^2)
9ceiling(nB) # 108
10z=(abs(muA-muB)-delta)/(sd*sqrt((1+1/kappa)/nB))
11(Power=2*(pnorm(z-qnorm(1-alpha))+pnorm(-z-qnorm(1-alpha)))-1)

References

Chow, S.-C., Shao, J., Wang, H., and Lokhnygina, Y. (2018). Sample Size Calculations in Clinical Research, Third Edition. Chapman & Hall/CRC.

PASS 2023 Power Analysis and Sample Size Software (2023). NCSS, LLC. Kaysville, Utah, USA, ncss.com/software/pass.