Assignment 4 :
Question 1:
Consider a repeated measures design model:
Y
ij
=
μ
..
+
ρ
i
+
τ
j
+
ε
ij
,
where the subject eﬀects are
ρ
i
independent
N
(0
,σ
2
ρ
),
the treatment eﬀects satisfy
∑
r
i
=1
τ
j
= 0,
the random errors
ε
ij
are independent
N
(0
,σ
2
) and
ρ
i
,ε
ij
are independent,
for
i
= 1
,...,s
and
j
= 1
,...,r
.
(a) Give the covariance structure associated to this model. That is give the
diﬀerent values taken by
σ
{
Y
ij
,Y
i
0
j
0
}
.
(b) Using the covariances from part (a), show that
1.
σ
2
{
Y
.j
}
=
σ
2
ρ
+
σ
2
s
.
2. for
j
6
=
j
0
:
σ
{
Y
.j
,
Y
.j
0
}
=
σ
2
ρ
s
3. for
j
6
=
j
0
:
σ
2
{
Y
.j

Y
.j
0
}
=
2
σ
2
s
(c) Let
j
6
=
j
0
. Using the results from part (b), argue that we can use
s
2
{
Y
.j

Y
.j
0
}
=
2
MSE
s
as a point estimate for
σ
2
{
Y
.j

Y
.j
0
}
.
Solution:
(a)
σ
{
Y
ij
,Y
i
0
j
0
}
=
σ
{
μ
..
+
ρ
i
+
τ
j
+
ε
ij
,μ
..
+
ρ
i
0
+
τ
j
0
+
ε
i
0
j
0
}
=
σ
{
ρ
i
,ρ
i
0
}
+
σ
{
ε
ij
,ε
i
0
j
0
}
=
σ
2
ρ
+
σ
2
,
if
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 Spring '11
 G.Lamothe
 Math, Power, Equals sign, Repeated measures design, Yij, noncentrality parameter

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