R²

172. R²

Proportion of variance explained by the model

𝑅^{2} = 1 - \frac{\sum_{𝑖 = 1}^{𝑛} {(𝑦_{𝑖} - {\hat{𝑦}}_{𝑖})}^{2}}{\sum_{𝑖 = 1}^{𝑛} {(𝑦_{𝑖} - {\bar{𝑦}}_{𝑖})}^{2}}

Example

Step 1: Fit the Regression Model

Step 2: Compute Total Sum of Squares (SST)

SST = \sum_{𝑖 = 1}^{𝑛} {(𝑦_{𝑖} - \bar{𝑦})}^{2}

Step 3: Compute Regression Sum of Squares (SSR)

SSR = \sum_{𝑖 = 1}^{𝑛} = {({\hat{𝑦}}_{𝑖} - \bar{𝑦})}^{2}

Step 4: Compute Residual Sum of Squares (SSE)

SSE = \sum_{𝑖 = 1}^{𝑛} = {(𝑦_{𝑖} - {\hat{𝑦}}_{𝑖})}^{2}

Step 5: Calculate $𝑅^{2}$

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

Adjusts the $𝑅^{2}$ value based on the number of predictors (penalty for adding non-informative variables)

Adj 𝑅^{2} = 1 - (1 - 𝑅^{2}) \frac{𝑛 - 1}{𝑛 - 𝑝 - 1}