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 Equivalence

This calculator is useful when we wish to test whether the proportions in two groups are equivalent, without concern of which group's proportion is larger. Suppose we collect a sample from a group 'A' and a group 'B'; that is we collect two samples, and will conduct a two-sample test. For example, we may wish to test whether a new product is equivalent to an existing, industry standard product. Here, the 'burden of proof', so to speak, falls on the new product; that is, equivalence is actually represented by the alternative, rather than the null hypothesis.

$H_0:|p_A-p_B|\ge\delta$
$H_1:|p_A-p_B|<\delta$

where $\delta$ is the superiority or non-inferiority margin and 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: $$n_A=\kappa n_B \;\text{ and }\; n_B=\left(\frac{p_A(1-p_A)}{\kappa}+p_B(1-p_B)\right) \left(\frac{z_{1-\alpha}+z_{1-\beta/2}}{|p_A-p_B|-\delta}\right)^2$$
$$1-\beta= 2\left[\Phi\left(z-z_{1-\alpha}\right)+\Phi\left(-z-z_{1-\alpha}\right)\right]-1 \quad ,\quad z=\frac{|p_A-p_B|-\delta}{\sqrt{\frac{p_A(1-p_A)}{n_A}+\frac{p_B(1-p_B)}{n_B}}}$$
where

### R Code

R code to implement these functions:

pA=0.65
pB=0.85
delta=0.05
kappa=1
alpha=0.05
beta=0.20
(nB=(pA*(1-pA)/kappa+pB*(1-pB))*((qnorm(1-alpha)+qnorm(1-beta/2))/(abs(pA-pB)-delta))^2)
ceiling(nB) # 136
z=(abs(pA-pB)-delta)/sqrt(pA*(1-pA)/nB/kappa+pB*(1-pB)/nB)
(Power=2*(pnorm(z-qnorm(1-alpha))+pnorm(-z-qnorm(1-alpha)))-1)

### References

Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series. page 91.

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