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be preferential to have different period effects in each stage. 

One simple extension would be to non-inferiority tests, from our present superiority testing framework. Non-inferiority tests, seeking to determine if new treatments are not clinically worse than an established control, would have hypotheses shifted by some factor from the ones presented here. Theorem 2.2 could easily be altered to accommodate this, and then popular methods for boundary determination in this setting applied. 

Here, we have worked under an idealised scenario, assuming the within patient variance to be known prior to trial commencement. Though this is a common assumption in group sequential theory, it does bring limitations, since often a good estimate for the key variance parameter cannot be provided at the design stage. In this instance group sequential t-tests would almost certainly be required. Furthermore, simulation is required to quantify error rates accurately in the case of small sample sizes. To explore this scenario we analysed the true familywise error rate under the global null hypothesis of a particular design motivated again by the TOMADO trial, but with L = 2 and n = 12. We found that provided restricted error maximum likelihood was utilised, there was very little inflation in the familywise error rate over the nominal level α . Details of this are provided in Appendix B. 

Moreover, we have only explored designing group sequential crossover trials. It is well known that if a final analysis is performed on data acquired in a sequential trial, not taking in to account the sequential nature, then biased treatment effects will be acquired. Extending established methodology for parameter estimation to our scenario will thus be important. 

Finally, we have implicitly assumed that there will be no patient drop out, and have not discussed the issue of patient recruitment rates. Though these are problems for all adaptive designs it is important to give them note. Owing to our need for one stages data to be analysed before the commencement of the following stage, it is likely the length of a trial using our approach would be longer for certain recruitment rates. It could be that recruitment is paused at interim, or that patients are continually recruited under the old scheme until results are available, which would lead to overrun and an increase in the expected number of observations and sample size. Thus this would be an important factor to consider when choosing an appropriate design for a trial. 

Nevertheless, for future crossover trials, consideration should be given to a group sequential approach. This may assist substantially in the efficient prioritisation of efficacious treatments. 

## APPENDIX A: FURTHER TECHNICAL DETAILS 

As discussed in Section 2, part (1) of Theorem 2.1 implies that 

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