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In
statistics Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, correlation or dependence is any statistical relationship, whether
causal Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
or not, between two
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s or
bivariate data In statistics, bivariate data is data on each of two variables, where each value of one of the variables is paired with a value of the other variable. Typically it would be of interest to investigate the possible association between the two variab ...
. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are '' linearly'' related. Familiar examples of dependent phenomena include the correlation between the
height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abo ...
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 economics, a demand curve is a graph depicting the relationship between the price of a certain commodity (the ''y''-axis) and the quantity of that commodity that is demanded at that price (the ''x''-axis). Demand curves can be used either for ...
. Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example, there is a causal relationship, because
extreme weather Extreme weather or extreme climate events includes unexpected, unusual, severe, or unseasonal weather; weather at the extremes of the historical distribution—the range that has been seen in the past. Often, extreme events are based on a locat ...
causes people to use more electricity for heating or cooling. However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship (i.e.,
correlation does not imply causation The phrase "correlation does not imply causation" refers to the inability to legitimately deduce a cause-and-effect relationship between two events or variables solely on the basis of an observed association or correlation between them. The id ...
). Formally, random variables are ''dependent'' if they do not satisfy a mathematical property of probabilistic independence. In informal parlance, ''correlation'' is synonymous with ''dependence''. However, when used in a technical sense, correlation refers to any of several specific types of mathematical operations between the tested variables and their respective expected values. Essentially, correlation is the measure of how two or more variables are related to one another. There are several
correlation coefficient A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two component ...
s, often denoted \rho or r, measuring the degree of correlation. The most common of these is the ''
Pearson correlation coefficient In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ...
'', which is sensitive only to a linear relationship between two variables (which may be present even when one variable is a nonlinear function of the other). Other correlation coefficients – such as '' Spearman's rank correlation'' – have been developed to be more
robust Robustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system’s functional body. In the same line ''robustness'' ca ...
than Pearson's, that is, more sensitive to nonlinear relationships.
Mutual information In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such a ...
can also be applied to measure dependence between two variables.


Pearson's product-moment coefficient

The most familiar measure of dependence between two quantities is the
Pearson product-moment correlation coefficient In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficien ...
(PPMCC), or "Pearson's correlation coefficient", commonly called simply "the correlation coefficient". It is obtained by taking the ratio of the covariance of the two variables in question of our numerical dataset, normalized to the square root of their variances. Mathematically, one simply divides the
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
of the two variables by the product of their
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
s.
Karl Pearson Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician and biostatistician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university st ...
developed the coefficient from a similar but slightly different idea by
Francis Galton Sir Francis Galton, FRS FRAI (; 16 February 1822 – 17 January 1911), was an English Victorian era polymath: a statistician, sociologist, psychologist, anthropologist, tropical explorer, geographer, inventor, meteorologist, proto- ...
. A Pearson product-moment correlation coefficient attempts to establish a line of best fit through a dataset of two variables by essentially laying out the expected values and the resulting Pearson's correlation coefficient indicates how far away the actual dataset is from the expected values. Depending on the sign of our Pearson's correlation coefficient, we can end up with either a negative or positive correlation if there is any sort of relationship between the variables of our data set. The population correlation coefficient \rho_ between two
random variables A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
X and Y with
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
s \mu_X and \mu_Y and
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
s \sigma_X and \sigma_Y is defined as: \rho_ = \operatorname(X,Y) = = , \quad \text\ \sigma_\sigma_>0. where \operatorname is the
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
operator, \operatorname means
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
, and \operatorname is a widely used alternative notation for the correlation coefficient. The Pearson correlation is defined only if both standard deviations are finite and positive. An alternative formula purely in terms of moments is: \rho_ =


Correlation and independence

It is a corollary of the
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality f ...
that the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of the Pearson correlation coefficient is not bigger than 1. Therefore, the value of a correlation coefficient ranges between −1 and +1. The correlation coefficient is +1 in the case of a perfect direct (increasing) linear relationship (correlation), −1 in the case of a perfect inverse (decreasing) linear relationship (anti-correlation), and some value in the
open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
(-1,1) in all other cases, indicating the degree of
linear dependence In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ar ...
between the variables. As it approaches zero there is less of a relationship (closer to uncorrelated). The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables. If the variables are
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...
, Pearson's correlation coefficient is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two variables. \begin X,Y \text \quad & \Rightarrow \quad \rho_ = 0 \quad (X,Y \text)\\ \rho_ = 0 \quad (X,Y \text)\quad & \nRightarrow \quad X,Y \text \end For example, suppose the random variable X is symmetrically distributed about zero, and Y=X^2. Then Y is completely determined by X, so that X and Y are perfectly dependent, but their correlation is zero; they are
uncorrelated In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, there ...
. However, in the special case when X and Y are jointly normal, uncorrelatedness is equivalent to independence. Even though uncorrelated data does not necessarily imply independence, one can check if random variables are independent if their
mutual information In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such a ...
is 0.


Sample correlation coefficient

Given a series of n measurements of the pair (X_i,Y_i) indexed by i=1,\ldots,n, the ''sample correlation coefficient'' can be used to estimate the population Pearson correlation \rho_ between X and Y. The sample correlation coefficient is defined as : r_ \quad \overset \quad \frac =\frac , where \overline and \overline are the sample
means Means may refer to: * Means LLC, an anti-capitalist media worker cooperative * Means (band), a Christian hardcore band from Regina, Saskatchewan * Means, Kentucky, a town in the US * Means (surname) * Means Johnston Jr. (1916–1989), US Navy ...
of X and Y, and s_x and s_y are the corrected sample standard deviations of X and Y. Equivalent expressions for r_ are : \begin r_ &=\frac \\ pt &=\frac. \end where s'_x and s'_y are the ''uncorrected'' sample standard deviations of X and Y. If x and y are results of measurements that contain measurement error, the realistic limits on the correlation coefficient are not −1 to +1 but a smaller range. For the case of a linear model with a single independent variable, the coefficient of determination (R squared) is the square of r_, Pearson's product-moment coefficient.


Example

Consider the
joint probability distribution Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
of and given in the table below. : For this joint distribution, the
marginal distribution In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables ...
s are: :\mathrm(X=x)= \begin \frac 1 3 & \quad \text x=0 \\ \frac 2 3 & \quad \text x=1 \end :\mathrm(Y=y)= \begin \frac 1 3 & \quad \text y=-1 \\ \frac 1 3 & \quad \text y=0 \\ \frac 1 3 & \quad \text y=1 \end This yields the following expectations and variances: :\mu_X = \frac 2 3 :\mu_Y = 0 :\sigma_X^2 = \frac 2 9 :\sigma_Y^2 = \frac 2 3 Therefore: : \begin \rho_ & = \frac \mathrm X-\mu_X)(Y-\mu_Y)\\ pt& = \frac \sum_ \\ pt& = \left(1-\frac 2 3\right)(-1-0)\frac + \left(0-\frac 2 3\right)(0-0)\frac + \left(1-\frac 2 3\right)(1-0)\frac = 0. \end


Rank correlation coefficients

Rank correlation 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 o ...
coefficients, such as
Spearman's rank correlation coefficient In statistics, Spearman's rank correlation coefficient or Spearman's ''ρ'', named after Charles Spearman and often denoted by the Greek letter \rho (rho) or as r_s, is a nonparametric measure of rank correlation (statistical dependence between ...
and Kendall's rank correlation coefficient (τ) measure the extent to which, as one variable increases, the other variable tends to increase, without requiring that increase to be represented by a linear relationship. If, as the one variable increases, the other ''decreases'', the rank correlation coefficients will be negative. It is common to regard these rank correlation coefficients as alternatives to Pearson's coefficient, used either to reduce the amount of calculation or to make the coefficient less sensitive to non-normality in distributions. However, this view has little mathematical basis, as rank correlation coefficients measure a different type of relationship than the
Pearson product-moment correlation coefficient In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficien ...
, and are best seen as measures of a different type of association, rather than as an alternative measure of the population correlation coefficient.Yule, G.U and Kendall, M.G. (1950), "An Introduction to the Theory of Statistics", 14th Edition (5th Impression 1968). Charles Griffin & Co. pp 258–270Kendall, M. G. (1955) "Rank Correlation Methods", Charles Griffin & Co. To illustrate the nature of rank correlation, and its difference from linear correlation, consider the following four pairs of numbers (x,y): :(0, 1), (10, 100), (101, 500), (102, 2000). As we go from each pair to the next pair x increases, and so does y. This relationship is perfect, in the sense that an increase in x is ''always'' accompanied by an increase in y. This means that we have a perfect rank correlation, and both Spearman's and Kendall's correlation coefficients are 1, whereas in this example Pearson product-moment correlation coefficient is 0.7544, indicating that the points are far from lying on a straight line. In the same way if y always ''decreases'' when x ''increases'', the rank correlation coefficients will be −1, while the Pearson product-moment correlation coefficient may or may not be close to −1, depending on how close the points are to a straight line. Although in the extreme cases of perfect rank correlation the two coefficients are both equal (being both +1 or both −1), this is not generally the case, and so values of the two coefficients cannot meaningfully be compared. For example, for the three pairs (1, 1) (2, 3) (3, 2) Spearman's coefficient is 1/2, while Kendall's coefficient is 1/3.


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 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 ...
. (See diagram above.) In the case of
elliptical distribution In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. Intuitively, in the simplified two and three dimensional case, the joint di ...
s it characterizes the (hyper-)ellipses of equal density; however, it does not completely characterize the dependence structure (for example, a
multivariate t-distribution In statistics, the multivariate ''t''-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization to random vectors of the Student's ''t''-distribution, which is a distribution applic ...
's degrees of freedom determine the level of tail dependence).
Distance correlation In statistics and in probability theory, distance correlation or distance covariance is a measure of dependence between two paired random vectors of arbitrary, not necessarily equal, dimension. The population distance correlation coefficient is zer ...
was introduced to address the deficiency of Pearson's correlation that it can be zero for dependent random variables; zero distance correlation implies independence. The Randomized Dependence Coefficient is a computationally efficient, copula-based measure of dependence between multivariate random variables. RDC is invariant with respect to non-linear scalings of random variables, is capable of discovering a wide range of functional association patterns and takes value zero at independence. For two binary variables, the
odds ratio An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due ...
measures their dependence, and takes range non-negative numbers, possibly infinity: . Related statistics such as Yule's ''Y'' and Yule's ''Q'' normalize this to the correlation-like range . The odds ratio is generalized by the logistic model to model cases where the dependent variables are discrete and there may be one or more independent variables. The
correlation ratio In statistics, the correlation ratio is a measure of the curvilinear relationship between the statistical dispersion within individual categories and the dispersion across the whole population or sample. The measure is defined as the ''ratio'' of tw ...
,
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
-based
mutual information In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such a ...
, total correlation, dual total correlation and
polychoric correlation In statistics, polychoric correlation{{Cite web, url=https://support.sas.com/documentation/cdl/en/procstat/65543/HTML/default/viewer.htm#procstat_corr_details14.htm, title=Base SAS(R) 9.3 Procedures Guide: Statistical Procedures, Second Edition, we ...
are all also capable of detecting more general dependencies, as is consideration of the copula between them, while the
coefficient of determination In statistics, the coefficient of determination, denoted ''R''2 or ''r''2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s). It is a statistic used i ...
generalizes the correlation coefficient to multiple regression.


Sensitivity to the data distribution

The degree of dependence between variables and does not depend on the scale on which the variables are expressed. That is, if we are analyzing the relationship between and , most correlation measures are unaffected by transforming to and to , where ''a'', ''b'', ''c'', and ''d'' are constants (''b'' and ''d'' being positive). This is true of some correlation
statistic A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hyp ...
s as well as their
population Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...
analogues. Some correlation statistics, such as the rank correlation coefficient, are also invariant to monotone transformations of the marginal distributions of and/or . Most correlation measures are sensitive to the manner in which and are sampled. Dependencies tend to be stronger if viewed over a wider range of values. Thus, if we consider the correlation coefficient between the heights of fathers and their sons over all adult males, and compare it to the same correlation coefficient calculated when the fathers are selected to be between 165 cm and 170 cm in height, the correlation will be weaker in the latter case. Several techniques have been developed that attempt to correct for range restriction in one or both variables, and are commonly used in meta-analysis; the most common are Thorndike's case II and case III equations. Various correlation measures in use may be undefined for certain joint distributions of and . For example, the Pearson correlation coefficient is defined in terms of moments, and hence will be undefined if the moments are undefined. Measures of dependence based on
quantile In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s are always defined. Sample-based statistics intended to estimate population measures of dependence may or may not have desirable statistical properties such as being unbiased, or asymptotically consistent, based on the spatial structure of the population from which the data were sampled. Sensitivity to the data distribution can be used to an advantage. For example,
scaled correlation In statistics, scaled correlation is a form of a coefficient of correlation applicable to data that have a temporal component such as time series. It is the average short-term correlation. If the signals have multiple components (slow and fast), sc ...
is designed to use the sensitivity to the range in order to pick out correlations between fast components of time series. By reducing the range of values in a controlled manner, the correlations on long time scale are filtered out and only the correlations on short time scales are revealed.


Correlation matrices

The correlation matrix of n random variables X_1,\ldots,X_n is the n \times n matrix C whose (i,j) entry is :c_:=\operatorname(X_i,X_j)=\frac,\quad \text\ \sigma_\sigma_>0. Thus the diagonal entries are all identically one. If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements o ...
of the standardized random variables X_i / \sigma(X_i) for i = 1, \dots, n. This applies both to the matrix of population correlations (in which case \sigma is the population standard deviation), and to the matrix of sample correlations (in which case \sigma denotes the sample standard deviation). Consequently, each is necessarily a
positive-semidefinite matrix In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a ...
. Moreover, the correlation matrix is strictly
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite fun ...
if no variable can have all its values exactly generated as a linear function of the values of the others. 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. A correlation matrix appears, for example, in one formula for the coefficient of multiple determination, a measure of goodness of fit in multiple regression. In
statistical modelling A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, ...
, correlation matrices representing the relationships between variables are categorized into different correlation structures, which are distinguished by factors such as the number of parameters required to estimate them. For example, in an exchangeable correlation matrix, all pairs of variables are modeled as having the same correlation, so all non-diagonal elements of the matrix are equal to each other. On the other hand, an
autoregressive In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
matrix is often used when variables represent a time series, since correlations are likely to be greater when measurements are closer in time. Other examples include independent, unstructured, M-dependent, and Toeplitz. In
exploratory data analysis In statistics, exploratory data analysis (EDA) is an approach of analyzing data sets to summarize their main characteristics, often using statistical graphics and other data visualization methods. A statistical model can be used or not, but prima ...
, the
iconography of correlations In exploratory data analysis, the iconography of correlations is a method which consists in replacing a correlation matrix by a diagram where the “remarkable” correlations are represented by a solid line (positive correlation), or a dotted line ...
consists in replacing a correlation matrix by a diagram where the “remarkable” correlations are represented by a solid line (positive correlation), or a dotted line (negative correlation).


Nearest valid correlation matrix

In some applications (e.g., building data models from only partially observed data) one wants to find the "nearest" correlation matrix to an "approximate" correlation matrix (e.g., a matrix which typically lacks semi-definite positiveness due to the way it has been computed). In 2002, Higham formalized the notion of nearness using the
Frobenius norm In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). Preliminaries Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m row ...
and provided a method for computing the nearest correlation matrix using the Dykstra's projection algorithm, of which an implementation is available as an online Web API. This sparked interest in the subject, with new theoretical (e.g., computing the nearest correlation matrix with factor structure) and numerical (e.g. usage the
Newton's method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real- ...
for computing the nearest correlation matrix) results obtained in the subsequent years.


Uncorrelatedness and independence of stochastic processes

Similarly for two stochastic processes \left\_ and \left\_: If they are independent, then they are uncorrelated. The opposite of this statement might not be true. Even if two variables are uncorrelated, they might not be independent to each other.


Common misconceptions


Correlation and causality

The conventional dictum that "
correlation does not imply causation The phrase "correlation does not imply causation" refers to the inability to legitimately deduce a cause-and-effect relationship between two events or variables solely on the basis of an observed association or correlation between them. The id ...
" means that correlation cannot be used by itself 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 ( tautologies), where no causal process exists. Consequently, 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.


Simple linear correlations

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 In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given ...
of Y given X, denoted \operatorname(Y \mid X), is not linear in X, the correlation coefficient will not fully determine the form of \operatorname(Y \mid X). The adjacent image shows
scatter plot A scatter plot (also called a scatterplot, scatter graph, scatter chart, scattergram, or scatter diagram) is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of data. ...
s of
Anscombe's quartet Anscombe's quartet comprises four data sets that have nearly identical simple descriptive statistics, yet have very different distributions and appear very different when graphed. Each dataset consists of eleven (''x'',''y'') points. They were ...
, a set of four different pairs of variables created by
Francis Anscombe Francis John Anscombe (13 May 1918 – 17 October 2001) was an English statistician. Born in Hove in England, Anscombe was educated at Trinity College at Cambridge University. After serving in the Second World War, he joined Rothamsted E ...
. The four y variables have the same mean (7.5), variance (4.12), correlation (0.816) and regression line (''y'' = 3 + 0.5''x''). 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. In this case the Pearson correlation coefficient does not indicate that there is an exact functional relationship: only the extent to which that relationship can be approximated by a linear relationship. In the third case (bottom left), the linear relationship is perfect, except for one
outlier In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter ar ...
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 In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate the largest amount of information as simply as possible. Statisticians commonly try to describe the observations in * a measure of ...
, cannot replace visual examination of the data. The examples are sometimes said to demonstrate that the Pearson correlation assumes that the data follow a
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...
, but this is only partially correct. The Pearson correlation can be accurately calculated for any distribution that has a finite
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements o ...
, which includes most distributions encountered in practice. However, the Pearson correlation coefficient (taken together with the sample mean and variance) is only a
sufficient statistic In statistics, a statistic is ''sufficient'' with respect to a statistical model and its associated unknown parameter if "no other statistic that can be calculated from the same sample provides any additional information as to the value of the par ...
if the data is drawn from a
multivariate normal distribution 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 ...
. As a result, the Pearson correlation coefficient fully characterizes the relationship between variables if and only if the data are drawn from a multivariate normal distribution.


Bivariate normal distribution

If a pair (X,Y) of random variables follows a
bivariate normal distribution 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 ...
, the conditional mean \operatorname(X \mid Y) is a linear function of Y, and the conditional mean \operatorname(Y \mid X) is a linear function of X. The correlation coefficient \rho_ between X and Y, along with the
marginal Marginal may refer to: * ''Marginal'' (album), the third album of the Belgian rock band Dead Man Ray, released in 2001 * ''Marginal'' (manga) * '' El Marginal'', Argentine TV series * Marginal seat or marginal constituency or marginal, in polit ...
means and variances of X and Y, determines this linear relationship: :\operatorname(Y\mid X) = \operatorname(Y) + \rho_ \cdot \sigma_Y\frac, where \operatorname(X) and \operatorname(Y) are the expected values of X and Y, respectively, and \sigma_X and \sigma_Y are the standard deviations of X and Y, respectively. The empirical correlation r is an
estimate Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is der ...
of the correlation coefficient \rho. A distribution estimate for \rho is given by\pi (\rho , r) = \frac (1 - r^2)^ \cdot (1 - \rho^2)^ \cdot (1 - r \rho )^ F\!\left(\frac,-\frac; \nu + \frac; \frac\right)where F is the
Gaussian hypergeometric function Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponym ...
and \nu = N-1 > 1 . This density is both a Bayesian posterior density and an exact optimal
confidence distribution In statistical inference, the concept of a confidence distribution (CD) has often been loosely referred to as a distribution function on the parameter space that can represent confidence intervals of all levels for a parameter of interest. Histori ...
density.


See also

*
Autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable as ...
*
Canonical correlation 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'' ...
*
Coefficient of determination In statistics, the coefficient of determination, denoted ''R''2 or ''r''2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s). It is a statistic used i ...
*
Cointegration Cointegration is a statistical property of a collection of time series variables. First, all of the series must be integrated of order ''d'' (see Order of integration). Next, if a linear combination of this collection is integrated of order less ...
*
Concordance correlation coefficient In statistics, the concordance correlation coefficient measures the agreement between two variables, e.g., to evaluate reproducibility or for inter-rater reliability. Definition The form of the concordance correlation coefficient \rho_c as :\rho_c ...
*
Cophenetic correlation In statistics, and especially in biostatistics, cophenetic correlation (more precisely, the cophenetic correlation coefficient) is a measure of how faithfully a dendrogram preserves the pairwise distances between the original unmodeled data points. ...
*
Correlation function A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables r ...
* Correlation gap *
Covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
*
Covariance and correlation In probability theory and statistics, the mathematical concepts of covariance and correlation are very similar. Both describe the degree to which two random variables or sets of random variables tend to deviate from their expected values in sim ...
* Cross-correlation * Ecological correlation *
Fraction of variance unexplained In statistics, the fraction of variance unexplained (FVU) in the context of a regression task is the fraction of variance of the regressand (dependent variable) ''Y'' which cannot be explained, i.e., which is not correctly predicted, by the exp ...
*
Genetic correlation In multivariate quantitative genetics, a genetic correlation (denoted r_g or r_a) is the proportion of variance that two traits share due to genetic causes, the correlation between the genetic influences on a trait and the genetic influences on a d ...
*
Goodman and Kruskal's lambda In probability theory and statistics, Goodman & Kruskal's lambda (\lambda) is a measure of proportional reduction in error in cross tabulation analysis. For any sample with a nominal independent variable and dependent variable (or ones that can be ...
*
Iconography of correlations In exploratory data analysis, the iconography of correlations is a method which consists in replacing a correlation matrix by a diagram where the “remarkable” correlations are represented by a solid line (positive correlation), or a dotted line ...
*
Illusory correlation In psychology, illusory correlation is the phenomenon of perceiving a relationship between variables (typically people, events, or behaviors) even when no such relationship exists. A false association may be formed because rare or novel occurren ...
* Interclass correlation *
Intraclass correlation In statistics, the intraclass correlation, or the intraclass correlation coefficient (ICC), is a descriptive statistic that can be used when quantitative measurements are made on units that are organized into groups. It describes how strongly u ...
*
Lift (data mining) In data mining and association rule learning, lift is a measure of the performance of a targeting model (association rule) at predicting or classifying cases as having an enhanced response (with respect to the population as a whole), measured ag ...
*
Mean dependence In probability theory, a random variable Y is said to be mean independent of random variable X if and only if its conditional mean E(Y , X = x) equals its (unconditional) mean E(Y) for all x such that the probability density/mass of X at x, f_X(x), ...
*
Modifiable areal unit problem __NOTOC__ The modifiable areal unit problem (MAUP) is a source of statistical bias that can significantly impact the results of statistical hypothesis tests. MAUP affects results when point-based measures of spatial phenomena are aggregated into ...
*
Multiple correlation In statistics, the coefficient of multiple correlation is a measure of how well a given variable can be predicted using a linear function of a set of other variables. It is the correlation between the variable's values and the best predictions th ...
*
Point-biserial correlation coefficient The point biserial correlation coefficient (''rpb'') is a correlation coefficient used when one variable (e.g. ''Y'') is dichotomous; ''Y'' can either be "naturally" dichotomous, like whether a coin lands heads or tails, or an artificially dichotomi ...
* Quadrant count ratio *
Spurious correlation In statistics, a spurious relationship or spurious correlation is a mathematical relationship in which two or more events or variables are associated but '' not'' causally related, due to either coincidence or the presence of a certain third, un ...
*
Statistical arbitrage In finance, statistical arbitrage (often abbreviated as ''Stat Arb'' or ''StatArb'') is a class of short-term financial trading strategies that employ mean reversion models involving broadly diversified portfolios of securities (hundreds to thousan ...
*
Subindependence In probability theory and statistics, subindependence is a weak form of independence. Two random variables ''X'' and ''Y'' are said to be subindependent if the characteristic function of their sum is equal to the product of their marginal characte ...


References


Further reading

* * *


External links


MathWorld page on the (cross-)correlation coefficient/s of a sample

Compute significance between two correlations
for the comparison of two correlation values. *
Proof that the Sample Bivariate Correlation has limits plus or minus 1


by Juha Puranen. * ttps://web.archive.org/web/20150407112430/http://www.biostat.katerynakon.in.ua/en/association/correlation.html Correlation analysis. Biomedical Statistics* R-Psychologis
Correlation
visualization of correlation between two numeric variables {{DEFAULTSORT:Correlation And Dependence Covariance and correlation Dimensionless numbers