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The arrays must be broadcastable except along the dimension given by " }, { "__type": "ParamRef", "__tag": 4071, "name": "axis" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "use_continuity", "annotation": "bool, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Whether a continuity correction (1/2) should be applied. 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Let " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "SX(u)" } ] }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "SY(u)" } ] }, { "__type": "Text", "__tag": 4046, "value": " be the survival functions of the distributions underlying " }, { "__type": "ParamRef", "__tag": 4071, "name": "x" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "ParamRef", "__tag": 4071, "name": "y" }, { "__type": "Text", "__tag": 4046, "value": ", respectively. 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The statistic of each axis-slice (e.g. row) of the input will appear in a corresponding element of the output. If " }, { "__type": "InlineCode", "__tag": 4051, "value": "None" }, { "__type": "Text", "__tag": 4046, "value": ", the input will be raveled before computing the statistic." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "method", "annotation": "{'auto', 'asymptotic', 'exact'} or `PermutationMethod` instance, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Selects the method used to calculate the " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "p" } ] }, { "__type": "Text", "__tag": 4046, "value": "-value. Default is 'auto'. The following options are available." } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "'asymptotic'" }, { "__type": "Text", "__tag": 4046, "value": ": compares the standardized test statistic against the normal distribution, correcting for ties." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "'exact'" }, { "__type": "Text", "__tag": 4046, "value": ": computes the exact " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "p" } ] }, { "__type": "Text", "__tag": 4046, "value": "-value by comparing the observed " }, { "__type": "InlineMath", "__tag": 4057, "value": "U" }, { "__type": "Text", "__tag": 4046, "value": " statistic against the exact distribution of the " }, { "__type": "InlineMath", "__tag": 4057, "value": "U" }, { "__type": "Text", "__tag": 4046, "value": " statistic under the null hypothesis. No correction is made for ties." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "'auto'" }, { "__type": "Text", "__tag": 4046, "value": ": chooses " }, { "__type": "InlineCode", "__tag": 4051, "value": "'exact'" }, { "__type": "Text", "__tag": 4046, "value": " when the size of one of the samples is less than or equal to 8 and there are no ties; chooses " }, { "__type": "InlineCode", "__tag": 4051, "value": "'asymptotic'" }, { "__type": "Text", "__tag": 4046, "value": " otherwise." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "CrossRef", "__tag": 4002, "value": "PermutationMethod", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.stats._resampling:PermutationMethod" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " instance. In this case, the p-value is computed using " }, { "__type": "CrossRef", "__tag": 4002, "value": "permutation_test", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.stats._resampling:permutation_test" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " with the provided configuration options and other appropriate settings." } ] } ] } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "nan_policy", "annotation": "{'propagate', 'omit', 'raise'}", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Defines how to handle input NaNs." } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "propagate" }, { "__type": "Text", "__tag": 4046, "value": ": if a NaN is present in the axis slice (e.g. row) along which the statistic is computed, the corresponding entry of the output will be NaN." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "omit" }, { "__type": "Text", "__tag": 4046, "value": ": NaNs will be omitted when performing the calculation. If insufficient data remains in the axis slice along which the statistic is computed, the corresponding entry of the output will be NaN." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "raise" }, { "__type": "Text", "__tag": 4046, "value": ": if a NaN is present, a " }, { "__type": "InlineCode", "__tag": 4051, "value": "ValueError" }, { "__type": "Text", "__tag": 4046, "value": " will be raised." } ] } ] } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "keepdims", "annotation": "bool, default: False", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The Mann-Whitney U test is a nonparametric test of the null hypothesis that the distribution underlying sample " }, { "__type": "ParamRef", "__tag": 4071, "name": "x" }, { "__type": "Text", "__tag": 4046, "value": " is the same as the distribution underlying sample " }, { "__type": "ParamRef", "__tag": 4071, "name": "y" }, { "__type": "Text", "__tag": 4046, "value": ". It is often used as a test of difference in location between distributions." } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "/scipy/stats/_mannwhitneyu.py", "item_line": 219, "item_type": "function", "aliases": [ "scipy.stats.mannwhitneyu" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "We follow the example from [4]_: nine randomly sampled young adults were\ndiagnosed with type II diabetes at the ages below.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "males = [19, 22, 16, 29, 24]\nfemales = [20, 11, 17, 12]\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nWe use the Mann-Whitney U test to assess whether there is a statistically\nsignificant difference in the diagnosis age of males and females.\nThe null hypothesis is that the distribution of male diagnosis ages is\nthe same as the distribution of female diagnosis ages. We decide\nthat a confidence level of 95% is required to reject the null hypothesis\nin favor of the alternative that the distributions are different.\nSince the number of samples is very small and there are no ties in the\ndata, we can compare the observed test statistic against the *exact*\ndistribution of the test statistic under the null hypothesis.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.stats import mannwhitneyu\nU1, p = mannwhitneyu(males, females, method=\"exact\")\nprint(U1)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\n`mannwhitneyu` always reports the statistic associated with the first\nsample, which, in this case, is males. This agrees with :math:`U_M = 17`\nreported in [4]_. The statistic associated with the second statistic\ncan be calculated:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "nx, ny = len(males), len(females)\nU2 = nx*ny - U1\nprint(U2)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nThis agrees with :math:`U_F = 3` reported in [4]_. The two-sided\n*p*-value can be calculated from either statistic, and the value produced\nby `mannwhitneyu` agrees with :math:`p = 0.11` reported in [4]_.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "print(p)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nThe exact distribution of the test statistic is asymptotically normal, so\nthe example continues by comparing the exact *p*-value against the\n*p*-value produced using the normal approximation.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "_, pnorm = mannwhitneyu(males, females, method=\"asymptotic\")\nprint(pnorm)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nHere `mannwhitneyu`'s reported *p*-value appears to conflict with the\nvalue :math:`p = 0.09` given in [4]_. The reason is that [4]_\ndoes not apply the continuity correction performed by `mannwhitneyu`;\n`mannwhitneyu` reduces the distance between the test statistic and the\nmean :math:`\\mu = n_x n_y / 2` by 0.5 to correct for the fact that the\ndiscrete statistic is being compared against a continuous distribution.\nHere, the :math:`U` statistic used is less than the mean, so we reduce\nthe distance by adding 0.5 in the numerator.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy.stats import norm\nU = min(U1, U2)\nN = nx + ny\nz = (U - nx*ny/2 + 0.5) / np.sqrt(nx*ny * (N + 1)/ 12)\np = 2 * norm.cdf(z) # use CDF to get p-value from smaller statistic\nprint(p)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nIf desired, we can disable the continuity correction to get a result\nthat agrees with that reported in [4]_.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "_, pnorm = mannwhitneyu(males, females, use_continuity=False,\n method=\"asymptotic\")\nprint(pnorm)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nRegardless of whether we perform an exact or asymptotic test, the\nprobability of the test statistic being as extreme or more extreme by\nchance exceeds 5%, so we do not consider the results statistically\nsignificant.\n\nSuppose that, before seeing the data, we had hypothesized that females\nwould tend to be diagnosed at a younger age than males.\nIn that case, it would be natural to provide the female ages as the\nfirst input, and we would have performed a one-sided test using\n``alternative = 'less'``: females are diagnosed at an age that is\nstochastically less than that of males.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "res = mannwhitneyu(females, males, alternative=\"less\", method=\"exact\")\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "print(res)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nAgain, the probability of getting a sufficiently low value of the\ntest statistic by chance under the null hypothesis is greater than 5%,\nso we do not reject the null hypothesis in favor of our alternative.\n\nIf it is reasonable to assume that the means of samples from the\npopulations are normally distributed, we could have used a t-test to\nperform the analysis.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.stats import ttest_ind\nres = ttest_ind(females, males, alternative=\"less\")\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "print(res)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nUnder this assumption, the *p*-value would be low enough to reject the\nnull hypothesis in favor of the alternative." } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 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"SigParam", "__tag": 4030, "name": "nan_policy", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "propagate" }, { "__type": "SigParam", "__tag": 4030, "name": "keepdims", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "False" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "mannwhitneyu" }, "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] Andreas Löffler,", " \"Über eine Partition der nat. Zahlen und ihr Anwendung beim U-Test\",", " Wiss. Z. Univ. Halle, XXXII'83 pp. 87-89.", ".. [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/" ], "qa": "scipy.stats._mannwhitneyu:mannwhitneyu", "arbitrary": [], "local_refs": [ "alternative", "axis", "keepdims", "method", "nan_policy", "res", "use_continuity", "x", "y" ] }