Sample partial correlation by residualization

Let $x^{(1)},\dots ,x^{(n)}\in {\mathbb{R}}^p$ be a sample, let $i\mathrm{\ne }j$, and let $S\subseteq \{1,\dots ,p\}\setminus \{i,j\}$. Let ${\mathbf{x}}_k\in {\mathbb{R}}^n$ be the sample column for coordinate $k$, and let ${\mathrm{\Pi }}_S$ be the orthogonal projection onto the span of the all-ones vector and $\{{\mathbf{x}}_s:s\in S\}$. When the sample covariance matrix of coordinates $\{i,j\}\cup S$ is positive definite, define the residual columns $R_{i\mid S}=(I-{\mathrm{\Pi }}_S){\mathbf{x}}_i,R_{j\mid S}=(I-{\mathrm{\Pi }}_S){\mathbf{x}}_j\mathrm{.}$ The sample partial correlation is ${\widehat{\rho }}_{ij\mid S}=\frac{R_{i\mid S}^T{R}_{j\mid S}}{\mathrm{\parallel }{R}_{i\mid S}{\mathrm{\parallel }}_2\mathrm{\parallel }{R}_{j\mid S}{\mathrm{\parallel }}_2}\mathrm{.}$ For $-1<r<1$, the Fisher transform is $g(r)=\frac{1}{2}\log \frac{1+r}{1-r}$, and the sample Fisher statistic is $Z_{ij\mid S}=g({\widehat{\rho }}_{ij\mid S})$. For a test using multiplier $\sqrt{n-\mathrm{\mid }S\mathrm{\mid }-3}$, require $n-\mathrm{\mid }S\mathrm{\mid }-3>0$ so that this factor is positive. Under independent samples from a nondegenerate Gaussian law, the stated covariance block is positive definite almost surely whenever $n-\mathrm{\mid }S\mathrm{\mid }-3>0$.

Gaussian partial correlation