# pingouin.wilcoxon

pingouin.wilcoxon(x, y, tail='two-sided')[source]

Wilcoxon signed-rank test. It is the non-parametric version of the paired T-test.

Parameters
x, yarray_like

First and second set of observations. x and y must be related (e.g repeated measures) and, therefore, have the same number of samples. Note that a listwise deletion of missing values is automatically applied.

tailstring

Specify whether to return ‘one-sided’ or ‘two-sided’ p-value. Can also be ‘greater’ or ‘less’ to specify the direction of the test. If tail='one-sided', the alternative of the test will be automatically detected by looking at the sign of the median of the differences between x and y. For instance, if np.median(x - y) > 0 and tail='one-sided', Pingouin will automatically set tail='greater' and vice versa.

Returns
statspandas.DataFrame
• 'W-val': W-value

• 'p-val': p-value

• 'RBC' : matched pairs rank-biserial correlation (effect size)

• 'CLES' : common language effect size

Notes

The Wilcoxon signed-rank test [1] tests the null hypothesis that two related paired samples come from the same distribution. In particular, it tests whether the distribution of the differences x - y is symmetric about zero. A continuity correction is applied by default (see scipy.stats.wilcoxon() for details).

The matched pairs rank biserial correlation [2] is the simple difference between the proportion of favorable and unfavorable evidence; in the case of the Wilcoxon signed-rank test, the evidence consists of rank sums (Kerby 2014):

$r = f - u$

The common language effect size is the proportion of pairs where x is higher than y. It was first introduced by McGraw and Wong (1992) [3]. Pingouin uses a brute-force version of the formula given by Vargha and Delaney 2000 [4]:

$\text{CL} = P(X > Y) + .5 \times P(X = Y)$

The advantage is of this method are twofold. First, the brute-force approach pairs each observation of x to its y counterpart, and therefore does not require normally distributed data. Second, the formula takes ties into account and therefore works with ordinal data.

When tail is 'less', the CLES is then set to $$1 - \text{CL}$$, which gives the proportion of pairs where x is lower than y.

Warning

Versions of Pingouin below 0.2.6 gave wrong two-sided p-values for the Wilcoxon test. P-values were accidentally squared, and therefore smaller. This issue has been resolved in Pingouin>=0.2.6. Make sure to always use the latest release.

References

1

Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics bulletin, 1(6), 80-83.

2

Kerby, D. S. (2014). The simple difference formula: An approach to teaching nonparametric correlation. Comprehensive Psychology, 3, 11-IT.

3

McGraw, K. O., & Wong, S. P. (1992). A common language effect size statistic. Psychological bulletin, 111(2), 361.

4

Vargha, A., & Delaney, H. D. (2000). A Critique and Improvement of the “CL” Common Language Effect Size Statistics of McGraw and Wong. Journal of Educational and Behavioral Statistics: A Quarterly Publication Sponsored by the American Educational Research Association and the American Statistical Association, 25(2), 101–132. https://doi.org/10.2307/1165329

Examples

Wilcoxon test on two related samples.

>>> import numpy as np
>>> import pingouin as pg
>>> x = [20, 22, 19, 20, 22, 18, 24, 20, 19, 24, 26, 13]
>>> y = [38, 37, 33, 29, 14, 12, 20, 22, 17, 25, 26, 16]
>>> pg.wilcoxon(x, y, tail='two-sided')
W-val       tail     p-val       RBC      CLES
Wilcoxon   20.5  two-sided  0.285765 -0.378788  0.395833


Compare with SciPy

>>> import scipy
>>> scipy.stats.wilcoxon(x, y, correction=True)
WilcoxonResult(statistic=20.5, pvalue=0.2857652190231508)


One-sided tail: one can either manually specify the alternative hypothesis

>>> pg.wilcoxon(x, y, tail='greater')
W-val     tail     p-val       RBC      CLES
Wilcoxon   20.5  greater  0.876244 -0.378788  0.395833

>>> pg.wilcoxon(x, y, tail='less')
W-val  tail     p-val       RBC      CLES
Wilcoxon   20.5  less  0.142883 -0.378788  0.604167


Or simply leave it to Pingouin, using the ‘one-sided’ argument, in which case Pingouin will look at the sign of the median of the differences between x and y and ajust the tail based on that:

>>> np.median(np.array(x) - np.array(y))
-1.5


The median is negative, so Pingouin will test for the alternative hypothesis that the median of the differences is negative (= less than 0).

>>> pg.wilcoxon(x, y, tail='one-sided')  # Equivalent to tail = 'less'
W-val  tail     p-val       RBC      CLES
Wilcoxon   20.5  less  0.142883 -0.378788  0.604167