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The calculation of the p-value depends on the keyword " }, { "__type": "ParamRef", "__tag": 4071, "name": "method" }, { "__type": "Text", "__tag": 4046, "value": ":" } ] }, { "__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": ": The p-value is approximated by using the limiting distribution of the test statistic." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "exact" }, { "__type": "Text", "__tag": 4046, "value": ": The exact p-value is computed by enumerating all possible combinations of the test statistic, see " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "2" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "InlineCode", "__tag": 4051, "value": "method='auto'" }, { "__type": "Text", "__tag": 4046, "value": ", the exact approach is used if both samples contain equal to or less than 20 observations, otherwise the asymptotic distribution is used." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If the underlying distribution is not continuous, the p-value is likely to be conservative (Section 6.2 in " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "3" }, { "__type": "Text", "__tag": 4046, "value": "). 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Must contain at least two observations." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "y", "annotation": "array_like", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "A 1-D array of observed values of the random variables " }, { "__type": "InlineMath", "__tag": 4057, "value": "Y_i" }, { "__type": "Text", "__tag": 4046, "value": ". Must contain at least two observations." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "method", "annotation": "{'auto', 'asymptotic', 'exact'}, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The method used to compute the p-value, see Notes for details. 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The function " }, { "__type": "InlineMath", "__tag": 4057, "value": "r(\\cdot)" }, { "__type": "Text", "__tag": 4046, "value": " here denotes the rank of the observed values " }, { "__type": "InlineMath", "__tag": 4057, "value": "x_i" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "y_j" }, { "__type": "Text", "__tag": 4046, "value": " within the pooled sample of size " }, { "__type": "InlineMath", "__tag": 4057, "value": "n + m" }, { "__type": "Text", "__tag": 4046, "value": ", with ties assigned mid-rank values:" } ] }, { "__type": "Math", "__tag": 4058, "value": "U = n \\sum_{i=1}^n (r(x_i)-i)^2 + m \\sum_{j=1}^m (r(y_j)-j)^2" } ], "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": "build-install/usr/lib/python3.14/site-packages/scipy/stats/_hypotests.py", "item_line": 1673, "item_type": "function", "aliases": [ "scipy.stats.cramervonmises_2samp" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "Suppose we wish to test whether two samples generated by\n``scipy.stats.norm.rvs`` have the same distribution. We choose a\nsignificance level of alpha=0.05.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy import stats\nrng = np.random.default_rng()\nx = stats.norm.rvs(size=100, random_state=rng)\ny = stats.norm.rvs(size=70, random_state=rng)\nres = stats.cramervonmises_2samp(x, y)\n", "execution_status": "success", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "res.statistic, res.pvalue\n", "execution_status": "failure", "out": "(np.float64(0.0897058823529413), np.float64(0.6443838640982176))\n" }, { "__type": "Text", "__tag": 4046, "value": "\nThe p-value exceeds our chosen significance level, so we do not\nreject the null hypothesis that the observed samples are drawn from the\nsame distribution.\n\nFor small sample sizes, one can compute the exact p-values:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "x = stats.norm.rvs(size=7, random_state=rng)\ny = stats.t.rvs(df=2, size=6, random_state=rng)\nres = stats.cramervonmises_2samp(x, y, method='exact')\n", "execution_status": "success", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "res.statistic, res.pvalue\n", "execution_status": "failure", "out": "(np.float64(0.2051282051282053), np.float64(0.291958041958042))\n" }, { "__type": "Text", "__tag": 4046, "value": "\nThe p-value based on the asymptotic distribution is a good approximation\neven though the sample size is small.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "res = stats.cramervonmises_2samp(x, y, method='asymptotic')\n", "execution_status": "success", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "res.statistic, res.pvalue\n", "execution_status": "failure", "out": "(np.float64(0.2051282051282053), np.float64(0.2803228231921372))\n" }, { "__type": "Text", "__tag": 4046, "value": "\nIndependent of the method, one would not reject the null hypothesis at the\nchosen significance level in this example." } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "anderson_ksamp", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.stats._morestats:anderson_ksamp" }, "kind": "module" }, "descriptions": [], "type": "func" }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "cramervonmises", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.stats._hypotests:cramervonmises" }, "kind": "module" }, "descriptions": [], "type": "func" }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "epps_singleton_2samp", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.stats._hypotests:epps_singleton_2samp" }, "kind": "module" }, "descriptions": [], "type": "func" }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "ks_2samp", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "ks_2samp" }, "kind": "module" }, "descriptions": [], "type": "func" } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "x", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "y", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "method", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "auto" }, { "__type": "SigParam", "__tag": 4030, "name": "axis", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "0" }, { "__type": "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": "cramervonmises_2samp" }, "references": [ ".. [1] https://en.wikipedia.org/wiki/Cramer-von_Mises_criterion", ".. [2] Anderson, T.W. (1962). On the distribution of the two-sample", " Cramer-von-Mises criterion. The Annals of Mathematical", " Statistics, pp. 1148-1159.", ".. [3] Conover, W.J., Practical Nonparametric Statistics, 1971." ], "qa": "scipy.stats._hypotests:cramervonmises_2samp", "arbitrary": [], "local_refs": [ "axis", "keepdims", "method", "nan_policy", "res", "x", "y" ] }