Thus P − 1 = Q , and This will be strictly positive provided But since t ≤ D . Thus, we have the required result. The R package groupSeqCrossover allows the determination, and exploration of, group sequential power family crossover trial designs. The function gsco is used to determine the design, taking inputs for the value of L , α , β , δ , σ 2 $_{e}$, Δ and sequence type ( "latin" or "williams" ). The function plot can then be used through S3 methods to plot power curves, the expected sample size, and the expected number of observations, for varying true treatment effects. Moreover, the function simulategsco can be used to simulate group sequential crossover trials in order to assess their operating characteristics. This is especially useful in the case of small sample size designs. The code below for example identifies the discussed design for Δ = 0 and then plots the expected sample size curve 4.1. R APPENDIX D: PROGRAMMES det$^{}$     1 ln    σ 2 b + 2 D − 1 D σ 2 e M T M N         = ( 1 ln $_{)}$2 t − 1 det$^{}$      σ 2 b + 2 D − 1 D σ 2 e M T M N      $_{}$, = ( 1 ln $_{)}$2 t − 1 det( N )det$^{{(}$ σ 2 b + 2 D − 1 D σ 2 e$^{)}$ − M T N − 1 M$^{}}$ , = ( 1 ln $_{)}$2 t − 1 σ 4( t − 1) e det( P ) 2 × det$^{}$         ( σ 2 b + 2 D − 1 D σ 2 e$^{)}$ − σ 2 e       − 1 . . − 1$^{}$      T    Q 0 0 Q$^{}$         − 1 . . − 1$^{}$               . det$^{}$       ( σ 2 b + 2 D − 1 D σ 2 e ) − σ 2 e      − 1 . − 1$^{}$    $^{T}$    Q 0 0 Q$^{}$        − 1 . − 1$^{}$            > 0 . det$^{}$         ( σ 2 b + 2 D − 1 D σ 2 e$^{)}$ − σ 2 e       − 1 . . − 1$^{}$      T    Q 0 0 Q$^{}$         − 1 . . − 1$^{}$               = det$^{}$         ( σ 2 b + 2 D − 1 D σ 2 e$^{)}$ − σ 2 e       − 1 /t . . − 1 /t$^{}$      T       − 1 . . − 1$^{}$               , = det$^{{(}$ σ 2 b + 2 D − 1 D σ 2 e$^{)}$ − σ 2 e $^{(}$2 t − 2 t )} , = σ 2 b + σ 2 e ( 2 t − 1 D$^{)}$ > 0 , 33 # Identify the design Delta.0 <- gsco(Delta = 0)