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I am working on comparisons between two time points 2013 and 2019, I have 180 subjects. I have a numeric "score" from 2013 and 2019. I calculated the average score at both time points and would like to assess significance. I've run a paired T-test on this, however when I check the normality assumption it is not there. So I moved onto using the Wilcoxon test for non parametric data. I suppose my main question is, reading a few things online it appears the T-test tests if the average score from 2013 is significantly different from the 2019 average. Does the Wilcoxon test do the same? A few of the things I've read seem to indicate it compares median?

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  • $\begingroup$ The term "Wilcoxon test" tends to be reserved for an unpaired test: Wilcoxon Mann-Whitney U. What exactly do you want to do? Do you indeed have pairing (such as before and after on the same subjects)? Is your lack of normality on the pairwise differences? // The Wilcoxon Mann-Whitney U test has a funky null hypothesis. There are advocates of using Wilcoxon as a test of means, however, when normality conditions are not met. $\endgroup$
    – Dave
    Commented Aug 11, 2021 at 18:54
  • $\begingroup$ Yes it is before and after (the same survey administered to the same 180 people once in 2013 and once in 2019), I have been using Wilcoxon signed-rank test, at least I thought I was with: wilcox.test(before1, after1, paired = TRUE). To assess the normality I conducted a Shapiro Wilks test on the differences (2013 scores-2019 scores) $\endgroup$
    – user228897
    Commented Aug 11, 2021 at 19:00
  • $\begingroup$ Normality testing with a test like Shapiro-Wilk is less helpful than one would hope. $\endgroup$
    – Dave
    Commented Aug 11, 2021 at 19:03
  • $\begingroup$ This was an interesting thread, thanks Dave. I'm wondering though, what is the appropriate path forward? $\endgroup$
    – user228897
    Commented Aug 11, 2021 at 19:17
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    $\begingroup$ Something else to keep in mind is that, if the normality assumptions are violated, differences in means might not even be interesting anymore or at least not all that is interesting. Consider $exp(1)$ and $N(1, 1)$. Those have the same mean and even the same variance, so testing either should not result in rejections of such equality, yet those are rather different distributions. $\endgroup$
    – Dave
    Commented Aug 11, 2021 at 21:54

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Wilcoxon rank test tells you whether there is a location shift between two samples. Wilcoxon signed-rank test tests is for one sample only and tests the null hypothesis $\mu = \mu_0$ for $\mu_0$ of your choice

So, if you wanted to run Wilcoxon rank test on your sample then you can see if the samples are shifted positively or negatively

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