# Comparison between groups with unequal sample sizes

I have calculated modulation indices (scalar) of neuronal responses for two categories of neurons $x, y$ recorded from two hemispheres $a, b$, with $n_{xa}=120, n_{xb}=80, n_{ya}=20, n_{yb}=20$. I would like to know whether each category of neurons has significantly different modulation indices between the two hemispheres. Since I am not sure whether modulation indices are normally distributed I took the conservative approach and chose a non-parametric Wilcoxon rank sum test. I found that ${xa}$ is significantly different to ${xb}$ but ${ya}$ is not different to ${yb}$.

However, given the small sample size of ${ya}$ and ${yb}$ I worry that the outcome might be a result of a Type II error.

To account for the unequal sample sizes I came up with the idea to draw 20 samples from ${xa}$ and 20 from ${xb}$ (with replacement) for e.g. 1000 bootstraps, use for each sample the Wilcoxon statistic and come up with a distribution of p-values. Then calculate whether this distribution is significantly different to p=0.05.

Do you find this approach valid? Any other ideas or suggestions would be appreciated.