correlation from partial correlations

Question

I know it is possible to get the partial correlation between $a$ and $b$ given $c$ when you know all the full correlations:

$$ r_{(a, b |c)} = \frac{r_{(a, b)}-r_{(a, c)}r_{(b, c)}} {\sqrt{(1-r_{(a, c)}^2)((1-r_{(b, c)}^2)}} $$

Presumably we can get the full correlations when we know all the partials, but I can't figure out the equations. Is it possible?

Answer 1

Partial correlation is the inverse of the covariance matrix... So Compute all the partials, write down the matrix and compute the inverse! :)

Answer 2

Some additional steps are necessary other than simply calculating the inverse. The actual steps are the following:

• Step 1: Get the negative of the matrix of partial correlations;
• Step 2: Set all the values in the diagonal of this new matrix to 1;
• Step 3: Calculate the inverse of this matrix (i.e., the modified matrix of partial correlations where all the off-diagonal elements have the inverse sign of the original matrix of partial correlations);
• Step 4: Scale the resulting inverse.

Below I present a reproducible example with R:

# Set seed for reproducibility
set.seed(1)

# Randomly sample the original random correlation matrix
R <- cov2cor(rWishart(n=1, df=6, Sigma=diag(5))[,,1])

# Get the partial correlation matrix using the "partial.r" function of the psych package
Rho <- psych::partial.r(R)

# Do the steps described above
# Step 1: Get the negative of the of matrix of partial correlations
temp <- -Rho

# Step 2: Set all the values in the diagonal of this new matrix to 1
diag(temp) <- 1

# Step 3: Calculate the inverse of the matrix
inv <- solve(temp)

# Step 4: Scale the resulting inverse
R_rec <- cov2cor(inv)

# Check if the solution is correct
# Due to computer precision, the matrices will not be identical, but
# you can round the results to your machine's precision and check
# that the procedure works as it should
round(R, .Machine$double.eps) == round(R_rec, .Machine$double.eps)