A part correlation is a measure of linear association between one variable and another after the variability in at least one other variable is removed from only one of the original variables. The traditional formula for the part correlation between, say, variables X and Y after controlling for Z in Y, denoted rX(Y∙Z), is

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

where r is the Pearson correlation between two variables.

rX(Y∙Z) is sometimes referred to as a first-order part 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 part correlation is really the correlation between X and the residual of Y after regressing ...