ST4241: Design and Analysis of Clinical Trials - The Department of ...
ST4241: Design and Analysis of Clinical Trials - The Department of ...
ST4241: Design and Analysis of Clinical Trials - The Department of ...
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Thus<br />
Var( ¯X·j − M j ) = 1 ∑<br />
Var(ɛ<br />
r 2 ij − 1 ∑<br />
ɛ<br />
k ij<br />
′)<br />
i∈S j j ′ ∈T i<br />
= 1 ∑<br />
[(1 − 1 r 2 k )2 + k − 1 ]σ<br />
k 2 e<br />
2<br />
i∈S j<br />
= k − 1<br />
rk σ2 e.<br />
Hence<br />
Var(a j ) = k − 1<br />
rkeff 2 σ2 e =<br />
σ2 e k − 1<br />
reff keff =<br />
σ2 e<br />
reff<br />
g − 1<br />
.<br />
g<br />
Similarly,<br />
Cov( ¯X·j − M j , ¯X·l − M l )<br />
= 1 ∑ ∑<br />
Cov(ɛ<br />
r 2 ij − 1 ∑<br />
ɛ<br />
k ij<br />
′, ɛ i ′ l − 1 ∑<br />
ɛ ′<br />
k i j<br />
′)<br />
i∈S j i ′ ∈S l j ′ ∈T i j ′ ∈T ′ i<br />
= 1 r λCov(ɛ 2 ij − 1 ∑<br />
ɛ<br />
k ij<br />
′, ɛ il − 1 ∑<br />
ɛ<br />
k ij<br />
′)<br />
j ′ ∈T i j ′ ∈T i<br />
= λ r 2 (−σ2 e<br />
k ).<br />
Hence<br />
Cov(a j , a l ) =<br />
λ<br />
r 2 eff 2 (−σ2 e<br />
k ) = − 1 σe<br />
2<br />
reff g .<br />
(iii) Show that, for any contrast ∑ g<br />
j=1 c ja j ,<br />
Var(<br />
g∑<br />
c j a j ) =<br />
j=1<br />
σ2 e<br />
reff<br />
∑<br />
c 2 j.<br />
j<br />
3