
118 Sample Size Calculations for Clustered and Longitudinal Outcomes
define δ
j
= E(∆
kij
) to be the probability of a subject having an observation
at t
j
, and δ
jj
0
= E(∆
kij
∆
kij
0
) to be the joint probability of a subject having
observations at both t
j
and t
j
0
.
Theorem 2 As n → ∞, the (K −1) ×(K −1) variance matrix
ˆ
W converges
to
W =
s
¯m
(1/r
1
− 1) −1 −1 ··· −1
−1 (1/r
2
− 1) −1 ··· −1
−1 −1 (1/r
3
− 1) ··· −1
··· ··· ··· ··· ···
−1 −1 −1 ··· (1/r
K−1
− 1)
.
Furthermore, we have W
−1
=
¯m
2
s
[diag(r) + r
−1
K
rr
0
]. Here s =
σ
2
P
m
j=1
P
m
j
0
=1
δ
jj
0
ρ
jj
0
, ¯m =
P
m
j=1
δ
j
, and r = (r
1
, . . . , r
K−1
)
0
.
Proof. See the Appendix of Zhang and Ahn [44].
From Theorem 2 we have
η
0
W
−1
η =
¯m
2