as follows As a final step we must compute the sum across sequences, i.e. over the index i . We can see instantly that we have confirmed the elements proposed to be 0 in ∑ | S$_{r}$ | i =1 n X T $_{s$_{ri}$}$Σ − 1 r X$_{s}$$_{ri}$ / | S$_{r}$ | are indeed so, and we therefore need only concentrate on the non-zero terms suggested; A , B , C and E . However, other than Π T $_{jr}$Σ − 1 r T$_{ks}$$_{ri}$ , all the elements above have been identified as independent of sequence s$_{ri}$ . Therefore computing the sum over the s$_{ri}$ ∈ S$_{r}$ can be done easily, and gives the forms proposed for A , B and C in the statement of the Lemma immediately, on multiplying through by n/ | S$_{r}$ | . But, by our imposed constraint that sequences be balanced for period we can also sum over the Π T $_{jr}$Σ − 1 r T$_{ks}$$_{ri}$ since exactly | S$_{r}$ | /r patients receive each treatment at each time period. This confirms the form proposed for E , and the proof is complete. Theorem 4.3. (Theorem 2.1 from Section 2) Let β = ( µ$_{0}$, π$_{2}$, . . . , π$_{D}$, τ$_{1}$, . . . , τ$_{D}$$_{−}$$_{1}$ ) $^{T}$. Consider an analysis to be performed after some number of stages l . Then 22 1 T $_{r}$Σ − 1 r 1$_{r}$ = ∑ j,k Σ − 1 rjk = rσ 2 e σ 2 $_{e}$( σ 2 e + rσ 2 $_{b}$) , Π T $_{jr}$Σ − 1 r 1$_{r}$ = 1 T $_{r}$Σ − 1 r Π$_{jr}$ = ∑ k Σ − 1 rjk I { j ≤ r } = σ 2 e σ 2 $_{e}$( σ 2 e + rσ 2 $_{b}$) I { j ≤ r $_{}}$, T T $_{js$_{ri}$}$Σ − 1 r 1$_{r}$ = 1 T $_{r}$Σ − 1 r T$_{js}$$_{ri}$ = ∑ k Σ − 1 rt$_{j}$k I { j ≤ r − 1 } = σ 2 e σ 2 $_{e}$( σ 2 e + rσ 2 $_{b}$) I { j ≤ r − 1 $_{}}$, Π T $_{jr}$Σ − 1 r Π$_{kr}$ = Σ − 1 rjk I { j ≤ r } I { k ≤ r } =$^{{}$ ( σ 2 e + rσ 2 $_{b}$)$_{I}$$_{{}$$_{j}$$_{=}$$_{k}$$_{}}$ − σ 2 b σ 2 $_{e}$( σ 2 e + rσ 2 $_{b}$) } I { j ≤ r } I { k ≤ r $_{}}$, Π T $_{jr}$Σ − 1 r T$_{ks}$$_{ri}$ = Σ − 1 rjt$_{k}$ I { j ≤ r } I { k ≤ r − 1 } =$^{{}$ ( σ 2 e + rσ 2 $_{b}$)$_{I}$$_{{}$$_{j}$$_{=}$$_{t}$$_{k}$$_{}}$ − σ 2 b σ 2 $_{e}$( σ 2 e + rσ 2 $_{b}$) } I { j ≤ r } I { k ≤ r − 1 $_{}}$. T T $_{js$_{ri}$}$Σ − 1 r T$_{ks}$$_{ri}$ = Σ − 1 rt$_{j}$t$_{k}$ I { j ≤ r − 1 } I { k ≤ r − 1 } =$^{{}$ ( σ 2 e + rσ 2 $_{b}$)$_{I}$$_{{}$$_{j}$$_{=}$$_{k}$$_{}}$ − σ 2 b σ 2 $_{e}$( σ 2 e + rσ 2 $_{b}$) } I { j ≤ r − 1 } I { k ≤ r − 1 $_{}}$, | S$_{r}$ | ∑ i =1 n | S$_{r}$ | Π T $_{jr}$Σ − 1 r T$_{ks}$$_{ri}$ = n | S$_{r}$ |$^{(}$ | S$_{r}$ | r [ 1 σ 2 $_{e}$( σ 2 e + rσ 2 $_{b}$)$^{{}$ σ 2 e + ( r − 1) σ 2 b } ] + { 1 − | S$_{r}$ | r } { − σ 2 b σ 2 $_{e}$( σ 2 e + rσ 2 $_{b}$)$^{})}$ , = n | S$_{r}$ |$^{{}$ | S$_{r}$ | r σ 2 e σ 2 $_{e}$( σ 2 e + rσ 2 $_{b}$)$^{}}$ , = n 1 r σ 2 e σ 2 $_{e}$( σ 2 e + rσ 2 $_{b}$) ,