content_image/326226.jpg content_image/326233.jpg content_image/326227.jpg content_image/326228.jpg content_image/326229.jpg content_image/326230.jpg content_image/326231.jpg content_image/326232.jpg
2. For the next part of the theorem, we re-order our vector of fixed effects such that 

and thus the ordering of the columns of the X$_{s}$$_{ri}$ is now 

<img src='content_image/326226.jpg'>

<img src='content_image/326233.jpg'>

We proceed by induction over the number of stages completed l , for general D . Now, we assume that at the l th interim analysis, the statement of the Theorem is true. Now, the covariance at this l th analysis is 

<img src='content_image/326227.jpg'>

It is this specifically we assume follows the required condition. Additionally, we assume that this covariance matrix can be computed, i.e. that A is invertible. We show that if we conduct another stage of the trial with t treatments remaining, 2 ≤ t ≤ q , then the new covariance matrix 

<img src='content_image/326228.jpg'>

has the required property for ˆ π$_{2}$$_{l}$, . . . , ˆ π$_{tl}$, ˆ τ$_{1}$$_{l}$, . . . , ˆ τ$_{t}$$_{−}$$_{1}$$_{l}$ as well as for ˆ π$_{2}$$_{l}$, . . . , ˆ π$_{tl}$, ˆ τ$_{1}$$_{l}$, . . . , ˆ τ$_{t}$$_{−}$$_{1}$$_{l}$ and ˆ π$_{t}$$_{+1}$$_{l}$, . . . , ˆ π$_{Dl}$ , ˆ τ$_{tl}$, . . . , ˆ τ$_{D}$$_{−}$$_{1}$$_{l}$ . Let 

Denote dim( T$_{l}$ ) = | T$_{l}$ | , and similarly for T ′ l and W$_{l}$ . 

By our assumptions, we can write 

<img src='content_image/326229.jpg'>

<img src='content_image/326230.jpg'>

<img src='content_image/326231.jpg'>

<img src='content_image/326232.jpg'>

where for example A − 1 T$_{l}$T$_{l}$ = cov( T$_{l}$ , T$_{l}$ | L$_{l}$$_{1}$, . . . , L$_{lD}$ ), and with A − 1 T$_{l}$T$_{l}$ and A − 1 T ′ $_{l}$T$_{l}$ = ( A − 1 T$_{l}$T ′ l ) T holding the required conditions for the fixed effects. Finally, as part of our inductive hypothesis we also assume 

