An answer to a previous question indicates that $R^2$ is " the true proportion of variance (of the response) explained by variation in the regressors." In the context of fixed X's, the definition of "the variance of the response" is not so clear to me. In experiment $j=1…L$ the response values are $Y_{i,j}$ $(i=1…n)$. The sequence $Y_{1,j}$ across experiments is i.i.d, and has a well-defined variance (equal to $\sigma^2$). However, $Y_{i,1}$ across observations are not i.i.d., as they have different expectations. So what would be an appropriate definition of $VAR(Y)$ in this context?
An answer to another question describes the parameter $\sigma^2$ estimated by $SSE/(n-2)$ as "residual variance". The answer indicates a ratio of $(1-r^2)$ between residual variance and $VAR(Y)$ (which is not defined there). Is $r^2$ here a parameter? In a random design $r^2$ is the population correlation between X and Y. How is $r^2$ defined in the fixed design context?
