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

CHAPTER 10 ST 520, A. TSIATIS and D. Zhang
Under the null hypothesis ∆ = 0, the distribution follows a standard normal, that is
T(t) ∆=0
∼ N(0, 1),
and is used a basis for a group-sequential test. For instance, if we considered a two-sided test,
we would reject the null hypothesis whenever
|T(t)| ≥ b(t),
where b(t) is some critical value or what we call a boundary value. If we only conducted one
analysis and wanted to construct a test at the α level of significance, then we would choose
b(t) = Zα/2. Since under the null hypothesis the distribution of T(t) is N(0, 1) then
PH0{|T(t)| ≥ Zα/2} = α.
If, however, the data were monitored at K different times, say, t1, . . .,tK, then we would want to
have the opportunity to reject the null hypothesis if the test statistic, computed at any of these
times, was sufficiently large. That is, we would want to reject H0 at the first time tj, j = 1, . . ., K
such that
|T(tj)| ≥ b(tj),
for some properly chosen set of boundary values b(t1), . . ., b(tK).
Note: In terms of probabilistic notation, using this strategy, rejecting the null hypothesis cor-
responds to the event
K�
{|T(tj)| ≥ b(tj)}.
j=1
Similarly, accepting the null hypothesis corresponds to the event
K�
{|T(tj)| < b(tj)}.
j=1
The crucial issue is how large should the treatment differences be during the interim analyses
before we reject H0; that is, how do we choose the boundary values b(t1), . . .,b(tK)? Moreover,
what are the consequences of such a strategy of sequential testing on the level and power of the
test and how does this affect sample size calculations?
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