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I have data from 100 individuals doing a 2AFC under a control and a manipulated setting. I want to compare the means. I initially used a paired t-test. My supervisor noticed the distributions were not normal and recommended me using the Mann–Whitney U test instead of the t-test. However, as far as I can tell, the Mann–Whitney U test does not allow for paired data. Or does it? What non-parametric test should I use? What test generalizes the paired t-test? In other words, I have a non-normal distribution and I want to test how different its mean is from zero.

LBogaardt
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1 Answers1

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The non-parametric analog of the paired $t$-test is the Wilcoxon signed rank test.

gung - Reinstate Monica
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  • My data are "True"/"False" answers, averaged for both control and experimental over 36 trials. Each individual, therefore, has two numbers between 0 and 1 indicating the 'proportion correct'. – LBogaardt Aug 24 '14 at 15:57
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    In that case, the Wilcoxon is not the ideal test to use. For each participant you have 36 or 72 data, ie >2. When you reduce your data to 2 points by averaging, you are throwing data away. You would have better power by using a GLMM or GEE logistic regression (my answer here: Difference between generalized linear models & generalized linear mixed models in SPSS discusses the difference). – gung - Reinstate Monica Aug 24 '14 at 16:20
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    +1 but I'd usually avoid referring to the signed rank test simply as "the Wilcoxon" without further adornment, since there's more than one test associated with the name Wilcoxon. – Glen_b Aug 25 '14 at 01:34
  • That's a reasonable point, @Glen_b. In my mind there is the Mann-Whitney U-test & the Wilcoxon test, but I should probably be clearer since many don't use the names that way (frustratingly, R has both listed under wilcox.test()). – gung - Reinstate Monica Aug 25 '14 at 02:18
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    Well, Wilcoxon deserves credit somewhere for the Mann-Whitney, since the rank sum test is his invention and that and the U-test are equivalent. Mann and Whitney's contribution is substantial (Wilcoxon's tables were very limited for starters, to the point of being not very useful, and the whole U-statistics thing is a big deal), but Wilcoxon certainly has priority. – Glen_b Aug 25 '14 at 02:22
  • Of course one can also argue that still other people deserve some credit as well, but Mann and Whitney certainly credit Wilcoxon in their paper. – Glen_b May 27 '20 at 10:32
  • I'm really sorry, but I don't think this answers the question. OP specifically asks for a non-parametric test for a mean. The Wilcoxon signed-rank test doesn't test means without further assumptions. In that sense, I find it also questionable to call it an "analog". A permutation test for the mean is probably more suitable if OP specifically wants to test the average. (Btw, I upvoted the answer because I think it's still useful). – COOLSerdash Jan 25 '24 at 16:59
  • That's a very interesting point, @COOLSerdash. I would argue that WSR is standardly considered the nonparametric test that's analogous to the paired t-test. I do think it's a reasonable answer to the title question, but it doesn't address some of the additional details in the text of the Q (which, in turn, end up not being a fully accurate description of the OP's actual situation...). Maybe I will add some more information later. (Also, no need to apologize ;-). – gung - Reinstate Monica Jan 25 '24 at 17:16