Sample Size, $n_B$

Power, $1-\beta$

Type I error rate, $\alpha$

X-axis

min

max

Documentation
Test 1 Mean
Compare 2 Means
Compare k Means
Test 1 Proportion
Compare 2 Proportions
Compare Paired Proportions
Compare k Proportions
Test Time-To-Event Data
Test Odds Ratio
Test Relative Incidence in Self Controlled Case Series Studies
Other

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

$H_0:\mu_A-\mu_B=0$
$H_1:\mu_A-\mu_B\neq0$
.
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

### R Code

R code to implement these functions:

muA=5
muB=10
kappa=1
sd=10
alpha=0.05
beta=0.20
(nB=(1+1/kappa)*(sd*(qnorm(1-alpha/2)+qnorm(1-beta))/(muA-muB))^2)
ceiling(nB) # 63
z=(muA-muB)/(sd*sqrt((1+1/kappa)/nB))
(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.