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Then d∗ is universally optimal for the estimation of direct effects over Ωt,n,p . Proof. 7), equality holds if and only if the following orthogonality condition is satisfied Td pr⊥ (1np )([P U Fd ]) = 0. 2) Td pr⊥ (1np )Fd = 0. 3) and Now Td pr⊥ (1np )Td is completely symmetric and has maximal trace if and only if d is equireplicate, a condition which is obviously met by the GLS d∗ . 2). 3) is equivalent to Zd = (np)−1 r d r¯ d , March 9, 2009 12:18 World Scientific Book - 9in x 6in Optimality of Balanced and Strongly Balanced Designs ws-book9x6 45 which, for equireplicate designs, reduces to zdss = r¯ds /t, 1 ≤ s, s ≤ t.

The next Lemma follows from the definitions of the relevant designs given above. 1. Let d1 , d2 , d3 and d4 be crossover designs with t treatments, which are, respectively, uniform on periods, uniform, balanced uniform and strongly balanced uniform. 1) where µ1 , µ2 , λ1 and λ2 are positive integers. 1) we have after simplification, Cd2 11 Cd2 12 Cd2 22 Cd3 11 Cd3 12 Cd3 22 Cd4 11 Cd4 12 Cd4 22 = µ1 pHt = Zd2 − t−1 µ1 (p − 1)Jt = Cd2 21 , = µ1 (p − 1 − p−1 )Ht , = µ1 pHt = −λ1 Ht = µ1 (p − 1 − p−1 )Ht , = µ1 pHt , = 0tt , = µ1 (p − 1 − p−1 )Ht .

Then d∗ maximizes tr(Cd11 ) and tr(Cd22 ) over Ωt,n,p . Proof. We present the proof for Cd22 ; the proof for Cd11 is similar. 10), for any d ∈ Ωt,n,p , t p−1 tr(Cd22 ) = n(p − 1) − n−1 n t s=1 i=1 t t p−1 n ¯ 2dsj − n−1 s=1 j=1 2 r¯ds s=1 s=1 j=1 n = n(p − 1) − p−1 t n ¯ 2dsj + (np)−1 m2dsi − p−1 (mdsi − (p − 1)−1 r¯ds )2 s=1 i=1 t + {(p − 1)−1 − p−1 } 2 r¯ds , s=1 p−1 as r¯ds = i=1 mdsi . 1, for the uniform design d∗ , md∗ si = (n/t) for all s, i and r¯d∗ s = n(p−1)/t for all s, 1 ≤ s ≤ t, 1 ≤ i ≤ p.