content_image/299792.jpg content_image/299793.jpg content_image/299794.jpg content_image/299795.jpg content_image/299796.jpg content_image/299798.jpg content_image/299797.jpg
that det( A − 1 T$_{l}$$_{T$_{l}$}$) > 0. 

With these definitions, our aim can then be stated to prove that 

<img src='content_image/299792.jpg'>

i.e. that the covariance between the fixed effects ˆ π$_{2}$$_{l}$, . . . , ˆ π$_{tl}$ and ˆ τ$_{1}$$_{l}$, . . . , ˆ τ$_{t}$$_{−}$$_{1}$$_{l}$ is l/ ( l + 1) that of its form at interim analysis l , and similarly that 

For brevity, from here we will write 

and similarly for T$_{l}$ , T ′ l and W$_{l}$ , for any l . 

<img src='content_image/299793.jpg'>

<img src='content_image/299794.jpg'>

We use the following identity, which requires only the invertibility of A to be valid (Henderson and Searle, 1981) 

Note that we can write 

<img src='content_image/299795.jpg'>

<img src='content_image/299796.jpg'>

since our general form for ∑ | S$_{t}$ | i $_{=1}$X T $_{s$_{ti}$}$Σ − 1 t X$_{s}$$_{ti}$ is only non-zero in a | T$_{l}$ | × | T$_{l}$ | block in the top left hand corner by Lemma 4.2. Therefore, provided A is invertible, we can always invert A + B to find the covariance matrix at the following interim analysis. Moreover, we have 

and 

<img src='content_image/299798.jpg'>

<img src='content_image/299797.jpg'>

