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

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594 b i o s t a t i s t i c s Tremmel t a b L E 3 t a b L E 4 True Treatment Average Sample Power rCI Effect θ Size per Group (%) Coverage (%) Bias ˆ ϑ – θ 0.00 114 2.1 97 –0.019 0.10 124 34 98 0.005 0.20 89 89 98 0.030 0.30 56 >99 99 0.022 0.40 43 100 99 0.007 assumptions: see text. that rigid statistical decision rules were not followed. In the absence of any clearly defined sample size recalculation rules, bias cannot be studied by simulation. The only other approach known to us is the double-conditioning approach by Coburger and Wassmer (11): bias is calculated based on recursive integration (20), after conditioning on past history, without speculating about the future. This means both restricting the sample space to all paths that stop for efficacy at k = 3, and “freezing” the first three sample sizes at the values that were actually used. Figure 1 shows this conditional bias; it is much more severe than the unconditional bias shown in Tables 3 and 4. The direction of the bias is conservative; it indicates that the estimator is likely to substantially underestimate the more extreme effect sizes. One could make the point that the boundaries that were actually in effect at the first two interim analyses were much wider than claimed Simulation Results for Probability Difference, Based on Rule 1 True Treatment Average Sample Power rCI Effect θ Size per Group (%) Coverage (%) Bias ˆ ϑ – θ 0.00 44 1.7 98 –0.006 0.10 55 16 98 –0.012 0.20 64 58 99 –0.001 0.30 56 89 99 0.0108 0.40 44 >99 99 0.0064 assumptions: see text. Simulation Results for Probability Difference, Based on Rule 2 by Pocock’s procedure, because the trial continued despite P values of
Inference After Adaptive GSD b i o s t a t i s t i c s 595 Bias θ′ − θ 0.25 0.2 0.15 0.1 0.05 place, making one’s own drug in development more worthwhile to pursue even at more conservative effect size assumptions. Or another important trial may mature, contributing valuable effect size information that one might want to use to update the power calculations of the ongoing trial. Or the emerging adverse events profile of the new drug is harsher than anticipated, which would increase the effect size deemed necessary to counterbalance the patients’ risk. On the other hand, this flexibility is a bane for full statistical inference. The openness in trial conduct has to be paid for by wider confidence intervals, and bias issues become intractable. Will those difficulties be resolved as statistical science advances, or are they of a more fundamental nature? This is an area of active research, and progress is being made. For instance, very recently, a method was developed that yields exact confidence bounds and median unbiased estimates when adaptive changes are restricted to the penultimate stage (22). On the other hand, for a different type of adaptive designs, it was shown convincingly that full statistical inference is not possible in principle (23). We believe that for the aGSD, the difficulties may be of this more fundamental nature. If one is willing to commit to rigid, prespecified sample size adjustment rules, these difficul- Drug Information Journal 0 −0.05 −0.1 −0.15 −0.2 −0.25 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 θ Bias - planned boundaries Bias - apparent boundaries ties disappear, and full statistical inference becomes possible. However, in this case, it is usually possible to construct a traditional GSD that is at least as powerful as the aGSD (24–26). In general, the aGSD should be less efficient than a traditional GSD because its predetermined weights will not be optimal. In our case of equally weighted stages, this would be the case if the segments are very unequal in terms of amount of information. If, as in this trial, the next interim analysis is scheduled without specifying a target number of PFS events, such an inequality is likely to happen for the time-to-event endpoint. For our case study, the corresponding traditional GSD would be the nonadaptive Pocock design. This design would yield powers of 2.5%, 40%, 95%, >99%, and >99% for the values of θ from Tables 3 and 4, at average sample sizes of 171, 149, 89, 53, and 40 per group. As can be seen, for θ of 0.2 or higher, the nonadaptive design achieves higher power with fewer patients. On the other hand, if there is no effect, the nonadaptive design spends considerably more patients than the aGSD with rule 1 or rule 2. Most likely, this is due to the implicit futility rule that we incorporated in rule 1 and rule 2. To decide which design is better overall, one faces the problem of comparing two different approaches with different operating character- F i g U R E 1 Conditional bias for estimate of probability differences, based on recursive integration.
Page 1 and 2: Lothar T. Tremmel, PhD Sr. Director
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