====== PCC ======

Partial Correlation Coefficient

[[Pearson|Pearson]] only measures the linear relationship between two variables without considering the effect that other possible variables might have. So when more than one input factor is under consideration, as it usually is, partial correlation coefficients (PCCs) can be used instead to provide a measure of the linear relationships between two variables when all linear effects of other variables are removed. PCC between an individual variable X<sub>i</sub> and Y can be obtained from the use of a sequence of regression models. Begin by constructing the following regression models:

Xhat<sub>i</sub> = c<sub>0</sub>+∑<sub>j≠i</sub>c<sub>j</sub>//X<sub>j</sub>,//

Yhat<sub>i</sub> = b<sub>0</sub>+∑<sub>j≠i</sub>b<sub>j</sub>//Y<sub>j</sub>.//

Then PCC is defined by the [[Pearson|Pearson]] correlation coefficient of X<sub>i</sub>−Xhat<sub>i</sub> and Y−Yhat.

PCC characterizes the strength of the linear relationship between two variables after a correction has been made for the linear effects of the other variables in the analysis. [[SRC|SRC]] on the other hand characterizes the effect on the output variable that results from perturbing an input variable by a fixed fraction of its standard deviation. Thus, PCC and [[SRC|SRC]] provide related, but not identical, measures of the variable importance. When input factors are uncorrelated, results from PCC and [[SRC|SRC]] are identical.

===== Reference =====

  * Francesca Campolongo, Andrea Saltelli, Tine Sørensen, and Stefano Tarantola. Hitchhiker’s guide to sensitivity analysis. In //Sensitivity analysis//, Wiley Ser. Probab. Stat., pages 15–47. Wiley, Chichester, 2000.