category is the conditional error probability method by Proschan and Hunsberger [23]. The conditional rejection probability is defined as
where and are the Stage 1 and Stage 2 sample sizes per group, respectively, is a standardized normal test statistic under the original design and is a standardized hypothesized effect size. When the above conditional probability is calculated under a non-zero (either
hypothesized effect size or observed), it becomes a conditional power.
If it is calculated under , then it becomes a conditional error
probability. Integrated over all possible realizations of , the conditional error probability becomes the overall type I error rate. One could reduce the type I error rate inflation resulting from data-driven sample size
adjustment by selecting a larger critical value at the final stage and agreeing to continue the trial only if the first stage p-value
(e.g. if is set to 0.15, which would translate into accepting the possibility of futility stop in case the p-value from the first stage ). However, these measures would only keep overall type I error rate below , but not give an exact -level procedure.
The key idea of the Proschan and Hunsberger method is to create an exact -level test at the end. They rewrite type I error rate as
, (3.4)
where could be any increasing function with range [0,1], with the integration in (3.4) over all possible realizations of under the null hypothesis. Proschan and Hunsberger suggest using the following circular function:
.
In the above equation, is the cut-off for the Stage 1 futility stop and the critical value for the Stage 1 test statistic is selected (given ) so that condition (3.4) holds. The function dictates how much conditional type I error rate to allow at the end of the study given the observed interim results .
When the design includes an unblinded SSR in addition to early stopping (efficacy and/or futility), the critical value for the final testing and new sample size per group for second stage are selected so that
and .
It is important to note that the rule lets users decide on how to select (e.g. how high the conditional power should be or whether to use empirical vs hypothesized effect size). Once the rule is set, the type I error rate is guaranteed at level . This method can also be viewed as a special case of the p-value combination principle.
Another method in the same category is by Mỹller and Schọfer [24]. The Mỹller and Schọfer method is a lot more general than the method by Proschan and Hunsberger and can be extended to multistage trials with adaptations other than sample size (e.g. hypotheses/endpoint changes).
The latter also allows an unplanned interim analysis to be inserted into a sequence of pre-planned interim analyses in a group-sequential trial.
Even though such extensions are beyond the scope of this chapter, the Mỹller and Schọfer principle is worth mentioning because even in case of a simple 2-stage procedure for SSR, it highlights a fundamental role of the conditional error function shared by the Proschan and Hunsberger method. That is, after the Stage 1 data have been observed, the
conditional error probability (or conditional rejection probabilities, CRP, in their terminology) given is all the information from Stage 1 that needs to be carried into Stage 2 of the study. As long as the Stage 2 in the new design has conditional type I error rate equal to the CRP
calculated from Stage 1, the overall type I error will be preserved.
Mỹller and Schọfer also recommend selecting Stage 2 sample size based on the conditional power argument [24]. When it comes to type I
error rate control, the core idea of both methods is that the stronger evidence is contained in , the less the burden of showing significance in the final test statistic.
Because of the extensive computations required of the conditional type I error approach, this approach is seldom used in practice. However, with software development for SSR on the rise, this statement is likely to change in the future. Another reason why these methods are seldom used in practice is probably due to the fact that they are too flexible. For example, the Mỹller and Schọfer principle allows not just SSR change at interim but other more extensive changes such as hypothesis/endpoint changes and recursive insertion of interim analyses. While such flexibility is valuable in theory, it is rarely preferred by regulatory reviewers,
especially for registration trials. Limited practical experience with such designs exist. One area where they may be of value is vaccine efficacy trial designs (see [34] for example). Because such trials typically rely on exact binomial distributions rather than a normal approximation for
testing the null hypothesis, the flexibility of the Mỹller and Schọfer
method could allow harnessing more precision in -spending when using discrete (binomial) boundaries for stopping and sample size calculations.