# What's the appropriate test to be used in this study?

I was asked by an acquaintance of mine for support in interpreting a medical study. I don't work in healthcare, thus of course this was not professional advice.

Two groups of people of size $$n_1$$ and $$n_2$$ are subject to a transplant, and for each individual in each group a certain measurement $$X$$ is taken before the transplant, after 6 month from the transplant and after 24 months. Thus I have 6 samples, 3 of size $$n_1$$ and 3 of size $$n_2$$. If I'm not mistaken, this is a longitudinal study.

Now, let's focus on group 1: if I test for differences between the measurements of X at time $$t_0$$ and at time $$t_0+\text{6 months}$$, it seems to me that this is a paired samples case. I don't see how we could consider the two samples to be independent, given that they are literally the same persons. Thus we should really be using the Wilcoxon Signed Rank test, and not the Mann-Withey test (which is what they used on the study). Am I correct?

• Did they take the different between before and after and compare that difference between the two groups? What exactly went into the Mann-Whitney test?
– Dave
Commented Jan 31, 2020 at 12:25
• @Dave don't consider the two groups for now. Let's just focus on group 1, and on the two set of measurements, at time $t_0$ and time $t_0$+ 6 months. Which test would you use to determine if there's a difference in distribution between the two sets of measurements, Mann-Withey or Wilcoxon? Commented Jan 31, 2020 at 19:35
• You would use the paired test, but I don't think that's the question the study wanted to investigate. What exactly went into the Mann-Whitney test?
– Dave
Commented Jan 31, 2020 at 20:03
• The Wilcoxon Test and the Mann-Whitney Test are equivalent and both test the hypothesis that two independent samples stem from the same distribution. Typically what you want to do in such a setting is test whether the mean/median of the distribution of differences is 0, which you could do using a Wilcoxon Signed Rank Test on the vector of differences (one value for each patient).
– elmo
Commented Feb 6, 2020 at 9:51
• Ok thank you, that clarifies things. In this case I would say the answer to your question is "yes" :-)
– elmo
Commented Feb 6, 2020 at 13:16

As you correctly pointed out, the Mann-Whitney U-Test (which is equivalent to the regular Wilcoxon Test) requires independence of the two samples, which is very unlikely in the described setting (not impossible however; imagine a situation in which your measure $$X$$ depends only on the organ about to be transplanted and is not influenced by e.g. other organs, the patient’s lifestyle, etc.).