Partial Correlations

A partial correlation is a measure of linear association between two variables after variability in at least one other variable is removed from both variables. The traditional formula for the partial correlation between, say, variables X and Y after controlling for Z, denoted rXY·Z, is

$$r_X{}_{Y\cdot Z}=\frac{r_{XY}-r_{XZ}\times r_{YZ}}{\sqrt{1-r_{YZ}^2}\times \sqrt{1-r_{YZ}^2}},$$

where r is the Pearson correlation between two variables.

rX(Y·Z) is sometimes referred to as a first-order partial correlation to note that the correlation only controls for one other variable. If controlling for two variables, it would be a second-order, and so on. Zero-order correlations do not control for any other variables.

While not obvious from Equation 1, the partial correlation is really the correlation between the residuals of X and Y after regressing both variables on Z. A path model representing this interpretation is shown in Figure 1, which assumes X, Y, and Z are standardized variables.

In Figure 1, the zero-order correlation between X and Y, rXY, can be found using the tracing rules for path diagrams. It is

$$r_{XY}=a\times b+c\times s_{E_X}^2\times r_{XY\cdot Z}\times s_{E_Y}^2\times d.$$

Figure 1 Path diagram of partial correlation. Variables X, Y, and Z are standardized

The values for paths a and b are the zero-order correlations of X with Z and Y with Z, respectively. To make the model identified (i.e., be able to find unique estimates for the paths), either c and d or the variances of Ex and Ey, $S_{E_X}^2$ and $S_{E_Y}^2$, need to be constrained. To make rXY·Z a correlation (as opposed to a covariance), constrain the variances of Ex and Ey to one (i.e., standardize these residuals). This makes the paths c and d equal the square root of the variance in X and Y not explained by Z (i.e., $c=\sqrt{1-r_{YZ}^2}$ and $d=\sqrt{1-r_{YZ}^2}$.

After substituting terms, Equation 2 now becomes

$$r_{XY}=r_{XZ}\times r_{YZ}+\sqrt{1-r_{YZ}^2}\times r_{XY\cdot Z}\times \sqrt{1-r_{YZ}^2}.$$

Equation 3 can be rearranged as

$$r_{XY\cdot Z}\times \sqrt{1-r_{XZ}^2}\times \sqrt{1-r_{YZ}^2}=r_{XY}-r_{XZ}\times r_{YZ}.$$

Dividing both sides of this rearrangement by $\sqrt{1-r_{XZ}^2}\sqrt{1-r_{YZ}^2}$ produces Equation 1.