Table 1 Formulas for computing partial variances and partial correlations

Covariance matrix:

cov(X _k , X _l ) = σ _kl

Σ = (σ _kl )

S= (s _kl )

Concentration matrix:

Ω = Σ^-1

Ω = (ω _kl )

Variances:

var(X _k ) = σ _kk = ${\sigma }_k^2$

σ _kk

s _kk

Partial variances

var(X _k |X_≠k) = ${\tilde{\sigma }}_{kk}$ = ${\tilde{\sigma }}_k^2$ = ${\omega }_{kk}^{-1}$

${\tilde{\sigma }}_{kk}$

${\tilde{s}}_{kk}$

Correlations:

corr(X _k , X _l ) = ρ _kl = σ _kl (σ _kk σ _ll )^-1/2

P= (ρ _kl )

R= (r _kl )

Partial correlations:

corr(X _k , X _l |X_{≠k, l}) = ${\tilde{\rho }}_{kl}=-{\omega }_{kl}{({\omega }_{kk}{\omega }_{ll})}^{-1/2}$

$\tilde{P}=\left({\tilde{\rho }}_{kl}\right)$

$\tilde{R}=\left({\tilde{r}}_{kl}\right)$