Recursive group-sequential boundary calibration

Inputs: The Axiom of Choice (The Axiom of Choice) and a canonical design from Canonical group-sequential alpha-spending design and a mode, either one-sided upper-boundary or symmetric two-sided.

Output: The look boundaries for the selected mode.

Set $t_0=0$, $R_0=\mathrm{\varnothing }$, and process $k=1,\dots ,K$ in order. Let $R_{k-1}$ be the union of the crossing events already defined at the earlier looks.

• If $A(t_k)=A(t_{k-1})$, set the boundary at look $k$ to $+\mathrm{\infty }$. Its crossing event is empty, so $R_k=R_{k-1}$.
• If $A(t_k)>A(t_{k-1})$ and the mode is one-sided, choose the real number $b_k$ satisfying $P_0(R_{k-1}\cup \{Z_k\ge b_k\})=A(t_k),$ and set $R_k=R_{k-1}\cup \{Z_k\ge b_k\}$.
• If $A(t_k)>A(t_{k-1})$ and the mode is symmetric two-sided, choose $c_k\ge 0$ satisfying $P_0(R_{k-1}\cup \{\mathrm{\mid }{Z}_k\mathrm{\mid }\ge c_k\})=A(t_k),$ and set $R_k=R_{k-1}\cup \{\mathrm{\mid }{Z}_k\mathrm{\mid }\ge c_k\}$.

The root-existence theorem Group-sequential boundaries exist uniquely and spend alpha exactly ↗ proves that every positive spending increment has a unique finite solution; the algorithm therefore makes no arbitrary boundary selection. The procedure specifies exact boundaries under the supplied joint null law. It does not prescribe a numerical integration package or tolerance.