
88 Sample Size Calculations for Clustered and Longitudinal Outcomes
denoted as χ
2
q
(λ
n
). Here Σ
0
is the limit of Σ
n
(
ˆ
β) as n → ∞. Under H
0
, we
have λ
n
= 0, i.e., the test statistics W has a central chi-square distribution,
denoted as χ
2
q
(0). Given the nominal type I error α, the critical value c
1−α
is the 100(1 − α)th percentile of the χ
2
q
(0) distribution. Together with the
nominal power, denoted by 1 − γ, the sample size can be obtained as the
solution to
Z
∞
c
1−α
dF (q, λ
n
) = 1 − γ,
where F (q, λ
n
) is the cumulative distribution function of χ
2
q
(λ
n
). In essence,
the Wald test statistics is constructed based on the approximate normal dis-
tribution of vector ...