Correlation



In statistics, correlation (often measured as a correlation coefficient, ϝ) indicates the strength and direction of a linear relationship between two random variables. That is in contrast with the usage of the term in colloquial speech, which denotes any relationship, not necessarily linear. In general statistical usage, correlation or co-relation refers to the departure of two random variables from independence. In this broad sense there are several coefficients, measuring the degree of correlation, adapted to the nature of the data.

Pearson's product-moment coefficient
A number of different coefficients are used for different situations. The best known is the efficient, which is obtained by dividing the covariance of the two variables by the product of their standard deviations. Despite its name, it was first introduced by Francis Galton.

Mathematical properties
The correlation coefficient ϝX, Y between two random variables X and Y with expected values μX and μY and standard deviations σX and σY is defined as:


 * $$\rho_{X,Y}={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ={E((X-\mu_X)(Y-\mu_Y)) \over \sigma_X\sigma_Y},$$

where E is the expected value operator and cov means covariance. A widely used alternative notation is
 * $$\mathrm{corr}(X,Y)=\rho_{X,Y} \,.$$

Since μX = E(X), σX2 = E[(X - E(X))2] = E(X2) − E2(X) and likewise for Y, and since $$E[(X-E(X))(Y-E(Y))]=E(XY)-E(X)E(Y)$$, we may also write


 * $$\rho_{X,Y}=\frac{E(XY)-E(X)E(Y)}{\sqrt{E(X^2)-E^2(X)}~\sqrt{E(Y^2)-E^2(Y)}}.$$

The correlation is defined only if both of the standard deviations are finite and both of them are nonzero. It is a corollary of the Cauchy–Schwarz inequality that the correlation cannot exceed 1 in absolute value.

The correlation is 1 in the case of an increasing linear relationship, −1 in the case of a decreasing linear relationship, and some value in between in all other cases, indicating the degree of linear dependence between the variables. The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables.

If the variables are independent then the correlation is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two variables. Here is an example: Suppose the random variable X is uniformly distributed on the interval from −1 to 1, and Y = X2. Then Y is completely determined by X, so that X and Y are dependent, but their correlation is zero; they are uncorrelated. However, in the special case when X and Y are jointly normal, uncorrelatedness is equivalent to independence.

A correlation between two variables is diluted in the presence of measurement error around estimates of one or both variables, in which case disattenuation provides a more accurate coefficient.

Sample correlation
If we have a series of n measurements of X  and Y  written as xi  and yi  where i = 1, 2, ..., n, then the efficient can be used to estimate the correlation of X  and Y. The Pearson coefficient is also known as the "sample correlation coefficient". The Pearson correlation coefficient is then the best estimate of the correlation of X and Y. The Pearson correlation coefficient is written:



r_{xy}=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{(n-1) s_x s_y}, $$

where $$\bar{x}$$ and $$\bar{y}$$ are the sample means of X and Y, sx  and sy  are the sample standard deviations of X  and Y  and the sum is from i = 1 to  n. As with the population correlation, we may rewrite this as



r_{xy}=\frac{\sum x_iy_i-n \bar{x} \bar{y}}{(n-1) s_x s_y}=\frac{n\sum x_iy_i-\sum x_i\sum y_i} {\sqrt{n\sum x_i^2-(\sum x_i)^2}~\sqrt{n\sum y_i^2-(\sum y_i)^2}}. $$

Again, as is true with the population correlation, the absolute value of the sample correlation must be less than or equal to 1. Though the above formula conveniently suggests a single-pass algorithm for calculating sample correlations, it is notorious for its numerical instability; the page for the efficient contains a numerically stable one-pass algorithm for computing Pearson product-moment correlation coefficients.

The square of the sample correlation coefficient, which is also known as the coefficient of determination, is the fraction of the variance in yi that is accounted for by a linear fit of xi  to yi. This is written


 * $$r_{xy}^2=1-\frac{s_{y|x}^2}{s_y^2},$$

where sy2 is the square of the error of a linear regression  of xi  on yi  by the equation y = a + bx:


 * $$s_{y|x}^2=\frac{1}{n-1}\sum_{i=1}^n (y_i-a-bx_i)^2,$$

and sy2 is just the variance of y:


 * $$s_y^2=\frac{1}{n-1}\sum_{i=1}^n (y_i-\bar{y})^2.$$

Note that since the sample correlation coefficient is symmetric in xi and yi, we will get the same value for a fit of yi  to xi :


 * $$r_{xy}^2=1-\frac{s_{x|y}^2}{s_x^2}.$$

This equation also gives an intuitive idea of the correlation coefficient for higher dimensions. Just as the above described sample correlation coefficient is the fraction of variance accounted for by the fit of a 1-dimensional linear submanifold to a set of 2-dimensional vectors (xi, yi ), so we can define a correlation coefficient for a fit of an m-dimensional linear submanifold to a set of n-dimensional vectors. For example, if we fit a plane z = a + bx + cy to a set of data (xi, yi , zi ) then the correlation coefficient of z  to x  and y  is


 * $$r^2=1-\frac{s_{z|xy}^2}{s_z^2}.$$

The distribution of the correlation coefficient has been examined by R. A. Fisher and A. K. Gayen.

Geometric interpretation
For centered data (i.e., data which have been shifted by the sample mean so as to have an average of zero), the correlation coefficient can also be viewed as the cosine of the angle between the two vectors of samples drawn from the two random variables.

Some practitioners prefer an uncentered (non-Pearson-compliant) correlation coefficient. See the example below for a comparison.

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 between two vectors (see dot product), the uncentered correlation coefficient is:


 * $$ \cos \theta = \frac { \bold{x} \cdot \bold{y} } { \left\| \bold{x} \right\| \left\| \bold{y} \right\| } = \frac { 2.93 } { \sqrt { 103 } \sqrt { 0.0983 } } = 0.920814711. $$

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


 * $$ \cos \theta = \frac { \bold{x} \cdot \bold{y} } { \left\| \bold{x} \right\| \left\| \bold{y} \right\| } = \frac { 0.308 } { \sqrt { 30.8 } \sqrt { 0.00308 } } = 1 = \rho_{xy}, $$

as expected.

Motivation for the form of the coefficient of correlation
Another motivation for correlation comes from inspecting the method of simple linear regression. As above, X is the vector of independent variables, $$x_i$$, and Y of the dependent variables, $$y_i$$, and a simple linear relationship between X and Y is sought, through a least-squares method on the estimate of Y:
 * $$ \ Y = X\beta + \varepsilon.\, $$

Then, the equation of the least-squares line can be derived to be of the form:



(Y - \bar{Y}) = \frac{n\sum x_iy_i-\sum x_i\sum y_i} {n\sum x_i^2-(\sum x_i)^2} (X - \bar{X}) $$

which can be rearranged in the form:

(Y - \bar{Y})=\frac{r s_y}{s_x} (X-\bar{X}) $$

where r has the familiar form mentioned above
 * $$ r=\frac{n\sum x_iy_i-\sum x_i\sum y_i} {\sqrt{n\sum x_i^2-(\sum x_i)^2}~\sqrt{n\sum y_i^2-(\sum y_i)^2}}.$$

Interpretation of the size of a correlation
Several authors have offered guidelines for the interpretation of a correlation coefficient. Cohen (1988), has observed, however, that all such criteria are in some ways arbitrary and should not be observed too strictly. This is because the interpretation of a correlation coefficient depends on the context and purposes. A correlation of 0.9 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.

Along this vein, it is important to remember that "large" and "small" should not be taken as synonyms for "good" and "bad" in terms of determining that a correlation is of a certain size. For example, a correlation of 1.0 or −1.0 indicates that the two variables analyzed are equivalent modulo scaling. Scientifically, this more frequently indicates a trivial result than a profound one. For example, consider discovering a correlation of 1.0 between how many feet tall a group of people are and the number of inches from the bottom of their feet to the top of their heads.

Sensitivity to the data distribution
The population version of Pearson's correlation coefficient is defined in terms of moments, and exists for any bivariate probability distribution for which the population covariance is defined and the marginal population variances are defined and non-zero. In the case of the bivariate normal distribution, the correlation coefficient characterizes the joint distribution as long as the marginal means and variances are known. For most other bivariate distributions this is not true. Nevertheless, the correlation coefficient is highly informative about the degree of linear dependence between two random quantities regardless of whether their joint distribution is normal.

The sample correlation coefficient is the maximum likelihood estimate of the population correlation coefficient for bivariate normal data, and is asymptotically unbiased and efficient, which roughly means that it is impossible to construct a more accurate estimate than the sample correlation coefficient if the data are normal and the sample size is moderate or large. For non-normal populations, the sample correlation coefficient remains approximately unbiased, but may not be efficient. The sample correlation coefficient is a consistent estimator of the population correlation coefficient as long as the sample means, variances, and covariance are consistent (which is guaranteed when the law of large numbers can be applied).

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. A stratified analysis is one way to accommodate a lack of bivariate normality due to clustering, assessing the effect of a risk factor on outcome while holding another variable constant.

Correlation measures other than Pearson's correlation have their own sensitivities to the data distribution. The population versions of correlation measures based on quantiles or ranks are always defined. Their sample-based estimates will be consistent as long as the underlying sample quantiles are consistent.

Most correlation measures in common use are invariant to location and scale transformations of the marginal distributions. That is, if we are analyzing the relationship between X and Y, the correlation is unaffected by transforming X to a+bX and Y to c+dY, where a, b, c, and d are constants. This is true of most correlation statistics as well as their population analogues.

Non-parametric correlation coefficients
The sample correlation coefficient is not robust, meaning that it is sensitive to outliers in a data set. Non-parametric correlation coefficients, such as Chi-square, Point biserial correlation, Spearman's ϝ, Kendall's τ, and Goodman and Kruskal's lambda may perform better than the sample correlation coefficient when outliers are present. These methods are often less precise than the sample correlation if no outliers are present. Note that in general these non-parametric statistics have different expected values from each other, and from the Pearson correlation coefficient, even for large samples. Since they estimate different population parameters, in general they cannot be directly compared. They generally should be viewed as alternative measures of association, rather than as alternative estimators of the population correlation coefficient.

Other measures of dependence among random variables
The information given by a correlation coefficient is not enough to define the dependence structure between random variables. The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the distribution is a multivariate normal distribution. (See diagram above.) In the case of elliptic distributions it characterizes the (hyper-)ellipses of equal density, however, it does not completely characterize the dependence structure (for example, the a multivariate t-distribution's degrees of freedom determine the level of tail dependence).

To get a measure for more general dependencies in the data (also nonlinear) it is better to use the correlation ratio which is able to detect almost any functional dependency, or the entropy-based mutual information/total correlation which is capable of detecting even more general dependencies. The latter are sometimes referred to as multi-moment correlation measures, in comparison to those that consider only 2nd moment (pairwise or quadratic) dependence.

The polychoric correlation is another correlation applied to ordinal data that aims to estimate the correlation between theorised latent variables.

One way to capture a more complete view of dependence structure is to consider a copula between them.

Correlation matrices
The correlation matrix of n random variables X1, ..., Xn is the n ×  n matrix whose i,j entry is corr(Xi, Xj). If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the covariance matrix of the standardized random variables Xi /SD(Xi) for i = 1, ..., n. Consequently it is necessarily a positive-semidefinite matrix.

The correlation matrix is symmetric because the correlation between $$X_i$$ and $$X_j$$ is the same as the correlation between $$X_j$$ and $$X_i$$.

Correlation and causality
The conventional dictum that "correlation does not imply causation" means that correlation cannot be used to infer a causal relationship between the variables. This dictum should not be taken to mean that correlations cannot indicate the potential existence of causal relations. However, the causes underlying the correlation, if any, may be indirect and unknown, and high correlations also overlap with identity relations, where no causal process exists. Consequently, establishing a correlation between two variables is not a sufficient condition to establish a causal relationship (in either direction).

A correlation between age and height in children is fairly causally transparent, but a correlation between mood and health in people is less so. Does improved mood lead to improved health; or does good health lead to good mood; or both? Or does some other factor underlie both? In other words, a correlation can be taken as evidence for a possible causal relationship, but cannot indicate what the causal relationship, if any, might be.

Correlation and linearity
The Pearson correlation coefficient indicates the strength of a linear relationship between two variables, but its value generally does not completely characterize their relationship. In particular, if the conditional mean of Y given X, denoted E(Y|X), is not linear in X, the correlation coefficient will not fully determine the form of E(Y|X).

The image on the right shows scatterplots of Anscombe's quartet, a set of four different pairs of variables created by Francis Anscombe. The four $$y$$ variables have the same mean (7.5), standard deviation (4.12), correlation (0.816) and regression line ($$y = 3 + 0.5x$$). However, as can be seen on the plots, the distribution of the variables is very different. The first one (top left) seems to be distributed normally, and corresponds to what one would expect when considering two variables correlated and following the assumption of normality. The second one (top right) is not distributed normally; while an obvious relationship between the two variables can be observed, it is not linear, and the Pearson correlation coefficient is not relevant. In the third case (bottom left), the linear relationship is perfect, except for one outlier which exerts enough influence to lower the correlation coefficient from 1 to 0.816. Finally, the fourth example (bottom right) shows another example when one outlier is enough to produce a high correlation coefficient, even though the relationship between the two variables is not linear.

These examples indicate that the correlation coefficient, as a summary statistic, cannot replace the individual examination of the data. Note that the examples are sometimes said to demonstrate that the Pearson correlation assumes that the data follow a normal distribution, but this is not correct.

If a pair (X, Y) of random variables follows a bivariate normal distribution, the conditional mean E(X|Y) is a linear function of Y, and the conditional mean E(Y|X) is a linear function of X. The correlation coefficient r between X and Y, along with the marginal means and variances of X and Y, determines this linear relationship:



E(Y|X) = EY + r\sigma_y\frac{X-EX}{\sigma_x}, $$

where EX and EY are the expected values of X and Y, respectively, and σx and σy are the standard deviations of X and Y, respectively.