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 Proportions: 2-Sample, 2-Sided Equality

This calculator is useful for tests concerning whether the proportions in 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 proportion in group A, $p_A$, is different from the proportion in group B, $p_B$. The hypotheses are

$H_0:p_A-p_B=0$
$H_1:p_A-p_B\neq0$
.
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: $$n_A=\kappa n_B \;\text{ and }\; n_B=\left(\frac{p_A(1-p_A)}{\kappa}+p_B(1-p_B)\right) \left(\frac{z_{1-\alpha/2}+z_{1-\beta}}{p_A-p_B}\right)^2$$
$$1-\beta= \Phi\left(z-z_{1-\alpha/2}\right)+\Phi\left(-z-z_{1-\alpha/2}\right) \quad ,\quad z=\frac{p_A-p_B}{\sqrt{\frac{p_A(1-p_A)}{n_A}+\frac{p_B(1-p_B)}{n_B}}}$$
where

### R Code

R code to implement these functions:

pA=0.65
pB=0.85
kappa=1
alpha=0.05
beta=0.20
(nB=(pA*(1-pA)/kappa+pB*(1-pB))*((qnorm(1-alpha/2)+qnorm(1-beta))/(pA-pB))^2)
ceiling(nB) # 70
z=(pA-pB)/sqrt(pA*(1-pA)/nB/kappa+pB*(1-pB)/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 89.