Open
Power Samplesize

Menu

Inputs

Calculate Sample Size Needed to Compare 2 Means: 2-Sample, 2-Sided Equality

This calculator is useful for tests concerning whether the means of two groups are different. Suppose the two groups are 'A' and 'B', and we collect a sample from both groups -- i.e. we have two samples. We perform a two-sample test to determine whether the mean in group A, μA\mu_A, is different from the mean in group B, μB\mu_B. The hypotheses are
H0:μA−μB=0H_0: \mu_A - \mu_B = 0
H1:μA−μB≠0H_1: \mu_A - \mu_B \neq 0
where the ratio between the sample sizes of the two groups is \kappa=\frac{n_A}{n_B}

Formulas

This calculator uses the following formulas to compute sample size and power, respectively: n_A=\kappa n_B \;\text{ and }\; n_B=\left(1+\frac{1}{\kappa}\right) \left(\sigma\frac{z_{1-\alpha/2}+z_{1-\beta}}{\mu_A-\mu_B}\right)^2 1-\beta= \Phi\left(z-z_{1-\alpha/2}\right)+\Phi\left(-z-z_{1-\alpha/2}\right) \quad ,\quad z=\frac{\mu_A-\mu_B}{\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

R Code

1R code to implement these functions:
2
3muA=5
4muB=10
5kappa=1
6sd=10
7alpha=0.05
8beta=0.20
9(nB=(1+1/kappa)*(sd*(qnorm(1-alpha/2)+qnorm(1-beta))/(muA-muB))^2)
10ceiling(nB) # 63
11z=(muA-muB)/(sd*sqrt((1+1/kappa)/nB))
12(Power=pnorm(z-qnorm(1-alpha/2))+pnorm(-z-qnorm(1-alpha/2)))

References

Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series. page 58.