Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Multivariate continuous distributions ▷ Multivariate normal distribution ▷ 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 partial correlation of $X$ and $Y$ controlling for $Z$ can be computed as

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

where $\rho_{XY}$, $\rho_{XZ}$ and $\rho_{YZ}$ are the pairwise correlations of the random variables $X$, $Y$ and $Z$.

Proof: In the first part of the proof, we will justify that we can work with a simplified form for $\Sigma$. In the second part of the proof, we will compute the partial correlation, as implied by its definition, from $\Sigma$.

1) We will denote the entries of $\Sigma$ as follows:

\[\label{eq:Sigma} \Sigma = \left[ \begin{matrix} \sigma_X^2 & \sigma_{XY} & \sigma_{XZ} \\ \sigma_{YX} & \sigma_Y^2 & \sigma_{YZ} \\ \sigma_{ZX} & \sigma_{ZY} & \sigma_Z^2 \end{matrix} \right]\]

where $\sigma_X^2$, $\sigma_Y^2$ and $\sigma_Z^2$ are the variances and $\sigma_{XY}$, $\sigma_{XZ}$ and $\sigma_{YZ}$ are the covariances of the entries of the random vector $V$. Define the following matrix:

\[\label{eq:C} C = \left[ \begin{matrix} \frac{1}{\sigma_X} & 0 & 0 \\ 0 & \frac{1}{\sigma_Y} & 0 \\ 0 & 0 & \frac{1}{\sigma_Z} \end{matrix} \right] \; .\]

With the linear transformation theorem for the multivariate normal distribution, the distribution of $W = CV$ is

\[\label{eq:W} W = CV = \left[ \begin{matrix} X/\sigma_X \\ Y/\sigma_Y \\ Z/\sigma_Z \end{matrix} \right] \sim \mathcal{N}(C \mu, C \Sigma C^\mathrm{T})\]

with

\[\label{eq:C-Sigma-C} C \Sigma C^\mathrm{T} = \left[ \begin{matrix} \frac{\sigma_X^2}{\sigma_X \sigma_X} & \frac{\sigma_{XY}}{\sigma_X \sigma_Y} & \frac{\sigma_{XZ}}{\sigma_X \sigma_Z} \\ \frac{\sigma_{YX}}{\sigma_Y \sigma_X} & \frac{\sigma_Y^2}{\sigma_Y \sigma_Y} & \frac{\sigma_{YZ}}{\sigma_Y \sigma_Z} \\ \frac{\sigma_{ZX}}{\sigma_Z \sigma_X} & \frac{\sigma_{ZY}}{\sigma_Z \sigma_Y} & \frac{\sigma_Z^2}{\sigma_Z \sigma_Z} \end{matrix} \right] = \left[ \begin{matrix} 1 & \rho_{XY} & \rho_{XZ} \\ \rho_{YX} & 1 & \rho_{YZ} \\ \rho_{ZX} & \rho_{ZY} & 1 \end{matrix} \right] \; .\]

Since correlation is invariant under linear transformation, the entries of $V$ and $W$ have the same pairwise correlations and partial correlations. Thus, for the above theorem, we can assume without loss of generality that

\[\label{eq:Sigma-wlog} \Sigma = \left[ \begin{matrix} 1 & \rho_{XY} & \rho_{XZ} \\ \rho_{YX} & 1 & \rho_{YZ} \\ \rho_{ZX} & \rho_{ZY} & 1 \end{matrix} \right] \; .\]

2) For deriving partial correlation, we define the residual variables

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

Note that $E^{(X)}$ and $E^{(Y)}$ can be written as

\[\label{eq:E-A-b} E = \left[ \begin{matrix} E^{(X)} \\ E^{(Y)} \end{matrix} \right] = \left[ \begin{matrix} 1 & 0 & -\beta_1^{(X)} \\ 0 & 1 & -\beta_1^{(Y)} \end{matrix} \right] \left[ \begin{matrix} X \\ Y \\ Z \end{matrix} \right] + \left[ \begin{matrix} -\beta_0^{(X)} \\ -\beta_0^{(Y)} \end{matrix} \right] = A V + b\]

with

\[\label{eq:A-b} A = \left[ \begin{matrix} 1 & 0 & -\beta_1^{(X)} \\ 0 & 1 & -\beta_1^{(Y)} \end{matrix} \right] \quad \text{and} \quad b = \left[ \begin{matrix} -\beta_0^{(X)} \\ -\beta_0^{(Y)} \end{matrix} \right] \; .\]

When two random variables are in a linear relationship of the form $Y = \beta_0 + \beta_1 X + E$, then the correlation is related to the slope parameter via the standard deviations:

\[\label{eq:corr-slope} \begin{split} \rho_{XY} = \frac{\sigma_X}{\sigma_Y} \beta_1 \quad \Leftrightarrow \quad \beta_1 = \frac{\sigma_Y}{\sigma_X} \rho_{XY} \; . \end{split}\]

This also holds in the finite-sample case. Thus, in the present case defined by \eqref{eq:Sigma-wlog}, we have

\[\label{eq:b1-XY} \beta_1^{(X)} = \frac{1}{1} \rho_{XZ} \quad \text{and} \quad \beta_1^{(Y)} = \frac{1}{1} \rho_{YZ} \; ,\]

such that

\[\label{eq:A-rev} A = \left[ \begin{matrix} 1 & 0 & -\rho_{XZ} \\ 0 & 1 & -\rho_{YZ} \end{matrix} \right] \; .\]

With that, we can obtain the distribution of the residual variables. Specifically, the linear transformation theorem for the multivariate normal distribution implies that the distribution of $E$ is

\[\label{eq:E} E = AV + b \sim \mathcal{N}(A \mu + b, A \Sigma A^\mathrm{T}) \; .\]

where

\[\label{eq:A-Sigma-A} \begin{split} A \Sigma A^\mathrm{T} &= \left[ \begin{matrix} 1 & 0 & -\rho_{XZ} \\ 0 & 1 & -\rho_{YZ} \end{matrix} \right] \left[ \begin{matrix} 1 & \rho_{XY} & \rho_{XZ} \\ \rho_{YX} & 1 & \rho_{YZ} \\ \rho_{ZX} & \rho_{ZY} & 1 \end{matrix} \right] \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ -\rho_{XZ} & -\rho_{YZ} \end{matrix} \right] \\ &= \left[ \begin{matrix} 1 - \rho_{XZ} \rho_{ZX} & \rho_{XY} - \rho_{XZ} \rho_{ZY} & \rho_{XZ} - \rho_{XZ} \\ \rho_{YX} - \rho_{YZ} \rho_{ZX} & 1 - \rho_{YZ} \rho_{ZY} & \rho_{YZ} - \rho_{YZ} \end{matrix} \right] \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ -\rho_{XZ} & -\rho_{YZ} \end{matrix} \right] \\ &= \left[ \begin{matrix} 1 - \rho_{XZ}^2 & \rho_{XY} - \rho_{XZ} \rho_{YZ} & 0 \\ \rho_{XY} - \rho_{XZ} \rho_{YZ} & 1 - \rho_{YZ}^2 & 0 \end{matrix} \right] \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ -\rho_{XZ} & -\rho_{YZ} \end{matrix} \right] \\ &= \left[ \begin{matrix} 1 - \rho_{XZ}^2 & \rho_{XY} - \rho_{XZ} \rho_{YZ} \\ \rho_{XY} - \rho_{XZ} \rho_{YZ} & 1 - \rho_{YZ}^2 \end{matrix} \right] = \left[ \begin{matrix} \sigma_{X \backslash Z}^2 & \sigma_{X,Y \backslash Z} \\ \sigma_{X,Y \backslash Z} & \sigma_{Y \backslash Z}^2 \end{matrix} \right] \; . \end{split}\]

With that, we can derive the partial correlation of $X$ and $Y$ controlling for $Z$. Specifically, dividing the covariance of the residual variable distribution by the product of its standard deviations, we get:

\[\label{eq:mvn-corr-part-qed} \begin{split} \mathrm{Corr}(X,Y \backslash Z) &= \mathrm{Corr}\left( E^{(X)}, E^{(Y)} \right) \\ &= \frac{\sigma_{X,Y \backslash Z}}{\sigma_{X \backslash Z} \sigma_{Y \backslash Z}} \\ &= \frac{\rho_{XY} - \rho_{XZ} \rho_{YZ}}{\sqrt{1-\rho_{XZ}^2} \sqrt{1-\rho_{YZ}^2}} \; . \end{split}\]
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Metadata: ID: P530 | shortcut: mvn-corrpart | author: JoramSoch | date: 2026-03-26, 17:37.