# Partial correlation

From AMS Glossary

## partial correlation

The correlation between the residuals of two random variables (variates) with respect to common regressors.

Denoting the regression function of two variates where the symbol

*y*and*z*with respect to a common set of regressors*x*_{1},*x*_{2}, · · ·*x*_{n}by*Y*and*Z*, the coefficient of partial correlation between*y*and*z*is defined as the coefficient of simple linear correlation between (*y*-*Y*) and (*z*-*Z*). To estimate the partial correlation, it is usually necessary to resort to sample approximations*Y*′ and*Z*′ of*Y*and*Z*. In that case, the estimate of the partial correlation is the sample value of the coefficient of simple, linear correlation between (*y*-*Y*′) and (*z*-*Z*′). In the simplest case in which*Y*′ and*Z*′ are taken as linear functions of a single variable*x*, the sample estimate*r*_{yz.x}of the partial correlation coefficient is given by the formula*r*_{uv}denotes the sample coefficient of linear correlation between any pair of variates*u*,*v*.*See*regression.