# Difference between coefficient of determination and explained variance

I'm trying to evaluate how good my model predictions w.r.t. the true values (paired). I came across two method coefficient of determination (https://en.wikipedia.org/wiki/Coefficient_of_determination) and explained variance (http://scikit-learn.org/stable/modules/model_evaluation.html#explained-variance-score) to do this but I can't really see what the difference between the two are and the pros and cons. The formulas seem about the same...

Can someone explain the difference and pros and cons?

Let $X$ be a random vector, and $Y$ a random variable that is modeled by a normal distribution with center $\mu+\Psi^\textrm{T}X$, in this case, the proportion of randomness $\rho_C^2$ equals the squared coefficient of determination, $R^2$.

Note the strong model assumptions: the center of the $Y$ distribution must be a linear function of $X$, and for any given $x$, the $Y$ distribution must be normal. In other situations, it is generally not justified to interpret $R^2$ as proportion of explained variance. What Does Explained Variance Explain?

They are the same. Using a bit of algebra, we can see that they are equivalent.

$R^2 = 1 - \dfrac{Sum\ of\ squared\ errors}{Total\ sum\ of\ squares}$

$Sum\ of\ Squared\ Errors = \sum(Y - \hat Y)^2$

$Total\ Sum\ of\ Squares = \sum(Y- \bar Y)^2$

$R^2 = 1 -\frac{\sum(Y - \hat Y)^2}{\sum(Y- \bar Y)^2}$

According to your source,

$Explained\ variance\ score\ = 1 - \dfrac{Var(Y- \hat Y)}{Var(Y)}$

$Explained\ variance\ score\ = 1 - \dfrac{\cfrac{\sum[(Y - \hat Y ) - E(Y - \hat Y)]^2}{n -1}}{\dfrac{\sum(Y- \bar Y)^2}{n-1}}$

$\sum[(Y - \hat Y ) - E(Y - \hat Y)]^2 = \sum(Y - \hat Y)^2$, i.e. SSE

So,

$Explained\ variance\ score\ = 1 - \dfrac{\dfrac{\sum(Y - \hat Y )^2}{n -1}}{\dfrac{\sum(Y- \bar Y)^2}{n-1}}$

$Explained\ variance\ score\ = 1 - \dfrac{(n-1)\sum(Y - \hat Y)^2}{(n-1)\sum(Y - \bar Y)^2}$

$Explained\ variance\ score\ = 1 - \dfrac{\sum(Y - \hat Y)^2}{\sum(Y - \bar Y)^2} = R^2$

Why are there two different names for the same thing? Because, statistics. Maybe there is some deeper reason, but often different terms for the same thing arise because of disciplinary differences.

• According to wiki this isn't true in general. "it is generally not justified to interpret $R^{2}$ as proportion of explained variance." I don't understand it but perhaps you will en.wikipedia.org/wiki/… – Hugh Oct 25 '16 at 0:30
• @Hugh only justified under the strict assumptions of ols, not for glm. I think the interpretation of R^2 as explained variance is somewhat controversial even in the case of ols. – paqmo Oct 25 '16 at 0:55
• @Hugh they are the same if the mean residue is zero, which is an assumption of OLS. Otherwise they are different things. See stackoverflow.com/questions/24378176/… – irene Jul 1 '20 at 9:28