Partial Correlation Interpretation

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?

Answer

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
 R =[1.000.290.450.291.000.930.450.931.00]
Then the cholesky-decomposition L is
 L =[XYZ]=[1.000.000.000.290.960.000.450.830.32]
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)

Attribution
Source : Link , Question Author : kvoigt , Answer Author : gung – Reinstate Monica

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