scipy.stats.mannwhitneyu¶
- scipy.stats.mannwhitneyu(x, y, use_continuity=True, alternative='two-sided', axis=0, method='auto')[source]¶
Perform the Mann-Whitney U rank test on two independent samples.
The Mann-Whitney U test is a nonparametric test of the null hypothesis that the distribution underlying sample x is the same as the distribution underlying sample y. It is often used as a test of of difference in location between distributions.
- Parameters
- x, yarray-like
N-d arrays of samples. The arrays must be broadcastable except along the dimension given by axis.
- use_continuitybool, optional
Whether a continuity correction (1/2) should be applied. Default is True when method is
'asymptotic'
; has no effect otherwise.- alternative{‘two-sided’, ‘less’, ‘greater’}, optional
Defines the alternative hypothesis. Default is ‘two-sided’. Let F(u) and G(u) be the cumulative distribution functions of the distributions underlying x and y, respectively. Then the following alternative hypotheses are available:
‘two-sided’: the distributions are not equal, i.e. F(u) ≠ G(u) for at least one u.
‘less’: the distribution underlying x is stochastically less than the distribution underlying y, i.e. F(u) > G(u) for all u.
‘greater’: the distribution underlying x is stochastically greater than the distribution underlying y, i.e. F(u) < G(u) for all u.
Under a more restrictive set of assumptions, the alternative hypotheses can be expressed in terms of the locations of the distributions; see [5] section 5.1.
- axisint, optional
Axis along which to perform the test. Default is 0.
- method{‘auto’, ‘asymptotic’, ‘exact’}, optional
Selects the method used to calculate the p-value. Default is ‘auto’. The following options are available.
'asymptotic'
: compares the standardized test statistic against the normal distribution, correcting for ties.'exact'
: computes the exact p-value by comparing the observed \(U\) statistic against the exact distribution of the \(U\) statistic under the null hypothesis. No correction is made for ties.'auto'
: chooses'exact'
when the size of one of the samples is less than 8 and there are no ties; chooses'asymptotic'
otherwise.
- Returns
- resMannwhitneyuResult
An object containing attributes:
- statisticfloat
The Mann-Whitney U statistic corresponding with sample x. See Notes for the test statistic corresponding with sample y.
- pvaluefloat
The associated p-value for the chosen alternative.
Notes
If
U1
is the statistic corresponding with sample x, then the statistic corresponding with sample y is U2 = `x.shape[axis] * y.shape[axis] - U1`.mannwhitneyu
is for independent samples. For related / paired samples, considerscipy.stats.wilcoxon
.method
'exact'
is recommended when there are no ties and when either sample size is less than 8 [1]. The implementation follows the recurrence relation originally proposed in [1] as it is described in [3]. Note that the exact method is not corrected for ties, butmannwhitneyu
will not raise errors or warnings if there are ties in the data.The Mann-Whitney U test is a non-parametric version of the t-test for independent samples. When the the means of samples from the populations are normally distributed, consider
scipy.stats.ttest_ind
.References
- 1(1,2)
H.B. Mann and D.R. Whitney, “On a test of whether one of two random variables is stochastically larger than the other”, The Annals of Mathematical Statistics, Vol. 18, pp. 50-60, 1947.
- 2
Mann-Whitney U Test, Wikipedia, http://en.wikipedia.org/wiki/Mann-Whitney_U_test
- 3
A. Di Bucchianico, “Combinatorics, computer algebra, and the Wilcoxon-Mann-Whitney test”, Journal of Statistical Planning and Inference, Vol. 79, pp. 349-364, 1999.
- 4(1,2,3,4,5,6,7)
Rosie Shier, “Statistics: 2.3 The Mann-Whitney U Test”, Mathematics Learning Support Centre, 2004.
- 5
Michael P. Fay and Michael A. Proschan. “Wilcoxon-Mann-Whitney or t-test? On assumptions for hypothesis tests and multiple interpretations of decision rules.” Statistics surveys, Vol. 4, pp. 1-39, 2010. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2857732/
Examples
We follow the example from [4]: nine randomly sampled young adults were diagnosed with type II diabetes at the ages below.
>>> males = [19, 22, 16, 29, 24] >>> females = [20, 11, 17, 12]
We use the Mann-Whitney U test to assess whether there is a statistically significant difference in the diagnosis age of males and females. The null hypothesis is that the distribution of male diagnosis ages is the same as the distribution of female diagnosis ages. We decide that a confidence level of 95% is required to reject the null hypothesis in favor of the alternative that that the distributions are different. Since the number of samples is very small and there are no ties in the data, we can compare the observed test statistic against the exact distribution of the test statistic under the null hypothesis.
>>> from scipy.stats import mannwhitneyu >>> U1, p = mannwhitneyu(males, females, method="exact") >>> print(U1) 17.0
mannwhitneyu
always reports the statistic associated with the first sample, which, in this case, is males. This agrees with \(U_M = 17\) reported in [4]. The statistic associated with the second statistic can be calculated:>>> nx, ny = len(males), len(females) >>> U2 = nx*ny - U1 >>> print(U2) 3.0
This agrees with \(U_F = 3\) reported in [4]. The two-sided p-value can be calculated from either statistic, and the value produced by
mannwhitneyu
agrees with \(p = 0.11\) reported in [4].>>> print(p) 0.1111111111111111
The exact distribution of the test statistic is asymptotically normal, so the example continues by comparing the exact p-value against the p-value produced using the normal approximation.
>>> _, pnorm = mannwhitneyu(males, females, method="asymptotic") >>> print(pnorm) 0.11134688653314041
Here
mannwhitneyu
’s reported p-value appears to conflict with the value \(p = 0.09\) given in [4]. The reason is that [4] does not apply the continuity correction performed bymannwhitneyu
;mannwhitneyu
reduces the distance between the test statistic and the mean \(\mu = n_x n_y / 2\) by 0.5 to correct for the fact that the discrete statistic is being compared against a continuous distribution. Here, the \(U\) statistic used is less than the mean, so we reduce the distance by adding 0.5 in the numerator.>>> import numpy as np >>> from scipy.stats import norm >>> U = min(U1, U2) >>> N = nx + ny >>> z = (U - nx*ny/2 + 0.5) / np.sqrt(nx*ny * (N + 1)/ 12) >>> p = 2 * norm.cdf(z) # use CDF to get p-value from smaller statistic >>> print(p) 0.11134688653314041
If desired, we can disable the continuity correction to get a result that agrees with that reported in [4].
>>> _, pnorm = mannwhitneyu(males, females, use_continuity=False, ... method="asymptotic") >>> print(pnorm) 0.0864107329737
Regardless of whether we perform an exact or asymptotic test, the probability of the test statistic being as extreme or more extreme by chance exceeds 5%, so we do not consider the results statistically significant.
Suppose that, before seeing the data, we had hypothesized that females would tend to be diagnosed at a younger age than males. In that case, it would be natural to provide the female ages as the first input, and we would have performed a one-sided test using
alternative = 'less'
: females are diagnosed at an age that is stochastically less than that of males.>>> res = mannwhitneyu(females, males, alternative="less", method="exact") >>> print(res) MannwhitneyuResult(statistic=3.0, pvalue=0.05555555555555555)
Again, the probability of getting a sufficiently low value of the test statistic by chance under the null hypothesis is greater than 5%, so we do not reject the null hypothesis in favor of our alternative.
If it is reasonable to assume that the means of samples from the populations are normally distributed, we could have used a t-test to perform the analysis.
>>> from scipy.stats import ttest_ind >>> res = ttest_ind(females, males, alternative="less") >>> print(res) Ttest_indResult(statistic=-2.239334696520584, pvalue=0.030068441095757924)
Under this assumption, the p-value would be low enough to reject the null hypothesis in favor of the alternative.