
48 Sample Size Calculations for Clustered and Longitudinal Outcomes
The expected value of T
w
under the null hypothesis is given by
µ
0
(T
w
) =
n
X
i=1
w
i
E(
P
m
i
j=1
Y
ij
|H
0
)
m
i
= 2p
0
− 1.
Under the null hypothesis, the variance of T
w
can be estimated by
σ
0
(T
w
)
2
=
n
X
i=1
w
2
i
V ar(
P
m
i
j=1
Y
ij
|H
0
)
m
2
i
= 4p
0
(1 − p
0
)
n
X
i=1
w
2
i
{1 + (m
i
− 1)ˆρ}
m
i
,
where ˆρ can be estimated by the ANOVA method [53].
The standardized weighted sign test statistic is defined by
Z
w
=
P
n
i=1
w
i
(m
+
i
− m
−
i
)/m
i
− (2p
0
− 1)
p
4p
0
(1 − p
0
)
P
n
i=1
w
2
i
{1 + (m
i
− 1)ˆρ}/m
i
,
which is asymptotically normal with mean 0 and variance 1. Hence, we reject
H
0
if the absolute value of Z
w
is larger than z
1−α/2
, which is the 100(1−α/2)th
percen ...