73  Partial Pearson Correlation

73.1 Definition

The Pearson Product Moment Partial Correlation (Yule 1897) between variables \(x\) and \(y\), controlled for variable \(z\) is defined as

\[ r_{xy.z} = \frac{r_{xy} - \left( r_{xz} \times r_{yz} \right)}{\sqrt{\left(1 - r_{xz}^2\right) \left( 1 - r_{yz}^2 \right)}} \tag{73.1}\]

where \(-1 \leq r_{xy.z} \leq +1\)

and

\[ \begin{cases}r_{xy} = \frac{\text{C}(xy)}{\sqrt{\text{V}(x) \text{V}(y)}} = \frac{\text{C}(xy)}{s_x s_y} \\r_{xz} = \frac{\text{C}(xz)}{\sqrt{\text{V}(x) \text{V}(z)}} = \frac{\text{C}(xz)}{s_x s_z} \\r_{yz} = \frac{\text{C}(yz)}{\sqrt{\text{V}(y) \text{V}(z)}} = \frac{\text{C}(yz)}{s_y s_z}\end{cases} \]

It is, however, also possible to control for more than just one variable. In case there are \(K\) variables \(z_i\) for \(i= 1, 2, …,K\), the Partial Pearson Correlation is defined as \(r_{e_x e_y}\) where

\[ \begin{cases}x = \alpha_x + \sum_{i=1}^{K} \beta_i z_i + e_x \\y = \alpha_y + \sum_{i=1}^{K} \gamma_i z_i + e_y\end{cases} \]

This implies that the Partial Pearson Correlation is, in fact, directly linked to the Multiple Regression equations in which both variables \(x\) and \(y\) are explained by the control variables \(z_i\).

73.2 R Module

73.2.1 Public website

The Partial Pearson Correlation module is available on the public website:

73.2.2 RFC

The multivariate Partial Pearson Correlation module can be found in RFC under the “Descriptive / Multivariate Descriptive Statistics”.

To compute the trivariate Partial Pearson Correlation on your local machine, the following script can be used in the R console:

z <- c(80,20,50,20,30)
y <- c(80,60,10,20,30)
x <- c(70,30,90,80,10)
(rho12 <- cor(x, y))
(rho23 <- cor(y, z))
(rho13 <- cor(x, z))
#Partial Correlation r(xy.z)
(rhoxy_z <- (rho12-(rho13*rho23))/(sqrt(1-(rho13*rho13)) * sqrt(1-(rho23*rho23))))
#Partial Correlation r(xz.y)
(rhoxz_y <- (rho13-(rho12*rho23))/(sqrt(1-(rho12*rho12)) * sqrt(1-(rho23*rho23))))
#Partial Correlation r(yz.x)
(rhoyz_x <- (rho23-(rho12*rho13))/(sqrt(1-(rho12*rho12)) * sqrt(1-(rho13*rho13))))
[1] -0.2496258
[1] 0.470871
[1] 0.3996413
[1] -0.5413762
[1] 0.6054065
[1] 0.6428553

To compute the Partial Pearson Correlations, the R code uses the cor function and the formulas from Equation 73.1.

It is also possible to compute the multivariate Partial Pearson Correlation Matrix on your local machine with the following R script:

library(corpcor)
A <- runif(150)
B <- -2*A + runif(150)
C <- 3*B + runif(150)
x <- cbind(A, B, C)
(r1 <- pcor.shrink(x)) #partial correlation matrix
(r0 <- cor(x)) #ordinary correlation matrix
Estimating optimal shrinkage intensity lambda (correlation matrix): 0.0089 

            A          B           C
A  1.00000000 -0.2839565 -0.07699398
B -0.28395654  1.0000000  0.91419667
C -0.07699398  0.9141967  1.00000000
attr(,"lambda")
[1] 0.008948936
attr(,"lambda.estimated")
[1] TRUE
attr(,"class")
[1] "shrinkage"
attr(,"spv")
         A          B          C 
0.22960286 0.03793588 0.04101836 
           A          B          C
A  1.0000000 -0.8848609 -0.8739970
B -0.8848609  1.0000000  0.9879927
C -0.8739970  0.9879927  1.0000000

To compute the Partial Pearson Correlation Matrix, the R code uses the pcor.shrink function from the corpcor library.

73.3 Purpose

Partial Correlations are used to investigate whether the relationship between two variables (\(x\) and \(y\)) depends on other variables (\(z_i\) for \(i = 1, 2, …, K\)). This is especially useful when one wishes to examine whether the (ordinary) Pearson Correlation \(r_{xy}\) represents a spurious result (i.e. affected by some common cause) or whether the association weakens after controlling for potential confounders.

73.4 Pros & Cons

73.4.1 Pros

Partial Correlations have the following advantages:

  • They can be used to compute correlations while controlling for other variables (which may avoid the spurious correlation problem).
  • They are closely linked to Multiple Linear Regression and provide interesting information.

73.4.2 Cons

Partial Correlations have the following disadvantages:

  • Most readers are not familiar with the statistical concept of Partial Correlations.
  • Partial Correlations are not featured in many software packages.

73.5 Example

Download the happystudent.csv dataset and use the R module shown below to:

  • upload the csv file
  • click on the Correlations tab
  • select all the variables from the uploaded file
  • change the “Type of Correlations” setting to “Partial”
Interactive Shiny app (click to load).
Open in new tab

The R module computes the multivariate Pearson and Partial Correlation matrices. The dataset contains information that was gathered through a survey in one of our statistics courses to investigate learning confidence (i.e. self-confidence of students to learn statistical concepts). For each student we obtained the following scores:

  • Happiness: measure of general happiness
  • Sport1: to what degree are students engaged in team sports?
  • Learning: degree of learning confidence
  • Depression: measure of depression (based on WHO)
  • Software: degree of software competence

The (unconditional) Pearson Correlation between Learning and Depression is (approximately) equal to -0.23 which seems to imply a weak, negative relationship (i.e. highly depressed students tend to have a lower learning confidence. Does this relationship remain if we control for the effects of all other variables?

The answer is no. The output of the R module shows that the Partial Pearson Correlation is (approximately) -0.067 (i.e. much closer to zero) implying that little linear association remains once the control variables are taken into account. This does not establish full statistical independence. To identify which variable primarily drives this reduction, one can compare the pairwise Pearson correlations between Learning and each of the remaining variables in the dataset.

73.6 Task

Based on the analysis from the previous problem, which variable is most closely related with learning confidence? Do you think that all the variables can be assumed to have a continuous distribution?