Table 2.

Possible Phase I-II/III Trial Outcomes, O. $x_S^{\mathit{opt}}$ is the truly optimal dose in terms of μ_A(x) and Δ is the desired improvement over μ_C. Column 2 gives the two trial decisions, the first row for selecting ${\widehat{x}}_s^{\mathit{opt}}$ and the second row for determining superiority, inferiority, or futility, with $A\left({\widehat{x}}_S^{\mathit{opt}}\right)\succ C$ indicating that $A\left({\widehat{x}}_S^{\mathit{opt}}\right)$ is declared superior to C, and $C\ge |A\left({\widehat{x}}_S^{\mathit{opt}}\right)$ indicating that the trial is stopped due to either superiority of C or futility.

O	Decision	Truth	Comments
1	$\begin{array}{l}{\widehat{x}}_S^{\text{opt}}=x_S^{\text{opt}}\\ A\left({\widehat{x}}_S^{\text{opt}}\right)\succ C\end{array}$	${\mu }_{A\left(x_S^{\text{opt}}\right)}>{\mu }_C+\mathrm{\Delta }$	This is the generalized power event at the optimal dose $x_S^{\text{opt}}$. The design correctly selects $x_S^{\text{opt}}$ as optimal and declares $A\left(x_S^{\text{opt}}\right)$ superior to C.
2	$\begin{array}{l}{\widehat{x}}_S^{\text{opt}}\ne x_S^{\text{opt}}\\ A\left({\widehat{x}}_S^{\text{opt}}\right)\succ C\end{array}$	$\begin{array}{l}{\mu }_{A\left({\widehat{x}}_S^{\mathit{opt}}\right)}>{\mu }_C+\mathrm{\Delta }\\ {\mu }_{A\left(x_S^{\text{opt}}\right)}>{\mu }_{A\left({\widehat{x}}_S^{\text{opt}}\right)}\end{array}$	This is a generalized power event in a case where the design correctly concludes $A\left({\widehat{x}}_S^{\text{opt}}\right)$ is superior to C but ${\widehat{x}}_S^{\text{opt}}$ is suboptimal, so it could have improved survival more had if it chosen the truly optimal dose $x_S^{\text{opt}}$.
3	$\begin{array}{l}{\widehat{x}}_S^{\text{opt}}=\text{any}{x}_j\\ C\ge A\left({\widehat{x}}_S^{\text{opt}}\right)\end{array}$	${\mu }_{A\left(x_S^{\text{opt}}\right)}\le {\mu }_C$	This is a correct conclusion, but the phase I-II/III design will require an increased sample size compared to the phase I-II → III design due to correctly switching to $x_S^{\text{opt}}$.
4	$\begin{array}{l}{\widehat{x}}_S^{\text{opt}}=\text{any}{x}_j\\ A\left({\widehat{x}}_S^{\text{opt}}\right)\succ C\end{array}$	${\mu }_{A\left(x_S^{\text{opt}}\right)}\le {\mu }_C$	This is a false positive conclusion. While the design may pick the best dose of A, it incorrectly concludes that A at that dose is superior to C.
5	$\begin{array}{l}{\widehat{x}}_S^{\text{opt}}\ne x_S^{\text{opt}}\\ A\left({\widehat{x}}_S^{\text{opt}}\right)\succ C\end{array}$	$\begin{array}{l}{\mu }_{A\left({\widehat{x}}_S^{\text{opt}}\right)}\le {\mu }_C\\ {\mu }_{A\left(x_S^{\text{opt}}\right)}⩾{\mu }_C+\mathrm{\Delta }\end{array}$	This is a disastrous false negative conclusion. The design chooses a suboptimal dose based on μS and incorrectly concludes $A\left({\widehat{x}}_S^{\text{opt}}\right)$ is inferior to C; instead of correctly selecting $x_S^{\text{opt}}$ and declaring $A\left(x_S^{\text{opt}}\right)$ superior to C.