Partial correlation

In probability theory and statistics, partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed.

Formally, the partial correlation between X and Y given a set of n controlling variables Z = {Z₁, Z₂, ..., Z_n}, written ρ_XY·Z, is the correlation between the residuals R_X and R_Y resulting from the linear regression of X with Z and of Y with Z, respectively. The first-order partial correlation (i.e. when n=1) is the difference between a correlation and the product of the removable correlations divided by the product of the coefficients of alienation of the removable correlations. The coefficient of alienation, and its relation with joint variance through correlation are available in Guilford (1973, pp. 344–345).

A simple way to compute the sample partial correlation for some data is to solve the two associated linear regression problems, get the residuals, and calculate the correlation between the residuals. Let X and Y be, as above, random variables taking real values, and let Z be the n-dimensional vector-valued random variable. If we write x_i, y_i and z_i to denote the ith of N i.i.d. samples of some joint probability distribution over real random variables X, Y and Z, solving the linear regression problem amounts to finding n-dimensional coefficient vectors ${\displaystyle \mathbf {w} _{X}^{*}}$ and ${\displaystyle \mathbf {w} _{Y}^{*}}$ such that

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