Statistical Inference After an Adaptive Group Sequential Design: A ...

596 b i o s t a t i s t i c s
Tremmel
istics regarding both expected sample size and
power. One possible solution can be found in
Bayesian decision theory (27, p. 204); the necessary
loss function could be constructed by assigning
monetary values for patients enrolled,
as well as for the type II error. This could be the
subject of future investigations.
Some of the advantages of classical GSD
hinge on strict adherence to predetermined decision
rules. In the typical DMB-driven clinical
trial, this may turn out to be an illusion, which
should lead to some of the same issues with statistical
inference as described above for the
aGSD. Indeed, this was the motivation behind
the development of the rCI (3, p. 189). In many
cases, the DMBs are fully unblinded to interim
results, which may inform their recommendation
of the timing of the next interim analysis,
as well as effect size assumptions and target
power (for an example, see Ref. 28). This may
impact not only the statistical inference, but
also the validity of the traditional GSD—a
much more severe problem. Granted, in the scenarios
investigated for results-driven timing of
interim analyses for traditional GSDs, the im-
pact on α was small (29). Nevertheless, it may
seem cleaner to use a design that fully “legalizes”
such “crimes” (and regulators ought to encourage
it)—in particular for open-label trials
such as our case study.
c o N c L U s i o N
There is a trade-off between flexibility in trial
conduct, and accuracy of statistical inference.
Generally, the flexible aGSD design will lead to
wider confidence intervals. In addition, the
openness of the trial causes theoretical difficulties
with some aspects of statistical inference
(in particular: bias) that are not all resolved.
There are cases when this trade-off may favor
the flexible approach—in particular, when the
trial is a randomized, open-label trial, and/or
when the size of a worthwhile effect depends on
future developments.
Acknowledgments—The author is indebted to Dr. C. K.
Chang for some early suggestions. The author also
owes thanks to two anonymous reviewers for their encouragement
and thorough questioning.
a P P E N D i x 1
P R o o F t H a t t H E P V a L U E b a s E D o N W R i g H t ( 1 2 ) i s U N i F o R M Ly
D i s t R i b U t E D F o R P o c o c k ’ s D E s i g N
Pocock’s procedure defines P crit = f(α) such that the probability (under H 0 ) that the smallest observed
P value is lower than P crit is α. For this case, Wright’s adjusted P value is α′ = f −1 (P min ), where P min is the
minimum P value actually observed, and α′ is the probability, under H 0 , to observe a minimum P value
as small as or smaller than P min .
α′ is a P value because it follows the uniform (0,1) distribution under H 0 : The function f −1 (.) is the
probability integral transformation of the null distribution of P min , that is, f −1 (.) says how likely it is, under
H 0 , to obtain a minimum P value that is even smaller than the observed minimum P value. Using
upper case for the random variables, prob(PMIN < P min ) = prob(A′ < α′) = α′, which defines the uniform
distribution.
a P P E N D i x 2
D E c o M P o s i t i o N o F t H E s t R a t i F i E D U N s q U a R E D M H s t a t i s t i c
The unsquared version Z MH of the Mantel-Haenszel statistic Q MH can be shown to be a weighted average
of the estimator ϑ ˆ = (πˆ 1 − πˆ 2 ) (30). For stage k, Z MH.k = (w Bk (πˆ B1k − πˆ B2k ) + w Ck (πˆ C1k − πˆ C2k ))/σ k , where
w Bk is the Mantel-Haenszel weight (n B1k * n B2k )/(n B1k + n B2k ), and the first index B designates the stratum
(B for Binet stage B, and C for Binet stage C). σ k is a function of the four margins of the 2 × 2 table: