Partialkorrelation

Partialkorrelation, or partial correlation, is a statistical measure of the strength and direction of the linear relationship between two random variables X and Y after removing the linear effects of one or more other variables Z. It aims to quantify the direct association between X and Y that is not explained by the control variables Z.

For the simple case of a single control variable Z, the partial correlation rXY·Z can be computed from the ordinary correlations rXY, rXZ, and rYZ as:

rXY·Z = (rXY − rXZ · rYZ) / sqrt((1 − rXZ^2) · (1 − rYZ^2)).

When multiple control variables Z are present, the partial correlation can be derived from the inverse of the correlation (or covariance) matrix of the variables {X, Y, Z}. Equivalently, it is the correlation between the residuals of X and Y after regressing each on Z.

In practice, a convenient way to obtain the partial correlation is to regress X on Z and Y on Z, obtain the residuals eX and eY, and then compute the correlation between eX and eY. This residual-based interpretation highlights that the partial correlation captures linear association not explained by the controls.

Interpretation and scope: a value near zero suggests little linear association between X and Y after accounting for Z; positive values indicate remaining positive association, and negative values indicate an inverse association. Partial correlation is a measure of linear relationships and does not in itself imply causation. It can be sensitive to outliers and requires a sufficient sample size, especially with many control variables. It is commonly used in graphical models, regression diagnostics, and covariate adjustment in scientific fields such as psychology, genomics, and finance.