= Solution
The correct denominator is the number of live deliveries $B_i=S_i+T_i+H_i$, not treatment cycles and not babies. Define $M_i=T_i+H_i$ and use $M_i\mid B_i\sim\operatorname{Bin}(B_i,q_i)$; this is the distinction in <denominators in grouped birth-outcome models>. The observed conditional multiple-delivery proportion is $661/2373=0.2786$ initially and $1147/4773=0.2403$ finally.
A constant logit has residual <deviance> $87.48$ on 13 degrees of freedom. A linear-year fit gives $\widehat\eta=-1.02533-0.020879u$, deviance $16.46$ on 12 degrees of freedom ($p=0.171$), and annual odds ratio $0.97934$. A quadratic term improves the deviance by $5.814$ on one degree of freedom ($p=0.0159$), whereas a further cubic improvement is only $1.390$ ($p=0.238$). A supported parsimonious binomial model is
$$
\boxed{\operatorname{logit}(\widehat q_i)=-0.999606-0.020027u_i-0.0016823u_i^2.}
$$
Its residual deviance is $10.651$ on 11 degrees of freedom ($p=0.473$), and Pearson statistic is $10.647$, with no detected excess dispersion. The binomial standard errors are $0.014445,0.002515,0.000699$; approximate 95% <Wald confidence intervals> are $[-1.02792,-0.97129]$, $[-0.02496,-0.01510]$, and $[-0.003052,-0.000313]$. The annual odds multiplier is $\exp\{\beta_1+\beta_2(2u+1)\}$, describing near-flat early behaviour and a stronger later reduction, rather than a uniform annual decrease.
As a secondary check, modelling the triplet-or-higher share alone among deliveries needs curvature: its quadratic logit has residual deviance $12.905$ on 11 degrees of freedom, compared with $92.834$ for a straight line. \b[Multiple delivery became less likely conditional on a live delivery, especially later in the series; higher-order multiples declined particularly strongly.] This conditional probability is separate from the probability of any successful live delivery per cycle.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-102-data-analysis.png]
{title=Guerry reconstruction and complete IVF analyses, with clearly labelled bus and village excerpts}
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