Generally, both of them can represent the linear relationship between x and y scale to [0,1]. Are they 99% very similar?

  • $\begingroup$ Not the same thing. See answers. $\endgroup$ – BruceET Jun 19 at 5:41

Computationally, R-sq in computer printouts is the square $r^2$of the (Pearson) correlation $r.$ Sometimes $r^2$ is called the 'coefficient of determination'. Because $-1 \le r \le 1$ we have $0 \le r^2 \le 1.$

Suppose we have the simple linear regression model $$Y_i = \beta_0 + \beta_1 x_i + e_i,$$ for $i = 1,2, \dots,n$ where $e_i \stackrel{iid}{\sim}\mathsf{Norm}(0, \sigma_e).$

In regression $r^2$ is sometimes referred to (intuitively) as the proportion of the variability in $Y$ that is explained by regression on $x.$ This interpretation roughly matches the equation $$S_{Y|x}^2 = \frac{n-1}{n-2}S_Y^2(1 - r^2),$$ where $S_Y^2$ is the sample variance of the $Y_i$ and $S_{Y|x}^2$ is the sum of squared residuals divided by $n-2,$ sometimes referred to as the variance about the regression line.

Thus if $r = \pm 1$ so that $r^2 = 1,$ then $S_{Y|x}^2 = 0$ and all of the $(x_i,Y_i)$-points lie precisely on the regression line. Also, if $r \approx 0,$ then $S_{Y|x}^2 \approx S_Y^2$ and the $x_i$ have no role to play in 'explaining' the $Y_i.$

Sometimes, the Pearson 'correlation coefficient' $r$ is said to express the 'linear component' of the association of $X_i$ and $Y_i.$

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    $\begingroup$ @mkt. Thanks for fixing typo. $\endgroup$ – BruceET Jun 19 at 5:59

R-squared or coefficient of determination generally has the value between 0 and 1. However cor(x,y) is between -1 and 1. Definition of R-squared for a multiple linear regression is the square of correlation between output (y) and predicted values(f). For the case of simple linear regression with an intercept and one explanatory variable it happens R-squared is as same as square of cor(x,y). I hope this helps you.

  • $\begingroup$ Definitely they are not same, but for simple linear regression : Y=ax+b (x is a single variable and not a vector), value of coefficient of determination is equal to square of correlation between x and y. $\endgroup$ – Masoud Norouzi Darabad Jun 19 at 5:19

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