ST 520 Statistical Principles of Clinical Trials - NCSU Statistics ...

CHAPTER 10 ST 520, A. TSIATIS and D. Zhang
information necessary to achieve a certain level of significance and power for a fixed sample
design and multiplying by an inflation factor. For designs with a maximum of K analyses after
equal increments of information, the inflation factor is a function of α (the significance level), β
(the type II error or one minus power), K, and Φ (the shape parameter of the boundary). We
denote this inflation factor by IF(α, K, Φ, β).
Let V denote the number of interim analyses conducted before a study is stopped. V is a discrete
integer-valued random variable that can take on values from 1, . . ., K. Specifically, for a K-look
group-sequential test with boundaries b1, . . .,bK, the event V = j (i.e. stopping after the j-th
interim analysis) corresponds to
(V = j) = {|T(t1)| < b1, . . ., |T(tj−1)| < bj−1, |T(tj)| ≥ bj}, j = 1, . . ., K.
The expected number of interim analyses for such a group-sequential test, assuming ∆ = ∆ ∗ is
given by
E∆∗(V ) =
K�
j=1
j × P∆∗(V = j).
Since each interim analysis is conducted after increments MI/K of information, this implies that
the average information before a study is stopped is given by
Since MI = I FS × IF(α, K, Φ, β), then
AI(α, K, Φ, β, ∆ ∗ ) = I FS
AI(∆ ∗ ) = MI
E∆∗(V ).
K
�� �
IF(α, K, Φ, β)
K
E∆∗(V )
Note: We use the notation AI(α, K, Φ, β, ∆ ∗ ) to emphasize the fact that the average information
depends on the level, power, maximum number of analyses, boundary shape, and alternative of
interest. For the most part we will consider the average information at the null hypothesis ∆ ∗ = 0
and the clinically important alternative ∆ ∗ = ∆A. However, other values of the parameter may
also be considered.
Using recursive numerical integration, the E∆∗(V ) can be computed for different sequential de-
signs at the null hypothesis, at the clinically important alternative ∆A, as well as other values
for the treatment difference. For instance, if we take K = 5, α = .05, power equal to 90%,
then under HA : ∆ = ∆A, the expected number of interim analyses for a Pocock design is
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