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I came across this formula in some online resources.

$$r_{12.34} = \frac {r_{12.3} - r_{14.3}r_{24.3}}{ \sqrt {(1- r_{14.3}^2 )(1-r_{24.3}^2 )}}$$

I can use this but I wanted a proof of the formula. Can you prove the formula or please direct me to any link with a proof?

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An example of how to derive formulas like this appears at – whuber Jan 11 '15 at 16:04

You can find a proof of the general case in Section 2.5.3 (pp. 42-43) of Anderson (1984). The proof covers about a page and half to obtain the general formula $$\rho_{ij\cdot q+1,...,p} = \frac {\rho_{ij\cdot q+2,...,p} - \rho_{i, q+1\cdot q+2,...,p} \rho_{j, q+1\cdot q+2,...,p}} { \sqrt{1 - \rho^2_{i,q+1\cdot q+2,...,p}} \sqrt{1 - \rho^2_{j,q+1\cdot q+2,...,p}} }.$$

Your formula follows on substitution and on a relabeling of indices if needed.

T.W. Anderson (1984) An Introduction to Multivariate Statistical Analysis. Second Edition. John Wiley & Sons.

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