Proof. See Appendix C. Note that part (1) of the above theorem implies for d$_{1}$, d$_{2}$ ∈ { 1 , . . . , D − 1 } . This is the familiar result for complete block sequences that there is no dependence upon the between patient variance σ 2 b (Jones and Kenward, 2014). Theorem 2.2. A group sequential crossover trial of the type considered, with D ≥ 2 , testing the D − 1 hypotheses H$_{0}$$_{d}$ : τ$_{d}$ ≤ 0 , H$_{1}$$_{d}$ : τ$_{d}$ > 0 , attains a maximal value of its familywise error rate for τ$_{1}$ = · · · = τ$_{D}$$_{−}$$_{1}$ = 0 . Proof. Theorem 2.1 implies that elements of the covariance matrix cov(ˆ τ$_{l}$ , ˆ τ$_{l}$ ) that differ from the case where no treatments have been dropped are exactly those corresponding to unstandardized test statistics no longer of importance. Consequently, the values of I$_{l,}$$_{(}$$_{ω}$$_{,}$$_{ψ}$$_{)}$ and Λ$_{(}$$_{ω}$$_{,}$$_{ψ}$$_{)}$ that differ from the case ω = ( L, . . . , L ) T are only ever those corresponding to limits of integration given by ( −∞ , ∞ ) in our computation of pr( ω$_{R}$ = ω , ψ$_{R}$ = ψ | τ ). By the marginal distribution properties of the multivariate normal distribution, we therefore need only consider one matrix Λ$_{(}$$_{ω}$$_{,}$$_{ψ}$$_{)}$ , and one set of vectors I$_{l,}$$_{(}$$_{ω}$$_{,}$$_{ψ}$$_{)}$ ( l = 1 , . . . , L ); exactly those given by the case ω = ( L, . . . , L ) $^{T}$. Denote these by Λ and I$_{l}$ , and set I = ( I T $_{1}$, . . . , I T $_{L}$) $^{T}$. For more information on this, see Appendix A. We now have Now, consider without loss of generality the probability we reject H$_{01}$ , and denote by Ω and Ψ the sets of all possible ω and ψ respectively. By integrating over all possible values of ω$_{2}$, . . . , ω$_{D}$$_{−}$$_{1}$ and ψ$_{2}$, . . . , ψ$_{D}$$_{−}$$_{1}$ , we have that the probability we reject each H$_{01}$ does not depend on the values of τ$_{2}$, . . . , τ$_{D}$$_{−}$$_{1}$ , i.e. on the other treatments tested L$_{lD}$$_{−}$$_{1}$ = 0 . cov (ˆ τ$_{d}$$_{1}$$_{l}$, ˆ τ$_{d}$$_{2}$$_{l}$ | L$_{l}$$_{1}$ = · · · = L$_{lD}$$_{−}$$_{1}$ = 0 , L$_{lD}$ = l ) = σ 2 e ln (1 + δ$_{d}$$_{1}$$_{d}$$_{2}$ ) , 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 / $^{2}$, Λ } d x$_{L}$$_{(}$$_{D}$$_{−}$$_{1)}$ . . . d x$_{11}$. pr (Reject H$_{01}$ | τ ) = ∑ { ψ ∈ Ψ: ψ$_{1}$ =1 } ∑ ω ∈ Ω pr( ω$_{R}$ = ω , ψ$_{R}$ = ψ | τ ) , 8