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  "body": "\n## Definition\n\nLet $X$ and $Y$ be real random variables and let $Z$ be a random vector in $\\R^k$, all with finite second moments.\nAssume that the [[covariance matrix|0V7G6WMFK0]] $\\Sigma_{ZZ}$ of $Z$ is [[invertible|0EB9N0S33C]].\nWrite $\\Sigma_{XZ}$ for the row vector of [[covariances|0KBFAWYRTE]] $\\operatorname{Cov}(X, Z_j)$, and similarly $\\Sigma_{YZ}$.\nThe residuals of the best linear predictions of $X$ and $Y$ from $Z$ are\n$$e_X = X - \\mathbb{E}X - \\Sigma_{XZ} \\Sigma_{ZZ}^{-1} (Z - \\mathbb{E}Z), \\qquad e_Y = Y - \\mathbb{E}Y - \\Sigma_{YZ} \\Sigma_{ZZ}^{-1} (Z - \\mathbb{E}Z).$$\nIf $\\operatorname{Var}(e_X) \u003e 0$ and $\\operatorname{Var}(e_Y) \u003e 0$, the *partial correlation* of $X$ and $Y$ given $Z$ is the [[correlation coefficient|0DNN9B6Q2F]]\n$$\\rho_{XY \\cdot Z} = \\operatorname{Corr}(e_X, e_Y).$$\n\n## Remarks\n\nExample: let $Z, U, V$ be uncorrelated with variance $1$ and mean $0$, and let $X = Z + U$, $Y = Z + V$.\nThen $\\Sigma_{XZ} = \\Sigma_{YZ} = 1$ and $\\Sigma_{ZZ} = 1$, so $e_X = U$ and $e_Y = V$, and $\\rho_{XY \\cdot Z} = 0$.\nYet $\\operatorname{Corr}(X, Y) = 1/2$.\nNon-example: if $X = Z$, then $e_X = 0$, so $\\operatorname{Var}(e_X) = 0$ and the partial correlation is not defined.\n\nFor a single control variable $Z$, with $\\rho_{XY}$, $\\rho_{XZ}$, $\\rho_{YZ}$ the pairwise correlations,\n$$\\rho_{XY \\cdot Z} = \\frac{\\rho_{XY} - \\rho_{XZ} \\rho_{YZ}}{\\sqrt{(1 - \\rho_{XZ}^2)(1 - \\rho_{YZ}^2)}}.$$\nIf $P = (p_{ij})$ is the inverse of the covariance matrix of $(X, Y, Z)$, then $\\rho_{XY \\cdot Z} = -p_{12} / \\sqrt{p_{11} p_{22}}$.\nThe sample partial correlation uses the sample covariances, or equivalently the residuals of least squares fits on the data.\n",
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