Design Single-stage Δ = − 0 . 25 Δ = 0 Δ = 0 . 25 Δ = 0 . 5 n 90 36 36 48 48 pr (Reject H$_{01}$ | τ = 0 ) 0.02 0.02 0.02 0.02 0.02 pr (Reject H$_{01}$ | τ = δ ) 0.80 0.85 0.83 0.90 0.83 pr (Reject H$_{0}$$_{d}$ for some d | τ = 0 ) 0.05 0.05 0.05 0.05 0.05 pr (Reject H$_{0}$$_{d}$ for some d | τ = δ ) 0.95 0.97 0.97 0.98 0.97 E ( N | τ = 0 ) 90.0 76.8 70.0 82.6 69.6 E ( N | τ = δ ) 90.0 100.3 95.7 110.7 98.9 E ( O | τ = 0 ) 360.0 269.3 240.3 283.1 244.5 E ( O | τ = δ ) 360.0 367.2 341.8 380.4 327.7 max N 90 108 108 144 144 max O 360 432 432 576 576 Table 1: Example Design Performance. Summary of the performance of the single-stage and considered group sequential designs. The number of decimal places displayed in each row indicates the number to which rounding was performed 4. DISCUSSION There is a long history on group sequential clinical trials. Very few however utilise a crossover design. This may at least in part be due to no formal proof existing for how to strongly control the familywise error rate of such a trial. Here, we provided such a proof and then explored the performance of several sequential designs for the TOMADO trial. The expected sample size of the sequential designs was observed to be far lower than that of the single-stage design for a large range of values of the true response rate on all experimental treatments. Unfortunately, but unsurprisingly given the trial is not stopped unless all experimental treatments are dropped, there are regions in which the sequential designs are less efficient. Indeed, this region includes some values of θ between 0 and δ , which may be more realistic observed treatment effects. However, for some considered designs this region is very small and does not include values near 0, which is notable for ethical reasons. This issue could even be further alleviated by utilising optimal stopping boundaries, as has been proposed for parallel arm designs (Wason and Jaki, 2012; Wason et al., 2012). Importantly, several of the designs always performed better than the single-stage design in terms of the expected number of observations required, which could be a significant factor in the cost and length of a trial. Consequently, we can conclude that a group sequential approach to a crossover trial improves efficiency in some circumstances. Several possible extensions to our work present themselves. For example, we assumed that period was reset in each trial stage. This could reflect a scenario where it is believed being enrolled in the trial will alter a patient’s behaviour. However, in some cases, such as to deal with seasonal effects, it would 12 number of observations. However, in this instance for Δ = 0, 0.5, the performance of the group sequential designs is better than the single-stage design across all values of θ .