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Test 1 Proportion

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Test Time-To-Event Data

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Test Relative Incidence in Self Controlled Case Series Studies

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

$H_1:p-p_0>\delta$

and $\delta$ is the superiority or non-inferiority margin.

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 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)))

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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