In statistics, the Pearson correlation coefficient (PCC, pronounced /ˈpɪərsən/ , also referred to as Pearson's r, the Pearson product-moment correlation coefficient PPMCC, the bivariate correlation,^{ [1] } or colloquially simply as the correlation coefficient^{ [2] }) is a measure of linear correlation between two sets of data. It is the covariance of two variables, divided by the product of their standard deviations; thus it is essentially a normalised measurement of the covariance, such that the result always has a value between −1 and 1. As with covariance itself, the measure can only reflect a linear correlation of variables, and ignores many other types of relationship or correlation. As a simple example, one would expect the age and height of a sample of teenagers from a high school to have a Pearson correlation coefficient significantly greater than 0, but less than 1 (as 1 would represent an unrealistically perfect correlation).
It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s, and for which the mathematical formula was derived and published by Auguste Bravais in 1844.^{ [lower-alpha 1] }^{ [6] }^{ [7] }^{ [8] }^{ [9] } The naming of the coefficient is thus an example of Stigler's Law.
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
Pearson's correlation coefficient, when applied to a population, is commonly represented by the Greek letter ρ (rho) and may be referred to as the population correlation coefficient or the population Pearson correlation coefficient.^{ [10] } Given a pair of random variables , the formula for ρ^{ [11] } is:^{ [12] }
| (Eq.1) |
where:
The formula for can be expressed in terms of mean and expectation. Since
the formula for can also be written as
| (Eq.2) |
where:
The formula for can be expressed in terms of uncentered moments. Since
the formula for can also be written as
Pearson's correlation coefficient, when applied to a sample, is commonly represented by and may be referred to as the sample correlation coefficient or the sample Pearson correlation coefficient.^{ [10] } We can obtain a formula for by substituting estimates of the covariances and variances based on a sample into the formula above. Given paired data consisting of pairs, is defined as:
| (Eq.3) |
where:
Rearranging gives us this formula for :
where are defined as above.
This formula suggests a convenient single-pass algorithm for calculating sample correlations, though depending on the numbers involved, it can sometimes be numerically unstable.
Rearranging again gives us this^{ [11] } formula for :
where are defined as above.
An equivalent expression gives the formula for as the mean of the products of the standard scores as follows:
where
Alternative formulae for are also available. For example. one can use the following formula for :
where:
Under heavy noise conditions, extracting the correlation coefficient between two sets of stochastic variables is nontrivial, in particular where Canonical Correlation Analysis reports degraded correlation values due to the heavy noise contributions. A generalization of the approach is given elsewhere.^{ [13] }
In case of missing data, Garren derived the maximum likelihood estimator.^{ [14] }
The absolute values of both the sample and population Pearson correlation coefficients are on or between 0 and 1. Correlations equal to +1 or −1 correspond to data points lying exactly on a line (in the case of the sample correlation), or to a bivariate distribution entirely supported on a line (in the case of the population correlation). The Pearson correlation coefficient is symmetric: corr(X,Y) = corr(Y,X).
A key mathematical property of the Pearson correlation coefficient is that it is invariant under separate changes in location and scale in the two variables. That is, we may transform X to a + bX and transform Y to c + dY, where a, b, c, and d are constants with b, d > 0, without changing the correlation coefficient. (This holds for both the population and sample Pearson correlation coefficients.) Note that more general linear transformations do change the correlation: see § Decorrelation of n random variables for an application of this.
The correlation coefficient ranges from −1 to 1. A value of 1 implies that a linear equation describes the relationship between X and Y perfectly, with all data points lying on a line for which Y increases as X increases. A value of −1 implies that all data points lie on a line for which Y decreases as X increases. A value of 0 implies that there is no linear correlation between the variables.^{ [15] }
More generally, note that (X_{i} − X)(Y_{i} − Y) is positive if and only if X_{i} and Y_{i} lie on the same side of their respective means. Thus the correlation coefficient is positive if X_{i} and Y_{i} tend to be simultaneously greater than, or simultaneously less than, their respective means. The correlation coefficient is negative (anti-correlation) if X_{i} and Y_{i} tend to lie on opposite sides of their respective means. Moreover, the stronger is either tendency, the larger is the absolute value of the correlation coefficient.
Rodgers and Nicewander^{ [16] } cataloged thirteen ways of interpreting correlation:
For uncentered data, there is a relation between the correlation coefficient and the angle φ between the two regression lines, y = g_{X}(x) and x = g_{Y}(y), obtained by regressing y on x and x on y respectively. (Here, φ is measured counterclockwise within the first quadrant formed around the lines' intersection point if r > 0, or counterclockwise from the fourth to the second quadrant if r < 0.) One can show^{ [17] } that if the standard deviations are equal, then r = sec φ − tan φ, where sec and tan are trigonometric functions.
For centered data (i.e., data which have been shifted by the sample means of their respective variables so as to have an average of zero for each variable), the correlation coefficient can also be viewed as the cosine of the angle θ between the two observed vectors in N-dimensional space (for N observations of each variable)^{ [18] }
Both the uncentered (non-Pearson-compliant) and centered correlation coefficients can be determined for a dataset. As an example, suppose five countries are found to have gross national products of 1, 2, 3, 5, and 8 billion dollars, respectively. Suppose these same five countries (in the same order) are found to have 11%, 12%, 13%, 15%, and 18% poverty. Then let x and y be ordered 5-element vectors containing the above data: x = (1, 2, 3, 5, 8) and y = (0.11, 0.12, 0.13, 0.15, 0.18).
By the usual procedure for finding the angle θ between two vectors (see dot product), the uncentered correlation coefficient is:
This uncentered correlation coefficient is identical with the cosine similarity. Note that the above data were deliberately chosen to be perfectly correlated: y = 0.10 + 0.01 x. The Pearson correlation coefficient must therefore be exactly one. Centering the data (shifting x by ℰ(x) = 3.8 and y by ℰ(y) = 0.138) yields x = (−2.8, −1.8, −0.8, 1.2, 4.2) and y = (−0.028, −0.018, −0.008, 0.012, 0.042), from which
as expected.
Several authors have offered guidelines for the interpretation of a correlation coefficient.^{ [19] }^{ [20] } However, all such criteria are in some ways arbitrary.^{ [20] } The interpretation of a correlation coefficient depends on the context and purposes. A correlation of 0.8 may be very low if one is verifying a physical law using high-quality instruments, but may be regarded as very high in the social sciences, where there may be a greater contribution from complicating factors.
Statistical inference based on Pearson's correlation coefficient often focuses on one of the following two aims:
We discuss methods of achieving one or both of these aims below.
Permutation tests provide a direct approach to performing hypothesis tests and constructing confidence intervals. A permutation test for Pearson's correlation coefficient involves the following two steps:
To perform the permutation test, repeat steps (1) and (2) a large number of times. The p-value for the permutation test is the proportion of the r values generated in step (2) that are larger than the Pearson correlation coefficient that was calculated from the original data. Here "larger" can mean either that the value is larger in magnitude, or larger in signed value, depending on whether a two-sided or one-sided test is desired.
The bootstrap can be used to construct confidence intervals for Pearson's correlation coefficient. In the "non-parametric" bootstrap, n pairs (x_{i}, y_{i}) are resampled "with replacement" from the observed set of n pairs, and the correlation coefficient r is calculated based on the resampled data. This process is repeated a large number of times, and the empirical distribution of the resampled r values are used to approximate the sampling distribution of the statistic. A 95% confidence interval for ρ can be defined as the interval spanning from the 2.5th to the 97.5th percentile of the resampled r values.
For pairs from an uncorrelated bivariate normal distribution, the sampling distribution of a certain function of Pearson's correlation coefficient follows Student's t-distribution with degrees of freedom n − 2. Specifically, if the underlying variables have a bivariate normal distribution, the variable
has a student's t-distribution in the null case (zero correlation).^{ [21] } This holds approximately in case of non-normal observed values if sample sizes are large enough.^{ [22] } For determining the critical values for r the inverse function is needed:
Alternatively, large sample, asymptotic approaches can be used.
Another early paper^{ [23] } provides graphs and tables for general values of ρ, for small sample sizes, and discusses computational approaches.
In the case where the underlying variables are not normal, the sampling distribution of Pearson's correlation coefficient follows a Student's t-distribution, but the degrees of freedom are reduced.^{ [24] }
For data that follow a bivariate normal distribution, the exact density function f(r) for the sample correlation coefficient r of a normal bivariate is^{ [25] }^{ [26] }^{ [27] }
where is the gamma function and is the Gaussian hypergeometric function.
In the special case when , the exact density function f(r) can be written as:
where is the beta function, which is one way of writing the density of a Student's t-distribution, as above.
Confidence intervals and tests can be calculated from a confidence distribution. An exact confidence density for ρ is^{ [28] }
where is the Gaussian hypergeometric function and .
In practice, confidence intervals and hypothesis tests relating to ρ are usually carried out using the Fisher transformation, :
F(r) approximately follows a normal distribution with
where n is the sample size. The approximation error is lowest for a large sample size and small and and increases otherwise.
Using the approximation, a z-score is
under the null hypothesis that , given the assumption that the sample pairs are independent and identically distributed and follow a bivariate normal distribution. Thus an approximate p-value can be obtained from a normal probability table. For example, if z = 2.2 is observed and a two-sided p-value is desired to test the null hypothesis that , the p-value is 2 Φ(−2.2) = 0.028, where Φ is the standard normal cumulative distribution function.
To obtain a confidence interval for ρ, we first compute a confidence interval for F():
The inverse Fisher transformation brings the interval back to the correlation scale.
For example, suppose we observe r = 0.3 with a sample size of n=50, and we wish to obtain a 95% confidence interval for ρ. The transformed value is arctanh(r) = 0.30952, so the confidence interval on the transformed scale is 0.30952 ± 1.96/√47, or (0.023624, 0.595415). Converting back to the correlation scale yields (0.024, 0.534).
The square of the sample correlation coefficient is typically denoted r^{2} and is a special case of the coefficient of determination. In this case, it estimates the fraction of the variance in Y that is explained by X in a simple linear regression. So if we have the observed dataset and the fitted dataset then as a starting point the total variation in the Y_{i} around their average value can be decomposed as follows
where the are the fitted values from the regression analysis. This can be rearranged to give
The two summands above are the fraction of variance in Y that is explained by X (right) and that is unexplained by X (left).
Next, we apply a property of least square regression models, that the sample covariance between and is zero. Thus, the sample correlation coefficient between the observed and fitted response values in the regression can be written (calculation is under expectation, assumes Gaussian statistics)
Thus
where
In the derivation above, the fact that
can be proved by noticing that the partial derivatives of the residual sum of squares (RSS) over β_{0} and β_{1} are equal to 0 in the least squares model, where
In the end, the equation can be written as:
where
The symbol is called the regression sum of squares, also called the explained sum of squares, and is the total sum of squares (proportional to the variance of the data).
The population Pearson correlation coefficient is defined in terms of moments, and therefore exists for any bivariate probability distribution for which the population covariance is defined and the marginal population variances are defined and are non-zero. Some probability distributions such as the Cauchy distribution have undefined variance and hence ρ is not defined if X or Y follows such a distribution. In some practical applications, such as those involving data suspected to follow a heavy-tailed distribution, this is an important consideration. However, the existence of the correlation coefficient is usually not a concern; for instance, if the range of the distribution is bounded, ρ is always defined.
Like many commonly used statistics, the sample statistic r is not robust,^{ [30] } so its value can be misleading if outliers are present.^{ [31] }^{ [32] } Specifically, the PMCC is neither distributionally robust,^{[ citation needed ]} nor outlier resistant^{ [30] } (see Robust statistics#Definition). Inspection of the scatterplot between X and Y will typically reveal a situation where lack of robustness might be an issue, and in such cases it may be advisable to use a robust measure of association. Note however that while most robust estimators of association measure statistical dependence in some way, they are generally not interpretable on the same scale as the Pearson correlation coefficient.
Statistical inference for Pearson's correlation coefficient is sensitive to the data distribution. Exact tests, and asymptotic tests based on the Fisher transformation can be applied if the data are approximately normally distributed, but may be misleading otherwise. In some situations, the bootstrap can be applied to construct confidence intervals, and permutation tests can be applied to carry out hypothesis tests. These non-parametric approaches may give more meaningful results in some situations where bivariate normality does not hold. However the standard versions of these approaches rely on exchangeability of the data, meaning that there is no ordering or grouping of the data pairs being analyzed that might affect the behavior of the correlation estimate.
A stratified analysis is one way to either accommodate a lack of bivariate normality, or to isolate the correlation resulting from one factor while controlling for another. If W represents cluster membership or another factor that it is desirable to control, we can stratify the data based on the value of W, then calculate a correlation coefficient within each stratum. The stratum-level estimates can then be combined to estimate the overall correlation while controlling for W.^{ [33] }
Variations of the correlation coefficient can be calculated for different purposes. Here are some examples.
The sample correlation coefficient r is not an unbiased estimate of ρ. For data that follows a bivariate normal distribution, the expectation E[r] for the sample correlation coefficient r of a normal bivariate is^{ [34] }
The unique minimum variance unbiased estimator r_{adj} is given by^{ [35] }
| (1) |
where:
An approximately unbiased estimator r_{adj} can be obtained^{[ citation needed ]} by truncating E[r] and solving this truncated equation:
| (2) |
An approximate solution^{[ citation needed ]} to equation ( 2 ) is:
| (3) |
where in ( 3 ):
Another proposed^{ [11] } adjusted correlation coefficient is:^{[ citation needed ]}
Note that r_{adj} ≈ r for large values of n.
Suppose observations to be correlated have differing degrees of importance that can be expressed with a weight vector w. To calculate the correlation between vectors x and y with the weight vector w (all of length n),^{ [36] }^{ [37] }
The reflective correlation is a variant of Pearson's correlation in which the data are not centered around their mean values.^{[ citation needed ]} The population reflective correlation is
The reflective correlation is symmetric, but it is not invariant under translation:
The sample reflective correlation is equivalent to cosine similarity:
The weighted version of the sample reflective correlation is
Scaled correlation is a variant of Pearson's correlation in which the range of the data is restricted intentionally and in a controlled manner to reveal correlations between fast components in time series.^{ [38] } Scaled correlation is defined as average correlation across short segments of data.
Let be the number of segments that can fit into the total length of the signal for a given scale :
The scaled correlation across the entire signals is then computed as
where is Pearson's coefficient of correlation for segment .
By choosing the parameter , the range of values is reduced and the correlations on long time scale are filtered out, only the correlations on short time scales being revealed. Thus, the contributions of slow components are removed and those of fast components are retained.
A distance metric for two variables X and Y known as Pearson's distance can be defined from their correlation coefficient as^{ [39] }
Considering that the Pearson correlation coefficient falls between [−1, +1], the Pearson distance lies in [0, 2]. The Pearson distance has been used in cluster analysis and data detection for communications and storage with unknown gain and offset^{ [40] }
For variables X = {x_{1},...,x_{n}} and Y = {y_{1},...,y_{n}} that are defined on the unit circle [0, 2π), it is possible to define a circular analog of Pearson's coefficient.^{ [41] } This is done by transforming data points in X and Y with a sine function such that the correlation coefficient is given as:
where and are the circular means of X and Y. This measure can be useful in fields like meteorology where the angular direction of data is important.
If a population or data-set is characterized by more than two variables, a partial correlation coefficient measures the strength of dependence between a pair of variables that is not accounted for by the way in which they both change in response to variations in a selected subset of the other variables.
It is always possible to remove the correlations between all pairs of an arbitrary number of random variables by using a data transformation, even if the relationship between the variables is nonlinear. A presentation of this result for population distributions is given by Cox & Hinkley.^{ [42] }
A corresponding result exists for reducing the sample correlations to zero. Suppose a vector of n random variables is observed m times. Let X be a matrix where is the jth variable of observation i. Let be an m by m square matrix with every element 1. Then D is the data transformed so every random variable has zero mean, and T is the data transformed so all variables have zero mean and zero correlation with all other variables – the sample correlation matrix of T will be the identity matrix. This has to be further divided by the standard deviation to get unit variance. The transformed variables will be uncorrelated, even though they may not be independent.
where an exponent of −+1⁄2 represents the matrix square root of the inverse of a matrix. The correlation matrix of T will be the identity matrix. If a new data observation x is a row vector of n elements, then the same transform can be applied to x to get the transformed vectors d and t:
This decorrelation is related to principal components analysis for multivariate data.
cor(x, y)
, or (with the P value also) with cor.test(x, y)
.pearsonr(x, y)
.pandas.DataFrame.corr
Correlation
function, or (with the P value) with CorrelationTest
.correlation_coefficient
function.In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. In the broadest sense correlation is any statistical association, though it commonly refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.
In statistics, Spearman's rank correlation coefficient or Spearman's ρ, named after Charles Spearman and often denoted by the Greek letter (rho) or as , is a nonparametric measure of rank correlation. It assesses how well the relationship between two variables can be described using a monotonic function.
In statistics, canonical-correlation analysis (CCA), also called canonical variates analysis, is a way of inferring information from cross-covariance matrices. If we have two vectors X = (X_{1}, ..., X_{n}) and Y = (Y_{1}, ..., Y_{m}) of random variables, and there are correlations among the variables, then canonical-correlation analysis will find linear combinations of X and Y which have maximum correlation with each other. T. R. Knapp notes that "virtually all of the commonly encountered parametric tests of significance can be treated as special cases of canonical-correlation analysis, which is the general procedure for investigating the relationships between two sets of variables." The method was first introduced by Harold Hotelling in 1936, although in the context of angles between flats the mathematical concept was published by Jordan in 1875.
In statistics, the Fisher transformation can be used to test hypotheses about the value of the population correlation coefficient ρ between variables X and Y. This is because, when the transformation is applied to the sample correlation coefficient, the sampling distribution of the resulting variable is approximately normal, with a variance that is stable over different values of the underlying true correlation.
Correction for attenuation is a statistical procedure developed by Charles Spearman in 1904 that is used to "rid a correlation coefficient from the weakening effect of measurement error", a phenomenon known as regression dilution. In measurement and statistics, the correction is also called disattenuation. The correction assures that the correlation across data units between two sets of variables is estimated in a manner that accounts for error contained within the measurement of those variables.
In statistics, the residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared estimate of errors (SSE), is the sum of the squares of residuals. It is a measure of the discrepancy between the data and an estimation model, such as a linear regression. A small RSS indicates a tight fit of the model to the data. It is used as an optimality criterion in parameter selection and model selection.
In statistics, simple linear regression is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable and finds a linear function that, as accurately as possible, predicts the dependent variable values as a function of the independent variable. The adjective simple refers to the fact that the outcome variable is related to a single predictor.
In statistics, a rank correlation is any of several statistics that measure an ordinal association—the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment of the ordering labels "first", "second", "third", etc. to different observations of a particular variable. A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess the significance of the relation between them. For example, two common nonparametric methods of significance that use rank correlation are the Mann–Whitney U test and the Wilcoxon signed-rank test.
In statistics, the Durbin–Watson statistic is a test statistic used to detect the presence of autocorrelation at lag 1 in the residuals from a regression analysis. It is named after James Durbin and Geoffrey Watson. The small sample distribution of this ratio was derived by John von Neumann. Durbin and Watson applied this statistic to the residuals from least squares regressions, and developed bounds tests for the null hypothesis that the errors are serially uncorrelated against the alternative that they follow a first order autoregressive process. Note that the distribution of this test statistic does not depend on the estimated regression coefficients and the variance of the errors.
In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters. A pivot quantity need not be a statistic—the function and its value can depend on the parameters of the model, but its distribution must not. If it is a statistic, then it is known as an ancillary statistic.
In probability theory and statistics, partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed. If we are interested in finding to what extent there is a numerical relationship between two variables of interest, using their correlation coefficient will give misleading results if there is another, confounding, variable that is numerically related to both variables of interest. This misleading information can be avoided by controlling for the confounding variable, which is done by computing the partial correlation coefficient. This is precisely the motivation for including other right-side variables in a multiple regression; but while multiple regression gives unbiased results for the effect size, it does not give a numerical value of a measure of the strength of the relationship between the two variables of interest.
A ratio distribution is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two random variables X and Y, the distribution of the random variable Z that is formed as the ratio Z = X/Y is a ratio distribution.
In statistics, the concordance correlation coefficient measures the agreement between two variables, e.g., to evaluate reproducibility or for inter-rater reliability.
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product
In probability theory and statistics, cokurtosis is a measure of how much two random variables change together. Cokurtosis is the fourth standardized cross central moment. If two random variables exhibit a high level of cokurtosis they will tend to undergo extreme positive and negative deviations at the same time.
In statistics, effective sample size is a notion defined for a sample from a distribution when the observations in the sample are correlated or weighted. In 1965, Leslie Kish defined it as the original sample size divided by the design effect to reflect the variance from the current sampling design as compared to what would be if the sample was a simple random sample
In the mathematical theory of probability, multivariate Laplace distributions are extensions of the Laplace distribution and the asymmetric Laplace distribution to multiple variables. The marginal distributions of symmetric multivariate Laplace distribution variables are Laplace distributions. The marginal distributions of asymmetric multivariate Laplace distribution variables are asymmetric Laplace distributions.
In mathematics and theoretical computer science, analysis of Boolean functions is the study of real-valued functions on or from a spectral perspective. The functions studied are often, but not always, Boolean-valued, making them Boolean functions. The area has found many applications in combinatorics, social choice theory, random graphs, and theoretical computer science, especially in hardness of approximation, property testing, and PAC learning.
Alternating conditional expectations (ACE) is an algorithm to find the optimal transformations between the response variable and predictor variables in regression analysis.
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