
128 Sample Size Calculations for Clustered and Longitudinal Outcomes
By Liang and Zeger [5], as n → ∞,
√
n
k
ˆa
k
− a
k
ˆ
β
k
− β
k
→ N(0, Σ
k
).
The variance matrix Σ
k
can be consistently estimated by
ˆ
Σ
k
=
ˆ
A
−1
(ˆa
k
,
ˆ
β
k
)
ˆ
V
k
ˆ
A
−1
(ˆa
k
,
ˆ
β
k
) where
ˆ
V
k
=
1
n
k
n
k
X
i=1
m
X
j=1
ˆ
kij
1
t
j
N
2
,
and ˆ
kij
= y
kij
− µ
kij
(ˆa
k
,
ˆ
β
k
).
To detect the difference in the rate of change between the two groups, we
reject the null hypothesis H
0
: β
1
= β
2
, in favor of H
1
: β
1
6= β
2
, if
ˆ
β
1
−
ˆ
β
2
p
ˆv
2
1
/n
1
+ ˆv
2
2
/n
2
> z
1−α/2
,
where ˆv
2
k
is the (2,2)th element of
ˆ
Σ
k
.
On the other hand, given the true difference d = β
2
− β
1
, to achieve a
power of 1 − γ with a type I error α, the required sample size is
n