## HyLown Power and Sample Size Calculators

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PowerAndSampleSize.com.

For either a non-inferiority (when $\delta>0$) or a superiority (when $\delta<0$) test, where

$H_0:p_0-p_1\le\delta$ versus $H_1:p_0-p_1>\delta$

where $\delta$ is the superiority or non-inferiority margin and the ratio between the sample sizes of the two groups is
$$\kappa=\frac{n_1}{n_2}$$

Sample Size

Power

X-axis

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This sample size calculator uses the following formula: $$n_1=\kappa n_2 \;\text{ and }\; n_2=\left(\frac{p_0(1-p_0)}{\kappa}+p_1(1-p_1)\right) \left(\frac{\Phi^{-1}(\alpha)+\Phi^{-1}(\beta)}{p_0-p_1-\delta}\right)^2$$ where

• $n$ is sample size
• $p_1$ is the rate, proportion, or probability to be tested
• $p_0$ is the comparison value
• $\kappa=n_1/n_2$ is the matching ratio
• $\Phi^{-1}$ 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 to implement this function:

p0=0.85
p1=0.65
delta=-0.10
kappa=1
alpha=0.05
beta=0.20
(n=ceiling((p0*(1-p0)/kappa+p1*(1-p1))*((qnorm(alpha)+qnorm(beta))/(p0-p1-delta))^2)) # 25

Reference: Chow, page 90