where I$_{τ}$$_{1}$ and Λ$_{τ}$$_{1}$ are the restrictions of I and Λ to rows and columns corresponding to experimental treatment d = 1 respectively. This final form for pr (Reject H$_{01}$ | τ ) is identical to that it would be in the case D = 2. Therefore to ascertain the τ giving the maximal familywise error rate of a trial with D ≥ 2, it suffices to consider which τ$_{∗}$ ≤ 0 maximises the probability H$_{01}$ is rejected in a trial with D = 2 initial treatments. For then, τ = ( τ$_{∗}$, . . . , τ$_{∗}$ ) T using this τ$_{∗}$ , will provide the maximum probability of rejecting at least one true H$_{0}$$_{d}$ for some d , i.e. the maximum familywise error rate. To see this, consider the familywise error rate for τ = ( τ$_{∗}$, . . . , τ$_{∗}$ ) $^{T}$. If one changes some individual element τ$_{d}$$_{1}$ of this vector, this does not effect the probability that H$_{0}$$_{d}$$_{2}$ is rejected for d$_{2}$ = d$_{1}$ , and it can only decrease the probability that H$_{0}$$_{d}$$_{1}$ is incorrectly rejected. Thus overall, straying from this τ = ( τ$_{∗}$, . . . , τ$_{∗}$ ) T can only decrease the familywise error rate. Thus, now consider all possible realisations of the test statistics of a trial with D = 2, and their associated values of ( ω , ψ ) = ( ω$_{1}$, ψ$_{1}$ ). We have Z = ( Z$_{11}$, . . . , Z$_{1}$$_{L}$ ) T ∈ R $^{L}$, with Z$_{1}$$_{L}$ = · · · = Z$_{1}$$_{ω}$$_{1}$ if the trial was stopped at stage ω$_{1}$ . Now consider increasing the value of the test statistics by some η > 0. All instances before where H$_{01}$ was rejected will still exceed the efficacy bound of that stage, or earlier, and so H$_{01}$ will still be rejected. Therefore, the probability of rejecting H$_{01}$ is at least as large as before. Thus, increasing the value of τ$_{1}$ ≤ 0 causes a non-decreasing change in the value of the type-I error rate. Therefore, the probability of rejecting H$_{01}$ is maximized by τ$_{1}$ = 0; implying in turn that the maximal familywise error rate of a trial with D ≥ 2 is given by τ = ( τ$_{1}$, . . . , τ$_{D}$$_{−}$$_{1}$ ) T = (0 , . . . , 0) $^{T}$. A trial will now be fully specified given values for D , L , σ 2 e and n , as well as choices for the S$_{r}$ , and the futility and efficacy boundaries, f$_{1}$, . . . , f$_{L}$ and e$_{1}$, . . . , e$_{L}$ respectively. Given these, Λ and I can be computed using the results above. Then, by Theorem 2.2 we can strongly control the familywise error 2.3. Design Characteristics 9 = ∑ { ψ ∈ Ψ: ψ$_{1}$ =1 } ∑ ω ∈ Ω ∫ u(1 ,ω$_{1}$,ψ$_{1}$ ) l(1 ,ω$_{1}$,ψ$_{1}$ ) . . . · · ·$^{∫}$ u( L,ω$_{D}$$_{−}$$_{1}$,ψ$_{D}$$_{−}$$_{1}$ ) l( L,ω$_{D}$$_{−}$$_{1}$,ψ$_{D}$$_{−}$$_{1}$ ) φ $^{{}$x , r ( τ , L ) ◦ I 1 / $^{2}$, Λ } d x$_{L}$$_{(}$$_{D}$$_{−}$$_{1)}$ . . . d x$_{11}$, = L ∑ ω$_{1}$ =1 ∫ u(1 ,ω$_{1}$, 1) l(1 ,ω$_{1}$, 1) . . . · · ·$^{∫}$ u( L,ω$_{1}$, 1) l( L,ω$_{1}$, 1) φ $^{{}$( x$_{11}$, . . . , x$_{L}$$_{1}$ ) $^{T}$, r ( τ$_{1}$, L ) ◦ I 1 / 2 τ$_{1}$ , Λ$_{τ}$$_{1}$ } d x$_{L}$$_{1}$ . . . d x$_{11}$,