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

Let $S_i$ be singleton deliveries among $C_i$ treatment cycles. Under a common-probability binomial distribution model, $S_i\sim \mathrm{Bin}(C_i,p)$ independently across years. The pooled maximum-likelihood estimator isThe Pearson chi-squared test of homogeneity compares singleton and non-singleton counts across all fourteen years, usingAll expected cells are large. One probability has been estimated, so the reference chi-squared distribution has 13 degrees of freedom. Numerically,The constant singleton-per-cycle probability is decisively rejected under the independent-binomial model. The observed proportion rises from $1712/18201=0.0941$ to $3626/23794=0.1524$. Dependence between repeated treatments or patient heterogeneity can invalidate literal binomial precision, but they should be addressed by an explicit variance model rather than ignored.

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.

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 $M_i\mid B_i\sim \mathrm{Bin}(B_i,q_i)$; 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 $\hat{\eta }=-1.02533-0.020879u$, 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 isIts 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. 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.