
Sample Size Determination for Outcomes from Three-Level Trials 217
6.5.3.1 Balanced Allocations
For a balanced design, it is assumed that Σ
k
X
k
Z
k
= N
(1,1)
1
so that i =
1, 2, . . . , N
3
, j = 1, . . . , N
2
and k = 1, 2, . . . , 4N
(1,1)
1
. Specifically, N
(1,1)
1
rep-
resents the level 1 sample size assigned to the (X = 1, Z = 1) arm, or the
(1,1) cell, in a 2-by-2 factorial table of X and Z within the second level units,
N
3
is the level 3 sample size in total, and N
2
is the level 2 sample size per
third-level units. In a balanced design, the sample sizes of the other cells are
identical to N
(1,1)
1
, i.e., N
(1,1)
1
= N
(1,0)
1
= N
(0,1)
1
= N
(0,0)
1
. The following test
statistic ...