= Solution
Use <grouped-binomial logistic regression>, $S_i\sim\operatorname{Bin}(C_i,p_i)$ and $\operatorname{logit}(p_i)=\eta_i$. Center the year as $u_i=i-7.5$. Maximizing $\ell=\sum_i[S_i\eta_i-C_i\ln(1+e^{\eta_i})]$ gives score $X^T(S-Cp)=0$ and information $X^T\operatorname{diag}\{C_ip_i(1-p_i)\}X$; these determine the estimates and working <covariance matrix>.
A straight-line logit fit is $\widehat\eta=-1.95965+0.045384u$, with naive binomial standard errors $0.005382$ and $0.001352$. Its yearly odds multiplier is $e^{0.045384}=1.04643$, with naive 95% <Wald confidence interval> $[1.04366,1.04921]$. However, its residual <deviance> is $58.20$ on 12 degrees of freedom ($p=4.8\times10^{-8}$), so those small standard errors do not describe an adequate independent-binomial fit.
Comparing polynomial mean structures gives residual deviances $1191.73,58.20,55.70,27.08$ for degrees zero through three, with respective residual degrees of freedom $13,12,11,10$. A quadratic term alone adds little; the cubic structure captures the middle-period acceleration and later leveling. Even the cubic independent-binomial fit has $p=0.00253$ for its residual <deviance goodness-of-fit test>. A useful working model is therefore cubic <quasibinomial regression>, retaining the logistic mean and estimating <overdispersion> rather than claiming an exact binomial law:
$$
\boxed{\widehat\eta(u)=-1.953614+0.062138u-0.0002830u^2-0.0005812u^3.}
$$
The Pearson estimate is $\widehat\phi=27.182/10=2.718$. Using standard errors multiplied by $\sqrt{\widehat\phi}$ and a $t_{10}$ reference, the approximate 95% coefficient intervals are $[-1.98294,-1.92429]$, $[0.04982,0.07445]$, $[-0.001697,0.001131]$, and $[-0.000980,-0.000182]$. An approximate quasi-deviance comparison of cubic against linear gives $F_{2,10}=5.72$ and $p=0.022$, supporting the added shape while acknowledging that only fourteen annual groups inform dispersion.
In this preferred mean model the odds change is not constant:
$$
\boxed{\frac{\mathrm{odds}(i+1)}{\mathrm{odds}(i)}
=\exp\{\beta_1+\beta_2(2u+1)+\beta_3(3u^2+3u+1)\}.}
$$
The fitted annual change is larger around the middle than at the ends; the straight-line 4.6% increase is a useful overall summary, not an exact yearly effect in the cubic model. The fitted curve, uncertainty band and residual plot appear in the figure. \b[Singleton success increased overall, but a constant log-odds trend with unit binomial dispersion fits poorly.] Quasibinomial intervals remain working-model intervals: serial dependence, changing patient mix and treatment practices cannot be separated from annual aggregates alone.
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