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The backfitting algorithm estimates each additive mean function by smoothing its partial residual, obtained after subtracting the current fits of all other terms, and cycling through terms until convergence. Centering each smooth separates its constant part from the intercept. For fixed quadratic smoothness penalties it is block coordinate minimization of a penalized least-squares objective; identifiability and positive definiteness of the constrained problem ensure a unique converged fit. Non-Gaussian generalized additive models can use weighted backfitting inside iteratively reweighted least squares.

The first model's residual deviance is $671.27$ on $197$ residual degrees of freedom, a ratio of about $3.41$. This is far above the scale expected from a well-fitting dispersion-one grouped binomial regression. It suggests an inadequate mean function, overdispersion, or both. The linear age and exposure restrictions may miss nonlinear associations; the generalized additive model allows those shapes to be estimated through penalized cubic regression splines rather than assumed.

The very small exposure Wald p-value in the first fit suggests an association, but does not establish that a linear logit effect is adequate. Nor does the nonsignificant linear age term exclude every possible nonlinear age effect. Allowing estimated scale also addresses the excessive residual variation, which can arise from clustered data because several births belong to the same woman. The second model's estimated scale $3.3506$ confirms that permitting smooth means has not removed all extra variation. The reason to proceed is poor initial fit and possible nonlinear effects, with dispersion-aware inference, not merely the wish to obtain more significant tests. These observational fits alone do not establish causation by the disaster.

For woman $i$, write $m_i$ for the number of births, $Y_i$ for the number with defects, $a_i$ for baseline age, and $e_i$ for exposure. Let $p_i$ be the common marginal defect probability for her births. The first fit is a grouped grouped-binomial logistic regression:It assumes independent women and, conditionally on their covariates, independent births with the same probability within each woman. The supplied birth totals are the binomial denominators; fitting the proportions with weights $m_i$ is equivalent to this grouped-binomial specification. The fitted linear predictor is $-1.74272+0.02030a_i+2.33028e_i$.

The second fit is a generalized additive model with the same logit link function and the mean specificationThe functions $f_a,f_e$ are penalized cubic regression splines with natural boundary conditions. Centering constraints such as ${\sum }_i{f}_a(a_i)={\sum }_i{f}_e(e_i)=0$ separate them from the intercept; the fitted centered intercept is $-0.2031$. Their second derivative roughness penalties control complexity.

The call estimates a common scale rather than fixing it at one. Its working variance specification for the counts isEquivalently the variance of $Y_i/m_i$ is $\varphi p_i(1-p_i)/m_i$. This is the working moment interpretation of an overdispersed binomial generalized additive model: the code uses binomial deviance for fitting and estimated scale for inference. With $\varphi \ne 1$, it is not an exact independent-binomial sampling model. Women are still treated as independent groups, but within-woman dependence or unobserved heterogeneity can motivate the extra dispersion. The quasibinomial regression interpretation states what the scaled analysis assumes without inventing a full probability distribution for its counts.

The residual smooth rises from negative values, arches above zero, then falls at high fitted values. This suggests a nonlinear altitude effect on the log mean. Since the fitted log mean in the linear model increases with altitude, the horizontal fitted-value axis is also an increasing rescaling of altitude. A generalized additive model with Quasi-Poisson regression is therefore a reasonable candidate:The centering constraint identifies the intercept. Here bs="cr" uses a penalized cubic regression spline. The plotted count residuals are deviance residuals, with the Q-Q panel using leverage-standardized versions. The quantile-quantile plot also shows tail discrepancies, but normality of count residuals is not a distributional assumption of this model: its main requirements are an adequate mean, variance relation, and independence. Investigate the labelled islands and possible excess variation as well.

The estimated smooth in the second plot increases fairly rapidly at low altitude, flattens around altitude 20 to 25, and is nearly level at the upper end. The pointwise bands widen where there are fewer observations. A penalized spline can describe a rise followed by a plateau without forcing a global parabola. A quadratic log mean has slope ${\beta }_1+2{\beta }_{2}a$, so negative curvature eventually forces a decline and its curvature is constant everywhere. A cubic regression spline allows different curvature over different ranges, with a penalty controlling unnecessary wiggles and natural linear tails outside its boundary knots. The displayed smooth supports this greater flexibility, but it is not by itself a formal rejection of every quadratic model. Select complexity and compare predictive performance using cross-validation, then reassess residuals and the Pearson dispersion estimator.

The proposed generalized additive model retains the binary response and logit link, but replaces the linear age effect by a smooth function:A penalized cubic regression spline represents $f$; frailty remains a parametric coefficient. The plotted age smooth is centered, so its vertical values represent an age contribution to log odds, not probabilities or the entire fitted predictor.

The plot gives no evidence that the nonlinear model improves the mean structure. The fitted curve is essentially a straight descending line, and the label s(age,1) reports effective degrees of freedom close to one, consistent with a linear effect. The widening bands near the ends reflect sparse data and increasing uncertainty, not detected curvature. A linear-age logistic regression is therefore adequate on this evidence and simpler to interpret. A formal comparative claim would need a suitable test or predictive/model-selection criterion, but the figure supplies no reason to retain extra nonlinear flexibility.

Both fits use the Gaussian generalized additive modelwhere $e_i$ is experience and $d_i$ education. Centering makes the additive decomposition identifiable. No experience-education interaction term is included.

The first fit uses local linear regression for each term. span=1 includes all observations in each nearest-neighbour neighbourhood, but distance weights and a moving target still vary: it does not make the fit exactly a global straight line. The second uses cubic smoothing splines, with separate tuning values spar=0.8 and spar=1.2. Increasing spar increases the curvature penalty and smooths more; spar is a monotone tuning parametrization, not the numerical $\lambda$ in part (b). The relation depends on the predictor scaling and design.

The plotted experience effect is increasing in both fits. The local fit gives a broadly smooth, nearly linear rise; the spline fit follows more of the steep low-experience rise and subsequent flattening, especially near the lower boundary. Education has a weaker increasing effect. The local estimate shows some curvature, while the spline fit at spar=1.2 is closer to a straight line. The clearest difference is the amount of smoothing and the experience curvature, rather than opposite effect directions. These are centered partial-effect plots with partial residuals, not scatterplots of wage against each predictor alone.

A smaller local span would allow the first fit to capture more experience curvature. A moderately smaller education spar could reveal nonlinear detail suppressed in the second fit; its small but significant nonparametric component in part (e) supports checking that possibility. Do not tune solely to follow every point, especially at sparse boundaries. Compare candidate spans or separate spline penalties using cross-validation or generalized cross-validation; increase smoothing if flexible fits introduce unsupported wiggles. See the documented monotone spar–penalty convention at stat.ethz.ch/R-manual/R-patched/library/stats/html/smooth.spline.html.

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).

Because no family is specified, the command fits a Gaussian identity-link generalized additive model:The $s_j$ are centered penalized regression splines with cubic regression bases; centering separates them from the intercept. The age curve in the original figure falls to a minimum near age 25, then rises markedly, particularly from about 40 onwards. Its effective degrees of freedom are $4.05$, reflecting curvature. The white-cell curve has effective degrees of freedom one and is nearly flat with a slight positive slope; the pointwise uncertainty bands are consistent with no substantial effect. These plots display centered partial mean contributions, not numbers of prescriptions by themselves.

A simpler mean model keeps the age regression spline and replaces the white-cell smooth by a linear term. A separate issue is that the response is a nonnegative count: a Poisson regression with logarithmic link function respects that support and positivity of the mean. A reasonable candidate isIf diagnostics show overdispersion, one can fitusing a negative binomial regression. Alternatively, if a Gaussian approximation is adequate, the minimal simplification is gam(npres ~ s(age, bs="cr") + wbc). These are candidates to compare by diagnostics and validation, rather than guaranteed improvements from the partial plots alone. Retain the curved age effect; consider a linear or removable white-cell effect, and assess an appropriate count family. Refit before interpreting the old curves on a new link scale.