Documentation

Test 1 Mean

Compare 2 Means

Compare k Means

Test 1 Proportion

Compare 2 Proportions

Compare Paired Proportions

Compare k Proportions

Test Time-To-Event Data

Test Odds Ratio

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

$H_1:\mu_A-\mu_B>\delta$

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:
$$
n_A=\kappa n_B \;\text{ and }\;
n_B=\left(1+\frac{1}{\kappa}\right)
\left(\sigma\frac{z_{1-\alpha}+z_{1-\beta}}{\mu_A-\mu_B-\delta}\right)^2$$

$$1-\beta=
\Phi\left(z-z_{1-\alpha}\right)+\Phi\left(-z-z_{1-\alpha}\right)
\quad ,\quad z=\frac{\mu_A-\mu_B-\delta}{\sigma\sqrt{\frac{1}{n_A}+\frac{1}{n_B}}}$$
where

- $\kappa=n_A/n_B$ is the matching ratio
- $\sigma$ is standard deviation
- $\Phi$ 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:

muA=5 muB=5 delta=5 kappa=1 sd=10 alpha=0.05 beta=0.20 (nB=(1+1/kappa)*(sd*(qnorm(1-alpha)+qnorm(1-beta))/(muA-muB-delta))^2) ceiling(nB) # 50 z=(muA-muB-delta)/(sd*sqrt((1+1/kappa)/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 61.

comments powered by Disqus