Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Multivariate continuous distributions ▷ Multivariate normal distribution ▷ Conditional and partial correlation

Theorem: Let $X$, $Y$ and $Z$ be random variables jointly following a multivariate normal distribution:

\[\label{eq:V} V = \left[ \begin{matrix} X \\ Y \\ Z \end{matrix} \right] \sim \mathcal{N}(\mu, \Sigma) \; .\]

Then, the conditional correlation of $X$ and $Y$ given $Z$ is equal to the partial correlation of $X$ and $Y$ controlling for $Z$:

\[\label{eq:mvn-ecpc} \mathrm{Corr}(X,Y|Z) = \mathrm{Corr}(X,Y \backslash Z) \; .\]

Proof: For a three-dimensional multivariate normal random vector

\[\label{eq:XYZ} \left[ \begin{matrix} X \\ Y \\ Z \end{matrix} \right] \sim \mathcal{N}(\mu, \Sigma) \; ,\]

the conditional correlation of $X$ and $Y$ given $Z$ is

\[\label{eq:mvn-corr-cond} \mathrm{Corr}(X,Y|Z) = \frac{\rho_{XY} - \rho_{XZ} \rho_{YZ}}{\sqrt{1-\rho_{XZ}^2} \sqrt{1-\rho_{YZ}^2}}\]

and the partial correlation of $X$ and $Y$ controlling for $Z$ is

\[\label{eq:mvn-corr-part} \mathrm{Corr}(X,Y \backslash Z) = \frac{\rho_{XY} - \rho_{XZ} \rho_{YZ}}{\sqrt{1-\rho_{XZ}^2} \sqrt{1-\rho_{YZ}^2}}\]

where $\rho_{XY}$, $\rho_{XZ}$ and $\rho_{YZ}$ are the pairwise correlations of the random variables $X$, $Y$ and $Z$. Thus, conditional and partial correlation are equivalent for the multivariate normal distribution:

\[\label{eq:mvn-ecpc-qed} \mathrm{Corr}(X,Y|Z) = \mathrm{Corr}(X,Y \backslash Z) \; .\]
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Metadata: ID: P547 | shortcut: mvn-ecpc | author: JoramSoch | date: 2026-08-28, 11:35.