content_image/305221.jpg content_image/305223.jpg content_image/305224.jpg content_image/305225.jpg content_image/305226.jpg
Similarly 

<img src='content_image/305221.jpg'>

Thus the covariance of the fixed effects at the ( l + 1)th analysis has the desired property. 

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 

<img src='content_image/305223.jpg'>

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 

<img src='content_image/305224.jpg'>

<img src='content_image/305225.jpg'>

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 

<img src='content_image/305226.jpg'>

