by Λ and I$_{l}$ , and set I = ( I T $_{1}$, . . . , I T $_{L}$). Explicitly, we have for any d, d$_{1}$, d$_{2}$ ∈ { 1 , . . . , D − 1 } and l$_{1}$ ≤ l$_{2}$ , l, l$_{1}$, l$_{2}$ ∈ { 1 , . . . , L } . For small sample sizes, simulation is required to accurately determine a designs performance. Since crossover trials are routinely conducted with small sample sizes, we here explore the impact this has upon the familywise error rate under the global null hypothesis. We determined a design corresponding to the TOMADO example that would require only 12 patients in each of two stages: the smallest allowable maximum sample size for a group sequential crossover trial with D = 4 treatments initially, given our restrictions on n . Taking Δ = 0 as an example, a trial with n = 12 and L = 2 with f$_{1}$ = 0 . 768 , f$_{2}$ = 2 . 036 and e$_{1}$ = 2 . 879 to 3 decimal places, would using our multivariate normal calculations have a maximal familywise error rate of α = 0 . 05 under the global null hypothesis, and β = 0 . 2 for δ = 2 . 2 when σ 2 e = 6 . 51. Ten-thousand of these trials were simulated in order to ascertain the true probability of rejecting H$_{0}$$_{d}$ for some d = 1 , 2 , 3, when τ = (0 , 0 , 0) $^{T}$. For simplicity, π$_{j}$ was set to 0 for j = 2 , . . . , D , and µ$_{0}$ was set to 0. Incorporating non-zero period effects however would not be expected to greatly effect the results. Whitehead et al. (2009) proposed a quantile substitution procedure for adapting the boundaries of a sequential trial to be more suitable to the case of unknown variance. We additionally considered employing this procedure. Given there is no consensus on how to determine the degrees of freedom when analysing using linear mixed models, we took the degrees of freedom at any analysis to be the classical decomposition of degrees of freedom in balanced, multilevel ANOVA designs (Pinheiro and Bates, 2009). Moreover, we also assessed the performance of the sequential design when the linear mixed model was fitted through either maximum likelihood or restricted error maximum likelihood estimation. Therefore in total these simulations were performed for each of four possible analysis procedures: maximum likelihood or restricted error maximum likelihood estimation, with or without boundary adjustment through quantile substitution. Thus, for each simulated study, patient response data for each stage l was randomly generated ac- cording to the distribution implied by their allocated treatment sequence (assigned according to the rules of the trial design), using the function rmvnorm (Genz et al., 2016) in R. The between person variance 18 APPENDIX B: SMALL SAMPLE SIZE PERFORMANCE cov( Z$_{d}$$_{1}$$_{l}$$_{1}$, Z$_{d}$$_{2}$$_{l}$$_{2}$ ) = Λ$_{d}$$_{1}$$_{+(}$$_{D}$$_{−}$$_{1)(}$$_{l}$$_{1}$$_{−}$$_{1)}$$_{,d}$$_{2}$$_{+(}$$_{D}$$_{−}$$_{1)(}$$_{l}$$_{2}$$_{−}$$_{1)}$ , = 1 2$^{(}$ l$_{1}$ l$_{2}$$^{)}$$^{1}$$^{/}$$^{2}$ (1 + δ$_{d}$$_{1}$$_{d}$$_{2}$ ) , I$_{dl}$ = ln 2 σ 2 e ,