= Canonical joint distribution for group sequential test statistics
A common approximation for group sequential <test statistic>[test statistics] represents the accumulating score by $S(I)=\theta I+B_I$, where $I$ is information and $B$ is standard <Brownian motion>. At fixed information levels $0<I_1<\cdots<I_J$, put $Z_j=S(I_j)/\sqrt{I_j}$. The <vector> $(Z_1,\ldots,Z_J)$ has a <multivariate normal distribution> with
$$
\mathbb E_\theta Z_j=\theta\sqrt{I_j},\quad \operatorname{Var}(Z_j)=1,\quad \operatorname{Corr}(Z_j,Z_k)=\sqrt{I_j/I_k}\quad(j\leq k).
$$
Indeed <independent random variables>[independent] <independent increments>[Brownian increments] give $\operatorname{Cov}(B_{I_j},B_{I_k})=I_j$. The statistics share observations, so later and earlier tests are correlated. For two equally sized stages, the interim/final correlation is $1/\sqrt2$, not zero. This joint law is used to calculate the <probability> of crossing any efficacy boundary in a <group sequential design>. It is exact for the Brownian score model; its use for a general trial statistic is a <normal approximation> requiring the appropriate information and independent-increment limit assumptions.
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