In this post, we perform a short numerical simulation to compare the statistical power of the Mann-Whitney U test and the Brunner-Munzel test under normality for various sample sizes and significance levels.

### Simulation design

We conduct a simulation according to the following scheme:

- Enumerate various pairs of the significance level \(\alpha\) and the sample size \(n\)
- Enumerate various effect sizes \(ES\) from \(0.1\) to \(2.0\)
- For each combination of the above parameters, we generate \(50\,000\) pairs of random samples of size \(n\): one from \(\mathcal{N}(0, 1)\) and one from \(\mathcal{N}(ES, 1)\). For each pair, we perform both statistical tests (one-tailed) and get the p-value. Next, we calculate the statistical power for each test based on the given value of \(\alpha\)

### Simulation results

Here are the results for some values of \(\alpha\) and \(n\):

As we can see, in the *presented* simulations,
the Brunner-Munzel test has higher statistical power than the Mann-Whitney U test
(especially for small \(\alpha\) and small \(n\)).
However, it’s just a single simulation, so we can’t derive a generic conclusion about which test is better.
In future posts, I will explore the behavior of these tests in more contexts.