Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 207 / 3

The pooled Wald test statistic isIn asymptotic calculations one replaces $\bar{p}$ by its probability limit $\bar{\pi }=(n_1{\pi }_1+n_0{\pi }_0)/(n_1+n_0)$.

At the boundary null ${\pi }_1={\pi }_0=\pi$, $\bar{\pi }=\pi$. Independence of the binomial proportions givesasymptotically, so $T\Rightarrow N(0,1)$. For the composite null $\delta <0$, the mean moves to the nonrejection side, making the boundary the least favorable case.

Under $\delta ={\delta }_A$,andThe central limit theorem therefore gives an asymptotic normal distribution with this mean and variance.

Let $n$ be the sample size in each arm, ${\pi }_{1A}={\pi }_0+{\delta }_A$, and ${\bar{\pi }}_A=({\pi }_{1A}+{\pi }_0)/2$. Solving the normal-approximation power equation givesrounded upward. The total sample size is $2n$.

Let $\pi$ be the new-treatment response probability in the trial's target population and let the fixed historical standard-of-care rate be ${\pi }_0$. The one-sided hypotheses areThe boundary $\pi ={\pi }_0$ determines the type-I error because rejection probability increases with $\pi$.

Let $X_1\sim \mathrm{Binomial}(22,{\pi }_0)$ be interim responses and independently $X_2\sim \mathrm{Binomial}(32,{\pi }_0)$ be second-stage responses. The design rejects immediately when $X_1\ge e_1=10$, continues when $r_1<X_1<e_1$, and after continuation rejects when $X_1+X_2\ge r=20$. Hence at ${\pi }_0=0.3$,This is the type-I error of the specified two-stage clinical trial design.