of sleep apnoea-hypopnoea using a four-treatment four-period crossover design. The normally distributed secondary endpoint, Epworth Sleepiness Scale, hoped to observe negative test statistics. Therefore, we consider the decrease as the endpoint in order to retain the same hypothesis tests as before H$_{0}$$_{d}$ : τ$_{d}$ ≤ 0 , H$_{1}$$_{d}$ : τ$_{d}$ > 0 , d = 1 , 2 , 3. The trial planned to recruit 90 patients, and utilising restricted error maximum likelihood estimation, the final analysis calculated that σ 2 e = 6 . 51. Taking this variance as the truth, the trial had a familywise error rate α = 0 . 05 for τ = ( τ$_{1}$, τ$_{2}$, τ$_{3}$ ) T = (0 , 0 , 0) $^{T}$, and β = 0 . 2 for H$_{11}$ at τ$_{1}$ = 1 . 11. Many methods exist for determining boundaries for a one-sided group sequential trial with parallel treatment arms. Here, we consider analogues of the power family boundaries of Pampallona and Tsiatis (1994). For this, values for the desired type-I and type-II error rates, a clinically relevant difference δ , the maximum number of stages L , the within person variance σ 2 $_{e}$, and a shape parameter Δ must be specified. A 2-dimensional grid search is then used to find the exact required maximal sample size. From this a suitable value of n is identified by rounding up to the nearest integer such that n is as required divisible by | S$_{2}$ | , . . . , | S$_{D}$ | . Utilizing Williams squares for our designs, n was forced to be divisible by 12. Taking α = 0 . 05, β = 0 . 2, δ = 1 . 11, σ 2 e = 6 . 51, L = 3, and Δ = − 0 . 25, 0, 0.5, 0.5 as examples, group sequential crossover trial designs were determined and compared to the single-stage design used by TOMADO. All computations were done in R (R Core Team, 2016) using the package groupSeqCrossover, available from the corresponding author upon request. Matlab (The Mathworks Inc., 2016) code employ- ing symbolic algebra is also available to return the matrices given by several of the equations in the text. Use of both the R and Matlab code is detailed in Appendix D. A summary of the performance of the designs is provided in Table 1, and their computed boundaries are displayed in Figure 1. We can see that, as is the case for two-arm parallel trial designs, there is a trend that larger values of Δ result in larger maximum sample sizes and lower expected sample sizes due to their larger stopping regions. However, this is not the case for Δ = 0 . 25 because of the requirement to round to a suitable integer value of n . Plots of the probability of rejecting H$_{01}$ , and rejecting H$_{0}$$_{d}$ for some d = 1 , 2 , 3, are provided for a range of values of θ when τ = ( θ, θ, θ ) T in Figure 2. The power curves are similar for all the designs, with the only differences a result of rounding in the group sequential designs to achieve suitable values of n . As is to be expected for group sequential designs, the maximum sample size and maximum number of observations is larger than for the single-stage design. However, the group sequential designs have lower expected sample sizes under the global null hypothesis ( τ = 0 = (0 , 0 , 0) $^{T}$); up to a maximum of 23% for Δ = 0 . 5. Though, this comes at the expense of an increased expected sample size under the global alternative hypothesis ( τ = δ = ( δ, δ, δ ) $^{T}$). From Figure 3, the expected sample sizes of the group sequential designs can be seen to be far lower than the single-stage design for more extreme values of θ . A similar statement holds for the expected 11