{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "There are three options for the null and corresponding alternative hypothesis that can be selected using the " }, { "__type": "ParamRef", "__tag": 4071, "name": "alternative" }, { "__type": "Text", "__tag": 4046, "value": " parameter." } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "less", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ": The null hypothesis is that F(x) >= G(x) for all x; the alternative is that F(x) < G(x) for at least one x. The statistic is the magnitude of the minimum (most negative) difference between the empirical distribution functions of the samples." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "greater", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ": The null hypothesis is that F(x) <= G(x) for all x; the alternative is that F(x) > G(x) for at least one x. The statistic is the maximum (most positive) difference between the empirical distribution functions of the samples." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "two-sided", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ": The null hypothesis is that the two distributions are identical, F(x)=G(x) for all x; the alternative is that they are not identical. The statistic is the maximum absolute difference between the empirical distribution functions of the samples." } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Note that the alternative hypotheses describe the " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "CDFs" } ] }, { "__type": "Text", "__tag": 4046, "value": " of the underlying distributions, not the observed values of the data. For example, suppose x1 ~ F and x2 ~ G. 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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": "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": "This test compares the underlying continuous distributions F(x) and G(x) of two independent samples. See Notes for a description of the available null and alternative hypotheses." } ] } ], "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/_stats_py.py", "item_line": 7775, "item_type": "function", "aliases": [ "scipy.stats.ks_2samp" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "Suppose we wish to test the null hypothesis that two samples were drawn\nfrom the same distribution.\nWe choose a confidence level of 95%; that is, we will reject the null\nhypothesis in favor of the alternative if the p-value is less than 0.05.\n\nIf the first sample were drawn from a uniform distribution and the second\nwere drawn from the standard normal, we would expect the null hypothesis\nto be rejected.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy import stats\nrng = np.random.default_rng()\nsample1 = stats.uniform.rvs(size=100, random_state=rng)\nsample2 = stats.norm.rvs(size=110, random_state=rng)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "stats.ks_2samp(sample1, sample2)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nIndeed, the p-value is lower than our threshold of 0.05, so we reject the\nnull hypothesis in favor of the default \"two-sided\" alternative: the data\nwere *not* drawn from the same distribution.\n\nWhen both samples are drawn from the same distribution, we expect the data\nto be consistent with the null hypothesis most of the time.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "sample1 = stats.norm.rvs(size=105, random_state=rng)\nsample2 = stats.norm.rvs(size=95, random_state=rng)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "stats.ks_2samp(sample1, sample2)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nAs expected, the p-value of 0.54 is not below our threshold of 0.05, so\nwe cannot reject the null hypothesis.\n\nSuppose, however, that the first sample were drawn from\na normal distribution shifted toward greater values. In this case,\nthe cumulative density function (CDF) of the underlying distribution tends\nto be *less* than the CDF underlying the second sample. Therefore, we would\nexpect the null hypothesis to be rejected with ``alternative='less'``:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "sample1 = stats.norm.rvs(size=105, loc=0.5, random_state=rng)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "stats.ks_2samp(sample1, sample2, alternative='less')\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nand indeed, with p-value smaller than our threshold, we reject the null\nhypothesis in favor of the alternative." } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "anderson_ksamp", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "anderson_ksamp" }, "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_1samp", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "ks_1samp" }, "kind": "module" }, "descriptions": [], "type": "func" }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "kstest", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "kstest" }, "kind": "module" }, "descriptions": [], "type": "func" } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "data1", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "data2", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "alternative", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "two-sided" }, { "__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": "ks_2samp" }, "references": [ ".. [1] Hodges, J.L. Jr., \"The Significance Probability of the Smirnov", " Two-Sample Test,\" Arkiv fiur Matematik, 3, No. 43 (1958), 469-486." ], "qa": "scipy.stats._stats_py:ks_2samp", "arbitrary": [], "local_refs": [ "alternative", "axis", "data1", "data2", "keepdims", "method", "nan_policy" ] }