Is the p-value of the Kruskal-Wallis test two-sided?

[ahmedhosny]

I have 2 (or more) non-normal groups of varying sizes, and so I looked at the Kruskal-Wallis test (when len(groups)>2) and the Mann-Whitney rank test (when len(groups)==2).

I know from the Mann-Whitney rank test docs [1] the the p-value returned is single-sided by default. However, I am unable to find documentation whether the returned p-value in Kruskal-Wallis test is two-sided or not. See docs here [2] and source code [3].

When comparing both tests head to head for len(group)==2, my experiments show that the Kruskal-Wallis p-value may be two-sided, one example here:

from scipy import stats
x = np.arange(1,41,1)
y = np.arange(2,42,2)
print ( stats.kruskal(x, y) ) 
print ( stats.mannwhitneyu(x, y) )

KruskalResult(statistic=0.02460383653043551, pvalue=0.8753582575468357)
MannwhitneyuResult(statistic=390.0, pvalue=0.4407715705950445)

While I understand they are different tests, the relationship between the p-values (Mann-Whitney p-value * 2 is almost equal to Kruskal-Wallis p-value) is very close and difficult to ignore. Any help would be much appreciated!

[1] https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.mannwhitneyu.html
[2] https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kruskal.html
[3] https://github.com/scipy/scipy/blob/v1.4.1/scipy/stats/stats.py#L6484-L6601

[josef-pkt]

see docstring description for alternative=None in mannwhitneyu

half the two-sided p-value or one-sided pvalue at larger tail is a historical artifact,
All other hypothesis test have correct definition of two-sided p-values

[ahmedhosny]

Thanks @josef-pkt - I can assume that Kruskal-Wallis returns two-sided p-value then.

[chrisb83]

The distribution of the test statistic of the Kruskal test is approximatred by a chi2-distribution under the null hypothesis and the p-value is computed as chi2.sf(h, df), e.g. it only considers deviations to one side. I would call this a one-sided p-value.

[mdhaber]

@ahmedhosny, I think @josef-pkt's answer was about the note in the mannwhitneyu documentation:

alternative: None: computes p-value half the size of the ‘two-sided’ p-value and a different U statistic. The default behavior is not the same as using ‘less’ or ‘greater’; it only exists for backward compatibility and is deprecated.

But you also asked:

However, I am unable to find documentation whether the returned p-value in Kruskal-Wallis test is two-sided or not.

As @chrisb83 mentioned, it is one-sided: the p-value is the survival function of the chi-square distribution evaluated at H.

Is gh-12565 a helpful addition to the documentation? I thought the statement about how it is calculated is more descriptive than simply saying "one-sided".