Given fixed futility boundaries, f$_{dl}$ , and efficacy bounds, e$_{dl}$ , the following stopping rules are used at each analysis l = 1 , ..., L , for each experimental treatment d = 1 , . . . , D − 1 satisfying f$_{dm}$ ≤ Z$_{dm}$ < e$_{dm}$ for m = 1 , . . . , l − 1 • if Z$_{dl}$ < f$_{dl}$ treatment d is dropped without rejecting H$_{0}$$_{d}$ , • if f$_{dl}$ ≤ Z$_{dl}$ < e$_{dl}$ the trial is continued with treatment d still present, • and if e$_{dl}$ ≤ Z$_{dl}$ treatment d is dropped and H$_{0}$$_{d}$ rejected. The control treatment, d = 0, remains present at every undertaken stage, and we only proceed to an additional stage if there is at least one experimental treatment remaining in the trial. It is convenient to take f$_{dl}$ = f$_{l}$ and e$_{dl}$ = e$_{l}$ for all d and l , as well as f$_{L}$ = e$_{L}$ in order to ensure the trial conforms to the desired maximum number of stages and so that a conclusion is made for each H$_{0}$$_{d}$ . Note that rejection of one treatment’s null hypothesis does not end the trial. Furthermore, with this formulation, once a treatment is dropped from the trial its standardised treatment effect is not tested at any future analyses. In what follows, we will make use of the vectors ω$_{R}$ = ( ω$_{R}$$_{1}$, . . . , ω$_{RD}$$_{−}$$_{1}$ ) T and ψ$_{R}$ = ( ψ$_{R}$$_{1}$, . . . , ψ$_{RD}$$_{−}$$_{1}$ ) $^{T}$. Here, ω$_{Rd}$ ∈ { 1 , . . . , L } is the analysis at which experimental treatment d was dropped from the trial. Moreover, ψ$_{Rd}$ ∈ { 0 , 1 } ; with ψ$_{Rd}$ = 1 if experimental treatment d was dropped for efficacy, and is 0 otherwise. Prior to a trials commencement ω$_{R}$ and ψ$_{R}$ are unknown random variables. However, the probability that the trial progresses according to some particular ω = ( ω$_{1}$, . . . , ω$_{D}$$_{−}$$_{1}$ ) T and ψ = ( ψ$_{1}$, . . . , ψ$_{D}$$_{−}$$_{1}$ ) $^{T}$, given a vector of true response rates τ = ( τ$_{1}$, . . . , τ$_{D}$$_{−}$$_{1}$ ) $^{T}$, can be computed us- ing multivariate normal integration. More specifically, given this particular ( ω , ψ ) pair the covariance between, and the information level of, the test statistics can be computed and the following integral evaluated (see Jennison and Turnbull (2000) or Wason (2015) for further details) pr( ω$_{R}$ = ω , ψ$_{R}$ = ψ | τ ) = ∫ u(1 ,ω$_{1}$,ψ$_{1}$ ) l(1 ,ω$_{1}$,ψ$_{1}$ ) . . . · · · ∫ u( L,ω$_{D}$$_{−}$$_{1}$,ψ$_{D}$$_{−}$$_{1}$ ) l( L,ω$_{D}$$_{−}$$_{1}$,ψ$_{D}$$_{−}$$_{1}$ ) φ $^{{}$x , r ( τ , L ) ◦ I 1 / 2 ( ω , ψ $_{)}$, Λ$_{(}$$_{ω}$$_{,}$$_{ψ}$$_{)}$ } d x$_{L}$$_{(}$$_{D}$$_{−}$$_{1)}$ . . . d x$_{11}$, where • x = ( x$_{11}$, . . . , x$_{1(}$$_{D}$$_{−}$$_{1)}$, . . . , x$_{L}$$_{1}$, . . . , x$_{L}$$_{(}$$_{D}$$_{−}$$_{1)}$ ) $^{T}$, • φ { x , µ , Λ } is the probability density function of a multivariate normal distribution with mean µ and covariance matrix Λ , evaluated at vector x , • r ( τ , L ) is the vector formed by repeating τ L times, 5