Trying to understand when to use which test, am I correct when saying both these tests, test whether two samples could be taken from the same distribution, but the difference is that the student's t-test assumes that the distributions that are tested have a normal distribution (Gauss distribution), while the Wilcoxon signed-rank test accept any type of distribution?

  • 1
    $\begingroup$ Are you referring to the paired or the independent sample situation? $\endgroup$ – Michael M Jun 13 '20 at 15:18

There are three t-tests:

  • One sample
  • Independent samples
  • Paired samples

Since Wilcoxon signed rank is about paired samples, I will assume you are comparing it to a paired sample t-test.

The paired sample t-test does not assume the distributions are normal, it assumes the differences are normal. Then it sees if the differences are different from 0 (usually; it is possible to change this); essentially. Wilcoxon, on the other hand, tests whether the mean ranks are different.

Neither one tests the rather general proposition that you state: "from the same population". For instance, suppose you have two samples, both from populations with mean = 10, but one with sd = 2, the other with sd = 100. Neither t nor Wilcoxon will show a difference:

x1 <- rnorm(1000, 10, 2)  #Normal, mean = 10, sd = 2, N = 1000
x2 <- rnorm(1000, 10, 10) #Normal, mean = 10, sd = 1-0, N = 1000

t.test(x1, x2, paired = TRUE) #t = -0.32, df = 999, p = 0.53
wilcox.test(x1, x2, paired = TRUE) #V = 243625 p = 0.40

You could even have much more different distributions:

x3 <- runif(1000, 0, 20)
t.test(x1, x3, paired = TRUE) #t = 1.25, df = 999, p = 0.21
wilcox.test(x1, x3, paired = TRUE) #V = 242525 p = 0.40

But if you do plot any pair of variables, you will see they are very different. E.g.

boxplot(x1, x2)
abline(a = 0, b = 1)

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.