{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The correlation coefficient is calculated as follows:" } ] }, { "__type": "Math", "__tag": 4058, "value": "r = \\frac{\\sum (x - m_x) (y - m_y)}\n {\\sqrt{\\sum (x - m_x)^2 \\sum (y - m_y)^2}}" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "m_x" }, { "__type": "Text", "__tag": 4046, "value": " is the mean of the vector x and " }, { "__type": "InlineMath", "__tag": 4057, "value": "m_y" }, { "__type": "Text", "__tag": 4046, "value": " is the mean of the vector y." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Under the assumption that x and y are drawn from independent normal distributions (so the population correlation coefficient is 0), the probability density function of the sample correlation coefficient r is (" }, { "__type": "FootnoteReference", "__tag": 4066, "label": "1" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "2" }, { "__type": "Text", "__tag": 4046, "value": "):" } ] }, { "__type": "Math", "__tag": 4058, "value": "f(r) = \\frac{{(1-r^2)}^{n/2-2}}{\\mathrm{B}(\\frac{1}{2},\\frac{n}{2}-1)}" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where n is the number of samples, and B is the beta function. 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In terms of the object " }, { "__type": "InlineCode", "__tag": 4051, "value": "dist" }, { "__type": "Text", "__tag": 4046, "value": " shown above, the p-value for a given r and length n can be computed as " } ] }, { "__type": "Code", "__tag": 4050, "value": "p = 2*dist.cdf(-abs(r))", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "When n is 2, the above continuous distribution is not well-defined. One can interpret the limit of the beta distribution as the shape parameters a and b approach a = b = 0 as a discrete distribution with equal probability masses at r = 1 and r = -1. More directly, one can observe that, given the data x = [x1, x2] and y = [y1, y2], and assuming x1 != x2 and y1 != y2, the only possible values for r are 1 and -1. Because abs(r') for any sample x' and y' with length 2 will be 1, the two-sided p-value for a sample of length 2 is always 1." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "For backwards compatibility, the object that is returned also behaves like a tuple of length two that holds the statistic and the p-value." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "Array API Standard Support" } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "pearsonr", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " has experimental support for Python Array API Standard compatible backends in addition to NumPy. 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The following combinations of backend and device (or other capability) are supported." } ] }, { "__type": "Code", "__tag": 4050, "value": "==================== ==================== ====================\nLibrary CPU GPU\n==================== ==================== ====================\nNumPy ✅ n/a \nCuPy n/a ✅ \nPyTorch ✅ ⛔ \nJAX ✅ ✅ \nDask ✅ n/a \n==================== ==================== ====================", "execution_status": null }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "See " }, { "__type": "InlineRole", "__tag": 4003, "value": "dev-arrayapi", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " for more information." } ] } ] } ], "title": [], "level": 0, "target": null }, "Warns": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "", "annotation": "`~scipy.stats.ConstantInputWarning`", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Raised if an input is a constant array. The correlation coefficient is not defined in this case, so " }, { "__type": "InlineCode", "__tag": 4051, "value": "np.nan" }, { "__type": "Text", "__tag": 4046, "value": " is returned." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "", "annotation": "`~scipy.stats.NearConstantInputWarning`", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Raised if an input is \"nearly\" constant. The array " }, { "__type": "InlineCode", "__tag": 4051, "value": "x" }, { "__type": "Text", "__tag": 4046, "value": " is considered nearly constant if " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm(x - mean(x)) < 1e-13 * abs(mean(x))" }, { "__type": "Text", "__tag": 4046, "value": ". 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The confidence interval is returned in a " }, { "__type": "InlineCode", "__tag": 4051, "value": "namedtuple" }, { "__type": "Text", "__tag": 4046, "value": " with fields " }, { "__type": "InlineRole", "__tag": 4003, "value": "low", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineRole", "__tag": 4003, "value": "high", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ". If " }, { "__type": "ParamRef", "__tag": 4071, "name": "method" }, { "__type": "Text", "__tag": 4046, "value": " is not provided, the confidence interval is computed using the Fisher transformation " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "1" }, { "__type": "Text", "__tag": 4046, "value": ". If " }, { "__type": "ParamRef", "__tag": 4071, "name": "method" }, { "__type": "Text", "__tag": 4046, "value": " is an instance of " }, { "__type": "CrossRef", "__tag": 4002, "value": "BootstrapMethod", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.stats._resampling:BootstrapMethod" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": ", the confidence interval is computed using " }, { "__type": "CrossRef", "__tag": 4002, "value": "scipy.stats.bootstrap", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "*", "kind": "api", "path": "scipy.stats._resampling:bootstrap" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " with the provided configuration options and other appropriate settings. In some cases, confidence limits may be NaN due to a degenerate resample, and this is typical for very small samples (~6 observations)." } ] } ] } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Pearson correlation coefficient and p-value for testing non-correlation." } ] } ], "title": [], "level": 0, "target": null }, "Receives": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Warnings": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Attributes": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Parameters": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "x", "annotation": "array_like", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Input array." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "y", "annotation": "array_like", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Input array." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "axis", "annotation": "int or None, default", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Axis along which to perform the calculation. Default is 0. If None, ravel both arrays before performing the calculation." } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "versionadded", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionadded 1.14.0" } ] } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "alternative", "annotation": "{'two-sided', 'greater', 'less'}, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Defines the alternative hypothesis. Default is 'two-sided'. 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": "Text", "__tag": 4046, "value": "'two-sided': the correlation is nonzero" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "'less': the correlation is negative (less than zero)" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "'greater': the correlation is positive (greater than zero)" } ] } ] } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "versionadded", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionadded 1.9.0" } ] } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "method", "annotation": "ResamplingMethod, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Defines the method used to compute the p-value. If " }, { "__type": "ParamRef", "__tag": 4071, "name": "method" }, { "__type": "Text", "__tag": 4046, "value": " is an instance of " }, { "__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": "/" }, { "__type": "CrossRef", "__tag": 4002, "value": "MonteCarloMethod", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.stats._resampling:MonteCarloMethod" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": ", the p-value is computed using " }, { "__type": "CrossRef", "__tag": 4002, "value": "scipy.stats.permutation_test", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "*", "kind": "api", "path": "scipy.stats._resampling:permutation_test" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": "/" }, { "__type": "CrossRef", "__tag": 4002, "value": "scipy.stats.monte_carlo_test", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "*", "kind": "api", "path": "scipy.stats._resampling:monte_carlo_test" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " with the provided configuration options and other appropriate settings. Otherwise, the p-value is computed as documented in the notes." } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "versionadded", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionadded 1.11.0" } ] } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The Pearson correlation coefficient " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "1" }, { "__type": "Text", "__tag": 4046, "value": " measures the linear relationship between two datasets. Like other correlation coefficients, this one varies between -1 and +1 with 0 implying no correlation. Correlations of -1 or +1 imply an exact linear relationship. Positive correlations imply that as x increases, so does y. Negative correlations imply that as x increases, y decreases." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This function also performs a test of the null hypothesis that the distributions underlying the samples are uncorrelated and normally distributed. (See Kowalski " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "3" }, { "__type": "Text", "__tag": 4046, "value": " for a discussion of the effects of non-normality of the input on the distribution of the correlation coefficient.) The p-value roughly indicates the probability of an uncorrelated system producing datasets that have a Pearson correlation at least as extreme as the one computed from these datasets." } ] } ], "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": 4369, "item_type": "function", "aliases": [ "scipy.stats.pearsonr" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy import stats\nx, y = [1, 2, 3, 4, 5, 6, 7], [10, 9, 2.5, 6, 4, 3, 2]\nres = stats.pearsonr(x, y)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "res\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nTo perform an exact permutation version of the test:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "rng = np.random.default_rng(7796654889291491997)\nmethod = stats.PermutationMethod(n_resamples=np.inf, random_state=rng)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "stats.pearsonr(x, y, method=method)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nTo perform the test under the null hypothesis that the data were drawn from\n*uniform* distributions:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "method = stats.MonteCarloMethod(rvs=(rng.uniform, rng.uniform))\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "stats.pearsonr(x, y, method=method)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nTo produce an asymptotic 90% confidence interval:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "res.confidence_interval(confidence_level=0.9)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nAnd for a bootstrap confidence interval:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "method = stats.BootstrapMethod(method='BCa', rng=rng)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "res.confidence_interval(confidence_level=0.9, method=method)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nIf N-dimensional arrays are provided, multiple tests are performed in a\nsingle call according to the same conventions as most `scipy.stats` functions:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "rng = np.random.default_rng(2348246935601934321)\nx = rng.standard_normal((8, 15))\ny = rng.standard_normal((8, 15))\nstats.pearsonr(x, y, axis=0).statistic.shape # between corresponding columns\nstats.pearsonr(x, y, axis=1).statistic.shape # between corresponding rows\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nTo perform all pairwise comparisons between slices of the arrays,\nuse standard NumPy broadcasting techniques. For instance, to compute the\ncorrelation between all pairs of rows:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "stats.pearsonr(x[:, np.newaxis, :], y, axis=-1).statistic.shape\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nThere is a linear dependence between x and y if y = a + b*x + e, where\na,b are constants and e is a random error term, assumed to be independent\nof x. For simplicity, assume that x is standard normal, a=0, b=1 and let\ne follow a normal distribution with mean zero and standard deviation s>0.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "rng = np.random.default_rng()\ns = 0.5\nx = stats.norm.rvs(size=500, random_state=rng)\ne = stats.norm.rvs(scale=s, size=500, random_state=rng)\ny = x + e\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "stats.pearsonr(x, y).statistic\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nThis should be close to the exact value given by\n\n" }, { "__type": "Code", "__tag": 4050, "value": "1/np.sqrt(1 + s**2)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nFor s=0.5, we observe a high level of correlation. In general, a large\nvariance of the noise reduces the correlation, while the correlation\napproaches one as the variance of the error goes to zero.\n\nIt is important to keep in mind that no correlation does not imply\nindependence unless (x, y) is jointly normal. Correlation can even be zero\nwhen there is a very simple dependence structure: if X follows a\nstandard normal distribution, let y = abs(x). Note that the correlation\nbetween x and y is zero. Indeed, since the expectation of x is zero,\ncov(x, y) = E[x*y]. By definition, this equals E[x*abs(x)] which is zero\nby symmetry. The following lines of code illustrate this observation:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "y = np.abs(x)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "stats.pearsonr(x, y)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nA non-zero correlation coefficient can be misleading. For example, if X has\na standard normal distribution, define y = x if x < 0 and y = 0 otherwise.\nA simple calculation shows that corr(x, y) = sqrt(2/Pi) = 0.797...,\nimplying a high level of correlation:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "y = np.where(x < 0, x, 0)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "stats.pearsonr(x, y)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nThis is unintuitive since there is no dependence of x and y if x is larger\nthan zero which happens in about half of the cases if we sample x and y.\n\nFor a more detailed example, see :ref:`hypothesis_pearsonr`." } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "hypothesis_pearsonr", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "hypothesis_pearsonr" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Extended example" } ] } ], "type": "ref" }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "kendalltau", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "kendalltau" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Kendall's tau, a correlation measure for ordinal data." } ] } ], "type": "func" }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "spearmanr", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "spearmanr" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Spearman rank-order correlation coefficient." } ] } ], "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": "alternative", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "two-sided" }, { "__type": "SigParam", "__tag": 4030, "name": "method", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "axis", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "0" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "pearsonr" }, "references": [ ".. [1] \"Pearson correlation coefficient\", Wikipedia,", " https://en.wikipedia.org/wiki/Pearson_correlation_coefficient", ".. [2] Student, \"Probable error of a correlation coefficient\",", " Biometrika, Volume 6, Issue 2-3, 1 September 1908, pp. 302-310.", ".. [3] C. J. Kowalski, \"On the Effects of Non-Normality on the Distribution", " of the Sample Product-Moment Correlation Coefficient\"", " Journal of the Royal Statistical Society. Series C (Applied", " Statistics), Vol. 21, No. 1 (1972), pp. 1-12." ], "qa": "scipy.stats._stats_py:pearsonr", "arbitrary": [], "local_refs": [ "alternative", "axis", "method", "result", "x", "y" ] }