Let
$$
v=p_1(1-p_1)+p_0(1-p_0),
\qquad I_j=\frac{m_j}{v},
$$
where $m_j$ is the cumulative sample size per arm. The <canonical joint distribution for group sequential test statistics> is
$$
\begin{pmatrix}Z_1\\Z_2\end{pmatrix}
\dot\sim N_2\!\left[
\begin{pmatrix}\delta\sqrt{I_1}\\\delta\sqrt{I_2}\end{pmatrix},
\begin{pmatrix}
1&\sqrt{I_1/I_2}\\
\sqrt{I_1/I_2}&1
\end{pmatrix}
\right].
$$
Thus $\operatorname{Corr}(Z_1,Z_2)=\sqrt{m_1/m_2}$. This correlation arises because the second statistic reuses all first-stage observations.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 2 / d / Solution

At information levels $I_1<\cdots<I_J$, standardized efficient test statistics have unit variances, means proportional to $\sqrt{I_j}$, and correlations $\operatorname{Corr}(Z_j,Z_k)=\sqrt{I_j/I_k}$ for $j\leq k$.

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