I heard that the power of a t-test used with unequal samples is limited by the size of the smaller sample. Can I take this to mean that the power of a t-test with unequal sample sizes is equal to the power of a t-test used on equal sample sizes where n equals the size of the smaller sample?

**Answer**

(*Note, by n, I usually mean the total sample size, so I interpret your last sentence to be ‘where \bf{.5}n equals the size of the smaller sample’*.)

No, not quite. Consider this simulation (conducted with R):

```
set.seed(9)
power1010 = vector(length=10000)
power9010 = vector(length=10000)
for(i in 1:10000){
n1a = rnorm(10, mean=0, sd=1)
n2a = rnorm(10, mean=.5, sd=1)
n1c = rnorm(90, mean=0, sd=1)
n2c = rnorm(10, mean=.5, sd=1)
power1010[i] = t.test(n1a, n2a, var.equal=T)$p.value
power9010[i] = t.test(n1c, n2c, var.equal=T)$p.value
}
mean(power1010<.05)
[1] 0.184
mean(power9010<.05)
[1] 0.323
```

What we see here is that when the **total** sample size is 20, with equal group sizes, n_1=n_2=10, power is 18\%; but when the **total** sample size is 100, but the smaller group has n_2=10, power is 32\%. Thus power can increase when the size of the larger group goes up even though the smaller sample size stays the same.

This answer is adapted from my answer here: How should one interpret the comparison of means from different sample sizes?, which you will probably want to read for more on this topic.

**Attribution***Source : Link , Question Author : Jimj , Answer Author : Community*