that det( A − 1 T$_{l}$$_{T$_{l}$}$) > 0. With these definitions, our aim can then be stated to prove that i.e. that the covariance between the fixed effects ˆ π$_{2}$$_{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 . We use the following identity, which requires only the invertibility of A to be valid (Henderson and Searle, 1981) Note that we can write since our general form for ∑ | S$_{t}$ | i $_{=1}$X T $_{s$_{ti}$}$Σ − 1 t X$_{s}$$_{ti}$ is only non-zero in a | T$_{l}$ | × | T$_{l}$ | block in the top left hand corner by Lemma 4.2. Therefore, provided A is invertible, we can always invert A + B to find the covariance matrix at the following interim analysis. Moreover, we have and 28 cov( W$_{l}$$_{+1}$ , W$_{l}$$_{+1}$ | L$_{l}$$_{+11}$, . . . , L$_{l}$$_{+1}$$_{D}$ ) = l l + 1 A − 1 W$_{l}$$_{W$_{l}$}$, cov( W ′ l $_{+1}$, W$_{l}$$_{+1}$ | L$_{l}$$_{+11}$, . . . , L$_{l}$$_{+1}$$_{D}$ ) = l l + 1 A − 1 W ′ $_{l}$W $_{l}$. cov( ˆ β$_{l}$ , ˆ β$_{l}$ | L$_{l}$$_{1}$, . . . , L$_{lD}$ ) = cov( ˆ β$_{l}$ , ˆ β$_{l}$ ) , ( A + UCV ) − 1 = A − 1 $^{{}$I − UCV A − $^{1}$( I + UCV A − $^{1}$) − 1 } . B =    B$_{T}$$_{l}$$_{+1}$$_{T}$$_{l}$$_{+1}$ 0 0 0$^{}$   =    I$_{|}$$_{T}$$_{l}$$_{+1}$$_{|}$ 0   $_{}$B$_{T}$$_{l}$$_{+1}$$_{T}$$_{l}$$_{+1}$ ( I$_{|}$$_{T}$$_{l}$$_{+1}$$_{|}$ 0$^{)}$ = UCV , BA − 1 =    B$_{T}$$_{l}$$_{+1}$$_{T}$$_{l}$$_{+1}$ 0 0 0$^{}$      A − 1 T$_{l}$T$_{l}$ A − 1 T$_{l}$T ′ l A − 1 T ′ $_{l}$T l A − 1 T ′ $_{l}$T ′ l    , =    B$_{T}$$_{l}$$_{+1}$$_{T}$$_{l}$$_{+1}$A − 1 T$_{l}$T$_{l}$ B$_{T}$$_{l}$$_{+1}$$_{T}$$_{l}$$_{+1}$A − 1 T$_{l}$T ′ l 0 0    , I$_{2}$$_{D}$$_{−}$$_{1}$ + BA − 1 =    B$_{T}$$_{l}$$_{+1}$$_{T}$$_{l}$$_{+1}$A − 1 T$_{l}$T$_{l}$ + I$_{|}$$_{T}$$_{l}$$_{|}$ B$_{T}$$_{l}$$_{+1}$$_{T}$$_{l}$$_{+1}$A − 1 T$_{l}$T ′ l 0 I$_{|}$$_{T}$ ′ $_{l}$|    ,