corrcoef

Correlation coefficient

Introduction

Correlation coefficient is a measure for linear relationship between two random variables. Let $X$ and $Y$ be random variables (real or complex). Their correlation coefficient is defined as

where $\sigma_{XY}$ is the covariance, and $\sigma_X$ and $\sigma_Y$ are the standard deviations. For a description of covariance, see the document page for the function cov.

• The variables $X$ and $Y$ are called correlated if $\rho_{XY}\neq 0$.
• It always holds that $-1\leq\rho_{XY}\leq 1$.
• Complete positive correlation means $\rho_{XY}=1$, which suggests that $(X,Y)$ fall along a line with positive slope.
• Complete negative correlation means $\rho_{XY}=-1$, which suggests that $(X,Y)$ fall along a line with negative slope.

The correlation coefficient $r$ obtained by corrcoef is actually a sample statistics, which gives an approximation of the coefficient $\rho_{XY}$ of the population. One can use $r$ to test the null hypothesis $H_0: \rho_{XY}=0$ against $H_a:\rho_{XY}\neq 0$. The null hypothesis suggests that $X$ and $Y$ are uncorrelated, whereas the alternative hypothesis suggests that they are somewhat linearly correlated. For the test statistic

the $p$-value is given by

If the $p$-value is less than a given significant level $\alpha$, the correlation is considered significant. A low $p$-value suggests that observing the null hypothesis $H_0$ is unlikely. A $100(1-\alpha)\%$ confidence interval for $\rho_{XY}$ is given by

where $a=\sqrt{\frac{2}{N-3}}\text{erf}^{-1}(1-\alpha)$.

Usage

R = corrcoef(A)

[R, P] = corrcoef(A)

[R, P, RL, RU] = corrcoef(A)

R = corrcoef(A, name_1, value_1, name_2, value_2)

[R, P] = corrcoef(A, name_1, value_1, name_2, value_2)

[R, P, RL, RU] = corrcoef(A, name_1, value_1, name_2, value_2)

• A should be an array of real or complex numbers, where ndims(A)==2.

• If A is empty or scalar, the outputs R, P, RL and RU are all NaN.

• If A is a vector, the outputs R, P, RL and RU are all equal to 1. If A contains NaN or Inf, they are all equal to NaN.

• If A is a matrix, each column of A contains observations of a random variable.

  • The output R is the correlation matrix where R(i,j) is the correlation coefficient of the i-th and j-th columns for $i\neq j$. The diagonal elements of R are all equal to 1.
  • The output P is a matrix having the same size as R, where P(i,j) is the $p$-value for R(i,j). The diagonal elements of P are all equal to 1.
  • The outputs RL and RU are matrices having the same size as R, where RL(i,j) and RU(i,j) give the lower and upper bounds, respectively, for the $(1-\alpha)\%$ confidence interval of R(i,j). The diagonal elements of RL and RU are equal to 1.
  • If the output R is complex, the outputs P, RL and RU are undefined. Therefore, there should be only one output argument.

• name_i is either 'alpha' or 'rows'.

  • When it is 'alpha', value_i should be a real number $\alpha$ with $0<\alpha<1$. It gives the significant level. The default value is $\alpha=0.05$.

  • When it is 'rows', value_i is either 'all' (default), 'complete' or 'pairwise':

    • all (default): NaN values are included in the computation of the output(s).
    • complete: When any row of A contains NaN, the whole row of A will be removed before computing any element of R. In other words, the number of rows of A will be reduced before computing R. (See Example 2 below.) The output R is NaN if the removal of NaN results in an empty matrix.
    • pairwise: When computing the correlation coefficient R(i,j), the $i$-th and $j$-th columns of A are examined. Any NaN appearing in one of columns and the corresponding element in the other column will be removed. (See Example 2 below.) The coefficient R(i,j) is NaN if the removal of NaN results in two empty columns.

R = corrcoef(A, B)

[R, P] = corrcoef(A, B)

[R, P, RL, RU] = corrcoef(A, B)

R = corrcoef(A, B, name_1, value_1, name_2, value_2)

[R, P] = corrcoef(A, B, name_1, value_1, name_2, value_2)

[R, P, RL, RU] = corrcoef(A, B, name_1, value_1, name_2, value_2)

• A and B are real or complex arrays having the same number of elements.

• A and B should be vectors. Otherwise, they will be reshaped to vectors.

• If A and B are empty, the outputs R, P, RL and RU are $2\times 2$ matrices of NaN.

• If A and B are scalars, and

  • are equal, then R, P, RL and RU are $2\times 2$ matrices of NaN.
  • are not equal, then R, P, RL and RU are $2\times 2$ matrices of 1.

• If A and B are vectors, the outputs R, P, RL and RU are $2\times 2$ matrices.

  • R is a $2\times 2$ matrix, where

    • R(1,2) and R(2,1) are the correlation coefficients.

  • P is a $2\times 2$ matrix, which is defined only when R is real, where

    • P(1,2) and P(2,1) are the $p$-values of R(1,2) and R(2,1), respectively.
    • The diagonal elements are all equal to 1.

  • RL is a $2\times 2$ matrix, which is defined only when R is real, where

    • RL(1,2) and RL(2,1) are the lower bounds of the $(1-\alpha)\%$ confidence intervals for R(1,2) and R(2,1), respectively.
    • The diagonal elements are all equal to 1.

  • RU is a $2\times 2$ matrix, which is defined only when R is real, where

    • RU(1,2) and RU(2,1) are the upper bounds of the $(1-\alpha)\%$ confidence intervals for R(1,2) and R(2,1), respectively.
    • The diagonal elements are all equal to 1.

• If the output R is complex, the outputs P, RL and RU are undefined. Therefore, there should be only one output argument.

• name_i is either 'alpha' or 'rows'.

  • When it is 'alpha', value_i should be a real number $\alpha$ with $0<\alpha<1$. It specifies the given significant level. The default value is $\alpha=0.05$.

  • When it is 'rows', value_i is either 'all' (default), 'complete' or 'pairwise':

    • all (default): NaN values are included in the computation of the output(s).
    • complete: When the $i$-th element of A or B contains NaN, the $i$-th elements of both A and B will be removed before computing any element of R. The output R is NaN if the removal of NaN results in empty arrays.
    • pairwise: This is different from complete in such a way that the removal of NaN is deferred to the stage when individual elements of R is computed.

Examples

Example 1

This example shows the correlation coefficient of $X$ and $Y$, where $Y=1+2X+\epsilon$ with noise $\epsilon\sim N(0,10)$ following normal distribution with zero mean and standard deviation of 10. The sample data show a linear relationship between $X$ and $Y$ with the 95% confidence interval of $[0.2123, 0.5467]$ as shown in the output below. The $p$-value is about $5.3940\times 10^{-5}$, which suggests that it is extremely unlikely that $X$ and $Y$ are uncorrelated.

Input

clear
x=sort(10*rand(1,100));
% y = 1 + 2x + noise.
y=1+2*x+10*randn(size(x));
plot(x,1+2*x);
hold('on')
scatter(x,y);
hold('off')
% Covariance of x and y is C(1,2) or C(2,1)
[R,P,RL,RU]=corrcoef(x,y)

Output

All variables cleared.

R = 
 1.000000000000000   0.392400272934828
 0.392400272934828   1.000000000000000

P = 
 1.000000000000000   0.000053940195081
 0.000053940195081   1.000000000000000

RL = 
 1.000000000000000   0.212348840869820
 0.212348840869820   1.000000000000000

RU = 
 1.000000000000000   0.546683054932463
 0.546683054932463   1.000000000000000

Example 2

This example demonstrates the difference between corrcoef(a,'rows','complete') and corrcoef(a,'rows','pairwise').

• When 'complete' is used, NaN elements in a(:,1) and the corresponding elements in a(:,2) and a(:,3) are removed before the computation of R1. Therefore, when R1(3,2) is computed, vectors of reduced length, namely, [11.2 12.8 13.0 14.1 16.3 17.4 19.1 20.9] and [21 22 23 24 26 27 29 30] are used although neither a(:,2) nor a(:,3) have NaN elements.
• When 'pairwise' is used, the removal of NaN is performed when individual elements of R2 is computed. Therefore, the computation of R2(3,2) uses all elements of a(:,2) and a(:,3).
• The results R1(3,2) and R2(3,2) are different due to different values of 'rows'.

Input

% Matrix of two columns
a=[
1.3 11.2 21;
2.5 12.8 22;
3.6 13.0 23;
4.1 14.1 24;
NaN 15.2 25;
6.1 16.3 26;
7.4 17.4 27;
NaN 18.4 28;
9.9 19.1 29;
10.1 20.9 30];
% Use 'complete'
R1=corrcoef(a,'rows','complete')
% Use 'pairwise'
R2=corrcoef(a,'rows','pairwise')

Output

R1 = 
 1.0000   0.9890   0.9962
 0.9890   1.0000   0.9955
 0.9962   0.9955   1.0000

R2 = 
 1.0000   0.9890   0.9962
 0.9890   1.0000   0.9957
 0.9962   0.9957   1.0000

References

• James T. McClave and Terry Sincich, A First Course in Statistics, 12th edition, Pearson, 2018.
• William H. Press et al., Numerical Recipes: The Art of Scientific Computing, 3rd edition, Cambridge University Press, 2007.
• Douglas C. Montgomery and George C. Runger, Applied Statistics and Probability for Engineers, 6th edition, Wiley, 2014.