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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 the proportion in group 'A', $p_A$, is non-inferior/superior to the proportion in group 'B', $p_B$.
We collect a sample from both groups, and thus will conduct a two-sample test.
The idea is that statistically significant differences between the proportions 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

$H_1:p_A-p_B>\delta$

where $\delta$ is the superiority or non-inferiority margin and the ratio between the sample sizes of the two groups is

This calculator uses the following formulas to compute sample size and power, respectively:
$$
n_A=\kappa n_B \;\text{ and }\;
n_B=\left(\frac{p_A(1-p_A)}{\kappa}+p_B(1-p_B)\right)
\left(\frac{z_{1-\alpha}+z_{1-\beta}}{p_A-p_B-\delta}\right)^2$$

$$1-\beta=
\Phi\left(z-z_{1-\alpha/2}\right)+\Phi\left(-z-z_{1-\alpha/2}\right)
\quad ,\quad z=\frac{p_A-p_B-\delta}{\sqrt{\frac{p_A(1-p_A)}{n_A}+\frac{p_B(1-p_B)}{n_B}}}$$
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 to implement these functions:

pA=0.85 pB=0.65 delta=-0.10 kappa=1 alpha=0.05 beta=0.20 (nB=(pA*(1-pA)/kappa+pB*(1-pB))*((qnorm(1-alpha)+qnorm(1-beta))/(pA-pB-delta))^2) ceiling(nB) # 25 z=(pA-pB-delta)/sqrt(pA*(1-pA)/nB/kappa+pB*(1-pB)/nB) (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 90.

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