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Use the backfitting algorithm to alternate conditional updates of the two additive functions. Initialize $\alpha =\overline{Y}$ and both centered function vectors at zero. With $S_e,S_d$ the smoothing matrices for the chosen experience and education spline penalties, one cycle isfollowed byThe centering transfers any constant component to the intercept; update $\alpha$ to the mean of $Y-f_e-f_d$ if necessary, which is $\overline{Y}$ under the stated constraints. Repeat until changes in the functions or objective are negligible. Each function is estimated by smoothing its partial residual, namely the response after subtracting all other currently fitted terms.

For fixed penalties these updates are block minimizations ofsubject to the centering constraints. Each step cannot increase the objective; with identifiable additive terms and a positive-definite constrained quadratic, backfitting converges to its unique minimizer. Highly dependent predictors can make the decomposition poorly identified or slow convergence. The Gaussian identity-link model needs these least-squares updates directly; non-Gaussian generalized additive models use weighted backfitting inside iteratively reweighted least squares. The package implementation also separates each spline's unpenalized linear component from its nonlinear component, giving the two ANOVA tables displayed in part (e).