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# corr

Linear or rank correlation

## Syntax

RHO = corr(X)
RHO = corr(X,Y)
[RHO,PVAL] = corr(X,Y)
[RHO,PVAL] = corr(X,Y,'name',value)

## Description

RHO = corr(X) returns a p-by-p matrix containing the pairwise linear correlation coefficient between each pair of columns in the n-by-p matrix X.

RHO = corr(X,Y) returns a p1-by-p2 matrix containing the pairwise correlation coefficient between each pair of columns in the n-by-p1 and n-by-p2 matrices X and Y.

The difference between corr(X,Y) and the MATLAB® function corrcoef(X,Y) is that corrcoef(X,Y) returns a matrix of correlation coefficients for the two column vectors X and Y. If X and Y are not column vectors, corrcoef(X,Y) converts them to column vectors.

[RHO,PVAL] = corr(X,Y) also returns PVAL, a matrix of p-values for testing the hypothesis of no correlation against the alternative that there is a nonzero correlation. Each element of PVAL is the p value for the corresponding element of RHO. If PVAL(i,j) is small, say less than 0.05, then the correlation RHO(i,j) is significantly different from zero.

[RHO,PVAL] = corr(X,Y,'name',value) specifies one or more optional name/value pairs. Specify name inside single quotes. The following table lists valid parameters and their values.

ParameterValues
type
• 'Pearson' (the default) computes Pearson's linear correlation coefficient

• 'Kendall' computes Kendall's tau

• 'Spearman' computes Spearman's rho

rows
• 'all' (the default) uses all rows regardless of missing values (NaNs)

• 'complete' uses only rows with no missing values

• 'pairwise'computes RHO(i,j) using rows with no missing values in column i or j

tail

— The alternative hypothesis against which to compute p-values for testing the hypothesis of no correlation

• 'both' — Correlation is not zero (the default)

• 'right' — Correlation is greater than zero

• 'left' — Correlation is less than zero

Using the 'pairwise' option for the rows parameter may return a matrix that is not positive definite. The 'complete' option always returns a positive definite matrix, but in general the estimates are based on fewer observations.

corr computes p-values for Pearson's correlation using a Student's t distribution for a transformation of the correlation. This correlation is exact when X and Y are normal. corr computes p-values for Kendall's tau and Spearman's rho using either the exact permutation distributions (for small sample sizes), or large-sample approximations.

corr computes p-values for the two-tailed test by doubling the more significant of the two one-tailed p-values.

## Examples

expand all

### Find Correlation Between Two Matrices

Find the correlation between two matrices and compare to the correlation between two column vectors.

Generate sample data.

```rng('default')
x = randn(30,4);
y = randn(30,4);
y(:,4) = sum(x,2); % introduce correlation```

Calculate the correlation between columns of X and Y.

`[r,p] = corr(x,y)`
```r =

-0.1686   -0.0363    0.2278    0.6901
0.3022    0.0332   -0.0866    0.2617
-0.3632   -0.0987   -0.0200    0.3504
-0.1365   -0.1804    0.0853    0.4908

p =

0.3731    0.8489    0.2260    0.0000
0.1045    0.8619    0.6491    0.1624
0.0485    0.6039    0.9166    0.0577
0.4721    0.3400    0.6539    0.0059```

Calculate the correlation between X and Y using corrcoef.

`[r,p] = corrcoef(x,y)`
```r =

1.0000    0.1252
0.1252    1.0000

p =

1.0000    0.1729
0.1729    1.0000```

MATLAB function corrcoef converts X and Y into column vectors before computing the correlation between them.

## References

[1] Gibbons, J.D. (1985) Nonparametric Statistical Inference, 2nd ed., M. Dekker.

[2] Hollander, M. and D.A. Wolfe (1973) Nonparametric Statistical Methods, Wiley.

[3] Kendall, M.G. (1970) Rank Correlation Methods, Griffin.

[4] Best, D.J. and D.E. Roberts (1975) "Algorithm AS 89: The Upper Tail Probabilities of Spearman's rho", Applied Statistics, 24:377-379.