Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 207 / 3

Under the boundary null $p={\pi }_0=0.20$, rejection requires both $X_1>r_1$ and $X_1+X_2\ge r$. Independence of the two binomial distributions givesThe rejection event is increasing in $p$, so this boundary value is the supremum over the composite null $p\le {\pi }_0$.

Writing $v_H=p_H(1-p_H)$ and $v_L=p_L(1-p_L)$, Neyman allocation minimizes the variance $v_H/n_H+v_L/n_L$ at fixed total sample size and gives

Minimizing the expected failures $n_H(1-p_H)+n_L(1-p_L)$ at a fixed variance gives the ethically optimal ratioIt allocates more patients to the dose with the larger response probability.

Neyman allocation optimizes information and may allocate more patients to the arm whose Bernoulli variance is nearer its maximum at $1/2$. The ethical allocation instead favors the arm with higher response probability. They coincide only for special parameter values; in general ethical gain is purchased with some loss of precision relative to Neyman allocation.

Targeting the Neyman ratio asymptotically minimizes the standard error for a fixed total sample size, so it cannot be worse than a fixed 1:1 design when the target probabilities are known. Targeting the ethical ratio can increase the standard error because it optimizes failures rather than information, although it may still outperform 1:1 for some probabilities.

In response-adaptive randomization, the probabilities are estimated during the trial. Delayed outcomes, time trends, unstable early estimates, and random final arm sizes complicate logistics and inference; naive Wald standard errors may also ignore adaptation.

Since $X_j\sim \mathrm{Poisson}(n_j{\lambda }_j)$, the rate MLE has asymptotic variance ${\lambda }_j/n_j$. The delta method for the logarithm givesIndependence of the arms therefore yieldsUnder $H_0:{\lambda }_T={\lambda }_C=\lambda$, the asymptotic variance of the log rate ratio is

With equal allocation and true alternative rates ${\lambda }_T,{\lambda }_C$, the asymptotic variance isThe normal approximation requires $|\log R{R}_A|/\sqrt{V_A}\ge z_{1-\alpha /2}+z_{1-\beta }$. Hencerounded upward. Writing ${\lambda }_T=R{R}_A{\lambda }_C$ expresses it using a specified control rate and the target rate ratio.