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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.

Formal definition

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).^[1]

Computation
Using linear regression

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 three real random variables X, Y and Z, solving the linear regression problem amounts to finding n-dimensional vectors \( \mathbf{w}_X^* \) and \( \mathbf{w}_Y^* \) such that

\( \mathbf{w}_X^* = \arg\min_{\mathbf{w}} \left\{ \sum_{i=1}^N (x_i - \langle\mathbf{w}, \mathbf{z}_i \rangle)^2 \right\} \)
\( \mathbf{w}_Y^* = \arg\min_{\mathbf{w}} \left\{ \sum_{i=1}^N (y_i - \langle\mathbf{w}, \mathbf{z}_i \rangle)^2 \right\} \)

with N being the number of samples and \( \langle\mathbf{v},\mathbf{w} \rangle \) the scalar product between the vectors \( \mathbf{v} \)and \( \mathbf{w} \). Note that in some formulations the regression includes a constant term, so the matrix \( \mathbf{z} \) would have an additional column of ones.

The residuals are then

\( r_{X,i} = x_i - \langle\mathbf{w}_X^*,\mathbf{z}_i \rangle \)
\( r_{Y,i} = y_i - \langle\mathbf{w}_Y^*,\mathbf{z}_i \rangle \)

and the sample partial correlation is then given by the usual formula for sample correlation , but between these new derived values.

\( \hat{\rho}_{XY\cdot\mathbf{Z}}=\frac{N\sum_{i=1}^N r_{X,i}r_{Y,i}-\sum_{i=1}^N r_{X,i}\sum_{i=1}^N r_{Y,i}} {\sqrt{N\sum_{i=1}^N r_{X,i}^2-\left(\sum_{i=1}^N r_{X,i}\right)^2}~\sqrt{N\sum_{i=1}^N r_{Y,i}^2-\left(\sum_{i=1}^N r_{Y,i}\right)^2}}. \)

Using recursive formula

It can be computationally expensive to solve the linear regression problems. Actually, the nth-order partial correlation (i.e., with |Z| = n) can be easily computed from three (n - 1)th-order partial correlations. The zeroth-order partial correlation ρ_XY·Ø is defined to be the regular correlation coefficient ρ_XY.

It holds, for any \( Z_0 \in \mathbf{Z} \):

\( \rho_{XY\cdot \mathbf{Z} } = \frac{\rho_{XY\cdot\mathbf{Z}\setminus\{Z_0\}} - \rho_{XZ_0\cdot\mathbf{Z}\setminus\{Z_0\}}\rho_{Z_0Y\cdot\mathbf{Z}\setminus\{Z_0\}}} {\sqrt{1-\rho_{XZ_0\cdot\mathbf{Z}\setminus\{Z_0\}}^2} \sqrt{1-\rho_{Z_0Y\cdot\mathbf{Z}\setminus\{Z_0\}}^2}}. \)

Naïvely implementing this computation as a recursive algorithm yields an exponential time complexity. However, this computation has the overlapping subproblems property, such that using dynamic programming or simply caching the results of the recursive calls yields a complexity of \mathcal{O}(n^3).

Note in the case where Z is a single variable, this reduces to:

\( \rho_{XY\cdot Z } = \frac{\rho_{XY} - \rho_{XZ}\rho_{ZY}} {\sqrt{1-\rho_{XZ}^2} \sqrt{1-\rho_{ZY}^2}}. \)

Using matrix inversion

In \( \mathcal{O}(n^3 \) ) time, another approach allows all partial correlations to be computed between any two variables Xi and Xj of a set V of cardinality n, given all others, i.e.,\( \mathbf{V} \setminus \{X_i,X_j\} \), if the correlation matrix (or alternatively covariance matrix) Ω = (ω_ij), where ω_ij = ρ_{X_iX_j}, is positive definite and therefore invertible. If we define P = Ω⁻¹, we have:

\( \rho_{X_iX_j\cdot \mathbf{V} \setminus \{X_i,X_j\}} = -\frac{p_{ij}}{\sqrt{p_{ii}p_{jj}}}. \)

Interpretation
Geometrical interpretation of partial correlation
Geometrical

Let three variables X, Y, Z (where x is the Independent Variable (IV), y is the Dependent Variable (DV), and Z is the "control" or "extra variable") be chosen from a joint probability distribution over n variables V. Further let v_i, 1 ≤ i ≤ N, be N n-dimensional i.i.d. samples taken from the joint probability distribution over V. We then consider the N-dimensional vectors x (formed by the successive values of X over the samples), y (formed by the values of Y) and z (formed by the values of Z).

It can be shown that the residuals R_X coming from the linear regression of X using Z, if also considered as an N-dimensional vector r_X, have a zero scalar product with the vector z generated by Z. This means that the residuals vector lives on a hyperplane S_z that is perpendicular to z.

The same also applies to the residuals R_Y generating a vector r_Y. The desired partial correlation is then the cosine of the angle φ between the projections r_X and r_Y of x and y, respectively, onto the hyperplane perpendicular to z.^[2]

As conditional independence test
See also: Fisher transformation

With the assumption that all involved variables are multivariate Gaussian, the partial correlation ρXY·Z is zero if and only if X is conditionally independent from Y given Z.[3] This property does not hold in the general case.

To test if a sample partial correlation \( \hat{\rho}_{XY\cdot\mathbf{Z}}\) vanishes, Fisher's z-transform of the partial correlation can be used:

\( z(\hat{\rho}_{XY\cdot\mathbf{Z}}) = \frac{1}{2} \ln\left(\frac{1+\hat{\rho}_{XY\cdot\mathbf{Z}}}{1-\hat{\rho}_{XY\cdot\mathbf{Z}}}\right). \)

The null hypothesis is \( H_0: \hat{\rho}_{XY\cdot\mathbf{Z}} = 0 \) , to be tested against the two-tail alternative \( H_A: \hat{\rho}_{XY\cdot\mathbf{Z}} \neq 0 \). We reject H0 with significance level α if:

\( \sqrt{N - |\mathbf{Z}| - 3}\cdot |z(\hat{\rho}_{XY\cdot\mathbf{Z}})| > \Phi^{-1}(1-\alpha/2), \)

where Φ(·) is the cumulative distribution function of a Gaussian distribution with zero mean and unit standard deviation, and N is the sample size. Note that this z-transform is approximate and that the actual distribution of the sample (partial) correlation coefficient is not straightforward. However, an exact t-test based on a combination of the partial regression coefficient, the partial correlation coefficient and the partial variances is available.[4]

The distribution of the sample partial correlation was described by Fisher.[5]
Semipartial correlation (part correlation)

The semipartial (or part) correlation statistic is similar to the partial correlation statistic. Both measure variance after certain factors are controlled for, but to calculate the semipartial correlation one holds the third variable constant for either X or Y, whereas for partial correlations one holds the third variable constant for both.[6] The semipartial correlation measures unique and joint variance while the partial correlation measures unique variance[clarification needed]. The semipartial (or part) correlation can be viewed as more practically relevant "because it is scaled to (i.e., relative to) the total variability in the dependent (response) variable." [7] Conversely, it is less theoretically useful because it is less precise about the unique contribution of the independent variable. Although it may seem paradoxical, the semipartial correlation of X with Y is always less than or equal to the partial correlation of X with Y
Use in time series analysis

In time series analysis, the partial autocorrelation function (sometimes "partial correlation function") of a time series is defined, for lag h, as

\( \phi(h)= \rho_{X_0X_h\cdot \{X_1,\dots,X_{h-1} \}}. \)

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Partial correlation