How to simulate significant data

Question

I would like to simulate significant distributions in R: For that I would like to specify a common M and SD (Group 1 in combined with Group 2), sample size and the effect size (Cohen's d) between two groups and get data that gets significant with a t-test when the effect size is sufficient. Can someone give me some advice on how to do this?

Answer 1

We'll going to assume the same SD in each group and equal sample sizes (if you want to relax these assumptions you'll have to do the algebra yourself!), so pooled SD=within-group SD. Cohen's D is (M2-M1)/SD, or (delta_M/SD), so delta_M = SD*Cohen_D. M1=M-delta_M/2, M2=M+delta_M/2.

M <- 2         # grand mean
C <- 2         # Cohen's D
S <- 1         # pooled SD
delta_M <- S*C
m <- c(M-delta_M/2,M+delta_M/2)     # specify means for each group
n <- c(20,20)   # sample size for each group
set.seed(101)
## generate Normal deviates for each group
grp1 <- rnorm(n[1],mean=m[1],sd=S)
grp2 <- rnorm(n[2],mean=m[2],sd=S)
## run t-test
t.test(grp1,grp2)

---

    Welch Two Sample t-test

data:  grp1 and grp2
t = -7.1805, df = 37.503, p-value = 1.511e-08
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -2.682429 -1.502160
sample estimates:
mean of x mean of y 
0.9026826 2.9949771