
Sample Size Determination for Correlated Outcomes Using GEE 93
and
V = lim
n→∞
V
n
= σ
2
η
0
0 η
1
0
0 η
0
σ
2
r
0 η
1
σ
2
r
η
1
0 η
2
0
0 η
1
σ
2
r
0 η
2
σ
2
r
,
where σ
2
r
= ¯r(1 − ¯r), µ
0
=
P
m
j=1
δ
j
, µ
k
= µ
−1
0
P
m
j=1
δ
j
t
k
j
for
k = 1, 2, η
0
=
P
m
j=1
P
m
j
0
=1
δ
jj
0
ρ
jj
0
, η
1
=
P
m
j=1
P
m
j
0
=1
δ
jj
0
ρ
jj
0
t
j
, η
2
=
P
m
j=1
P
m
j
0
=1
δ
jj
0
ρ
jj
0
t
j
t
j
0
. Defining σ
2
t
= µ
2
− µ
2
1
, it can be shown after a few
steps of algebra that
A
−1
=
1
µ
0
σ
2
r
σ
2
t
µ
2
σ
2
r
0 −µ
1
σ
2
r
0
0 µ
2
0 −µ
1
−µ
1
σ
2
r
0 σ
2
r
0
0 −µ
1
0 1
.
Thus we can obtain a closed-form expression of the (4,4)th component of
Σ = A
−1
V A
−1
,
σ
2
4
=
σ
2
s
2
t
µ
2
0
σ
2
r
σ
4
t
, (4.8)
where s
2
t
= η
2
− 2µ
1
η
1
+ µ
2
1
η
0
=
P
m
j=1
P
m
j
0
=1
δ
jj
0
ρ
jj
0
(t
j
− µ
1
)(t
j
0
− µ
1
). The
closed-form sample size formula is
n =
σ
2
s
2
t
(z
1−α/2
+ z
1−γ
)
2