
Sample Size Determination for Outcomes from Three-Level Trials 227
ϕ can be computed as
N
3
=
f
2
z
2
α,φ,p
N
2
N
1
(p
1
− p
0
)
2
(6.48)
from the power function
ϕ = Φ
(
|p
1
− p
0
|
p
N
1
N
2
N
3
/f
2
− Φ
−1
(1 − α/2)
p
2¯p (1 − ¯p)
p
p
0
(1 − p
0
) + p
1
(1 − p
1
)
)
From Equation (6.48), sample size N
2
for given N
1
, and N
3
can be imme-
diately obtained as
N
2
=
f
2
z
2
α,φ,p
N
3
N
1
(p
1
− p
0
)
2
since f
2
= 1 + (N
1
− 1)ρ
1
− N
1
ρ
2
in (6.10) is not a function of N
2
. In other
words, the power is invariant over the product of N
3
N
2
Equation (6.48) should
be solved to obtain N
1
.
6.7.2.2 Unbalanced Allocations
For an unbalanced design with Σ
j
X
j
= λN
(0)
2
for λ > 0 and i =
1, 2, . . . , N
3
, j = 1, 2, . . . (1 + λ)N
(0)
2
, and k = 1,