Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Correlation ▷ Partial correlation

Definition: Let $X$, $Y$ and $Z$ be random variables, such that there are linear relationships

\[\label{eq:xz-yz} \begin{split} X &= \beta_0^{(X)} + \beta_1^{(X)} Z + E^{(X)} \\ Y &= \beta_0^{(Y)} + \beta_1^{(Y)} Z + E^{(Y)} \end{split}\]

with the residual random variables

\[\label{eq:ex-ey} \begin{split} E^{(X)} &= X - \beta_0^{(X)} - \beta_1^{(X)} Z \\ E^{(Y)} &= Y - \beta_0^{(Y)} - \beta_1^{(Y)} Z \end{split}\]

Then, the partial correlation of $X$ and $Y$ controlling for $Z$ is defined as the correlation of the residual variables:

\[\label{eq:corr-part} \mathrm{Corr}(X,Y \backslash Z) = \mathrm{Corr}\left( E^{(X)}, E^{(Y)} \right) \; .\]
 
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Metadata: ID: D227 | shortcut: corr-part | author: JoramSoch | date: 2026-03-26, 10:25.