Is it still Wilcxon test recommended for non-nomal data in the U.S.?

Question

This is a general and longstanding question in elementary statistics (, not a question about JMP itself). It comes from my personal interest. This is not a question from users.

I see the following suggestion in JMP's Statistics Knowledge Portal.

https://www.jmp.com/en_ch/statistics-knowledge-portal/t-test/two-sample-t-test.html

What if you know the underlying measurements are not normally distributed? Or what if your sample size is large and the test for normality is rejected? In this situation, you can use nonparametric analyses. These types of analyses do not depend on an assumption that the data values are from a specific distribution. For the two-sample t­-test, the Wilcoxon rank sum test is a nonparametric test that could be used.

I personally think that Welch's test is OK also for non-normality data.

* If the sample size is large (or not small), Welch's test is robust to non-normality.
* If you are interested in the shift of location parameters, the Wilcoxon test needs the more strict assumption (i.e., F_1(x) = F_2(x-theta)) for population distributions than Welch's test.

I myself would like to recommend Welch's test (without testing variances or normality) if analysts are interested only in the difference of means, and they don't care about other parameters.

And I recommend Student's t test if your H0 is "two distributions are identital" or if your experiment is a randomized control experiment (because standard t-test is an approximation of permutation test with using data score. See Lehmann's Testing Statistical Hypotheses).

The Wilcoxon test is often described as a nonparametric test, but it depends heaviliy on the assumptions on population distributions if your H0 is "Two location parameters are equal". In that sense, when your H0 is "H0:Two location parameters are equal", Welch's test is more nonparametric and Wilcoxon's test is more parametric.

In Bradley Joens-san's Design of Experiments book, I think the robustness of Welch's t-test is emphasized, and they recommend variable transformation (in manufacturing domain).

If you also want to read a philosophical critic for this kind of "flow chart" or "cook-book" decisions, See the chapter 1 in McElreath,R.(2020:2nd ed.) Statistical Rethinking (2nd ed.), CRC Press.

yusuke_ono · Answer

Thank you for your advice.

I didn't care about power when I wrote the question in my first entry. In Hollander and Wolfe's Nonparametric Statiscal Methods, they set the following "location-shift model".

G(t) = F(t - Delta) (p.107, (4.2))

Their H0 is "H0: Delta = 0" under the above assumption. Although this framework is introduced in many cook books (including Hollander and Wolfe), I personally think this location-shift assumption is too strict and strong in practice.

(Note:Actually, Wilcoxon's test is invariant for monotone transformation, we can weaken the above assumption.)

Welch's test doesn't need the above location-shift assumption.

If we can assume the location-shift assumption, then we can compare powers of Welch's t-test(or usual t-test) and Wilcoxon's test. The Wilcoxon test is more powerful than t-test when the distribution is logistic distribution, and almost the same power when the distribution is normal (although it's a little bit lower).

Note that,if your H0 is "Two distributions are identital", then both Wilcoxon's test and standard t-test become valid.

yusuke_ono · Answer

Thank you for your suggestion.

Inspired by your experiment, I asked the following question. I don't believe ChatGPT, but I think their reply to the following question is reasonable. (We always need to ask statisitcal questions to statisticians!)

yusuke_ono · Answer

Thank you  for your reply.  I have the same opinion and impression for Welch's test.