Partial residual ($r_{ij}=Y_i-\hat{\alpha }-{\sum }_{k\ne j}{\hat{f}}_k(X_{ik})$)

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.