Partial Correlation Matrix

Computes the partial correlation matrix of a set of variables x while controlling for another set of variables y, based on a covariance/correlation matrix or on raw data.

Usage

corPart(m, x, y)

Arguments

m

a numeric matrix, either:

• a square, symmetric covariance or correlation matrix, or

• a data matrix (observations in rows, variables in columns)

The two are told apart by symmetry, not by shape alone - a data matrix with as many rows as columns would otherwise be mistaken for a correlation matrix.

x

integer vector of indices specifying the variables of interest for which partial correlations are computed

y

integer vector of indices specifying the control variables (conditioning set)

Value

a symmetric numeric matrix containing the partial correlations among variables in x, adjusted for variables in y. Row and column names correspond to colnames(m)[x].

Details

Partial correlations are read off the precision matrix. Let \(K\) be the inverse of the joint covariance matrix of \((x, y)\); then

$$\rho_{ij \cdot y} = - K_{ij} / \sqrt{K_{ii} K_{jj}}$$

for \(i, j\) in \(x\). This is algebraically equivalent to forming the Schur complement \(\Sigma_{xx} - \Sigma_{xy}\Sigma_{yy}^{-1}\Sigma_{yx}\) and scaling it to unit diagonal, but needs a single inversion instead of two.

Because the result is scaled to unit diagonal, it makes no difference whether m is a covariance or a correlation matrix.

Numerical considerations

• The joint submatrix of x and y must be invertible. Near-singularity from collinearity among the control variables is detected via the reciprocal condition number, not merely by a failure of base::solve(), which succeeds and returns nonsense well before the matrix is numerically singular.

• x and y must not overlap.

• For raw data, correlations are computed pairwise, which can produce a non-positive-definite matrix when values are missing.

See also

stats::cor(), stats::cov()

Other assoc.continuous: corPolychor(), findCorrX(), hoeffdingD(), keepSig(), pearsonCor(), spearmanCor()

Examples

# Simulated data
set.seed(1)
X <- matrix(rnorm(100 * 5), ncol = 5)
colnames(X) <- paste0("V", 1:5)

# Partial correlations of V1, V2 controlling for V3, V4
corPart(X, x = 1:2, y = 3:4)
#>              V1           V2
#> V1  1.000000000 -0.002908161
#> V2 -0.002908161  1.000000000

# Using a correlation matrix directly
C <- cor(X)
corPart(C, x = 1:2, y = 3:4)
#>              V1           V2
#> V1  1.000000000 -0.002908161
#> V2 -0.002908161  1.000000000

# a single variable of interest is allowed and returns a 1x1 matrix
corPart(C, x = 1, y = 3:4)
#>    V1
#> V1  1