# Pseudo median from Wilcoxon signed rank test larger than sample median?

I am left wondering, why I can be in a situation where the pseudo median (population estimator) of a Wilcoxon signed rank test, can be larger than the simple differences between sample medians? I am performing a paired signed rank test, since my data is dependent.

I have uploaded my dataset here: Dataset as a .csv file.

I am using R to test this, with the following "standard" code:

wilcox.test(SARD, SBP, alternative = "two.sided", mu=0, conf.int=T, conf.level = 0.99, paired = TRUE)


The sample medians are: GICS=22.1% and SBP=20.5%, i.e. indicating that the simple difference is 1.6%. When running the wilcoxon test, I am getting a pseudo median LARGER than this of 1.9% (statistically significant at 1%) which I do not intuitively understand how is even possible..

Can someone help my understanding?

• Because the pseudo-median is not the same as the median? What kind of information are you seeking? (It's difficult to guess what you need and to guess what you might find intuitive) – Glen_b May 7 at 13:10
• Incidentally I am getting an error when I try to download your data. It may be a problem my end but it's probably worth double-checking that you can download it. – Glen_b May 7 at 13:13
• I updated the link, to a working one. What I would like to postulate is, how much better on average the SBP is, relative to GICS, by looking at the median value (lower is better). – Philip May 7 at 13:21
• I don't follow what you mean, sorry (particularly by "on average"); if you want to look at medians, why not do that? I have attempted an intuitive explanation of how it's possible for the two to differ. – Glen_b May 7 at 13:27

Consider the following data:

 y    x
1   1.4
2   1.6
9    3


The difference in medians m(y)-m(x) = 0.4

The pair-differences $$y_i-x_i$$ are -0.4, 0.4 and 6, and the pairwise averages (including the self-averages) are 0, 2.8, 3.2, -0.4, 0.4, 6, for which the median is 1.6; hence the pseudo-median (1.6) is quite different to the difference in medians (0.4) for this example.

How did I construct it? Mainly making the x-sample not too far from symmetric (/mildly skew) and the y-sample more heavily skew in order to get a skewed set of pair-differences; this pulls the pair-averages up relative to the differences in median.

My guess is you'll probably see some skewness in your pair-differences as well.

Looking at your data, this is indeed the case -- the pair differences are somewhat skew, with more of a tail to the right.

• Alright, I can actually see your point; so no matter the statistical "power" of the Wilcoxon signed rank test, the pseudo median is not necessarily bounded between the two sample medians; because skewness of the samples plays a role? Hope I understood it correctly. :) – Philip May 7 at 13:38
• No, even with near symmetry you wouldn't have the pseudo-median of pair differences bounded between the two medians, but instead it should be close to the difference in medians (consider samples with medians of 100 and 101 and a small spread around those; the pseudo-median of pair differences will not be anywhere near 100, but it might be more-or-less near 1). But with fairly asymmetric pair-differences, you don't generally expect them to be close. None of this is much related to power (at least not directly). – Glen_b May 7 at 13:40
• Okay, that actually makes much more sense. Naturally, I was led to believe that it would be bounded. Do you have any literature I can study, specific to my question? – Philip May 7 at 13:47
• I am quite sure it will be bounded (it's a function of the data, it should be possible to establish some kind of bounds from the data), but it cannot be bounded by the two medians themselves (i.e. the medians cannot in general be upper and lower bounds for the pseudo-median). Even when the difference in medians exactly equals the pseudo-median, the two medians won't be likely to bracket their own difference (you probably meant to say something else instead). – Glen_b May 7 at 13:51