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For my research, I am interested in calculating the multiple correlation coefficient(s) as a function of the "simple" (zero-order) correlations in a 4x4 (and larger) correlation matrix.

For 3 variables in a 3x3 correlation matrix (say with $x$, $y$ and $z$) the multiple correlation coefficient for $y$ is (Cohen, Cohen, Aiken & West, 2003, p. 70):

$$ r_{y.xz} = \sqrt{R_{y.xz}^2} = \sqrt{\frac {r_{xy}^2 + r_{zy}^2 - 2r_{xy}r_{zy}r_{xz}}{1-r_{xz}^2}} $$ where the $r$s are the respective zero-order correlations, and analogous for $r_{x.yz}$ and $r_{z.xy}$. I have not been able to extend this to 4 and more variables.

I am aware of the general matrix notation (https://en.wikipedia.org/wiki/Multiple_correlation) but I am looking for an expression similar to the one above for 4 variables, and more.

Can anyone help? Thanks in advance.

Cohen, Jacob, Patricia Cohen, Stephen G. West, and Leona S. Aiken (2003), Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences, 3rd ed. Mahwah, NJ: Lawrence Erlbaum Associates.

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  • $\begingroup$ for 4 variables, and more. The formula will be very cumbersome. Why do you need that? $\endgroup$ – ttnphns Apr 10 at 10:04
  • $\begingroup$ @ttnphns, I expected it to be a long equation, and that's fine. Without going into too many specifics, I need it in my research where I can't enter a matrix but can only work with an expression like I give above for three variables. $\endgroup$ – user244248 Apr 10 at 12:24

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