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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 difference in location between distributions. Parameters ---------- x, y : array-like N-d arrays of samples. The arrays must be broadcastable except along the dimension given by `axis`. use_continuity : bool, 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*. Note that the mathematical expressions in the alternative hypotheses above describe the CDFs of the underlying distributions. The directions of the inequalities appear inconsistent with the natural language description at first glance, but they are not. For example, suppose *X* and *Y* are random variables that follow distributions with CDFs *F* and *G*, respectively. If *F(u) > G(u)* for all *u*, samples drawn from *X* tend to be less than those drawn from *Y*. 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. axis : int, optional Axis along which to perform the test. Default is 0. method : {'auto', 'asymptotic', 'exact'} or `PermutationMethod` instance, 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 :math:`U` statistic against the exact distribution of the :math:`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 or equal to 8 and there are no ties; chooses ``'asymptotic'`` otherwise. * `PermutationMethod` instance. In this case, the p-value is computed using `permutation_test` with the provided configuration options and other appropriate settings. Returns ------- res : MannwhitneyuResult An object containing attributes: statistic : float The Mann-Whitney U statistic corresponding with sample `x`. See Notes for the test statistic corresponding with sample `y`. pvalue : float 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, consider `scipy.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, but `mannwhitneyu` will not raise errors or warnings if there are ties in the data. If there are ties and either samples is small (fewer than ~10 observations), consider passing an instance of `PermutationMethod` as the `method` to perform a permutation test. The Mann-Whitney U test is a non-parametric version of the t-test for independent samples. When the means of samples from the populations are normally distributed, consider `scipy.stats.ttest_ind`. See Also -------- scipy.stats.wilcoxon, scipy.stats.ranksums, scipy.stats.ttest_ind References ---------- .. [1] 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] 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 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 :math:`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 :math:`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 :math:`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 :math:`p = 0.09` given in [4]_. The reason is that [4]_ does not apply the continuity correction performed by `mannwhitneyu`; `mannwhitneyu` reduces the distance between the test statistic and the mean :math:`\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 :math:`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. 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