Compare 2 Means 2-Sample, 1-Sided | Power and Sample Size Calculators

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, $\mu_A$, is different from the mean in group B, $\mu_B$. The hypotheses are

$H_0:\mu_A=\mu_B$
$H_1:\mu_A\lt \mu_B$.
or $H_0:\mu_A=\mu_B$
$H_1:\mu_A\gt \mu_B$.
where the ratio between the sample sizes of the two groups is
$$\kappa=\frac{n_B}{n_A}$$

Formulas

This calculator uses the following formulas to compute sample size and power, respectively: $$n_A=\left(\sigma_A^2+\sigma_B^2/\kappa\right)\left(\frac{z_{1-\alpha}+z_{1-\beta}}{\mu_A-\mu_B}\right)^2$$
$$n_B=\kappa n_A$$
$$1-\beta=\Phi\left(\frac{|\mu_A-\mu_B|\sqrt{n_A}}{\sqrt{\sigma_A^2+\sigma_B^2/\kappa}}-z_{1-\alpha}\right)$$ where

$\kappa=n_A/n_B$ is the matching ratio
$\sigma$ is standard deviation
$\sigma_A$ is standard deviation in Group "A"
$\sigma_B$ is standard deviation in Group "B"
$\Phi$ is the standard Normal distribution function
$\Phi^{-1}$ is the standard Normal quantile function
$\alpha$ is Type I error
$\beta$ is Type II error, meaning $1-\beta$ is power

R Code

R code to implement these functions:

muA=132.86
muB=127.44
kappa=2
sdA=15.34
sdB=18.23
alpha=0.05
beta=0.20
(nA=(sdA^2+sdB^2/kappa)*((qnorm(1-alpha)+qnorm(1-beta))/(muA-muB))^2)
ceiling(nA) # 85
z=(muA-muB)/sqrt(sdA^2+sdB^2/kappa)*sqrt(nA)
(Power=pnorm(z-qnorm(1-alpha)))
## Note: Rosner example on p.303 is for 2-sided test.
## These formulas give the numbers in that example
## after dividing alpha by 2, to get 2-sided alpha.

References

Rosner B. 2010. Fundamentals of Biostatistics. 7th Ed. Brooks/Cole. page 302 and 303.

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