[R] Wilcoxon signed-rank test

Tue Aug 22 18:20:30 CEST 2017

Hi,

I am using wilcox.test function to test the difference between the means of
two samples. The data points are paired, so I am using a paired test.

There is one strange case. Sample A has a higher mean than a sample B.
However, wilcox.test function says that sample B has a significantly higher
"mean rank" than sample A. How is it possible?

Here is the code (data file is attached):
df <- read.table("wilcox_data.txt", head=TRUE)
mean(df$A)
[1] 0.7987849
mean(df$B)
[1] 0.7977966
mean(df$C)
[1] 0.6350737

wilcox.test(df$B, df$A, paired=TRUE, alternative="greater")
        Wilcoxon signed rank test with continuity correction

data:  df$B and df$A
V = 134300, p-value = 3.299e-05
alternative hypothesis: true location shift is greater than 0

wilcox.test(df$C, df$A, paired=TRUE, alternative="greater")
        Wilcoxon signed rank test with continuity correction

data:  df$C and df$A
V = 41423, p-value = 1
alternative hypothesis: true location shift is greater than 0

The p-value of the first test is rather low (3.299e-05), which indicates
that the alternative hypothesis is true - sample B has a higher "mean rank"
than sample A. Just to make sure I am not doing a dumb mistake, I added a
third variable C to this example, which is much smaller than A or B. As
expected, the second test has p-value = 1, which means that "mean rank" of
C is lower than A (null hypothesis is true).

I am afraid, I am not very strong in statistics, but I would very much
appreciate if someone could explain me in simple words:
1) Wikipedia says that Wilcoxon signed-rank test is used to test whether
population "mean ranks" differ. What is exactly the definition of "mean
rank"? https://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test
2) How can the mean of a variable A be bigger than the mean of variable B,
but the "mean rank" of variable B is significantly bigger than "mean rank"
of variable A.

There is a small chance that this is because of a bug in wilcox.test
function, but it is probably more likely that this paradox is because of
some statistics phenomena that I don't understand.

Best regards,
Karolis Uziela

P. S. I have another strange example, where the difference between A and B
is much smaller than the difference between A and C, but the significance
of the "mean rank" difference between A and B is much larger then the
significance of mean rank difference between A and C. For simplicity
reasons, I didn't add that example here, but I guess that the answer to the
above question will be related to this one.
-------------- next part --------------
A B C
0.493579934773664 0.511836168474316 0.520749710683309
0.630184025170146 0.58786433109054 0.458946430077942
0.633512186023312 0.493175938647652 0.66308680696517
0.491989020132133 0.520090183585035 0.420844896432205
0.0399860725369978 -0.0301629556552143 -0.161169807765403
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0.954025476347433 0.977016137944116 0.750624074377985
0.862950985422539 0.868732350639475 0.730996449953475
0.659015458019471 0.631657908519014 0.649268814371901
0.944446851459376 0.951663741829449 0.887377555251177
0.791126281396265 0.806829715682826 0.520987050223108
0.790981668919217 0.832554187306653 0.816906470482764
0.952134063327928 0.950267697811188 0.888111039865367
0.979447757413485 0.965082926636581 0.978260925492216
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