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Is it correct to use the Wilcoxon signed ranks test for relative differences $\frac{Wx-Wy}{Wy}$, instead of absolute differences $Wx-Wy$? In some cases the relative difference is more relevant, for example if we want to compare the change in house prices I would argue that the percentage difference is a much better indicator than the absolute difference.

I need to compare the reaction time of participants before/after, and some participants have a really slow reaction time, meaning that even a (relatively) small improvement has a huge impact (since the absolute difference is big) while if someone with a good reaction time improves by 30% it gets lost in the data.

My question is that what is the correct way to compare relative differences?

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    $\begingroup$ Good question! We also see relative differences in epidemiology in the form of attributable risk, avoidable risk, population attributable risk, and population avoidable risk. $\endgroup$
    – Alexis
    Mar 22, 2022 at 17:53

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Yes, that's fine.

As long as you have one sample (either to begin with, or formed by taking differences between two columns of data, relative or otherwise), then the signed-rank test is appropriate. Usually the only assumptions of this single sample are that the r.v.s are iid and (usually) that they're continuous random variables. When these assumptions are satisfied, then you can find rejection regions by using quantiles of the asymptotic sampling distribution of the test statistic under the null hypothesis, or the from the exact sampling distribution.

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  • $\begingroup$ I'm sorry, my phrasing is a bit misleading. What I wanted to know if I could use Wilcoxon signed ranks test on a set of matched samples to prove if the reaction time is shorter after a certain type of training (+calculate the confidence interval). My question is if it's correct to use the relative differences between the two samples when testing the null hypothesis (that the median of the population of differences between the paired data is zero), or the test (using two-sample paired data) only works with absolute differences. $\endgroup$
    – gubi
    Mar 22, 2022 at 22:50
  • $\begingroup$ @gubi "or formed by taking differences between two columns of data, relative or otherwise" $\endgroup$
    – Taylor
    Mar 23, 2022 at 16:51
  • $\begingroup$ Thank You for the answer! I also found this (graphpad.com/guides/prism/latest/statistics/…) article about comparing ratios that recommends transforming the measurements to their logarithm, performing a paired samples t-test and calculating the antilogarithm of the result. This also could be used in my case to compare relative differences, although I could not find any other resource using it. Is this a valid method? $\endgroup$
    – gubi
    Apr 4, 2022 at 11:08

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