Canonical joint distribution for group sequential test statistics

A common approximation for group sequential 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 $(Z_1,\dots ,Z_J)$ has a multivariate normal distribution withIndeed independent Brownian increments give $\mathrm{Cov}(B_{I_j},B_{I_k})=I_j$. The statistics share observations, so later and earlier tests are correlated. For two equally sized stages, the interim/final correlation is $1/\sqrt{2}$, 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.