Pearson correlation coefficient


In statistics, the Pearson correlation coefficient, also referred to as Pearson's r, the Pearson product-moment correlation coefficient, or the bivariate correlation, is a statistic that measures linear correlation between two variables X and Y. It has a value between +1 and −1. A value of +1 is total positive linear correlation, 0 is no linear correlation, and −1 is total negative linear correlation.

Naming and history

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. The naming of the coefficient is thus an example of Stigler's Law.

Definition

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 of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.

For a population

Pearson's correlation coefficient when applied to a population is commonly represented by the Greek letter ρ and may be referred to as the population correlation coefficient or the population Pearson correlation coefficient. Given a pair of random variables, the formula for ρ is:
where:
The formula for can be expressed in terms of mean and expectation. Since
the formula for can also be written as
where:
The formula for can be expressed in terms of uncentered moments. Since
the formula for can also be written as

For a sample

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. 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:
where:
Rearranging gives us this formula for :
where:
This formula suggests a convenient single-pass algorithm for calculating sample correlations, but, depending on the numbers involved, it can sometimes be numerically unstable.
Rearranging again gives us this formula for :
where:
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. One can use the following formula for :
where:

Practical issues

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.
In case of missing data, Garren derived the maximum likelihood estimator.

Mathematical properties

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, or to a bivariate distribution entirely supported on a line. The Pearson correlation coefficient is symmetric: corr = corr.
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, without changing the correlation coefficient. Note that more general linear transformations do change the correlation: see for an application of this.

Interpretation

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.
More generally, note that is positive if and only if Xi and Yi lie on the same side of their respective means. Thus the correlation coefficient is positive if Xi and Yi tend to be simultaneously greater than, or simultaneously less than, their respective means. The correlation coefficient is negative if Xi and Yi 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 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, and, obtained by regressing y on x and x on y respectively. One can show that if the standard deviations are equal, then, where sec and tan are trigonometric functions.
For centered data, the correlation coefficient can also be viewed as the cosine of the angle θ between the two observed vectors in N-dimensional space
Both the uncentered 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 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: and.
By the usual procedure for finding the angle θ between two vectors, the uncentered correlation coefficient is:
This uncentred correlation coefficient is identical with the cosine similarity.
Note that the above data were deliberately chosen to be perfectly correlated:. The Pearson correlation coefficient must therefore be exactly one. Centering the data yields and, from which
as expected.

Interpretation of the size of a correlation

Several authors have offered guidelines for the interpretation of a correlation coefficient. However, all such criteria are in some ways arbitrary. 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.

Inference

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.

Using a permutation test

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:
  1. Using the original paired data, randomly redefine the pairs to create a new data set, where the i′ are a permutation of the set. The permutation i′ is selected randomly, with equal probabilities placed on all n! possible permutations. This is equivalent to drawing the i′ randomly without replacement from the set. In bootstrapping, a closely related approach, the i and the i′ are equal and drawn with replacement from ;
  2. Construct a correlation coefficient r from the randomized data.
To perform the permutation test, repeat steps and a large number of times. The p-value for the permutation test is the proportion of the r values generated in step 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.

Using a bootstrap

The bootstrap can be used to construct confidence intervals for Pearson's correlation coefficient. In the "non-parametric" bootstrap, n pairs 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.

Testing using Student's ''t''-distribution

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 are white and have a bivariate normal distribution, the variable
has a Student's t-distribution in the null case. This holds approximately in case of non-normal observed values if sample sizes are large enough. For determining the critical values for r the inverse function is needed:
Alternatively, large sample, asymptotic approaches can be used.
Another early paper 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 white, the sampling distribution of Pearson's correlation coefficient follows a Student's t-distribution, but the degrees of freedom are reduced.

Using the exact distribution

For data that follow a bivariate normal distribution, the exact density function f for the sample correlation coefficient r of a normal bivariate is
where is the gamma function and is the Gaussian hypergeometric function.
In the special case when, the exact density function f 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.

Using the Fisher transformation

In practice, confidence intervals and hypothesis tests relating to ρ are usually carried out using the Fisher transformation, :
F 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 = 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 = 0.30952, so the confidence interval on the transformed scale is 0.30952 ± 1.96/, or. Converting back to the correlation scale yields.

In least squares regression analysis

The square of the sample correlation coefficient is typically denoted r2 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 Yi 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 and that is unexplained by X.
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
Thus
where
In the derivation above, the fact that
can be proved by noticing that the partial derivatives of the residual sum of squares over β0 and β1
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.

Sensitivity to the data distribution

Existence

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.

Sample size

Like many commonly used statistics, the sample statistic r is not robust, so its value can be misleading if outliers are present. Specifically, the PMCC is neither distributionally robust, nor outlier resistant. 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.

Variants

Variations of the correlation coefficient can be calculated for different purposes. Here are some examples.

Adjusted correlation coefficient

The sample correlation coefficient is not an unbiased estimate of. For data that follows a bivariate normal distribution, the expectation for the sample correlation coefficient of a normal bivariate is
The unique minimum variance unbiased estimator adj is given by
where:
An approximately unbiased estimator adj can be obtained by truncating and solving this truncated equation:
An approximate solution to equation is:
where in :
Another proposed adjusted correlation coefficient
is:
Note that adj ≈ for large values of .

Weighted correlation coefficient

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,
The reflective correlation is a variant of Pearson's correlation in which the data are not centered around their mean values. 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 coefficient

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. 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.

Pearson's distance

A distance metric for two variables X and Y known as Pearson's distance can be defined from their correlation coefficient as
Considering that the Pearson correlation coefficient falls between , the Pearson distance lies in . The Pearson distance has been used in cluster analysis and data detection for communications and storage with unknown gain and offset

Circular correlation coefficient

For variables X = and Y = that are defined on the unit circle it is possible to define a circular analog of Pearson's coefficient. 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.

Partial correlation

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.

Decorrelation of ''n'' random 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.
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 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.

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