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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?

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

M. GENTILI
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    I like the nature of this answer but believe it's incomplete, because the inverse of a *covariance* matrix cannot, in general, possibly contain correlation coefficients. See https://en.wikipedia.org/wiki/Partial_correlation#Using_matrix_inversion. – whuber Jan 22 '20 at 23:23
  • I guess you are right... in general. Because the cov. matrix needs to be invertible. In this case you only have (a,b,c), so my guess is that if the cov. matrix is not invertible (for instance you have only 2 samples to work with), than also partial correlation will be 1. because the residuals you would get from the regression would be 0. – M. GENTILI Jan 23 '20 at 15:48
  • I read you post [here](https://stats.stackexchange.com/questions/140080/why-does-inversion-of-a-covariance-matrix-yield-partial-correlations-between-ran), I have few questions, can I contact you? if so... how? – M. GENTILI Jan 23 '20 at 15:49