Similarly Now, as the base case consider having completed one stage of the trial, and proceeding to complete another with any number of treatments t = 2 , . . . , D remaining. Then, in this instance By the previous result of this theorem this is indeed invertible and has the desired property, and moreover by Lemma 4.4 det( A − 1 T$_{l}$$_{T$_{l}$}$) > 0. The proof is then complete. Lemma 4.4. Consider the matrix from part (1) of Theorem 4.3; the case L$_{l}$$_{1}$ = · · · = L$_{lD}$$_{−}$$_{1}$ = 0 , L$_{ld}$ = l Now consider restricting to the columns and rows corresponding to for some t = 2 , . . . , D . The determinant of this matrix, cov ( T$_{l}$ , T$_{l}$ | L$_{l}$$_{1}$ = · · · = L$_{lD}$$_{−}$$_{1}$ = 0 , L$_{lD}$ = l ) , is strictly positive for any t . Proof. We have, by part (1) of Theorem 4.3 31 cov( W ′ l $_{+1}$, W$_{l}$$_{+1}$ ) = A − 1 W ′ $_{$_{l}$W$_{l}$}${ I$_{|}$$_{W}$$_{l}$$_{|}$ − (1 + l ) − $^{1}$I$_{|}$$_{W}$$_{l}$$_{|}$ } , = A − 1 W ′ $_{l}$W$_{l}$ { l l + 1 I$_{|}$$_{W}$$_{l}$$_{|}$$^{}}$ , = l l + 1 A − 1 W ′ $_{$_{l}$W$_{l}$}$. Thus the covariance of the fixed effects at the ( l + 1)th analysis has the desired property. A = n | S$_{D}$ | | S$_{D}$ | ∑ i =1 X T $_{s$_{Di}$}$Σ − 1 D X$_{s}$$_{Di}$ .   ln | S$_{D}$ | | S$_{D}$ | ∑ i =1 X T $_{s$_{Di}$}$Σ − 1 D X$_{s}$$_{Di}$$^{}$  − 1 . T$_{l}$ = (ˆ µ$_{0}$$_{l}$, ˆ π$_{2}$$_{l}$, ˆ π$_{3}$$_{l}$, . . . , ˆ π$_{tl}$, ˆ τ$_{1}$$_{l}$, ˆ τ$_{2}$$_{l}$, . . . , ˆ τ$_{t}$$_{−}$$_{1}$$_{l}$ ) $^{T}$, cov( T$_{l}$ , T$_{l}$ | L$_{l}$$_{1}$ = · · · = L$_{lD}$$_{−}$$_{1}$ = 0 , L$_{lD}$ = l ) = 1 ln    σ 2 b + 2 D − 1 D σ 2 e M T M N   $_{}$,