Fig. 3

Optimization results. Maximal expected utility u^∗, corresponding optimal design parameters \( {\delta}^{\ast }=\left({d}_2^{\ast },{HR}_{go}^{\ast}\right) \), \( {\delta}^{\ast }=\left({d}_2^{\ast },{HR}_{go}^{\ast },{\lambda}^{\ast}\right) \) or \( {\delta}^{\ast }=\left({d}_2^{\ast },{HR}_{go}^{\ast },{\alpha}_{CI}^{\ast}\right) \), expected probability to go to phase III \( {p}_{go}^{\ast } \), expected probability of a successful program sP^∗, expected estimate used for sample size calculation \( {\varepsilon}_2^{\ast } \), expected number of events in phase III when going to phase III \( {d}_3^{\ast } \) and expected total number of events of program d^∗ in the optimal design, for c₂ = 0.75, c₃ = 1, c₀₂ = 100, c₀₃ = 150 in $ 10⁵, ξ₂ = ξ₃ = 0.7, 1 − β = 0.9, α = 0.025 (one sided), for program set-ups \( S\left({\hat{\theta}}_2^{s_1},{\hat{\theta}}_2^{s_2}\right) \), s₁, s₂ = λ, α_CI or u (that is \( S\left({\hat{\theta}}_2^u,{\hat{\theta}}_2^u\right) \): black circle; \( S\left({\hat{\theta}}_2^u,{\hat{\theta}}_2^{\lambda}\right) \), \( S\left({\hat{\theta}}_2^{\lambda },{\hat{\theta}}_2^{\lambda}\right) \): green cross; \( S\left({\hat{\theta}}_2^u,{\hat{\theta}}_2^{\alpha_{CI}}\right) \), \( S\left({\hat{\theta}}_2^{\alpha_{CI}},{\hat{\theta}}_2^{\alpha_{CI}}\right) \): violet triangle), benefit scenarios bs 1–7, and weights for the prior distribution w = 0.3, 0.6, 0.9, where the yellow line indicates \( \exp \left(-\mathrm{E}\left[{\hat{\theta}}_2\right]\right) \). Note that the symbols used to show the program characteristics of both multiplicatively and additively adjusted program set-ups, i.e., green crosses and violet triangles, appear as stars when plotted on top of each other