Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 102 / 2 / b

Use grouped-binomial logistic regression, $S_i\sim \mathrm{Bin}(C_i,p_i)$ and $\mathrm{logit}(p_i)={\eta }_i$. 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 $X^T\mathrm{diag}\{C_i{p}_i(1-p_i)\}X$; these determine the estimates and working covariance matrix.

A straight-line logit fit is $\hat{\eta }=-1.95965+0.045384u$, 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 ($p=4.8\times 10^{-8}$), 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:The Pearson estimate is $\hat{\varphi }=27.182/10=2.718$. Using standard errors multiplied by $\sqrt{\hat{\varphi }}$ 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: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. 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.