# Power of the t-test under unequal sample sizes

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?

(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)
 0.184

mean(power9010<.05)
 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.