17.2 Fitted Values and Residuals

• A fitted (or predicted) value is calculated as

\[\hat{y}_{i}=\hat{\beta}_{0}+\hat{\beta}_{1}x_{i,1}+\hat{\beta}_{2}x_{i,2}+\ldots+\hat{\beta}_{k}x_{i,k}\] - A residual (error) term is calculated as \[e_{i}=y_{i}-\hat{y}_{i},\] the difference between an actual and a predicted value of \(y\). A plot of residuals versus fitted values ideally should resemble a horizontal random band. Departures from this form indicates difficulties with the model and/or data. - Other residual analyses can be done exactly as we did in simple regression. For instance, we might wish to examine a normal probability plot (NPP) of the residuals. Additional plots to consider are plots of residuals versus each \(x\) variable separately. This might help us identify sources of curvature or nonconstant variance.