7
$\begingroup$

I was calculating a correlation between two variables (A and B) which revealed these variables are highly correlated. I know that one variable is also highly correlated with another one (C), therefore I did a partial correlation between A and B controlling for C. Now I receive a even higher correlation between A and B than I did before. - How can I interpret this?

$\endgroup$
  • 1
    $\begingroup$ A correlation coefficient is something always standardized. When standardized, a "part" can become bigger than a "whole". It can even change sign! See here $\endgroup$ – ttnphns Feb 16 '13 at 11:50
10
$\begingroup$

For understanding this I always prefer the cholesky-decomposition of the correlation-matrix.
Assume the correlation-matrix R of the three variable $X.Y.Z$ as $$ \text{ R =} \left[ \begin{array} {rrr} 1.00& -0.29& -0.45\\ -0.29& 1.00& 0.93\\ -0.45& 0.93& 1.00 \end{array} \right] $$ Then the cholesky-decomposition L is $$ \text{ L =} \left[ \begin{array} {rrr} X\\ Y \\ Z \end{array} \right] = \left[ \begin{array} {rrr} 1.00& 0.00& 0.00\\ -0.29& 0.96& 0.00\\ -0.45& 0.83& 0.32 \end{array} \right] $$ The matrix L gives somehow the coordinates of the three variables in an euclidean space if the variables are seen as vectors from the origin, where the x-axis is identified with the variable/vector X and so on.

Then the correlations of X and Y is $\newcommand{\corr}{\rm corr} \corr(X,Y)=x_1 y_1 + x_2 y_2 + x_3 y_3 $ and we see immediately it it $\corr(X,Y)=-0.29 $ because of the zeros and the unit-factor. We see also immediately the correlation $\corr(X,Z)=-0.45$ again because of the zeros and the unit-cofactor. However, the correlation between Y and Z is $\corr(Y,Z) = -0.29 \cdot -0.45 + 0.96 \cdot 0.83$ The partial correlation (after X is removed) is that part for which no variance in the X-variable is present, so $\corr(Y,Z)._X = 0.96 \cdot 0.83 $. Now imagine, the value $0.83$ would be $-0.83$ instead. Then the partial correlation would be negative and the correlation between Y and Z were $ 0.29 \cdot 0.45 - 0.96 \cdot 0.83$

What we see is, that the partial correlations are partly independent from the overall correlations (though within some bounds)

$\endgroup$
6
$\begingroup$

@Gottfried Helms has given you a good answer. If you're looking for a slightly more intuitively-accessible interpretation, the standard answer is this: Imagine regressing A onto C and B onto C, and in both cases saving the residuals. The partial correlation of A and B controlling for C is the correlation between those two sets of residuals. In other words, it indexes the strength of the linear association between the portion of the variability in A and B that cannot be accounted for by recourse to variability in C. This can be contrasted with the part (or semi-partial) correlation in which the residuals for one of A or B is correlated with the other full variable. For an example of how it can be used, showing that the partial correlation between A and B controlling for C is zero can be part of an argument that the relationship between A and B is fully mediated by C (although this approach will only work in the simplest case, see Baron & Kenny (1986), and Kenny's mediation webpage). If you want a little more information about these topics, I discuss it here, there's a decent Wikipedia page, and I'm particularly fond of this webpage.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.