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 ...