Multiple correlation coefficient of a simple linear regression

Question

I'm having a bit of a hard time understanding why the 'multiple correlation' coefficient within a simple linear regression (i.e., 1 predictor) isn't identical to the coefficient of determination. I understand that they're respectively denoted by $R$ and $R^2$, and thus the transformation from one to another is should be $\sqrt{R^2}$...but my mental barrier is that there is no combination of predictors to produce a the $R$ and therefore the two would be equivalent. Many thanks in advance.

Answer 1

If you have a linear regression model which regresses an outcome variable Y on a single predictor variable X, then you can only talk about a simple coefficient of determination $R^2$ which describes the percentage of variability in the Y values which can be accounted for by X. The coefficient $R^2$ can be shown to be equal to the squared value of the (Pearson) correlation coefficient $r$ between Y and X: $R^2$ = $r^2$. Recall that $r$ can range between -1 (perfect negative association between Y and X) and 1 (perfect positive association between Y and X).

If the linear regression model has multiple predictors, then it makes sense to talk about a multiple coefficient of determination $R^2$ which describes the percentage of variability in the Y which can be accounted for by the multiple predictors. The multiple correlation coefficient is simply defined as the square root of the multiple coefficient of determination and is a measure of how well Y can be predicted using a linear combination of the multiple predictors. By virtue of how it is defined, the multiple correlation coefficient can range between 0 and 1, where 1 indicates that Y can be perfectly predicted by a linear combination of the multiple predictors (which includes an intercept) and 0 means that the multiple predictors cannot be used to predict Y. See https://en.m.wikipedia.org/wiki/Multiple_correlation for more details.