Compare 2 Proportions 2-Sample Equivalence | Power and Sample Size Calculators

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

$\kappa=n_A/n_B$ is the matching ratio
$\Phi$ 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
$\delta$ is the testing margin

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.