This article is devoted to the asymptotic study of adaptive group sequential designs in the case of randomized clinical trials with binary treatment, binary outcome and no covariate. By adaptive design, we mean in this setting a clinical trial 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. Prior to collection of the data, the trial protocol specifies the parameter of scientific interest. In the estimation framework, the trial protocol also a priori specifies the confidence level to be used in constructing frequentist confidence intervals for the latter parameter and the related inferential method, which will be based on the maximum likelihood principle. In the testing framework, the trial protocol also a priori specifies the null and alternative hypotheses regarding the latter parameter, the wished type I and type II errors, the rule for determining the maximal statistical information to be accrued, and the frequentist testing procedure, including conditions for early stopping. Furthermore, we assume that the protocol specifies a user-supplied optimal unknown choice of randomization scheme, and we will focus on that randomization scheme which minimizes the asymptotic variance of the maximum likelihood estimator of the parameter of interest.

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-statistics 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 validates the theory. It notably shows in the estimation framework that the confidence intervals we obtain achieve the desired coverage even for moderate sample sizes. In addition, it shows in the testing framework that type I error control at the prescribed level is guaranteed, and that all sampling procedures only suffer from a very slight increase of the type II error.

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."



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