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

partialCor(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

column indices (whole numbers, no duplicates) of the variables of interest for which partial correlations are computed

y

column indices (whole numbers, no duplicates) of the control variables (conditioning set); must not overlap with x

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

Only the variables in y are controlled for. Let \(S\) be the joint covariance matrix of the selected variables. The residual covariance matrix of x after linear adjustment for y is the Schur complement

$$V = S_{xx} - S_{xy} S_{yy}^{-1} S_{yx}.$$

The result has entries \(V_{ij}/\sqrt{V_{ii} V_{jj}}\). With complete raw data, this equals the correlation matrix of the residuals from regressing each variable in x on all variables in y, including an intercept. Adding another variable to x does not change the correlations between the previously selected variables.

Normalizing the inverse of the joint covariance matrix would also control for the other variables in x. That is a different quantity when length(x) > 2.

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, and every selected variable must have positive variance. Near-singularity from collinearity 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. The condition number is taken on the correlation scale, so variables measured in very different units are not mistaken for collinear ones.

• For raw data only the selected columns enter stats::cov().

• 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: findCor(), hoeffdingD(), keepSig(), pearsonCor(), polychorCor(), 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
partialCor(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)
partialCor(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
partialCor(C, x = 1, y = 3:4)
#>    V1
#> V1  1