Kolmogorov-Smirnov test vs. Wilcoxon signed-rank test for two samples

Question

I would like to ask a question about the usage of Kolmogorov-Smirnov test and Wilcoxon signed-rank test to show that two samples have the same distribution (specifically, log-normal). Specifically, I have a sequence of measurements (at least 1000) and I would like to show that measurements have log-normal distributions. To do that, I want compute estimated mean and variance and use it in one of the tests.

In the literature, authors suggest to use Kolmogorov-Smirnov test. However, I would like to ask what if I use Wilcoxon signed-rank test to show that two samples have the same distribution? Can I use the resulting p-value to decide whether to reject $H_0$ hypothesis or not?

Thank you for your answer.

Answer 1

Nothing in your question indicates that a paired test like the Wilcoxon signed-rank test is appropriate.

KS would be a fairly standard to use for a two-sample distribution comparison. Wilcoxon is for checking location and can lack the sensitivity to distributions with the same location that differ in other ways, such as scale. Compare a standard normal and $t_{2.1}$ in a simulation. Here's some R code.

N <- 1000
times <- 1000
p.ks <- p.wi <- rep(NA,times)
set.seed(12)
for (i in 1:times){

    y0 <- rt(N,2.1)
    y1 <- rnorm(N,0,1)

    p.wi[i] <- wilcox.test(y0,y1)$p.value
    p.ks[i] <- ks.test(y0,y1)$p.value
}
length(p.ks[p.ks<=0.05])/times
length(p.wi[p.wi<=0.05])/times

Wilcoxon can pick up on differences other than median when you have very different shapes, however, such as if you take:

y0 <- rexp(N,1)
y1 <- rnorm(N,log(2,base=exp(1)),1)

Run that in the same simulation. KS winds up being more sensitive, but Wilcoxon still does well. If you drop the sample size down to 100, KS kick's Wilcoxon's butt.

In summary, use KS (or some other general distribution comparison, such as Anderson-Darling) when you're interested in overall distribution differences, and use Wilcoxon when you are specifically interested in location differences.

In your case, I would use KS to tell if your two samples come from the same population distribution. That KS test alone won't tell you if that distribution is lognormal, however, though KS can compare an empirical distribution to a population distribution. Be careful about specifying the parameters of the population distribution, however, as it is poor practice to use the estimated parameters, tempting as that seems. You should know beforehand what parameters your population distribution follows.