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?

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

Leave a Comment