1. INTRODUCTION The efficiency of crossover trials often makes them the best design for a clinical trial. Administering multiple treatments to patients reduces the standard error of the estimated treatment effects compared to a parallel trial design with an equal number of patients. Therefore, whilst restrictions to their use exist; such as a requirement for patients to begin each new treatment period in a comparable state to those completed, crossover trials are the design of choice in many settings (Jones and Kenward, 2014; Senn, 2002), resulting in them accounting for 22% of all published trials in December 2000 for example (Mills et al., 2009). In a parallel design setting, group sequential methods are frequently utilized to improve a clinical trials efficiency (Jennison and Turnbull, 2000). These designs incorporate interim analyses which allow for early rejection of null hypotheses; efficacy stopping, or early stopping for lack of benefit; futility stopping. This way, the expected sample size required can be reduced over the more classical single-stage approach. Moreover, multi-arm multi-stage designs, which allow multiple experimental treatments to share a control group, can increase efficiency even further (Parmar et al., 2014). Group sequential methods are not frequently used in crossover trial settings however, in particular ones with multiple experimental treatments. Hauck et al. (1997) investigated the performance of group sequential trials for average bioequivalence employing an AB/BA crossover design, whilst Jennison and Turnbull (2000) provided one possible analysis method for a group sequential AB/BA crossover with a normally distributed endpoint. No one, to the best of our knowledge, has explored group sequential theory for crossover trials with more than one experimental treatment being compared to a shared control. Thus, one possible explanation for the lack of group sequential crossover trials may be that there is not yet available a formal proof of how to strongly control the familywise error rate of such a trial with multiple experimental treatments; since such a proof is usually required for regulatory approval (Wason et al., 2014). In comparison to a proof for a parallel multi-arm multi-stage design (Magirr et al., 2012), proving strong control of the familywise error rate is complicated here due to difficulties associated with the covariance structure implied by mixed model analysis. As has been remarked, multiple testing corrections for mixed models are only presently available for certain specific circumstances (Bender and Lange, 2001). Extension to this setting is particularly significant though given the noted advantages of comparing multiple experimental treatments to a shared control, both in terms of trial management and sample size (Parmar et al., 2014). Potential exists, given a proof, for the efficiency of crossover trial designs to be improved. In this work, we begin by providing such a proof for a linear mixed model using period and treatment as fixed effects, and individuals as random effects. Following this, using the four treatment, four-period TOMADO trial (Quinnell et al., 2014) as an example, we explore and discuss the efficiency gains that group sequential 2