Index: The Book of Statistical ProofsModel Selection ▷ Goodness-of-fit measures ▷ R-squared ▷ Derivation of R² and adjusted R²

Theorem: Given a linear regression model

$\label{eq:rsq-mlr} y = X\beta + \varepsilon, \; \varepsilon_i \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2)$

with $n$ independent observations and $p$ independent variables,

1) the coefficient of determination is given by

$\label{eq:R2} R^2 = 1 - \frac{\mathrm{RSS}}{\mathrm{TSS}}$

2) the adjusted coefficient of determination is

$\label{eq:R2-adj} R^2_{\mathrm{adj}} = 1 - \frac{\mathrm{RSS}/(n-p)}{\mathrm{TSS}/(n-1)}$

where the residual and total sum of squares are

$\label{eq:SS} \begin{split} \mathrm{RSS} &= \sum_{i=1}^{n} (y_i - \hat{y}_i)^2, \quad \hat{y} = X\hat{\beta} \\ \mathrm{TSS} &= \sum_{i=1}^{n} (y_i - \bar{y})^2\;, \quad \bar{y} = \frac{1}{n} \sum_{i=1}^n y_i \\ \end{split}$

where $X$ is the $n \times p$ design matrix and $\hat{\beta}$ are the ordinary least squares estimates.

Proof: The coefficient of determination $R^2$ is defined as the proportion of the variance explained by the independent variables, relative to the total variance in the data.

1) If we define the explained sum of squares as

$\label{eq:ESS} \mathrm{ESS} = \sum_{i=1}^{n} (\hat{y}_i - \bar{y})^2 \; ,$

then $R^2$ is given by

$\label{eq:R2-s1} R^2 = \frac{\mathrm{ESS}}{\mathrm{TSS}} \; .$

which is equal to

$\label{eq:R2-s2} R^2 = \frac{\mathrm{TSS}-\mathrm{RSS}}{\mathrm{TSS}} = 1 - \frac{\mathrm{RSS}}{\mathrm{TSS}} \; ,$

because $\mathrm{TSS} = \mathrm{ESS} + \mathrm{RSS}$.

2) Using \eqref{eq:SS}, the coefficient of determination can be also written as:

$\label{eq:R2'} R^2 = 1 - \frac{\sum_{i=1}^{n} (y_i - \hat{y}_i)^2}{\sum_{i=1}^{n} (y_i - \bar{y})^2} = 1 - \frac{\frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2}{\frac{1}{n} \sum_{i=1}^{n} (y_i - \bar{y})^2} \; .$

If we replace the variance estimates by their unbiased estimators, we obtain

$\label{eq:R2-adj'} R^2_{\mathrm{adj}} = 1 - \frac{\frac{1}{n-p} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2}{\frac{1}{n-1} \sum_{i=1}^{n} (y_i - \bar{y})^2} = 1 - \frac{\mathrm{RSS}/\mathrm{df}_r}{\mathrm{TSS}/\mathrm{df}_t}$

where $\mathrm{df}_r = n-p$ and $\mathrm{df}_t = n-1$ are the residual and total degrees of freedom.

This gives the adjusted $R^2$ which adjusts $R^2$ for the number of explanatory variables.

Sources:

Metadata: ID: P8 | shortcut: rsq-der | author: JoramSoch | date: 2019-12-06, 11:19.