Table 1.

Possible relationships between ${\widehat{x}}_{ET}^{\mathit{opt}}$, $x_{ET}^{\mathit{opt}}$, ${\widehat{x}}_S^{\mathit{opt}}$, and $x_S^{\mathit{opt}}$, including comments related to the phase I-II → phase III and phase I-II/III designs. ${\widehat{x}}_{ET}^{\mathit{opt}}\stackrel{?}{=}{x}_{ET}^{\mathit{opt}}$ refers to whether or not the optimal dose is selected at the end of phase I-II based on ϕ, and $x_{ET}^{\mathit{opt}}\stackrel{?}{=}{x}_S^{\mathit{opt}}$ refers to whether the optimal dose based on μ_S is the same as that based on ϕ

${\widehat{x}}_{ET}^{\text{opt}}\stackrel{?}{=}{x}_{ET}^{\text{opt}}$	$x_{ET}^{\text{opt}}\stackrel{?}{=}{x}_S^{\text{opt}}$	Comments
${\widehat{x}}_{ET}^{\text{opt}}=x_{ET}^{\text{opt}}$	$x_{ET}^{\text{opt}}=x_S^{\text{opt}}$	The optimal dose in terms of μS was selected in phase I-II, so it is not desirable to switch doses at stage 2. In this scenario, the phase I-II/III design cannot provide an improvement over phase I-II → III.
${\widehat{x}}_{ET}^{\text{opt}}=x_{ET}^{\text{opt}}$	$x_{ET}^{\text{opt}}\ne x_S^{\text{opt}}$	The dose selected in phase I-II is optimal in terms of ϕ but is not optimal in terms of μS. This illustrates the advantage of the phase I-II/III design over phase I-II → III design.
${\widehat{x}}_{ET}^{\text{opt}}\ne x_{ET}^{\text{opt}}$	$x_{ET}^{\text{opt}}=x_S^{\text{opt}}$	The dose selected in phase I-II is suboptimal based on ϕ, but the optimal doses in terms of ϕ and μS are identical. This scenario illustrates the advantage of the phase I-II/III design over phase I-II → III design.
${\widehat{x}}_{ET}^{\text{opt}}\ne x_{ET}^{\text{opt}}$	$x_{ET}^{\text{opt}}\ne x_S^{\text{opt}}$	The dose selected in phase I-II is suboptimal based on ϕ, but the optimal doses in terms of ϕ and μS are not identical. This scenario illustrates the advantage of the phase I-II/III design over phase I-II → III design.