Regression residual

A regression residual is the observed response minus its fitted value. Residual plots diagnose failures of assumptions such as constant variance, linear mean structure, and independent errors.

A systematic curved pattern in regression residuals plotted against fitted values suggests a misspecified conditional mean. Positive residuals at both ends and negative residuals in the middle motivate trying quadratic regression. The diagnostic does not by itself identify the error probability distribution or prove heteroscedasticity.

In an additive mean model, the partial residual for component $j$ subtracts every fitted term except that component. It equals the full regression residual plus the fitted component. Plotting it against that component's predictor shows the conditional pattern after accounting for other fitted effects, and supplies the response for a backfitting update.

In a normal linear model, the $i$th regression residual has variance ${\sigma }^2(1-h_{ii})$, where $h_{ii}$ is a diagonal entry of the hat matrix. Replacing $\sigma$ by the residual standard deviation $s$ gives the standardized residual used in the normal Q-Q plot of R's linear-model diagnostic plots.