
Sample Size Determination for Clustered Outcomes 33
has a standard normal distribution with mean 0 and variance 1 for large n.
Under the alternative hypothesis (H
1
: p = p
1
), ˆσ
2
m
converges to σ
2
m
, where
σ
2
m
= p
1
(1 − p
1
)
1
E[m + {1 + (m − 1)ˆρ}
−1
]
.
The required sample size against the alternative hypothesis H
1
: p = p
1
for a two–sided significance level of α and power of 1 − β is
n
m
=
p
1
(1 − p
1
)(z
1−α/2
+ z
1−β
)
2
(p
0
− p
1
)
2
1
E[m + {1 + (m − 1)ˆρ}
−1
]
.
The sample size (n
m
) under minimum variance estimate is always smaller
than or equal to n
u
and n
c
.
2.3.2.4 Example
We use the data of Hujoel et al. [23] as a pilot data to illustrate sample
size calculation for clustered ...