designs could bring in a crossover setting. 2. METHODS The trial is assumed to have D ≥ 2 treatments initially, indexed d = 0 , . . . , D − 1. Treatments d = 1 , . . . , D − 1 are experimental, to be compared to the control d = 0. A maximum of L stages are planned for the trial. At each stage patients are allocated to each of a set of treatment sequences, which specify an order in which a patient receives treatments. The sequences used at each stage are determined by the number of treatments remaining in the trial at that stage. Without loss of generality, we will assume that if a treatment, or treatments, are dropped, it is treatment D − 1 dropped first, then D − 2, and so on, since treatments can always be re-labelled at each interim analysis. Then, we denote by S$_{r}$ = { s$_{ri}$ : i = 1 , . . . , | S$_{r}$ |} , r = 2 , . . . , D , the set of sequences for patient treatment allocation when r treatments remain in the trial, with each S$_{r}$ written in the form assuming it is exactly treatments d = 0 , . . . , r − 1 that remain. We further constrain each S$_{r}$ to contain only complete block sequences that are balanced for period. Specifically, complete block allocation requires all sequences to contain each treatment remaining in the trial exactly once, and period balance requires an equal number of patients to receive each treatment remaining in the trial in each period. These constraints allow the use of the popular Latin and Williams squares (Jones and Kenward, 2014). A fixed group size n is used for each stage of the trial, and is chosen such that at every stage each sequence is used an equal number of times. Thus n must be divisible by the lowest common multiple of | S$_{2}$ | , . . . , | S$_{D}$ | . Designing the trial in this manner ensures each treatment is considered equally. Outcome data is assumed to be normally distributed, and a linear mixed model is used for analysis, given by or where • Y is the vector of responses, containing the values of the y$_{ijkl}$ ; the response for individual i , in period j , on sequence k , in stage l , • β is the vector of fixed effects, of length 2 D − 1, consisting of – µ$_{0}$ the mean response on treatment 0 in period 1, an intercept term, 3 y$_{ijkl}$ = µ$_{0}$ + π$_{j}$ + τ$_{d}$$_{[}$$_{j,k,l}$$_{]}$ + s$_{ikl}$ + ϵ$_{ijkl}$, Y = Xβ + Zb + ϵ , 2.1. Notation, Hypotheses and Analysis