– π$_{j}$ the fixed period effect for period j , with the identifiability constraint π$_{1}$ = 0. Note that period is reset to 1 for each new stage of the trial. That is, the first period of stage 2 is treated as period 1 rather than period D + 1, and similarly for later stages. Thus, we have exactly D − 1 non-zero period effects given our restriction to complete block sequences. – τ$_{d}$$_{[}$$_{j,k,l}$$_{]}$ is the fixed direct treatment effect for an individual in period j , on sequence k , in stage l , with the identifiability constraint τ$_{0}$ = 0, • X is the matrix linking the fixed effects to the vector of responses, • b is the vector of random effects, consisting of the s$_{ikl}$ ; the random effect for individual i , on sequence k , in stage l , • Z is the matrix linking the random effects to the vector of responses, • ϵ is the vector of residuals, consisting of the ϵ$_{ijkl}$ ; the residual for individual i , in period j , on sequence k , in stage l . Additionally, denoting by σ 2 b > 0 and σ 2 e > 0 the between and within subject variances respectively, we take cov( s$_{i}$$_{1}$$_{k}$$_{1}$$_{l}$$_{1}$, s$_{i}$$_{2}$$_{k}$$_{2}$$_{l}$$_{2}$ ) = σ 2 $_{b}$δ$_{i}$$_{1}$$_{i}$$_{2}$ δ$_{k}$$_{1}$$_{k}$$_{2}$δ$_{l}$$_{1}$$_{l}$$_{2}$, cov( ϵ$_{i}$$_{1}$$_{j}$$_{1}$$_{k}$$_{1}$$_{l}$$_{1}$, ϵ$_{i}$$_{2}$$_{j}$$_{2}$$_{k}$$_{2}$$_{l}$$_{2}$ ) = σ 2 $_{e}$δ$_{i}$$_{1}$$_{i}$$_{2}$ δ$_{j}$$_{1}$$_{j}$$_{2}$δ$_{k}$$_{1}$$_{k}$$_{2}$δ$_{l}$$_{1}$$_{l}$$_{2}$, where δ$_{ij}$ is the Kronecker Delta function. Incorporation of fixed effects for period and treatment only, and our chosen covariance structure above, are the conventional choices for a crossover trial (Jones and Kenward, 2014). We test D − 1 hypotheses. Since we are interested in testing the efficacy of experimental treatments in comparison to a control, we consider the case of one-sided alternative hypotheses H$_{0}$$_{d}$ : τ$_{d}$ ≤ 0 , H$_{1}$$_{d}$ : τ$_{d}$ > 0 , for d = 1 , . . . , D − 1. At each interim analysis the above model is used to compute an estimate, ˆ β$_{l}$ ( l = 1 , . . . , L ), for β through the standard maximum likelihood estimator of a linear mixed model ˆ β$_{l}$ = ( X $^{T}$Σ − $^{1}$X ) − $^{1}$X $^{T}$Σ − $^{1}$Y ∼ M V N $^{{}$β , ( X $^{T}$Σ − $^{1}$X ) − 1 $^{}}$, where Σ = Z cov( b , b ) Z T + cov( ϵ , ϵ ) (Fitzmaurice et al., 2011). From this we acquire ˆ τ$_{l}$ = (ˆ τ$_{1}$$_{l}$, . . . , ˆ τ$_{D}$$_{−}$$_{1}$$_{l}$ ) $^{T}$, which consists of the maximum likelihood estimates for each τ$_{d}$ . Then, each ˆ τ$_{dl}$ is standardized to give D − 1 test statistics Z$_{dl}$ = ˆ τ$_{dl}$I 1 / 2 dl , d = 1 , . . . , D − 1, with I$_{dl}$ = { var(ˆ τ$_{dl}$ ) } − 1 the information level for treatment d at interim analysis l . Since ˆ τ$_{l}$ is estimated via a normal linear model we know that E ( Z$_{dl}$ ) = τ$_{d}$I 1 / 2 dl (Jennison and Turnbull, 2000). 4