{ "id": "epxwnt92m9", "slug": "pearson-correlation-test", "title": "Pearson correlation test", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.multivariate-analysis", "label": "Multivariate association and dimension reduction", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.multivariate-analysis" }, { "id": "physical.math.statistics.multivariate-analysis.alpha-mo-to-w", "label": "Titles Mo to W", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.multivariate-analysis.alpha-mo-to-w" } ], "geo": [ { "id": "geo.nongeo.t1800.physical.math.statistics.multivariate-analysis", "label": "Non-geographic · 1800 to 1945: Multivariate association and dimension reduction", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics.multivariate-analysis", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1800", "label": "Non-geographic · 1800 to 1945", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800" }, { "id": "geo.nongeo.t1800.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical" }, { "id": "geo.nongeo.t1800.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math" }, { "id": "geo.nongeo.t1800.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics" }, { "id": "geo.nongeo.t1800.physical.math.statistics.multivariate-analysis", "label": "Multivariate association and dimension reduction", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics.multivariate-analysis" } ] } ], "excerpt": "The Pearson correlation test is a statistical test of whether the population correlation between two continuous variables differs from zero, exact under bivariate normality and otherwise approximate.", "snippet": "The Pearson correlation test is a statistical test of whether the population correlation between two continuous variables differs from zero, exact under bivariate normality and otherwise approximate.", "node": "physical.math.statistics.multivariate-analysis.alpha-mo-to-w", "markdown": "# Pearson correlation test\n\nThe Pearson correlation test is a statistical test of whether the population correlation coefficient ρ between two continuous variables differs from zero. The coefficient itself is \\( r = \\sum (x - m_{x}) \\cdot (y - m_{y}) / \\sqrt{\\sum (x - m_{x})^{2} \\, \\sum (y - m_{y})^{2}} \\), where \\( m_{x} \\) and \\( m_{y} \\) are the sample means.[1](https://scipy.github.io/devdocs/reference/generated/scipy.stats.pearsonr.html) The null hypothesis is H₀: ρ = 0 (no linear association) against Hₐ: ρ ≠ 0.[2](https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Introductory_Statistics_2e_%28OpenStax%29/13%3A_Linear_Regression_and_Correlation/13.03%3A_Testing_the_Significance_of_the_Correlation_Coefficient) A key interpretive limit is that zero correlation implies independence only when (x, y) is jointly normal; otherwise two variables can be uncorrelated yet dependent.[1](https://scipy.github.io/devdocs/reference/generated/scipy.stats.pearsonr.html)\n\n| Key fact | Value |\n|---|---|\n| Hypotheses | H₀: ρ = 0 vs Hₐ: ρ ≠ 0 (one-sided versions also used)[2](https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Introductory_Statistics_2e_%28OpenStax%29/13%3A_Linear_Regression_and_Correlation/13.03%3A_Testing_the_Significance_of_the_Correlation_Coefficient) |\n| Test statistic | \\( t = r\\sqrt{(n-2)/(1-r^{2})} \\), t distribution with \\( n-2 \\) degrees of freedom under \\( H_{0} \\)[3](https://online.stat.psu.edu/stat415/book/export/html/830) |\n| Exact null distribution of r | Beta distribution on (−1, 1) with shape parameters \\( a = b = n/2 - 1 \\), exact for \\( n > 2 \\) under the bivariate-normal null[4](https://docs.scipy.org/doc/scipy-1.17.0/tutorial/stats/hypothesis_pearsonr.html) |\n| Fisher z variance | \\( \\mathrm{Var}(z) \\approx 1/(n-3) \\), nearly independent of \\( \\rho \\)[3](https://online.stat.psu.edu/stat415/book/export/html/830) |\n| Sample size example | \\( n = 112 \\) gives 90% power to detect \\( \\rho = 0.3 \\) (two-sided, \\( \\alpha = 0.05 \\))[5](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/PASS/Pearsons_Correlation_Tests.pdf) |\n| Software | R `cor.test()` and `scipy.stats.pearsonr` both implement the test; SciPy's `pearsonr` offers exact, permutation, and Fisher-z options[6](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/cor.test.html) • [1](https://scipy.github.io/devdocs/reference/generated/scipy.stats.pearsonr.html) |\n| Relation to regression | The test is identical to the t-test of whether a linear regression slope differs from zero[7](http://psychclassics.yorku.ca/Fisher/Methods/chap6.htm) |\n\n## How it works\n\nUnder bivariate normality, testing \\( H_{0} \\colon \\rho = 0 \\) is equivalent to testing independence of X and Y. The test statistic is\n\n\\[ t = \\frac{r\\sqrt{n-2}}{\\sqrt{1-r^2}} \\]\n\nwhich follows a t distribution with \\( n - 2 \\) degrees of freedom when \\( \\rho = 0 \\).[3](https://online.stat.psu.edu/stat415/book/export/html/830) Equivalently, r itself has an exact null distribution: the beta distribution on (−1, 1) with shape parameters \\( a = b = n/2 - 1 \\), exact for \\( n > 2 \\) under the bivariate-normal null with independent observations; for \\( n = 2 \\) the distribution is degenerate and the two-sided p-value is 1.[4](https://docs.scipy.org/doc/scipy-1.17.0/tutorial/stats/hypothesis_pearsonr.html) • [1](https://scipy.github.io/devdocs/reference/generated/scipy.stats.pearsonr.html) In density form, provided \\( \\rho = 0 \\),\n\n\\[ g(r) = \\frac{\\Gamma[(n-1)/2]}{\\Gamma(1/2)\\,\\Gamma[(n-2)/2]}\\,(1-r^2)^{(n-4)/2}, \\quad -1 < r < 1. \\]\n\n[3](https://online.stat.psu.edu/stat415/book/export/html/830)\n\nFor values of ρ other than zero, r's distribution is skewed, so Fisher's z-transformation, z = ½{logₑ(1+r) − logₑ(1−r)}, is used; its standard error is approximately 1/√(n−3) and is practically independent of the population correlation.[7](http://psychclassics.yorku.ca/Fisher/Methods/chap6.htm) • [8](http://www2.psych.purdue.edu/~gfrancis/Classes/PSY201/Lecture24.pdf) For the specific test of \\( \\rho = 0 \\) the t statistic is used without the transform.[8](http://www2.psych.purdue.edu/~gfrancis/Classes/PSY201/Lecture24.pdf) The z scale is preferred for confidence intervals, for testing \\( \\rho = \\rho_{0} \\), and for comparing or combining correlations.[9](https://online.stat.psu.edu/stat509/lesson/18/18.1) Fisher noted that the test of \\( \\rho = 0 \\) is identical to the test of whether the linear regression coefficient differs from zero.[7](http://psychclassics.yorku.ca/Fisher/Methods/chap6.htm)\n\n## How it is done\n\nA practitioner first checks assumptions: both variables measured on an interval or ratio scale, an approximately linear relationship, independent observations, and, for small samples, approximate bivariate normality. Only linearity is required for r as a descriptive measure; normality is required for the test itself.[10](https://tuos-bio-data-skills.github.io/intro-stats-book/correlation-tests.html) • [11](https://quarto.wessa.net/testcorr.html)\n\nThe test then proceeds by computing r, forming \\( t \\) with \\( n - 2 \\) degrees of freedom, and reading the p-value; critical-value tables offer a shortcut, using \\( r = \\sqrt{t^{2}/(t^{2} + \\mathrm{df})} \\).[12](https://www.medcalc.org/en/manual/correlation-table.php) Exact p-values come from the beta null distribution, equivalently from the t statistic with \\( n - 2 \\) degrees of freedom under bivariate normality; Fisher-z inference is an approximation, and it can differ from the exact result in small samples.[1](https://scipy.github.io/devdocs/reference/generated/scipy.stats.pearsonr.html) • [4](https://docs.scipy.org/doc/scipy-1.17.0/tutorial/stats/hypothesis_pearsonr.html) In R, `cor.test(x, y)` reports t, df, the p-value, and a 95% confidence interval based on Fisher's Z transform when there are at least 4 complete pairs; the `alternative` argument accepts \"two.sided\", \"greater\", or \"less\".[6](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/cor.test.html) In SciPy, `pearsonr` offers permutation or [Monte Carlo](https://www.edgepedia.org/monte-carlo) p-values through its `method` parameter and returns a two-sided p-value of 1 when \\( n = 2 \\).[1](https://scipy.github.io/devdocs/reference/generated/scipy.stats.pearsonr.html)\n\n## Origin\n\nThe coefficient's algebra descends from the product-moment method: the standard calculation is the one termed the method of \"product moments\".[13](https://gwern.net/doc/psychology/1904-spearman.pdf)\n\nThe sampling theory came later. R. A. Fisher derived the exact frequency distribution of the correlation coefficient in samples from a normal population in his 1915 [Biometrika](https://www.edgepedia.org/biometrika) paper, using a geometrical argument in 2n-dimensional space; Fisher states the problem was drawn to his attention by H. E. Soper's 1913 article.[14](https://doi.org/10.1093/biomet/10.4.507) The z-transformation appears in Statistical Methods for Research Workers, although some sources credit it to the 1915 paper.[7](http://psychclassics.yorku.ca/Fisher/Methods/chap6.htm) • [11](https://quarto.wessa.net/testcorr.html) The test of significance for the correlation coefficient was treated as a problem in its own right by Egon S. Pearson in a 1931 [Journal of the American Statistical Association](https://www.edgepedia.org/journal-of-the-american-statistical-association) paper.[15](https://doi.org/10.1080/01621459.1931.10503208) The surrounding framework of errors of the first and second kind was introduced by Neyman and E. S. Pearson in their 1928 Biometrika papers.[16](https://doi.org/10.1093/biomet/20a.3-4.263)\n\n## Variants\n\nOne-sided alternatives are standard options in software.[6](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/cor.test.html) Resampling variants exist but carry a caveat often missed in practice: the permutation test is exact for Type I error only when the metric ρ is used to test \\( H_{0} \\colon F_{XY} = F_{X} \\cdot F_{Y} \\) (independence), and it is exact for testing \\( H_{0} \\colon \\rho = 0 \\) only under bivariate normality or certain elliptical families; this non-distribution-free behavior was established by Andrew F. Hayes in a 1996 Psychological Methods paper.[17](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0216287) • [18](https://doi.org/10.1037/1082-989x.1.2.184) The studentized permutation test is exact when those nulls coincide and asymptotically valid otherwise.[17](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0216287) A surrogate bootstrap test handles the general null \\( \\rho = \\rho_{0} \\), with \\( B = 500 \\) or 1000 replicates usually sufficient.[17](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0216287)\n\nUnder nonnormality, simulation work by Bishara and Hittner found that heavy-tailed distributions inflate the Pearson estimate by up to +0.14, that with \\( n \\geq 20 \\) transforming to rankit scores before computing Pearson minimized both error types, and that with \\( n \\leq 10 \\) and extreme nonnormality the permutation test often performed best.[19](https://pmc.ncbi.nlm.nih.gov/articles/PMC5965513/) • [20](https://bpb-us-w2.wpmucdn.com/blogs.cofc.edu/dist/7/881/files/2021/06/Bishara-Hittner-2012.pdf) The t-based and Fisher-z tests of Spearman's \\( \\rho = 0 \\) have been shown to be theoretically incorrect, because ranks cannot be bivariate normal; in R, `cor.test` computes an exact p-value for eligible untied samples and otherwise uses an approximation, and a studentized permutation test in the R package `perk` controls Type I error even at \\( n = 10 \\).[21](https://pmc.ncbi.nlm.nih.gov/articles/PMC11029148/) Kendall's tau, introduced as a new measure of rank correlation by M. G. Kendall in his 1938 Biometrika paper, is a further rank-based alternative.[22](https://doi.org/10.1093/biomet/30.1-2.81) Contrary to a common belief, Pearson's r is not ruled out for nonlinear monotonic associations, and in simulations Kendall's tau and Spearman's ρ outperformed r in power at low correlations while r was more efficient in other settings.[23](https://www.tandfonline.com/doi/pdf/10.1080/00031305.2021.2004922) Modern nonparametric independence tests go further: Julian D. Karch and colleagues found in a 2024 Multivariate Behavioral Research simulation study that distance correlation and the HHG-Pearson test are substantially more powerful than Pearson's, Spearman's, and Kendall's tests for many nonlinear relationships.[24](https://doi.org/10.1080/00273171.2024.2347960) For time series, where ordinary tests are miscalibrated under serial dependence, [M. S. Bartlett](https://www.edgepedia.org/m-s-bartlett)'s 1935 paper addressed effective-sample-size adjustments, and the truncated time-shift test provably controls false positives when one series is stationary.[25](https://doi.org/10.2307/2342284) • [26](https://journals.plos.org/plosbiology/article?id=10.1371%2Fjournal.pbio.3002758)\n\n## Applications\n\nPower for the test can be computed exactly from the cumulative distribution of r under a nonzero ρ, using the algorithm of William C. Guenther's 1977 American Statistician paper on desk calculation of these probabilities.[27](https://doi.org/10.1080/00031305.1977.10479195) • [5](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/PASS/Pearsons_Correlation_Tests.pdf) The numbers are sobering for small effects: for a two-sided test with \\( \\alpha = 0.05 \\), \\( \\rho_{0} = 0 \\), and alternative \\( \\rho_{1} = 0.3 \\), power is 0.86524 at \\( n = 100 \\), and the required sample size for 90% power is 112.[5](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/PASS/Pearsons_Correlation_Tests.pdf) Several authors argue that testing \\( \\rho = 0 \\) is often the wrong question, and an alternative is to choose n for a desired interval width; for example, \\( n = 300 \\) yields a 95% CI of width 0.20 for a Spearman coefficient with planning value 0.4.[28](https://ww2.amstat.org/meetings/proceedings/2016/data/assets/pdf/389762.pdf) In software, R `cor.test()` and SciPy's `pearsonr` both implement the test, the latter offering exact, permutation, and Fisher-z options.[6](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/cor.test.html) • [1](https://scipy.github.io/devdocs/reference/generated/scipy.stats.pearsonr.html)\n\n## Limitations and alternatives\n\nThe \\( t(n-2) \\) null distribution is exact for independent bivariate-normal pairs under \\( H_{0} \\colon \\rho = 0 \\); independence without normality does not give an exact t distribution, and in a general context with dependent or non-Gaussian data the classical Pearson test is not well calibrated, with simulations showing false positive rates well above the nominal 5% level. In one worked example the usual Pearson test gave \\( p = 9.25\\% \\) while the corrected test in the `robusTest` package gave \\( p = 1.74\\% \\) on the same data.[29](https://export.arxiv.org/pdf/2211.08784v1.pdf) Early claims that the null distribution of r survives gross distortion of the bivariate normal surface have since been disproved.[17](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0216287) Outliers are a further weakness: Pearson is the maximum-likelihood estimator of ρ only under bivariate normality and is sensitive to outliers and nonlinear relationships.[30](https://cran.r-project.org/web/packages/smartcor/vignettes/theory.html) Dichotomizing continuous variables attenuates the coefficient badly: with true \\( \\rho = 0.7 \\) reduced to a binary variable, the [phi coefficient](https://www.edgepedia.org/phi-coefficient) falls to roughly 0.45–0.50, while the polychoric correlation, whose maximum-likelihood estimation was treated in Ulf Olsson's 1979 Psychometrika paper, recovers 0.70.[30](https://cran.r-project.org/web/packages/smartcor/vignettes/theory.html) • [31](https://doi.org/10.1007/bf02296207)\n\nInterpretation is a further failure mode. A Penn State STAT 509 text notes of the SAS PROC CORR test of \\( \\rho = 0 \\) that it \"generally is not very useful\", recommending a confidence interval instead.[9](https://online.stat.psu.edu/stat509/lesson/18/18.1) Relying solely on p-values for interpretation can lead to incorrect conclusions; the coefficient r itself serves as the effect size.[32](https://www.ncbi.nlm.nih.gov/sites/books/NBK606101/) When many correlations are tested at once, naive use of sample correlations or Fisher-z statistics produces many false positives; a \"sandwich estimator\" that de-correlates the samples, combined with the Benjamini-Hochberg procedure, asymptotically controls the false discovery rate.[33](https://arxiv.org/pdf/1703.08843) As a simpler guard, the HHG-Pearson test combines Pearson's p-value with the HHG test's via \\( p = 2 \\cdot \\min(p_{\\mathrm{HHG}}, p_{\\mathrm{Pearson}}) \\), a Bonferroni-style combination with family-wise error control.[24](https://doi.org/10.1080/00273171.2024.2347960)\n\n## References\n\n1. [scipy.stats.pearsonr, SciPy documentation](https://scipy.github.io/devdocs/reference/generated/scipy.stats.pearsonr.html)\n2. [13.03: Testing the Significance of the Correlation Coefficient (stats.libretexts.org)](https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Introductory_Statistics_2e_%28OpenStax%29/13%3A_Linear_Regression_and_Correlation/13.03%3A_Testing_the_Significance_of_the_Correlation_Coefficient)\n3. [Lesson 15: Tests Concerning Regression and Correlation (Penn State STAT 415)](https://online.stat.psu.edu/stat415/book/export/html/830)\n4. [Pearson's Correlation, SciPy hypothesis testing tutorial](https://docs.scipy.org/doc/scipy-1.17.0/tutorial/stats/hypothesis_pearsonr.html)\n5. [Pearson's Correlation Tests (PASS power/sample-size procedure)](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/PASS/Pearsons_Correlation_Tests.pdf)\n6. [R: Test for Association/Correlation Between Paired Samples (cor.test)](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/cor.test.html)\n7. [Statistical Methods for Research Workers, Chapter 6 (R. A. Fisher, 1925), 'The Correlation Coefficient'](http://psychclassics.yorku.ca/Fisher/Methods/chap6.htm)\n8. [PSY 201 Lecture 24: Hypothesis testing for correlations (Purdue, G. Francis)](http://www2.psych.purdue.edu/~gfrancis/Classes/PSY201/Lecture24.pdf)\n9. [18.1 - Pearson Correlation Coefficient | STAT 509 (Penn State)](https://online.stat.psu.edu/stat509/lesson/18/18.1)\n10. [Chapter 11 Correlation tests, Introductory Biostatistics with R](https://tuos-bio-data-skills.github.io/intro-stats-book/correlation-tests.html)\n11. [131 Testing Correlations – Statistical Analysis for Small and Big Data](https://quarto.wessa.net/testcorr.html)\n12. [Critical Values of the Pearson Correlation Coefficient (r) | MedCalc](https://www.medcalc.org/en/manual/correlation-table.php)\n13. [The Proof and Measurement of Association between Two Things (C. Spearman, 1904, American Journal of Psychology)](https://gwern.net/doc/psychology/1904-spearman.pdf)\n14. [R. A. FISHER (1915). FREQUENCY DISTRIBUTION OF THE VALUES OF THE CORRELATION COEFFIENTS IN SAMPLES FROM AN INDEFINITELY LARGE POPU;ATION. Biometrika.](https://doi.org/10.1093/biomet/10.4.507)\n15. [Egon S. Pearson (1931). The Test of Significance for the Correlation Coefficient. Journal of the American Statistical Association.](https://doi.org/10.1080/01621459.1931.10503208)\n16. [J. NEYMAN, E. S. PEARSON (1928). ON THE USE AND INTERPRETATION OF CERTAIN TEST CRITERIA FOR PURPOSES OF STATISTICAL INFERENCE. Biometrika.](https://doi.org/10.1093/biomet/20a.3-4.263)\n17. [A robust Pearson correlation test for a general point null using a surrogate bootstrap distribution (PLOS One)](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0216287)\n18. [Andrew F. Hayes (1996). Permutation test is not distribution-free: Testing H₀: ρ = 0.. Psychological Methods.](https://doi.org/10.1037/1082-989x.1.2.184)\n19. [Reducing Bias and Error in the Correlation Coefficient Due to Nonnormality (Bishara & Hittner)](https://pmc.ncbi.nlm.nih.gov/articles/PMC5965513/)\n20. [Testing the significance of a correlation with nonnormal data: Comparison of Pearson, Spearman, transformation, and resampling approaches (Bishara & Hittner, 2012, Psychological Methods)](https://bpb-us-w2.wpmucdn.com/blogs.cofc.edu/dist/7/881/files/2021/06/Bishara-Hittner-2012.pdf)\n21. [A robust Spearman correlation coefficient permutation test](https://pmc.ncbi.nlm.nih.gov/articles/PMC11029148/)\n22. [M. G. KENDALL (1938). A NEW MEASURE OF RANK CORRELATION. Biometrika.](https://doi.org/10.1093/biomet/30.1-2.81)\n23. [Myths About Linear and Monotonic Associations: Pearson's r, Spearman's ρ, and Kendall's τ (van den Heuvel, The American Statistician)](https://www.tandfonline.com/doi/pdf/10.1080/00031305.2021.2004922)\n24. [Julian D. Karch, Andres F. Perez-Alonso, Wicher P. Bergsma (2024). Beyond Pearson’s Correlation: Modern Nonparametric Independence Tests for Psychological Research. Multivariate Behavioral Research.](https://doi.org/10.1080/00273171.2024.2347960)\n25. [M. S. Bartlett (1935). Some Aspects of the Time-Correlation Problem in Regard to Tests of Significance. Journal Of The Royal Statistical Society.](https://doi.org/10.2307/2342284)\n26. [A rigorous and versatile statistical test for correlations between stationary time series (PLOS Biology)](https://journals.plos.org/plosbiology/article?id=10.1371%2Fjournal.pbio.3002758)\n27. [William C. Guenther (1977). Desk Calculation of Probabilities for the Distribution of the Sample Correlation Coefficient. The American Statistician.](https://doi.org/10.1080/00031305.1977.10479195)\n28. [Simplified Tools for Sample Size Determination for Correlation Coefficient Inference (May, Ketchum, Looney, JSM 2016)](https://ww2.amstat.org/meetings/proceedings/2016/data/assets/pdf/389762.pdf)\n29. [robusTest: corrected (robust) versions of Pearson, Kendall, Spearman correlation tests (arXiv preprint)](https://export.arxiv.org/pdf/2211.08784v1.pdf)\n30. [Choosing the Right Correlation Method: Theory and Rationale (smartcor vignette, CRAN)](https://cran.r-project.org/web/packages/smartcor/vignettes/theory.html)\n31. [Ulf Olsson (1979). Maximum Likelihood Estimation of the Polychoric Correlation Coefficient. Psychometrika.](https://doi.org/10.1007/bf02296207)\n32. [Correlation (Coefficient, Partial, and Spearman Rank) and Regression Analysis (StatPearls, NCBI Bookshelf)](https://www.ncbi.nlm.nih.gov/sites/books/NBK606101/)\n33. [Testing independence with high-dimensional covariates (max-type test, sandwich estimator, multiple testing of Pearson correlations)](https://arxiv.org/pdf/1703.08843)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Multivariate association and dimension reduction › Titles Mo to W*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "http://www2.psych.purdue.edu/~gfrancis/Classes/PSY201/Lecture24.pdf" ], "url": "https://www.edgepedia.org/pearson-correlation-test", "markdown_url": "https://www.edgepedia.org/pearson-correlation-test.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Pearson correlation test\", Edgepedia (EdgeChat), https://www.edgepedia.org/pearson-correlation-test. Edgepedia Community License 1.0.", "credit_md": "\"[Pearson correlation test](https://www.edgepedia.org/pearson-correlation-test)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/pearson-correlation-test](https://www.edgepedia.org/pearson-correlation-test). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Pearson correlation test\", Edgepedia (EdgeChat), https://www.edgepedia.org/pearson-correlation-test. Edgepedia Community License 1.0.", "speakable": "The Pearson correlation test is a statistical test of whether the population correlation between two continuous variables differs from zero, exact under bivariate normality and otherwise approximate." }