partialcorr
Linear or rank partial correlation coefficients
Syntax
Description
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.rho
= partialcorr(___,Name,Value
)
Examples
Compute Partial Correlation Coefficients
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 -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 Correlations with Controlled Variables
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 Paired Partial Correlation Coefficients
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
One-Tailed Partial Correlation Test
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
x
— Data matrix
matrix
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
y
— Data matrix
matrix
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
z
— Data matrix
matrix
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 Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: 'Type','Spearman','Rows','complete'
computes
Spearman partial correlations using only the data in rows that contain
no missing values.
Type
— Type of partial correlations
'Pearson'
(default) | 'Spearman'
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
— 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 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'
Tail
— Alternative hypothesis
'both'
(default) | 'right'
| 'left'
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
rho
— Sample linear partial correlation coefficients
matrix
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 inx
.If you input
x
andz
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 inx
, controlled for the variables inz
.If you input
x
,y
, andz
matrices,rho
is a px-by-py matrix, where the (i,j)th entry is the sample linear partial correlation between the ith column inx
and the jth column iny
, controlled for the variables inz
.
If the covariance matrix of [x,z]
is
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).
pval
— p-values
matrix
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.
References
[1] Stuart, Alan, K. Ord, and S. Arnold. Kendall's Advanced Theory of Statistics. 6th edition, Volume 2A, Chapter 28, Wiley, 2004.
[2] Fisher, Ronald A. "The Distribution of the Partial Correlation Coefficient." Metron 3 (1924): 329-332
Extended Capabilities
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool
or accelerate code with Parallel Computing Toolbox™ ThreadPool
.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
Version History
Introduced before R2006a
See Also
corr
| tiedrank
| corrcoef
| partialcorri
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