Coefficient of Determination

Coefficient of determination ($R^2)$ measures the proportion of the variance in the response that is explained by the model.

Coefficient determination close to 1 implies a good fit of the model, i.e., the model is accurate. Meanwhile, $R^2$ value close to 0 implies the opposite.

Warning

Use $R^2$ only if the number of observations is substantially larger than the number of coefficients.

Formula

$$R^2=1-\frac{SSE}{SST}=\frac{SSR}{SST}$$

Where

• SSE: Sum of Squares Error
• SSR: Sum of Squares Regression
• SST: Sum of Squares Total

Relation with F statistic

Relation of $R^2$ with $F$ is given in the following formula:

$$R^2=1-{\left[1+F\left(\frac{p-1}{n-p}\right)\right]}^{-1}$$

This formula is derived from the $F$ formula with simple algebra.

$F$ can also be obtained from $R$ using the following formula:

$$F=\frac{(SST-SSE)/(p-1)}{SSE/(n-p)}$$

Adjusted $R^2$

TODO what’s this

$$R_{\mathrm{a}}^2=1-\left[\frac{(n-1)}{n-(p+1)}\right]\left(1-R^2\right)$$