rate to α for this design using the following sum of integrals Additionally, suppose that we wish to power this trial to reject a particular null hypothesis, without loss of generality H$_{01}$ , at some clinically relevant difference τ$_{1}$ = δ . The type-II error rate β for H$_{11}$ is then given by Moreover, denoting by N and O the total number of patients and observations required by the trial respectively, we can compute the expected sample size, E ( N | τ ), or expected number of observations, E ( O | τ ), for any τ , according to Here, N( ω , ψ ) and O( ω , ψ ) are functions that give the number of patients and observations respectively, required by a trial that progresses according to ( ω , ψ ). Specifically As an example of how to design a group sequential crossover trial with strong control of the familywise error rate, we will make use of the TOMADO crossover randomized controlled trial (Quinnell et al., 2014). This open-label trial compared three experimental treatments to a single control for the treatment where I { ω$_{d}$ ≥ l } = 1 if ω$_{d}$ ≥ l , and is 0 otherwise. 3. EXAMPLE: TOMADO α = ∑ { ψ ∈ Ψ:Σ$_{d}$ ψ$_{d}$> 0 } ∑ 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 (0 , L ( D − 1)) , Λ } d x$_{L}$$_{(}$$_{D}$$_{−}$$_{1)}$ . . . d x$_{11}$, β = 1 − 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}$$_{1}$ ) $^{T}$, r ( δ, L ) ◦ I 1 / 2 τ$_{1}$ , Λ$_{τ}$$_{1}$ } d x$_{L}$$_{1}$ . . . d x$_{11}$. E ( N | τ ) = ∑ ψ ∈ Ψ ∑ ω ∈ Ω pr( ω$_{R}$ = ω , ψ$_{R}$ = ψ | τ )N( ω , ψ ) , E ( O | τ ) = ∑ ψ ∈ Ψ ∑ ω ∈ Ω pr( ω$_{R}$ = ω , ψ$_{R}$ = ψ | τ )O( ω , ψ ) . N( ω , ψ ) = n max { d =1 ,...,D − 1 $_{}}$ω$_{d}$ , O( ω , ψ ) = n L ∑ l =1 ( D − 1 ∑ d =1 I { ω$_{d}$ ≥ l } + 1$^{)}$ , 10