Let $X_1\sim\operatorname{Binomial}(22,\pi_0)$ be interim responses and independently $X_2\sim\operatorname{Binomial}(32,\pi_0)$ be second-stage responses. The design rejects immediately when $X_1\geq e_1=10$, continues when $r_1<X_1<e_1$, and after continuation rejects when $X_1+X_2\geq r=20$. Hence at $\pi_0=0.3$,
$$
\alpha=
\sum_{x=10}^{22}b(22,0.3,x)
+\sum_{x=8}^{9}b(22,0.3,x)
\sum_{y=20-x}^{32}b(32,0.3,y).
$$
This is the type-I error of the specified <two-stage clinical trial design>.

Solution

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

A two-stage clinical trial design permits early stopping after an interim analysis for futility or efficacy and otherwise continues to a prespecified final decision boundary.


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 Incoming links: Two-stage clinical trial design