# partialcorr

Linear or rank partial correlation coefficients

## Syntax

``rho = partialcorr(x)``
``rho = partialcorr(x,z)``
``rho = partialcorr(x,y,z)``
``rho = partialcorr(___,Name,Value)``
``````[rho,pval] = partialcorr(___)``````

## Description

example

````rho = partialcorr(x)` returns the sample linear partial correlation coefficients between pairs of variables in `x`, controlling for the remaining variables in `x`.```

example

````rho = partialcorr(x,z)` returns the sample linear partial correlation coefficients between pairs of variables in `x`, controlling for the variables in `z`.```

example

````rho = partialcorr(x,y,z)` returns the sample linear partial correlation coefficients between pairs of variables in `x` and `y`, controlling for the variables in `z`.```

example

````rho = partialcorr(___,Name,Value)` returns the sample linear partial correlation coefficients with additional options specified by one or more name-value pair arguments, using input arguments from any of the previous syntaxes. For example, you can specify whether to use Pearson or Spearman partial correlations, or specify how to treat missing values.```

example

``````[rho,pval] = partialcorr(___)``` also returns a matrix `pval` of p-values for testing the hypothesis of no partial correlation against the one- or two-sided alternative that there is a nonzero partial correlation.```

## Examples

collapse all

Compute partial correlation coefficients between pairs of variables in the input matrix.

Load the sample data. Convert the genders in `hospital.Sex` to numeric group identifiers.

```load hospital; hospital.SexID = grp2idx(hospital.Sex);```

Create an input matrix containing the sample data.

`x = [hospital.SexID hospital.Age hospital.Smoker hospital.Weight];`

Each row in `x` contains a patient’s gender, age, smoking status, and weight.

Compute partial correlation coefficients between pairs of variables in `x`, while controlling for the effects of the remaining variables in `x`.

`rho = partialcorr(x)`
```rho = 4×4 1.0000 -0.0105 0.0273 0.9421 -0.0105 1.0000 0.0419 0.0369 0.0273 0.0419 1.0000 0.0451 0.9421 0.0369 0.0451 1.0000 ```

The matrix `rho` indicates, for example, a correlation of 0.9421 between gender and weight after controlling for all other variables in `x`. You can return the $p$-values as a second output, and examine them to confirm whether these correlations are statistically significant.

For a clearer display, create a table with appropriate variable and row labels.

```rho = array2table(rho, ... 'VariableNames',{'SexID','Age','Smoker','Weight'},... 'RowNames',{'SexID','Age','Smoker','Weight'}); disp('Partial Correlation Coefficients')```
```Partial Correlation Coefficients ```
`disp(rho)`
``` SexID Age Smoker Weight ________ ________ ________ ________ SexID 1 -0.01052 0.027324 0.9421 Age -0.01052 1 0.041945 0.036873 Smoker 0.027324 0.041945 1 0.045106 Weight 0.9421 0.036873 0.045106 1 ```

Test for partial correlation between pairs of variables in the input matrix, while controlling for the effects of a second set of variables.

Load the sample data. Convert the genders in `hospital.Sex` to numeric group identifiers.

```load hospital; hospital.SexID = grp2idx(hospital.Sex);```

Create two matrices containing the sample data.

```x = [hospital.Age hospital.BloodPressure]; z = [hospital.SexID hospital.Smoker hospital.Weight];```

The `x` matrix contains the variables to test for partial correlation. The `z` matrix contains the variables to control for. The measurements for `BloodPressure` are contained in two columns: The first column contains the upper (systolic) number, and the second column contains the lower (diastolic) number. `partialcorr` treats each column as a separate variable.

Test for partial correlation between pairs of variables in `x`, while controlling for the effects of the variables in `z`. Compute the correlation coefficients.

`[rho,pval] = partialcorr(x,z)`
```rho = 3×3 1.0000 0.1300 0.0462 0.1300 1.0000 0.0012 0.0462 0.0012 1.0000 ```
```pval = 3×3 0 0.2044 0.6532 0.2044 0 0.9903 0.6532 0.9903 0 ```

The large values in `pval` indicate that there is no significant correlation between age and either blood pressure measurement after controlling for gender, smoking status, and weight.

For a clearer display, create tables with appropriate variable and row labels.

```rho = array2table(rho, ... 'VariableNames',{'Age','BPTop','BPBottom'},... 'RowNames',{'Age','BPTop','BPBottom'}); pval = array2table(pval, ... 'VariableNames',{'Age','BPTop','BPBottom'},... 'RowNames',{'Age','BPTop','BPBottom'}); disp('Partial Correlation Coefficients')```
```Partial Correlation Coefficients ```
`disp(rho)`
``` Age BPTop BPBottom ________ _________ _________ Age 1 0.13 0.046202 BPTop 0.13 1 0.0012475 BPBottom 0.046202 0.0012475 1 ```
`disp('p-values')`
```p-values ```
`disp(pval)`
``` Age BPTop BPBottom _______ _______ ________ Age 0 0.20438 0.65316 BPTop 0.20438 0 0.99032 BPBottom 0.65316 0.99032 0 ```

Test for partial correlation between pairs of variables in the `x` and `y` input matrices, while controlling for the effects of a third set of variables.

Load the sample data. Convert the genders in `hospital.Sex` to numeric group identifiers.

```load hospital; hospital.SexID = grp2idx(hospital.Sex);```

Create three matrices containing the sample data.

```x = [hospital.BloodPressure]; y = [hospital.Weight hospital.Age]; z = [hospital.SexID hospital.Smoker];```

`partialcorr` can test for partial correlation between the pairs of variables in `x` (the systolic and diastolic blood pressure measurements) and `y` (weight and age), while controlling for the variables in `z` (gender and smoking status). The measurements for `BloodPressure` are contained in two columns: The first column contains the upper (systolic) number, and the second column contains the lower (diastolic) number. `partialcorr` treats each column as a separate variable.

Test for partial correlation between pairs of variables in `x` and `y`, while controlling for the effects of the variables in `z`. Compute the correlation coefficients.

`[rho,pval] = partialcorr(x,y,z)`
```rho = 2×2 -0.0257 0.1289 0.0292 0.0472 ```
```pval = 2×2 0.8018 0.2058 0.7756 0.6442 ```

The results in `pval` indicate that, after controlling for gender and smoking status, there is no significant correlation between either of a patient’s blood pressure measurements and that patient’s weight or age.

For a clearer display, create tables with appropriate variable and row labels.

```rho = array2table(rho, ... 'RowNames',{'BPTop','BPBottom'},... 'VariableNames',{'Weight','Age'}); pval = array2table(pval, ... 'RowNames',{'BPTop','BPBottom'},... 'VariableNames',{'Weight','Age'}); disp('Partial Correlation Coefficients')```
```Partial Correlation Coefficients ```
`disp(rho)`
``` Weight Age ________ ________ BPTop -0.02568 0.12893 BPBottom 0.029168 0.047226 ```
`disp('p-values')`
```p-values ```
`disp(pval)`
``` Weight Age _______ _______ BPTop 0.80182 0.2058 BPBottom 0.77556 0.64424 ```

Test the hypothesis that pairs of variables have no correlation, against the alternative hypothesis that the correlation is greater than 0.

Load the sample data. Convert the genders in `hospital.Sex` to numeric group identifiers.

```load hospital; hospital.SexID = grp2idx(hospital.Sex);```

Create three matrices containing the sample data.

```x = [hospital.BloodPressure]; y = [hospital.Weight hospital.Age]; z = [hospital.SexID hospital.Smoker];```

`partialcorr` can test for partial correlation between the pairs of variables in `x` (the systolic and diastolic blood pressure measurements) and `y` (weight and age), while controlling for the variables in `z` (gender and smoking status). The measurements for `BloodPressure` are contained in two columns: The first column contains the upper (systolic) number, and the second column contains the lower (diastolic) number. `partialcorr` treats each column as a separate variable.

Compute the correlation coefficients using a right-tailed test.

`[rho,pval] = partialcorr(x,y,z,'Tail','right')`
```rho = 2×2 -0.0257 0.1289 0.0292 0.0472 ```
```pval = 2×2 0.5991 0.1029 0.3878 0.3221 ```

The results in `pval` indicate that `partialcorr` does not reject the null hypothesis of nonzero correlations between the variables in `x` and `y`, after controlling for the variables in `z`, when the alternative hypothesis is that the correlations are greater than 0.

For a clearer display, create tables with appropriate variable and row labels.

```rho = array2table(rho, ... 'RowNames',{'BPTop','BPBottom'},... 'VariableNames',{'Weight','Age'}); pval = array2table(pval, ... 'RowNames',{'BPTop','BPBottom'},... 'VariableNames',{'Weight','Age'}); disp('Partial Correlation Coefficients')```
```Partial Correlation Coefficients ```
`disp(rho)`
``` Weight Age ________ ________ BPTop -0.02568 0.12893 BPBottom 0.029168 0.047226 ```
`disp('p-values')`
```p-values ```
`disp(pval)`
``` Weight Age _______ _______ BPTop 0.59909 0.1029 BPBottom 0.38778 0.32212 ```

## Input Arguments

collapse all

Data matrix, specified as an n-by-px matrix. The rows of `x` correspond to observations, and the columns correspond to variables.

Data Types: `single` | `double`

Data matrix, specified as an n-by-py matrix. The rows of `y` correspond to observations, and the columns correspond to variables.

Data Types: `single` | `double`

Data matrix, specified as an n-by-pz matrix. The rows of `z` correspond to observations, and columns correspond to variables.

Data Types: `single` | `double`

### Name-Value Pair Arguments

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`.

Example: `'Type','Spearman','Rows','complete'` computes Spearman partial correlations using only the data in rows that contain no missing values.

Type of partial correlations to compute, specified as the comma-separated pair consisting of `'Type'` and one of the following.

 `'Pearson'` Compute Pearson (linear) partial correlations. `'Spearman'` Compute Spearman (rank) partial correlations.

Example: `'Type','Spearman'`

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

 `'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`.

Example: `'Rows','complete'`

Alternative hypothesis to test against, specified as the comma-separated pair consisting of `'Tail'` and one of the following.

 `'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.

Example: `'Tail','right'`

## Output Arguments

collapse all

Sample linear partial correlation coefficients, returned as a matrix.

• If you input only an `x` matrix, `rho` is a symmetric px-by-px matrix. The (i,j)th entry is the sample linear partial correlation between the i-th and j-th columns in `x`.

• If you input `x` and `z` matrices, `rho` is a symmetric px-by-px matrix. The (i,j)th entry is the sample linear partial correlation between the ith and jth columns in `x`, controlled for the variables in `z`.

• If you input `x`, `y`, and `z` matrices, `rho` is a px-by-py matrix, where the (i,j)th entry is the sample linear partial correlation between the ith column in `x` and the jth column in `y`, controlled for the variables in `z`.

If the covariance matrix of `[x,z]` is

`$S=\left(\begin{array}{cc}{S}_{xx}& {S}_{xz}\\ {S}_{xz}{}^{T}& {S}_{zz}\end{array}\right)\text{\hspace{0.17em}},$`

then the partial correlation matrix of `x`, controlling for `z`, can be defined formally as a normalized version of the covariance matrix: Sxx – (SxzSzz–1SxzT).

p-values, returned as a matrix. Each element of `pval` is the p-value for the corresponding element of `rho`.

If `pval(i,j)` is small, then the corresponding partial correlation `rho(i,j)` is statistically significantly different from 0.

`partialcorr` computes p-values for linear and rank partial correlations using a Student's t distribution for a transformation of the correlation. This is exact for linear partial correlation when `x` and `z` are normal, but is a large-sample approximation otherwise.

 Stuart, Alan, K. Ord, and S. Arnold. Kendall's Advanced Theory of Statistics. 6th edition, Volume 2A, Chapter 28, Wiley, 2004.

 Fisher, Ronald A. "The Distribution of the Partial Correlation Coefficient." Metron 3 (1924): 329-332