# Statistical Power in t tests with Unequal Group Sizes

When performing Student’s t-test to compare difference in means between two group, it is a useful exercise to determine the effect of unequal sample sizes in the comparison groups on power. Large imbalances generally will not have adequate statistical power to detect even large effect sizes associated with a factor, leading to a high Type II error rate as shown in the figure below:

To jusity this reasoning I performed a power analysis for different group sizes. I considered the following group sizes:

1. n1 = 28, n2 = 1406: n1 represents 2% of the entire sample size of 1434
2. n1 = 144, n2 = 1290: n1 represents 10% of the entire sample size of 1434
3. n1 = 287, n2 = 1147: n1 represents 20% of the entire sample size of 1434
4. n1 = 430, n2 = 1004: n1 represents 30% of the entire sample size of 1434
5. n1 = 574, n2 = 860: n1 represents 40% of the entire sample size of 1434
6. n1 = 717, n2 = 717: equal size groups (this is optimal because it leads to the highest power for a given effect size)

In the figure above we plotted the power curves for the $t$-test, as a function of the effect size, assuming a Type I error rate of 5%.

## Code

Here is the code used to produce the above plot