Linear or rank correlation

`[`

specifies options using one or more name-value pair arguments in addition to the
input arguments in the previous syntaxes. For example,
`rho`

,`pval`

] = corr(___,`Name,Value`

)`'Type','Kendall'`

specifies computing Kendall's tau
correlation coefficient.

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

Generate sample data.

```
rng('default')
X = randn(30,4);
Y = randn(30,4);
```

Introduce correlation between column two of the matrix `X`

and column four of the matrix `Y`

.

Y(:,4) = Y(:,4)+X(:,2);

Calculate the correlation between columns of `X`

and `Y`

.

[rho,pval] = corr(X,Y)

`rho = `*4×4*
-0.1686 -0.0363 0.2278 0.3245
0.3022 0.0332 -0.0866 0.7653
-0.3632 -0.0987 -0.0200 -0.3693
-0.1365 -0.1804 0.0853 0.0279

`pval = `*4×4*
0.3731 0.8489 0.2260 0.0802
0.1045 0.8619 0.6491 0.0000
0.0485 0.6039 0.9166 0.0446
0.4721 0.3400 0.6539 0.8837

As expected, the correlation coefficient between column two of `X`

and column four of `Y`

, `rho(2,4)`

, is the highest, and it represents a high positive correlation between the two columns. The corresponding *p*-value, `pval(2,4)`

, is zero to the four digits shown. Because the *p*-value is less than the significance level of `0.05`

, it indicates rejection of the hypothesis that no correlation exists between the two columns.

Calculate the correlation between `X`

and `Y`

using `corrcoef`

.

[r,p] = corrcoef(X,Y)

`r = `*2×2*
1.0000 -0.0329
-0.0329 1.0000

`p = `*2×2*
1.0000 0.7213
0.7213 1.0000

The MATLAB® function `corrcoef`

, unlike the `corr`

function, converts the input matrices `X`

and `Y`

into column vectors, `X(:)`

and `Y(:)`

, before computing the correlation between them. Therefore, the introduction of correlation between column two of matrix `X`

and column four of matrix `Y`

no longer exists, because those two columns are in different sections of the converted column vectors.

The value of the off-diagonal elements of `r`

, which represents the correlation coefficient between `X`

and `Y`

, is low. This value indicates little to no correlation between `X`

and `Y`

. Likewise, the value of the off-diagonal elements of `p`

, which represents the *p*-value, is much higher than the significance level of `0.05`

. This value indicates that not enough evidence exists to reject the hypothesis of no correlation between `X`

and `Y`

.

Test alternative hypotheses for positive, negative, and nonzero correlation between the columns of two matrices. Compare values of the correlation coefficient and *p*-value in each case.

Generate sample data.

```
rng('default')
X = randn(50,4);
Y = randn(50,4);
```

Introduce positive correlation between column one of the matrix `X`

and column four of the matrix `Y`

.

Y(:,4) = Y(:,4)+0.7*X(:,1);

Introduce negative correlation between column two of `X`

and column two of `Y`

.

Y(:,2) = Y(:,2)-2*X(:,2);

Test the alternative hypothesis that the correlation is greater than zero.

[rho,pval] = corr(X,Y,'Tail','right')

`rho = `*4×4*
0.0627 -0.1438 -0.0035 0.7060
-0.1197 -0.8600 -0.0440 0.1984
-0.1119 0.2210 -0.3433 0.1070
-0.3526 -0.2224 0.1023 0.0374

`pval = `*4×4*
0.3327 0.8405 0.5097 0.0000
0.7962 1.0000 0.6192 0.0836
0.7803 0.0615 0.9927 0.2298
0.9940 0.9397 0.2398 0.3982

As expected, the correlation coefficient between column one of `X`

and column four of `Y`

, `rho(1,4)`

, has the highest positive value, representing a high positive correlation between the two columns. The corresponding *p*-value, `pval(1,4)`

, is zero to the four digits shown, which is lower than the significance level of `0.05`

. These results indicate rejection of the null hypothesis that no correlation exists between the two columns and lead to the conclusion that the correlation is greater than zero.

Test the alternative hypothesis that the correlation is less than zero.

[rho,pval] = corr(X,Y,'Tail','left')

`rho = `*4×4*
0.0627 -0.1438 -0.0035 0.7060
-0.1197 -0.8600 -0.0440 0.1984
-0.1119 0.2210 -0.3433 0.1070
-0.3526 -0.2224 0.1023 0.0374

`pval = `*4×4*
0.6673 0.1595 0.4903 1.0000
0.2038 0.0000 0.3808 0.9164
0.2197 0.9385 0.0073 0.7702
0.0060 0.0603 0.7602 0.6018

As expected, the correlation coefficient between column two of `X`

and column two of `Y`

, `rho(2,2)`

, has the negative number with the largest absolute value (`-0.86`

), representing a high negative correlation between the two columns. The corresponding *p*-value, `pval(2,2)`

, is zero to the four digits shown, which is lower than the significance level of `0.05`

. Again, these results indicate rejection of the null hypothesis and lead to the conclusion that the correlation is less than zero.

Test the alternative hypothesis that the correlation is not zero.

[rho,pval] = corr(X,Y)

`rho = `*4×4*
0.0627 -0.1438 -0.0035 0.7060
-0.1197 -0.8600 -0.0440 0.1984
-0.1119 0.2210 -0.3433 0.1070
-0.3526 -0.2224 0.1023 0.0374

`pval = `*4×4*
0.6654 0.3190 0.9807 0.0000
0.4075 0.0000 0.7615 0.1673
0.4393 0.1231 0.0147 0.4595
0.0120 0.1206 0.4797 0.7964

The *p*-values, `pval(1,4)`

and `pval(2,2)`

, are both zero to the four digits shown. Because the *p*-values are lower than the significance level of `0.05`

, the correlation coefficients `rho(1,4)`

and `rho(2,2)`

are significantly different from zero. Therefore, the null hypothesis is rejected; the correlation is not zero.

`X`

— Input matrixmatrix

Input matrix, specified as an *n*-by-*k*
matrix. The rows of `X`

correspond to observations, and the
columns correspond to variables.

**Example: **`X = randn(10,5)`

**Data Types: **`single`

| `double`

`Y`

— Input matrixmatrix

Input matrix, specified as an
*n*-by-*k*_{2}
matrix when `X`

is specified as an
*n*-by-*k*_{1}
matrix. The rows of `Y`

correspond to observations, and the
columns correspond to variables.

**Example: **`Y = randn(20,7)`

**Data Types: **`single`

| `double`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

`corr(X,Y,'Type','Kendall','Rows','complete')`

returns
Kendall's tau correlation coefficient using only the rows that contain no missing
values.`'Type'`

— Type of correlation`'Pearson'`

(default) | `'Kendall'`

| `'Spearman'`

Type of correlation, specified as the comma-separated pair consisting
of `'Type'`

and one of these values.

Value | Description |
---|---|

`'Pearson'` | Pearson's Linear Correlation Coefficient |

`'Kendall'` | Kendall's Tau Coefficient |

`'Spearman'` | Spearman's Rho |

`corr`

computes the *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`

come from a
normal distribution. `corr`

computes the
*p*-values for Kendall's tau and Spearman's rho
using either the exact permutation distributions (for small sample
sizes) or large-sample approximations.

**Example: **`'Type','Spearman'`

`'Rows'`

— Rows to use in computation`'all'`

(default) | `'complete'`

| `'pairwise'`

Rows to use in computation, specified as the comma-separated pair
consisting of `'Rows'`

and one of these values.

Value | Description |
---|---|

`'all'` | Use all rows of the input regardless of missing
values (`NaN` s). |

`'complete'` | Use only rows of the input with no missing values. |

`'pairwise'` | Compute `rho(i,j)` using rows with
no missing values in column `i` or
`j` . |

The `'complete'`

value, unlike the
`'pairwise'`

value, always produces a positive
definite or positive semidefinite `rho`

. Also, the
`'complete'`

value generally uses fewer
observations to estimate `rho`

when rows of the input
(`X`

or `Y`

) contain missing
values.

**Example: **`'Rows','pairwise'`

`'Tail'`

— Alternative hypothesis`'both'`

(default) | `'right'`

| `'left'`

Alternative hypothesis, specified as the comma-separated pair
consisting of `'Tail'`

and one of the values in the
table. `'Tail'`

specifies the alternative hypothesis
against which to compute *p*-values for testing the
hypothesis of no correlation.

Value | Description |
---|---|

`'both'` | Test the alternative hypothesis that the correlation
is not `0` . |

`'right'` | Test the alternative hypothesis that the correlation
is greater than `0` |

`'left'` | Test the alternative hypothesis that the correlation
is less than `0` . |

`corr`

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

**Example: **`'Tail','left'`

`rho`

— Pairwise linear correlation coefficientmatrix

Pairwise linear correlation coefficient, returned as a matrix.

If you input only a matrix

`X`

,`rho`

is a symmetric*k*-by-*k*matrix, where*k*is the number of columns in`X`

. The entry`rho(a,b)`

is the pairwise linear correlation coefficient between column*a*and column*b*in`X`

.If you input matrices

`X`

and`Y`

,`rho`

is a*k*_{1}-by-*k*_{2}matrix, where*k*_{1}and*k*_{2}are the number of columns in`X`

and`Y`

, respectively. The entry`rho(a,b)`

is the pairwise linear correlation coefficient between column*a*in`X`

and column*b*in`Y`

.

`pval`

— matrix

*p*-values, returned as a matrix. Each element of
`pval`

is the *p*-value for the
corresponding element of `rho`

.

If `pval(a,b)`

is small (less than
`0.05`

), then the correlation
`rho(a,b)`

is significantly different from zero.

Pearson's linear correlation coefficient is the most commonly
used linear correlation coefficient. For column
*X _{a}* in matrix

$$rho(a,b)=\frac{{\displaystyle \sum _{i=1}^{n}({X}_{a,i}-{\overline{X}}_{a})({Y}_{b,i}-{\overline{Y}}_{b})}}{{\left\{{\displaystyle \sum _{i=1}^{n}{({X}_{a,i}-{\overline{X}}_{a})}^{2}\text{\hspace{0.17em}}{\displaystyle \sum _{j=1}^{n}{({Y}_{b,j}-{\overline{Y}}_{b})}^{2}}}\right\}}^{1/2}}\text{\hspace{0.17em}},$$

where *n* is the length of each column.

Values of the correlation coefficient can range from `–1`

to
`+1`

. A value of `–1`

indicates perfect
negative correlation, while a value of `+1`

indicates perfect
positive correlation. A value of `0`

indicates no correlation
between the columns.

Kendall's tau is based on counting the number of
(*i,j*) pairs, for *i<j*, that are
concordant—that is, for which $${X}_{a,i}-{X}_{a,j}$$ and $${Y}_{b,i}-{Y}_{b,j}$$ have the same sign. The equation for Kendall's tau includes an
adjustment for ties in the normalizing constant and is often referred to as
tau-b.

For column *X _{a}* in matrix

$$\tau =\frac{2K}{n(n-1)},$$

where $$K={\displaystyle \sum _{i=1}^{n-1}{\displaystyle \sum _{j=i+1}^{n}{\xi}^{*}({X}_{a,i},{X}_{a,j},{Y}_{b,i},{Y}_{b,j})}},$$ and

$${\xi}^{*}({X}_{a,i},{X}_{a,j},{Y}_{b,i},{Y}_{b,j})=\{\begin{array}{ccc}1& \text{if}& ({X}_{a,i}-{X}_{a,j})({Y}_{b,i}-{Y}_{b,j})>0\\ 0& \text{if}& ({X}_{a,i}-{X}_{a,j})({Y}_{b,i}-{Y}_{b,j})=0\\ -1& \text{if}& ({X}_{a,i}-{X}_{a,j})({Y}_{b,i}-{Y}_{b,j})<0\end{array}\text{\hspace{0.17em}}.$$

Values of the correlation coefficient can range from `–1`

to
`+1`

. A value of `–1`

indicates that one
column ranking is the reverse of the other, while a value of `+1`

indicates that the two rankings are the same. A value of `0`

indicates no relationship between the columns.

Spearman's rho is equivalent to Pearson's Linear Correlation Coefficient applied to the
rankings of the columns *X _{a}* and

If all the ranks in each column are distinct, the equation simplifies to:

$$rho(a,b)=1-\frac{6{\displaystyle \sum {d}^{2}}}{n({n}^{2}-1)}\text{\hspace{0.17em}},$$

where *d* is the difference between the ranks of the two columns,
and *n* is the length of each column.

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
two column vectors `X`

and `Y`

. If
`X`

and `Y`

are not column vectors,
`corrcoef(X,Y)`

converts them to column vectors.

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

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

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

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

Calculate with arrays that have more rows than fit in memory.

This function supports tall arrays for out-of-memory data with the limitation:

Only the `'Pearson'`

type is supported.

For more information, see Tall Arrays for Out-of-Memory Data (MATLAB).

`corrcoef`

| `corrcov`

| `partialcorr`

| `tiedrank`

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