# Power calculations

I am using a toy example to explain power calculations for a t-test

Let's say we have two populations where we measure a certain parameter. Population A has a mean of 0 and population B has a mean of 3, both with a sd of 2. The effect size is 1.5

I use the pwr package in R to do the power calculation as such

library(pwr)

pwr.t.test(n = NULL, d = 2/3, sig.level = 0.05,
power = 0.9, type = "two.sample",
alternative = "two.sided")

Two-sample t test power calculation

n = 48.26427
d = 0.6666667
sig.level = 0.05
power = 0.9
alternative = two.sided

NOTE: n is number in *each* group


So, R is suggesting 48 observations per group, to get a power of 0.9

If I repeat the calculation using G*Power, however, I get this, which suggests 11 observations per group!

Finally, if I use this online tool from the University of British Columbia I get 10.

I can understand that rounding errors may bring you from 10 to 11... but why is R giving me such a high value?

Indeed, if I do a little simulation, it looks like 10 or 11 observation give you a power of ~90%

n <- 10

res <- sapply(1:1000, function(x)
{
a <- rnorm(n, 0, 2)
b <- rnorm(n, 3, 2)
res <- t.test(a, b, alternative = "two.sided")
res$p.value }) table(res<=0.05) FALSE TRUE 115 885 <-- 88.5% power  n = 48 always gives me a power of 100%... am I missing something? • In R, you use an effect size of 2/3 = 0.66 whereas in G*Power you use 1.5. If you use$d=1.5$in pwr.t.test, I get an n of$10.4$which, rounded up is$11\$ and coincides with G*Power. – COOLSerdash Apr 25 at 11:30
• +1 for a very nicely written question. – Stephan Kolassa Apr 25 at 11:32

In your example using R, you're using an effect size of $$d=2/3\approx0.6667$$ whereas in G*Power, you're using an effect size of $$d=1.5$$. Repeating the calculations in R using an effect size of $$1.5$$ results in comparable results:

library(pwr)

pwr.t.test(n = NULL, d = 3/2, sig.level = 0.05,
power = 0.9, type = "two.sample",
alternative = "two.sided")

Two-sample t test power calculation

n = 10.40147
d = 1.5
sig.level = 0.05
power = 0.9
alternative = two.sided

NOTE: n is number in *each* group


As you would round the sample size up, both R and G*Power suggest using a sample size of $$11$$ per group.

• I knew it was a silly mistake! I wrote 2/3 instead of 3/2... thank you for spotting that! – nico Apr 25 at 11:46