Table 1

Formulas for computing partial variances and partial correlations

Definition

True value

Estimate

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 =

σ_kk

s_kk

Partial variances

var(X_k|X_≠k) = = =

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}) =

Index i runs from 1 to n (sample size), and indices k and l run from 1 to p (dimension). A tilde denotes a "partial" quantity.

Opgen-Rhein and Strimmer BMC Systems Biology 2007 1:37   doi:10.1186/1752-0509-1-37