On the almost sure convergence of adaptive allocation procedures

TL;DR

This paper establishes general almost sure convergence results for a broad class of adaptive treatment allocation procedures, including both continuous and discontinuous rules, enhancing theoretical understanding of adaptive designs in clinical trials.

Contribution

It provides a unified mathematical framework proving convergence for various adaptive allocation methods, including those previously only explored via simulations.

Findings

Almost sure convergence of treatment allocation proportions.

Applicability to both continuous and discontinuous allocation rules.

Includes analysis of designs like Atkinson's biased coin and Pocock-Simon's minimization.

Abstract

In this paper, we provide some general convergence results for adaptive designs for treatment comparison, both in the absence and presence of covariates. In particular, we demonstrate the almost sure convergence of the treatment allocation proportion for a vast class of adaptive procedures, also including designs that have not been formally investigated but mainly explored through simulations, such as Atkinson's optimum biased coin design, Pocock and Simon's minimization method and some of its generalizations. Even if the large majority of the proposals in the literature rely on continuous allocation rules, our results allow to prove via a unique mathematical framework the convergence of adaptive allocation methods based on both continuous and discontinuous randomization functions. Although several examples of earlier works are included in order to enhance the applicability, our approach provides substantial insight for future suggestions, especially in the absence of a prefixed target and for designs characterized by sequences of allocation rules.