content_image/295017.jpg content_image/295025.jpg content_image/295026.jpg
<img src='content_image/295017.jpg'>

was set to σ 2 b = 10 . 12; the value ascertained in the final analysis of the TOMADO trial data. Follow- ing this, our linear mixed model was fitted on all accumulated data (with either maximum likelihood or restricted error maximum likelihood estimation according to the particular analysis procedure being considered) and Z$_{dl}$ = ˆ τ$_{dl}$ ˆ I 1 / 2 dl determined for d = 1 , 2 , 3, where ˆ I 1 / 2 dl is the observed Fisher information for ˆ τ$_{dl}$ . Then, each Z$_{dl}$ was compared to e$_{l}$ and f$_{l}$ and our stopping rules applied (with e$_{l}$ and f$_{l}$ ad- justed using quantile substitution if the analysis procedure under consideration so dictated). If for some d = 1 , 2 , 3, f$_{l}$ ≤ Z$_{dl}$ < e$_{l}$ , the trial proceeded to the following stage and the process was repeated. In each instance, simulations in which H$_{0}$$_{d}$ was rejected for some d = 1 , 2 , 3 were recorded in order to ascertain true rejection rates. 

The performance of these procedures is displayed in Table 2. We observe that when maximum likelihood estimation is utilised and the boundaries are not adjusted using the procedure of Whitehead et al. (2009), there is substantial inflation in the familywise error rate under the global null hypothesis to 0.077. However, when restricted error maximum likelihood estimation is used, there is only negligible inflation if adjustment of the boundaries is employed. A program to perform this analysis is available. Its use is detailed in Appendix D. 

## APPENDIX C: TECHNICAL PROOFS 

Lemma 4.1. Element pq of Σ − 1 r is given by 

<img src='content_image/295025.jpg'>

Proof. We demonstrate this by verifying Σ$_{rps}$Σ − 1 rsq = δ$_{pq}$ . From the chosen covariance for a patient’s responses, we have that element pq of Σ$_{r}$ is given by 

<img src='content_image/295026.jpg'>

where Z$_{rpq}$ , ( p = 1; q = 1 , . . . , r ), is the pq th element of Z$_{r}$ ; the random effects design matrix for a single 

