Partial Correlation

Partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed. The partial correlation between X and Y given a set of n controlling variables Z = {Z1, Z2,…,Zn}, is the correlation between the residuals R_X and R_Y resulting from the linear regression of X with Z and of Y with Z. The first order partial correlation is nothing else than a difference between a correlation and the product of the removable correlations divided by the product of the coefficient of alienation of the removable correlations. A simple way to compute the partial correlation for some data is to solve the two associated linear regression problems, get the residuals and calculate the correlation between the residuals.

The semi-partial correlation statistic is similar to the partial correlation statistic. Both measure variance after certain factors are controlled for, but to calculate the semi-partial correlation one holds the third variable constant for either X or Y. Whereas for partial correlations one holds the third variable constant for both. This correlation is viewed as ore practically relevant. It is less theoretically useful however, because it is less precise about the unique contribution of the independent variables.