
202 Sample Size Calculations for Clustered and Longitudinal Outcomes
the maximum likelihood estimate. Specifically, for the gth arm,
ˆη
g
=
N
3
P
i=1
N
2
P
j=1
N
1
P
k=1
T
k
−
¯
T
Y
ijk
−
¯
Y
g
,
N
3
P
i=1
N
2
P
j=1
N
1
P
k=1
T
k
−
¯
T
2
=
N
3
P
i=1
N
2
P
j=1
N
1
P
k=1
T
k
−
¯
T
Y
ijk
−
¯
Y
g
,
N
3
N
2
N
1
V ar
p
(T )
, (6.18)
where: (1)
¯
Y
g
(g = 0, 1) is the overall mean of the outcome Y for the gth
arm; (2)
¯
T =
P
N
1
k=1
T
k
/N
1
is the “mean” time point; and (3) V ar
p
(T ) =
P
N
1
k=1
T
k
−
¯
T
2
/N
1
is the “population variance” of the time variable T . The
variance of
ˆ
δ
(f)
can be obtained as
V ar(
ˆ
δ
(f)
) = V ar (ˆη
1
− ˆη
0
) =
2(1 − ρ
1
)σ
2
N
3
N
2
N
1
V ar
p
(T )
. (6.19)
It is worth noting that the variance of
ˆ
δ depends only on the residual
variance