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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 odds of an outcome in group 'A', $p_A(1-p_A)$, is non-inferior/superior to the odds of the outcome in group 'B', $p_B(1-p_B)$, where $p_A$ and $p_B$ are the probabilities of the outcome in the two groups.
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.
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

$H_1:\ln(OR)>\delta$

where $\delta$ is the superiority or non-inferiority margin on the log scale, 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:
$$
n_A=\kappa n_B \;\text{ and }\;
n_B=\left(\frac{1}{\kappa p_A(1-p_A)}+\frac{1}{p_B(1-p_B)}\right)
\left(\frac{z_{1-\alpha}+z_{1-\beta}}{\ln(OR)-\delta}\right)^2$$

$$1-\beta=
\Phi\left(z-z_{1-\alpha}\right)+\Phi\left(-z-z_{1-\alpha}\right)
\quad ,\quad z=\frac{(\ln(OR)-\delta)\sqrt{n_B}}{\sqrt{\frac{1}{\kappa p_A(1-p_A)}+\frac{1}{p_B(1-p_B)}}}$$

where $$OR=\frac{p_A(1-p_B)}{p_B(1-p_A)}$$ and
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=0.40 pB=0.25 delta=0.20 kappa=1 alpha=0.05 beta=0.20 (OR=pA*(1-pB)/pB/(1-pA)) # 2 (nB=(1/(kappa*pA*(1-pA))+1/(pB*(1-pB)))*((qnorm(1-alpha)+qnorm(1-beta))/(log(OR)-delta))^2) ceiling(nB) # 242 z=(log(OR)-delta)*sqrt(nB)/sqrt(1/(kappa*pA*(1-pA))+1/(pB*(1-pB))) (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 107.

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