Partial correlation analysis:

Partial correlation analysis is aimed at finding correlation between two variables after removing the effects of other variables. This type of analysis helps spot spurious correlations (i.e. correlations explained by the effect of other variables) as well as to reveal hidden correlations - i.e correlations masked by the effect of other variables.

The central concept in partial correlation analysis is the partial correlation coefficient r_xy.z between variables x and y , adjusted for a third variable z . Both x and y are presumed to be linearly related to z :

x = Az + B + dx; y = Cz + D + dy;

The partial correlation coefficient r_xy.z is defined as the correlation coefficient between residuals d_x and d_y in this model.

The partial correlation coefficient r_xy.z between x and y adjusted for z may be computed from the pairwise values of the correlation between variables x , y , and z (r_xy, r_yz, r_xz) :

rxy.z =  rxy - rxz ryz ? (1-rxz2)(1-ryz2)

The r_xy.z takes on values between -1 and 1.

There are also generalizations of the partial correlation coefficient for the case of adjustment for more than one variable.

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