Calculate Sample Size Needed to Test 1 Proportion: 1-Sample Non-Inferiority or Superiority

This calculator is useful for the types of tests known as non-inferiority and superiority tests. Whether the null hypothesis represents 'non-inferiority' or 'superiority' depends on the context and whether the non-inferiority/superiority margin, $\delta$, is positive or negative. In this setting, we wish to test whether a proportion, $p$, is non-inferior/superior to a reference value, $p_0$. The idea is that statistically significant differences between the proportion and the reference value may not be of interest unless the difference is greater than a threshold, $\delta$. This is particularly popular in clinical studies, where the margin is chosen based on clinical judgement and subject-domain knowledge. The hypotheses to test are

$H_0:p-p_0\le\delta$
$H_1:p-p_0>\delta$
and $\delta$ is the superiority or non-inferiority margin.

Formulas

This calculator uses the following formulas to compute sample size and power, respectively: $$n=p(1-p)\left(\frac{z_{1-\alpha}+z_{1-\beta}}{p-p_0-\delta}\right)^2$$
$$1-\beta= \Phi\left(z-z_{1-\alpha}\right)+\Phi\left(-z-z_{1-\alpha}\right) \quad ,\quad z=\frac{p-p_0-\delta}{\sqrt{\frac{p(1-p)}{n}}}$$ where

• $n$ is sample size
• $p_0$ is the comparison value
• $\Phi$ 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:

p=0.5
p0=0.3
delta=-0.1
alpha=0.05
beta=0.20
(n=p*(1-p)*((qnorm(1-alpha)+qnorm(1-beta))/(p-p0-delta))^2)
ceiling(n) # 18
z=(p-p0-delta)/sqrt(p*(1-p)/n)
(Power=pnorm(z-qnorm(1-alpha))+pnorm(-z-qnorm(1-alpha)))

References

Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series. page 86.

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