= Solution
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>:
$$
Y_i\mid a_i,e_i,m_i\sim\operatorname{Bin}(m_i,p_i),\qquad
\log\frac{p_i}{1-p_i}=\beta_0+\beta_a a_i+\beta_e e_i.
$$
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 specification
$$
\log\frac{p_i}{1-p_i}=\beta_0+f_a(a_i)+f_e(e_i),\qquad E(Y_i)=m_ip_i.
$$
The 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 is
$$
\boxed{\operatorname{Var}(Y_i)=\phi m_ip_i(1-p_i),\qquad\widehat\phi=3.3506.}
$$
Equivalently the variance of $Y_i/m_i$ is $\phi p_i(1-p_i)/m_i$. 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 $\phi\ne1$, 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.
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