I need assistance to figure out if the following statement is true:
The proportion of variation in the dependent variable explained by fitting the simple
linear regression model does not depend on which variable is treated as the dependent
variable
My Thoughts:
In one sense the equation $y = m * x + c$ can be written as $x = y/m - c/m$
So it seems the above statement is true. But I am not sure. If this is a valid
argument.
The statement is true since the proportion of explained variance, $R^2$, in simple linear regression equals the squared Pearson correlation coefficient between $X$ and $Y$, $r_{X,Y}^2$. Due to the fact that $r^2$ is symmetric measure, i.e., $r_{X,Y}^2 = r_{Y,X}^2$, therefore it does not matter whether $X$ or $Y$ is treated as dependent variable.