Randomized Urn Models revisited using Stochastic Approximation

TL;DR

This paper connects stochastic approximation techniques with randomized urn models in clinical trials, providing new convergence and normality results under weaker assumptions, especially for complex multi-arm trial models.

Contribution

It reformulates urn models as stochastic approximation algorithms, deriving new asymptotic properties and extending results to multi-arm clinical trial models with dependent updates.

Findings

Established almost sure convergence of the normalized procedures.

Derived asymptotic normality under less restrictive conditions.

Extended analysis to multi-arm clinical trials with dependent urn updates.

Abstract

This paper presents the link between stochastic approximation and clinical trials based on randomized urn models investigated in Bai and Hu (1999,2005) and Bai, Hu and Shen (2002). We reformulate the dynamics of both the urn composition and the assigned treatments as standard stochastic approximation (SA) algorithms with remainder. Then, we derive the a.s. convergence and the asymptotic normality (CLT) of the normalized procedure under less stringent assumptions by calling upon the ODE and SDE methods. As a second step, we investigate a more involved family of models, known as multi-arm clinical trials, where the urn updating depends on the past performances of the treatments. By increasing the dimension of the state vector, our SA approach provides this time a new asymptotic normality result.