# Multiple correlation coefficient from zero-order correlations (4 or more variables)

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.

• for 4 variables, and more. The formula will be very cumbersome. Why do you need that? – ttnphns Apr 10 at 10:04
• @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. – user244248 Apr 10 at 12:24