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Test Relative Incidence in Self Controlled Case Series Studies

This calculator is useful for tests concerning whether the proportions in two groups are different.
Suppose the two groups are 'A' and 'B', and we collect a sample from both groups -- i.e. we have two samples.
We perform a two-sample test to determine whether the proportion in group A, $p_A$, is different from the proportion in group B, $p_B$.
The hypotheses are

$H_1:p_A-p_B\neq0$

where the ratio between the sample sizes of the two groups is

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

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

R code to implement these functions:

pA=0.65 pB=0.85 kappa=1 alpha=0.05 beta=0.20 (nB=(pA*(1-pA)/kappa+pB*(1-pB))*((qnorm(1-alpha/2)+qnorm(1-beta))/(pA-pB))^2) ceiling(nB) # 70 z=(pA-pB)/sqrt(pA*(1-pA)/nB/kappa+pB*(1-pB)/nB) (Power=pnorm(z-qnorm(1-alpha/2))+pnorm(-z-qnorm(1-alpha/2)))

Chow S, Shao J, Wang H. 2008.Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series.page 89.

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