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I have two dependent samples that I want to compare using the Wilcoxon signed rank test. I've now run into a problem with a few of my variables. The difference is significant so I want to know the direction of the difference. Most instructions I found state that you can use the median for interpreting the direction of difference. However, the medians are the same. So I read that I could use the mean rank as well and interpret it in the following way:

A) variable_1 < variable_2: mean rank = 34.17

B) variabel_1 > variable_2: mean rank = 28.94

-> the value for A) is higher than for B) so variable_2 is higher than variable_1

In most cases, a high mean rank correlates with a high sum of ranks but not all the time. Can I still use the mean rank to interpret my results or could you recommend me another measure I could use for interpretation?

Example:

variable_1: Mdn = 4, mean rank = 34.17, sum of ranks = 410.00

variable_2: Mdn = 4, mean rank = 28.94, sum of ranks = 1360.00

If I can clarify anything, let me know. This is my first time posting on this forum and my first time working with statistics :)

Thanks so much for your help!

Eileen

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2 Answers 2

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The Wilcoxon signed-rank test tests whether the probability that a randomly chosen pair of observations sums to a positive number with probability 0.5, i.e., whether or not it is equally likely that their sum is positive as it is negative. If $s$ is the average of the signed ranks, the estimate of this probability is $\frac{s}{n+1} - \frac{1}{2}$. So interpret the signed rank test in terms of a probability, estimated by this estimator.

See section 7.2.1 of BBR.

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It is recommended to use rank-biserial correlation with the signed-rank test to estimate the effect size. You did not provide all the information needed to calculate the correlation. Try to calculate it following the example on the Wikipedia page. Then, add the necessary information to the question and I could check the calculation. The information needed is the sum of the signed ranks and the number of pairs in your sample.

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    $\begingroup$ At present it is more of a comment than an answer by our standards. Can you expand on it? $\endgroup$
    – T.E.G.
    Jan 30, 2021 at 20:48
  • $\begingroup$ Hi, thank you for your comment! Could you please elaborate on why I should estimate the effect size? For now I only want to interpret the direction of difference (without using the median because I don't have unique medians). Can I use the rank-biserial correlation for that? $\endgroup$
    – Eileen Fee
    Jan 31, 2021 at 9:07

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