Sample Size, $n_A$

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, 1-Sided

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.