Targeting the Optimal Design in Randomized Clinical Trials with Binary Outcomes and No Covariate: Theoretical Study

This article is devoted to the asymptotic study of adaptive group sequential designs in the case of randomized clinical trials (RCTs) with binary treatment, binary outcome and no covariate. By adaptive design, we mean in this setting a RCT design that allows the investigator to dynamically modify its course through data-driven adjustment of the randomization probability based on data accrued so far, without negatively impacting on the statistical integrity of the trial. By adaptive group sequential design, we refer to the fact that group sequential testing methods can be equally well applied on top of adaptive designs. We obtain that, theoretically, the adaptive design converges almost surely to the targeted unknown randomization scheme. In the estimation framework, we obtain that our maximum likelihood estimator of the parameter of interest is a strongly consistent estimator, and it satisfies a central limit theorem. We can estimate its asymptotic variance, which is the same as that it would feature had we known in advance the targeted randomization scheme and independently sampled from it. Consequently, inference can be carried out as if we had resorted to independent and identically distributed (iid) sampling. In the testing framework, we obtain that the multidimensional t-statistic that we would use under iid sampling still converges to the same canonical distribution under adaptive sampling. Consequently, the same group sequential testing can be carried out as if we had resorted to iid sampling. Furthermore, a comprehensive simulation study that we undertake in a companion article validates the theory. A three-sentence take-home message is “Adaptive designs do learn the targeted optimal design and inference, and testing can be carried out under adaptive sampling as they would under the targeted optimal randomization probability iid sampling. In particular, adaptive designs achieve the same efficiency as the fixed oracle design. This is confirmed by a simulation study, at least for moderate or large sample sizes, across a large collection of targeted randomization probabilities.'”