Independant Two-Samples t-test with N > 30

Question

I have two groups of data (N > 30 in each group). I want to know if the two groups differ from each other. So I thought about making an independent t-test. But then I remembered that the groups must be normally distributed in order to perform a t-test. So I performed a Shapiro test and it appears that both groups are not normally distributed.

My question is: Am I still allowed to perform a t-test knowing that my groups are not normally distributed and that I have N > 30 in each group?

Thank you for your time.

Edit: When I do a QQ-plot on R with package ggpubr, both groups look like this (group 1 first, then group 2):

It seems that there are some flat "levels" in both cases.

Here are the histograms (group 1 first, then group 2):

Answer 1

Here are fictitious data, sampled using R, somewhat similar to your data.

set.seed(1234)
x1 = c(rnorm(50,100, 2), rnorm(140, 125, 5), rnorm(40,155,3))
x2 = c(rnorm(50,105, 2), rnorm(150, 125, 5), rnorm(40,156,2))
summary(x1); length(x1); sd(x1)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  95.31  118.30  124.70  125.01  131.94  163.76 
[1] 230
[1] 18.12978
summary(x2); length(x2); sd(x2)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  98.53  117.40  124.34  125.83  130.64  160.12 
[1] 240
[1] 16.01616

par(mfrow=c(1,2))
 hist(x1, br=5, col="skyblue2")
 hist(x2, br=5, col="skyblue2")
par(mfrow=c(1,1))

x = c(x1,x2)
g = c(rep(1,230),rep(2,240))
boxplot(x~g, notch=T, col="skyblue2", horizontal=T)

Notches in the sides of boxplots are nonparametric CIs roughly calibrated so that overlapping CIs suggest no significant difference in location.

Shapes of the samples are similar, so a nonparametric 2-sample Wilcoxon rank sum test should test for a significant difference in location. The P-value $0.75 > 0.05 = 5\%$ indicates no significant difference.

wilcox.test(x~g)

        Wilcoxon rank sum test 
        with continuity correction

data:  x by g
W = 27134, p-value = 0.7518
alternative hypothesis: 
 true location shift is not equal to 0

If these were the actual data, I would stop here. Of course, your data may show different results.

For the reasons you mention, I would not trust a 2-sample t test to give reliable results, but for such large sample sizes its P-value may be roughly correct. Again, no indication of a significant difference.

t.test(x~g, var.eq=T)  # pooled t test

        Two Sample t-test

data:  x by g
t = -0.51847, df = 468, p-value = 0.6044
alternative hypothesis: 
 true difference in means is not equal to 0
95 percent confidence interval:
 -3.914830  2.280273
sample estimates:
mean in group 1 mean in group 2 
       125.0149        125.8322 

Even though I would not trust a pooled t statistic to have Student's t distribution in these circumstances, the t statistic does seem a reasonable way to measure a difference between means of the two samples.

A permutation test using the pooled t statistic as metric, does not find a significant difference, its P-value is about the same as for the pooled t test. No significant difference.

set.seed(2021)
t.obs = t.test(x~g, var.eq=T)$stat
t.prm = replicate(10^5, t.test(x~sample(g), var.eq=T)$stat)
mean(abs(t.prm) > abs(t.obs))
[1] 0.6094  # P-value of simulated permutation test

hdr = "Simulated Permutation Dist'n"
hist(t.prm, prob=T, col="skyblue2", main=hdr)
abline(v = c(t.obs,-t.obs), col="red", lwd=2)