I understand that for simple linear regression, the sample correlation coefficient is the square root of the $R^2$. But that's just for a simple (i.e., single variable) regression $Y=\beta_0+\beta_1X+\varepsilon$.
How about multiple regression, e.g., $Y=\beta_0+\beta_1X_1 + \beta_2X_2+\varepsilon$? Is there any relationship between the correlations $corr(Y, X_1)$, $corr(Y, X_2)$ and the regression $R^2$?
Best Answer
For two predictors, it is easy to write out the equation in algebraic form:
$R^2 = \frac{r^2_{x1,y} + r^2_{x2,y} - 2r_{x1,y}r_{x2,y}r_{x1,x2}}{1-r^2_{x1,x2}}$.
As pointed out by @gung, you also need to know the correlation between $x1$ and $x2$.
EDIT: Just a quick example (in R) to illustrate this equation:
gives the exact same answer of
0.2928677
.